🧠:<think>Okay, so I have this problem where I need to evaluate the safety and sustainability of a new bio-based material called BioFiber compared to a conventional synthetic material, SynFiber. There are two main parts: mechanical strength analysis and environmental impact evaluation. Let me tackle each part step by step.Starting with the first part, the mechanical strength analysis. Both BioFiber and SynFiber have tensile strengths that follow normal distributions. For BioFiber, the mean tensile strength is 150 MPa with a standard deviation of 10 MPa. For SynFiber, the mean is 160 MPa with a standard deviation of 15 MPa. The task is to find the probability that a randomly selected sample of BioFiber has a tensile strength greater than a randomly selected sample of SynFiber.Hmm, okay. So, I think this is a problem where we have two independent normal distributions, and we need to find the probability that one is greater than the other. I remember that when dealing with two independent normal variables, their difference is also normally distributed. So, let me denote X as the tensile strength of BioFiber and Y as that of SynFiber. Then, X ~ N(150, 10²) and Y ~ N(160, 15²). We need to find P(X > Y).To find this probability, I can consider the difference D = X - Y. Since X and Y are independent, the distribution of D will be normal with mean μ_D = μ_X - μ_Y = 150 - 160 = -10 MPa. The variance of D will be Var(D) = Var(X) + Var(Y) = 10² + 15² = 100 + 225 = 325. So, the standard deviation σ_D is sqrt(325). Let me calculate that. sqrt(325) is approximately 18.0278 MPa.Therefore, D ~ N(-10, 18.0278²). We need to find P(D > 0), which is the probability that X - Y > 0, or X > Y.To find this probability, I can standardize D. So, Z = (D - μ_D)/σ_D = (0 - (-10))/18.0278 ≈ 10/18.0278 ≈ 0.5547.Now, I need to find P(Z > 0.5547). Using standard normal distribution tables or a calculator, I can find the area to the right of Z = 0.5547. Alternatively, I can compute 1 - Φ(0.5547), where Φ is the cumulative distribution function.Looking up Z = 0.55 in the standard normal table, the value is approximately 0.7088. For Z = 0.56, it's about 0.7123. Since 0.5547 is between 0.55 and 0.56, I can interpolate. The difference between 0.55 and 0.56 is 0.01, and 0.5547 is 0.0047 above 0.55. So, the corresponding probability increase would be roughly (0.7123 - 0.7088) * (0.0047 / 0.01) ≈ 0.0035 * 0.47 ≈ 0.0016. Therefore, Φ(0.5547) ≈ 0.7088 + 0.0016 ≈ 0.7104.Thus, P(Z > 0.5547) = 1 - 0.7104 ≈ 0.2896. So, approximately 28.96% chance that a randomly selected BioFiber sample has a higher tensile strength than a SynFiber sample.Wait, let me double-check my calculations. The Z-score was 0.5547, which is about 0.55. The cumulative probability for 0.55 is indeed around 0.7088, so 1 - 0.7088 is 0.2912. Hmm, my interpolation gave me 0.2896, which is close. Maybe I should use a more precise method or a calculator for better accuracy. Alternatively, using a calculator, the exact value for Z = 0.5547 is approximately 0.7104, so 1 - 0.7104 = 0.2896. Yeah, that seems consistent. So, roughly 28.96% probability.Moving on to the second part, the environmental impact evaluation. The production of BioFiber emits 30 kg of CO2 per ton, while SynFiber emits 80 kg per ton. The company plans to produce 500 tons of material. The regulatory target is a 35% reduction in CO2 emissions compared to using only SynFiber.First, I need to calculate the total emissions if only SynFiber were used. That would be 500 tons * 80 kg/ton = 40,000 kg of CO2.A 35% reduction means the target emissions are 40,000 kg * (1 - 0.35) = 40,000 * 0.65 = 26,000 kg of CO2.Now, let’s denote the proportion of BioFiber in the production mix as p, and the proportion of SynFiber as (1 - p). Since the total production is 500 tons, the amount of BioFiber is 500p tons, and SynFiber is 500(1 - p) tons.The total CO2 emissions would then be: 500p * 30 + 500(1 - p) * 80.We need this total to be less than or equal to 26,000 kg.So, setting up the inequality:500p * 30 + 500(1 - p) * 80 ≤ 26,000Let me compute each term:500p * 30 = 15,000p500(1 - p) * 80 = 40,000(1 - p) = 40,000 - 40,000pAdding them together: 15,000p + 40,000 - 40,000p = (15,000p - 40,000p) + 40,000 = (-25,000p) + 40,000So, the inequality becomes:-25,000p + 40,000 ≤ 26,000Subtract 40,000 from both sides:-25,000p ≤ -14,000Divide both sides by -25,000, remembering to reverse the inequality sign:p ≥ (-14,000)/(-25,000) = 14,000/25,000 = 0.56So, p ≥ 0.56, meaning the proportion of BioFiber needs to be at least 56%. Therefore, the maximum allowable proportion of SynFiber is 1 - 0.56 = 0.44, or 44%.Wait, let me verify that again. If p is the proportion of BioFiber, then to meet the target, p must be at least 56%. Therefore, the maximum proportion of SynFiber is 44%. That makes sense because if we use more BioFiber, which emits less CO2, we can reduce total emissions.Let me plug p = 0.56 back into the total emissions equation to check:Total emissions = 500*0.56*30 + 500*0.44*80Compute each term:500*0.56 = 280 tons of BioFiber: 280*30 = 8,400 kg500*0.44 = 220 tons of SynFiber: 220*80 = 17,600 kgTotal emissions: 8,400 + 17,600 = 26,000 kg, which matches the target. So, if p is 56%, emissions are exactly 26,000 kg. Therefore, any higher proportion of BioFiber would result in lower emissions, which is acceptable, but the maximum allowable proportion of SynFiber is 44%.So, summarizing:1. The probability that a randomly selected BioFiber sample has a higher tensile strength than a SynFiber sample is approximately 28.96%.2. The maximum allowable proportion of SynFiber in the production mix to meet the emission reduction target is 44%.I think that covers both parts of the problem. I should probably present the answers clearly.Final Answer1. The probability is boxed{0.29} (rounded to two decimal places).2. The maximum allowable proportion of SynFiber is boxed{0.44}.</think>

🧠:<think>Okay, so I have this problem about a botanical garden shaped like an ellipse. The event planner wants to set up a circular stage at one of the foci, and then plant a flower bed around it with a 5-meter distance from the stage's edge. I need to figure out the coordinates of the foci, the maximum radius for the stage, and then the area of the flower bed. Hmm, let me break this down step by step.First, the garden is an ellipse with a major axis of 120 meters and a minor axis of 80 meters. Since the center is at the origin, the standard equation of the ellipse should be (x²/a²) + (y²/b²) = 1, where a is the semi-major axis and b is the semi-minor axis. So, the major axis is 120, which means a = 60 meters, and the minor axis is 80, so b = 40 meters.Now, to find the foci of the ellipse. I remember that for an ellipse, the distance from the center to each focus is given by c, where c² = a² - b². Let me compute that. So, a² is 60² = 3600, and b² is 40² = 1600. Therefore, c² = 3600 - 1600 = 2000. Taking the square root, c = sqrt(2000). Hmm, sqrt(2000) can be simplified. 2000 is 100*20, so sqrt(100*20) = 10*sqrt(20). And sqrt(20) is 2*sqrt(5), so c = 10*2*sqrt(5) = 20*sqrt(5). Let me calculate that numerically to get an idea. sqrt(5) is approximately 2.236, so 20*2.236 ≈ 44.72 meters. So, the foci are located at (±44.72, 0) since the major axis is along the x-axis.So, the coordinates of the foci are (44.72, 0) and (-44.72, 0). The planner wants to set up the stage at one of these foci. Now, the stage is a circle, and it needs to be entirely within the ellipse. So, the maximum radius of the stage would be the distance from the focus to the nearest point on the ellipse. Hmm, wait, actually, since the ellipse is symmetric, the closest point from the focus would be along the major axis towards the center.Wait, actually, the ellipse's vertices are at (±60, 0). So, the distance from the focus at (44.72, 0) to the center is 44.72 meters, and the distance from the focus to the vertex is 60 - 44.72 = 15.28 meters. So, if the stage is placed at the focus, the maximum radius it can have without going outside the ellipse would be 15.28 meters. Because beyond that, it would go beyond the ellipse's boundary.But wait, is that correct? Let me think. If the stage is a circle centered at (44.72, 0), then the maximum radius it can have without crossing the ellipse is the minimum distance from the focus to the ellipse along any direction, not just along the major axis. Because if you go in another direction, say along the y-axis, the ellipse is narrower.So, maybe the maximum radius isn't just 15.28 meters, but something smaller. Hmm, this complicates things. I need to find the maximum radius such that the entire circle is inside the ellipse. So, the circle must satisfy the ellipse equation for all points on the circle.Let me denote the center of the circle as (c, 0), where c = 20*sqrt(5) ≈ 44.72. The equation of the circle is (x - c)² + y² = r². We need this circle to be entirely inside the ellipse (x²/a²) + (y²/b²) ≤ 1.So, for all points (x, y) on the circle, the inequality must hold. Let's substitute x and y from the circle into the ellipse equation.From the circle: (x - c)² + y² = r² => y² = r² - (x - c)².Substitute into the ellipse equation:x²/a² + [r² - (x - c)²]/b² ≤ 1.We need this inequality to hold for all x, y on the circle. Let's simplify this expression.First, expand [r² - (x - c)²]:= r² - (x² - 2cx + c²)= r² - x² + 2cx - c².So, substitute back into the ellipse equation:x²/a² + (r² - x² + 2cx - c²)/b² ≤ 1.Multiply through by a²b² to eliminate denominators:b²x² + a²(r² - x² + 2cx - c²) ≤ a²b².Expand the terms:b²x² + a²r² - a²x² + 2a²c x - a²c² ≤ a²b².Combine like terms:(b² - a²)x² + 2a²c x + (a²r² - a²c² - a²b²) ≤ 0.This is a quadratic inequality in terms of x. For the inequality to hold for all x on the circle, the quadratic must be ≤ 0 for all x. However, since the quadratic is in x, and the circle is a closed curve, this might not be straightforward. Maybe another approach is needed.Alternatively, perhaps the maximum radius occurs where the circle is tangent to the ellipse. So, the point of tangency would be where both the ellipse and the circle have the same tangent line. That might be a way to find the maximum r.Let me set up the equations:Ellipse: x²/60² + y²/40² = 1.Circle: (x - c)² + y² = r², where c = 20√5.To find the points of intersection, substitute y² from the circle into the ellipse equation:x²/3600 + (r² - (x - c)²)/1600 = 1.Multiply both sides by 4800 (the least common multiple of 3600 and 1600) to eliminate denominators:4800*(x²/3600) + 4800*((r² - (x - c)²)/1600) = 4800*1.Simplify:(4800/3600)x² + (4800/1600)(r² - (x - c)²) = 4800.Which simplifies to:(4/3)x² + 3(r² - (x² - 2cx + c²)) = 4800.Expand the terms:(4/3)x² + 3r² - 3x² + 6cx - 3c² = 4800.Combine like terms:(4/3 - 3)x² + 6cx + (3r² - 3c² - 4800) = 0.Compute 4/3 - 3 = 4/3 - 9/3 = -5/3.So:(-5/3)x² + 6cx + (3r² - 3c² - 4800) = 0.Multiply through by -3 to eliminate fractions:5x² - 18cx + (-9r² + 9c² + 14400) = 0.So, 5x² - 18cx + (9c² - 9r² + 14400) = 0.For the circle and ellipse to be tangent, this quadratic equation must have exactly one solution, meaning the discriminant is zero.Discriminant D = [(-18c)]² - 4*5*(9c² - 9r² + 14400) = 0.Compute D:= 324c² - 20*(9c² - 9r² + 14400)= 324c² - 180c² + 180r² - 288000= (324 - 180)c² + 180r² - 288000= 144c² + 180r² - 288000.Set D = 0:144c² + 180r² - 288000 = 0.Divide through by 36 to simplify:4c² + 5r² - 8000 = 0.So, 5r² = 8000 - 4c².Therefore, r² = (8000 - 4c²)/5.We know c = 20√5, so c² = (20√5)² = 400*5 = 2000.Substitute c² into the equation:r² = (8000 - 4*2000)/5 = (8000 - 8000)/5 = 0/5 = 0.Wait, that can't be right. That would mean r = 0, which doesn't make sense. Did I make a mistake somewhere?Let me check my steps.Starting from the discriminant:D = (-18c)^2 - 4*5*(9c² - 9r² + 14400) = 0.Compute:= 324c² - 20*(9c² - 9r² + 14400)= 324c² - 180c² + 180r² - 288000= 144c² + 180r² - 288000.Set equal to zero:144c² + 180r² = 288000.Divide by 36:4c² + 5r² = 8000.So, 5r² = 8000 - 4c².r² = (8000 - 4c²)/5.But c² = 2000, so:r² = (8000 - 8000)/5 = 0.Hmm, that suggests that the only solution is r = 0, which is not helpful. Maybe my approach is flawed.Alternatively, perhaps the maximum radius is simply the distance from the focus to the nearest point on the ellipse, which is along the major axis, as I initially thought. So, the distance from the focus to the vertex is a - c = 60 - 44.72 ≈ 15.28 meters. So, the maximum radius is 15.28 meters.But wait, if I place a circle of radius 15.28 at the focus, does it stay entirely within the ellipse? Because along the major axis, yes, it just reaches the vertex. But what about in other directions? For example, along the minor axis, the ellipse is narrower. So, the circle might extend beyond the ellipse in other directions.Wait, let's check that. If the circle is centered at (44.72, 0) with radius 15.28, then the topmost point of the circle would be (44.72, 15.28). Let's see if this point is inside the ellipse.Plug into the ellipse equation:x²/60² + y²/40² = (44.72²)/(60²) + (15.28²)/(40²).Compute 44.72²: approximately (44.72)^2 ≈ 2000 (since c² = 2000). So, 2000/3600 ≈ 0.5556.15.28² ≈ (15.28)^2 ≈ 233.5. 233.5 / 1600 ≈ 0.1459.Adding them together: 0.5556 + 0.1459 ≈ 0.7015, which is less than 1. So, the point (44.72, 15.28) is inside the ellipse. Therefore, the circle with radius 15.28 is entirely within the ellipse.Wait, but if I go in the opposite direction along the major axis, the circle would reach the vertex at (60, 0), which is on the ellipse. So, that point is on the ellipse. So, the circle touches the ellipse at (60, 0) and is entirely inside elsewhere.Therefore, the maximum radius is indeed 15.28 meters, which is a - c = 60 - 20√5.Wait, 20√5 is approximately 44.72, so 60 - 44.72 ≈ 15.28. So, exact value is 60 - 20√5.So, part 1 answer is coordinates of foci at (±20√5, 0), and maximum radius is 60 - 20√5 meters.Now, moving on to part 2. The planner wants to plant a circular flower bed around the stage, with a 5-meter distance from the edge of the stage to the edge of the flower bed. So, the flower bed is another circle, concentric with the stage, but with a radius 5 meters larger.Wait, no. Wait, the flower bed is around the stage, so the distance from the edge of the stage to the edge of the flower bed is 5 meters. So, if the stage has radius r, the flower bed has radius r + 5.But wait, actually, the distance between the edges is 5 meters, so if the stage is at radius r, the flower bed is at radius r + 5. So, the area of the flower bed would be the area between the stage and the flower bed, which is π(R² - r²), where R = r + 5.But wait, actually, the flower bed is a circular region around the stage, so it's just a larger circle with radius r + 5. So, the area is π(R)^2, where R = r + 5. But wait, no, the flower bed is the area between the stage and the outer circle. So, it's the area of the larger circle minus the area of the stage.But the problem says "the area of the flower bed." So, if the flower bed is the region between the stage and the outer circle, then yes, it's π(R² - r²). But if the flower bed is just the outer circle, then it's πR². But the wording says "a circular flower bed around the stage, such that the distance from the edge of the stage to the edge of the flower bed is always 5 meters." So, it's the area between the two circles, I think.But let me clarify. If the stage is a circle of radius r, and the flower bed is another circle around it with a 5-meter gap, then the flower bed's radius is r + 5. So, the area of the flower bed would be π(r + 5)² - πr² = π[(r + 5)² - r²] = π(10r + 25).Alternatively, if the flower bed is just the outer ring, then that's correct. But sometimes, "flower bed" might refer to the entire outer circle, but given the context, it's more likely the area around the stage, so the annular region.But let me check the problem statement again: "plant a circular flower bed around the stage, such that the distance from the edge of the stage to the edge of the flower bed is always 5 meters." So, the flower bed is a circular region around the stage, with a uniform 5-meter border. So, yes, it's an annulus with inner radius r and outer radius r + 5.Therefore, the area is π[(r + 5)² - r²] = π(10r + 25).We already found r = 60 - 20√5. So, let's compute 10r + 25.10r = 10*(60 - 20√5) = 600 - 200√5.Add 25: 600 - 200√5 + 25 = 625 - 200√5.Therefore, the area is π*(625 - 200√5) square meters.Alternatively, we can factor it as π*(625 - 200√5). But maybe we can write it as π*(25² - 200√5). Alternatively, factor out 25: π*25*(25 - 8√5). But perhaps it's fine as is.Alternatively, if we calculate the numerical value, but since the problem doesn't specify, probably leave it in exact form.So, to recap:1. Foci coordinates: (±20√5, 0). Maximum radius: 60 - 20√5 meters.2. Area of the flower bed: π*(625 - 200√5) square meters.Wait, let me double-check the area calculation. If the flower bed is the annulus, then yes, it's π[(r + 5)^2 - r^2] = π(10r + 25). With r = 60 - 20√5, then 10r = 600 - 200√5, plus 25 is 625 - 200√5. So, area is π*(625 - 200√5). That seems correct.Alternatively, if the flower bed is just the outer circle, then the area would be π*(r + 5)^2, but that would include the stage area, which is probably not intended. The problem says "plant a circular flower bed around the stage," so it's likely the area around the stage, hence the annulus.Therefore, I think that's the correct approach.Final Answer1. The coordinates of the foci are boxed{(pm 20sqrt{5}, 0)} and the maximum radius of the stage is boxed{60 - 20sqrt{5}} meters.2. The area of the flower bed is boxed{pi (625 - 200sqrt{5})} square meters.</think>

🧠:<think>Alright, so I have this problem where a humanitarian worker is trying to negotiate the release of 5 hostages. The kidnappers initially wanted 1,000,000, but they can reduce the ransom by providing supplies like food, medicine, and clothing. Each type of supply has a certain value in terms of ransom reduction. The goal is to figure out how many units of each supply to provide to minimize the total ransom paid while getting all 5 hostages released.First, I need to understand the problem thoroughly. The humanitarian worker can provide up to 3 types of supplies: food, medicine, and clothing. Each unit of these supplies reduces the ransom by a specific amount: food by 10,000, medicine by 20,000, and clothing by 5,000. The kidnappers will release one hostage for every 200,000 reduction in ransom. So, to release all 5 hostages, the total ransom reduction needs to be at least 1,000,000 (since 5 hostages * 200,000 each = 1,000,000). But wait, the initial ransom is 1,000,000. If we reduce it by 1,000,000, that would mean the ransom becomes 0. But that doesn't make sense because the problem says "minimize the total ransom paid." So, actually, the total ransom paid would be the initial amount minus the total ransom reduction from the supplies. So, the total ransom paid is 1,000,000 minus the total reduction from supplies. Therefore, to minimize the ransom paid, we need to maximize the total ransom reduction from the supplies, subject to the constraint that the total reduction is at least 1,000,000. But wait, the maximum possible reduction is limited by the supplies we can provide.Wait, hold on. Let me clarify. The initial ransom is 1,000,000. The humanitarian worker wants to reduce this ransom by providing supplies. For each 200,000 reduction, one hostage is released. So, to release all 5 hostages, the total ransom reduction must be at least 5 * 200,000 = 1,000,000. Therefore, the total ransom reduction from supplies must be at least 1,000,000. However, the supplies can only provide a certain amount of reduction. So, the problem is to find the combination of food, medicine, and clothing that provides at least 1,000,000 in ransom reduction, while minimizing the total ransom paid, which is 1,000,000 minus the total ransom reduction.But wait, actually, the total ransom paid is the initial amount minus the total reduction. So, to minimize the ransom paid, we need to maximize the total ransom reduction. But the maximum possible ransom reduction is limited by the supplies. So, the problem is to maximize the total ransom reduction, subject to the constraints on the number of units of each supply.But hold on, the problem says "minimize the total ransom paid while securing the release of all 5 hostages." So, the total ransom reduction needs to be at least 1,000,000, and we need to find the combination of supplies that achieves this with the minimal possible ransom paid, which is equivalent to maximizing the ransom reduction beyond 1,000,000 as much as possible, but actually, since the ransom reduction can't exceed the supplies, we just need to reach at least 1,000,000.Wait, perhaps I'm overcomplicating. Let me rephrase.The total ransom reduction needed is 1,000,000 to release all 5 hostages. The supplies can provide a certain amount of reduction. So, the problem is to find the minimal number of supplies (in terms of cost?) that can provide at least 1,000,000 in ransom reduction. But no, the cost isn't given. Wait, the supplies have a value in terms of ransom reduction per unit, but the cost of each unit isn't specified. So, perhaps the problem is simply to find the combination of food, medicine, and clothing that provides at least 1,000,000 in ransom reduction, while minimizing the total ransom paid, which is 1,000,000 minus the total ransom reduction. But since the total ransom reduction can't exceed the maximum possible from the supplies, we need to see if the maximum possible ransom reduction is at least 1,000,000.Wait, let's calculate the maximum possible ransom reduction from each supply:- Food: 40 units * 10,000 = 400,000- Medicine: 25 units * 20,000 = 500,000- Clothing: 50 units * 5,000 = 250,000Total maximum ransom reduction: 400,000 + 500,000 + 250,000 = 1,150,000So, the maximum possible ransom reduction is 1,150,000, which is more than the required 1,000,000. Therefore, it's possible to release all 5 hostages by providing the maximum units of each supply, but perhaps we can do it with less, thereby paying more ransom, but the goal is to minimize the ransom paid, which would require maximizing the ransom reduction. Wait, no, the goal is to minimize the total ransom paid, which is 1,000,000 minus the ransom reduction. So, to minimize the ransom paid, we need to maximize the ransom reduction. Therefore, the optimal solution is to provide as much as possible of the supplies that give the highest ransom reduction per unit.Wait, but each unit of medicine gives 20,000 reduction, which is higher than food (10,000) and clothing (5,000). So, to maximize the ransom reduction, we should prioritize providing as much medicine as possible, then food, then clothing.But let's formalize this as an optimization problem.Let me define the decision variables:Let x = number of food units providedy = number of medicine units providedz = number of clothing units providedObjective: Maximize the total ransom reduction, which is 10,000x + 20,000y + 5,000zSubject to:x ≤ 40y ≤ 25z ≤ 50And the total ransom reduction must be at least 1,000,000:10,000x + 20,000y + 5,000z ≥ 1,000,000Also, x, y, z ≥ 0 and integers (since you can't provide a fraction of a unit)But wait, the problem says "the humanitarian worker can provide up to 3 different types of supplies," but doesn't specify whether they have to provide all three or can choose to provide fewer. So, x, y, z can be zero or more, up to their maximums.But since we want to maximize the ransom reduction, we should use as much as possible of the most valuable supplies first.So, medicine gives the highest reduction per unit, so we should use all 25 units of medicine first. That gives 25 * 20,000 = 500,000 reduction.Then, food gives next highest, so use all 40 units: 40 * 10,000 = 400,000. Total so far: 900,000.We still need 100,000 more to reach 1,000,000. So, we can use clothing: each unit gives 5,000. So, 100,000 / 5,000 = 20 units. So, z = 20.Therefore, total ransom reduction is 500,000 + 400,000 + 100,000 = 1,000,000.Therefore, the total ransom paid is 1,000,000 - 1,000,000 = 0.Wait, but that seems too good. The problem says "the humanitarian worker has managed to convince them to reduce the ransom in exchange for supplies." So, perhaps the ransom can't be reduced below zero, but in this case, it's exactly zero. So, the minimum total ransom paid is 0.But let me check if there's a cheaper way, but since we're maximizing the ransom reduction, which is equivalent to minimizing the ransom paid, and we've already achieved the maximum possible reduction needed, which is exactly 1,000,000, so the ransom paid is zero.But wait, the maximum possible ransom reduction is 1,150,000, so if we provide all supplies, the ransom reduction would be 1,150,000, which would make the ransom paid negative, which doesn't make sense. So, the ransom reduction can't exceed 1,000,000 because that's the initial ransom. So, we only need to provide enough supplies to reduce the ransom by 1,000,000.Therefore, the optimal solution is to provide 25 units of medicine, 40 units of food, and 20 units of clothing, which gives exactly 1,000,000 reduction, making the ransom paid zero.But let me double-check the calculations:25 * 20,000 = 500,00040 * 10,000 = 400,00020 * 5,000 = 100,000Total: 500,000 + 400,000 + 100,000 = 1,000,000Yes, that's correct.Alternatively, could we use less of the higher value supplies and more of the lower ones to achieve the same reduction? For example, instead of using all 25 medicine, use fewer and compensate with more food or clothing. But since medicine gives the highest reduction per unit, using less of it would require more units of food or clothing, which might not be possible due to their lower per-unit reduction.Wait, let's see. Suppose we use 24 units of medicine: 24 * 20,000 = 480,000Then, we need 1,000,000 - 480,000 = 520,000 more.If we use 40 units of food: 40 * 10,000 = 400,000Then, we still need 120,000 more, which would require 24 units of clothing (24 * 5,000 = 120,000)Total supplies: 24 medicine, 40 food, 24 clothing. Total ransom reduction: 480,000 + 400,000 + 120,000 = 1,000,000But in this case, we're using more clothing (24 vs 20) but less medicine (24 vs 25). However, the total ransom reduction is the same, but the total units provided are more (24+40+24=88 vs 25+40+20=85). But since the problem doesn't specify minimizing the number of units, just the ransom paid, which is the same in both cases (0). So, either way, the ransom paid is zero.But wait, the problem says "the humanitarian worker can provide up to 3 different types of supplies." So, they don't have to provide all three, but in this case, providing all three is necessary to reach the required reduction.Alternatively, could we provide only medicine and clothing? Let's see.Let me try to see if we can use only medicine and clothing.Let y be the number of medicine units, z be clothing.We need 20,000y + 5,000z ≥ 1,000,000Subject to y ≤25, z ≤50To minimize the number of units, we should maximize y.So, y=25: 25*20,000=500,000Then, z needs to be (1,000,000 - 500,000)/5,000 = 100,000 /5,000=20So, z=20, which is within the limit of 50.So, total units: 25+20=45, which is less than the previous 85 when including food. But wait, in this case, we're not using food at all. So, the total ransom reduction is 500,000 + 100,000=600,000, which is less than 1,000,000. Wait, no, 25*20,000=500,000, and 20*5,000=100,000, total 600,000, which is less than 1,000,000. So, that's not enough.Wait, no, I think I made a mistake. If we use only medicine and clothing, and set y=25, then z needs to be (1,000,000 - 500,000)/5,000=100,000/5,000=20. So, z=20. So, total ransom reduction is 500,000 + 100,000=600,000, which is insufficient. Therefore, we need more.Wait, that can't be. Because 25 medicine gives 500,000, and 20 clothing gives 100,000, total 600,000, which is less than 1,000,000. So, we need more. So, perhaps we need to use more clothing.Wait, but if we set y=25, z=50 (max clothing), then total ransom reduction is 500,000 + 250,000=750,000, still less than 1,000,000. Therefore, we can't reach 1,000,000 with only medicine and clothing. So, we need to include food as well.Therefore, the minimal way is to use all 25 medicine, all 40 food, and 20 clothing, which gives exactly 1,000,000 reduction.Alternatively, could we use less food and more clothing? Let's see.Suppose we use 25 medicine, 30 food, and then see how much more we need.25*20,000=500,00030*10,000=300,000Total so far: 800,000Need 200,000 more, which would require 40 units of clothing (40*5,000=200,000). But we can only provide up to 50 units of clothing, so 40 is acceptable.So, in this case, x=30, y=25, z=40.Total units: 30+25+40=95, which is more than the previous 85, but the ransom reduction is the same.But since the problem is to minimize the ransom paid, which is the same in both cases (0), it doesn't matter. However, if we had to minimize the number of units, this would be worse.But the problem doesn't specify minimizing the number of units, just the ransom paid. So, any combination that provides at least 1,000,000 in ransom reduction is acceptable, but we need to find the one that minimizes the ransom paid, which is achieved by maximizing the ransom reduction. However, since the maximum possible ransom reduction is 1,150,000, which is more than needed, but we only need 1,000,000, so we can stop at that.Wait, but if we provide more than 1,000,000 in ransom reduction, the ransom paid would be negative, which isn't possible. So, the maximum ransom reduction we can use is 1,000,000.Therefore, the optimal solution is to provide exactly 1,000,000 in ransom reduction, which can be achieved by various combinations, but the one that uses the least number of units would be providing as much of the highest value supplies as possible.So, the optimal solution is to provide 25 units of medicine, 40 units of food, and 20 units of clothing, which gives exactly 1,000,000 in ransom reduction, making the total ransom paid 0.But let me formalize this as an optimization problem.Decision variables:x = number of food unitsy = number of medicine unitsz = number of clothing unitsObjective function:Minimize total ransom paid = 1,000,000 - (10,000x + 20,000y + 5,000z)But since we want to minimize this, it's equivalent to maximizing (10,000x + 20,000y + 5,000z), subject to:10,000x + 20,000y + 5,000z ≥ 1,000,000x ≤40y ≤25z ≤50x, y, z ≥0 and integersBut since we can't have negative ransom paid, the maximum ransom reduction is capped at 1,000,000.Therefore, the optimal solution is to provide the minimal number of supplies that achieve exactly 1,000,000 in ransom reduction, using as much as possible of the highest value supplies.So, the solution is x=40, y=25, z=20.But wait, let me check if using more of the higher value supplies beyond what's needed is possible. For example, if we use more medicine beyond 25, but we can't because the maximum is 25. Similarly, food is maxed at 40, and clothing at 50.Wait, but if we use all 25 medicine, all 40 food, and all 50 clothing, the total ransom reduction would be 500,000 + 400,000 + 250,000=1,150,000, which is more than needed. But since the ransom can't be negative, the total ransom paid would be 1,000,000 - 1,150,000= -150,000, which doesn't make sense. Therefore, the maximum ransom reduction we can use is 1,000,000.Therefore, the optimal solution is to provide exactly 1,000,000 in ransom reduction, which can be achieved by various combinations, but the one that uses the least number of units is providing 25 medicine, 40 food, and 20 clothing.So, the answer is:x=40, y=25, z=20Total ransom paid= 0But let me check if there's a way to provide less than 25 medicine and still reach 1,000,000. For example, using 24 medicine, which gives 480,000, then needing 520,000 more. If we use 40 food, that gives 400,000, leaving 120,000, which requires 24 clothing. So, total units: 24+40+24=88, which is more than 25+40+20=85. So, the previous solution is better in terms of fewer units, but since the problem doesn't specify minimizing units, just ransom paid, both are acceptable, but the first one uses fewer units, which might be preferable, but it's not required.Therefore, the optimal solution is to provide 25 medicine, 40 food, and 20 clothing, resulting in a total ransom reduction of 1,000,000, making the total ransom paid 0.</think>

🧠:<think>Okay, so I have this problem about Alex planning a photoshoot with mirrors and lights. It's in a 3D studio space, which is 10x8x6 meters. The first part is about finding the equation of a mirror plane, and the second part is about reflecting a light off that mirror. Let me tackle them one by one.Starting with the first question: I need to find the equation of the plane representing the mirror. The mirror is inclined at 45° to the floor, which is the plane z=0. It also intersects the line y=x at the point (2,2,0). Hmm, okay.First, I remember that the angle between two planes is determined by the angle between their normal vectors. Since the mirror is inclined at 45° to the floor, which is the xy-plane (z=0), the angle between the mirror's plane and the floor is 45°. So, the dihedral angle between the two planes is 45°.The floor has a normal vector of (0,0,1). Let the normal vector of the mirror's plane be (a,b,c). The angle θ between the two planes is given by the dot product formula:cosθ = |n1 • n2| / (|n1||n2|)Here, θ is 45°, n1 is (0,0,1), and n2 is (a,b,c). So,cos45° = |(0*a + 0*b + 1*c)| / (sqrt(0²+0²+1²) * sqrt(a²+b²+c²))√2/2 = |c| / sqrt(a² + b² + c²)Let me square both sides to eliminate the square roots:( (√2)/2 )² = (c²) / (a² + b² + c²)(2/4) = c² / (a² + b² + c²)1/2 = c² / (a² + b² + c²)Multiply both sides by denominator:a² + b² + c² = 2c²So, a² + b² = c²That's one condition on the normal vector.Now, the mirror plane also intersects the line y=x at (2,2,0). So, the point (2,2,0) lies on the mirror plane. Let me denote the equation of the mirror plane as:ax + by + cz + d = 0Since (2,2,0) is on the plane:2a + 2b + 0 + d = 0So, 2a + 2b + d = 0Which simplifies to:a + b + d/2 = 0But I don't know d yet. Maybe I can find another condition.Also, the line y=x lies on the mirror plane? Wait, no, the mirror intersects the line y=x at (2,2,0). So, the mirror plane contains the point (2,2,0) and is inclined at 45° to the floor. Hmm.Wait, maybe the line y=x is in the floor plane (z=0). So, the mirror plane intersects this line at (2,2,0). So, the mirror plane contains the point (2,2,0) and is inclined at 45° to the floor.I need another condition to find the normal vector. Maybe the mirror plane also contains another point or has a certain orientation.Alternatively, since the mirror is inclined at 45°, and it's intersecting the line y=x at (2,2,0), perhaps the mirror plane is such that it's rising at 45° from that point.Wait, maybe I can think of the mirror plane as containing the line y=x, z=0, but that's not necessarily true because it only intersects the line at a single point. So, the mirror plane is not containing the entire line y=x, just intersecting it at (2,2,0).Hmm, maybe I can parametrize the mirror plane. Let me think about the direction of the mirror.Since it's inclined at 45°, the normal vector makes a 45° angle with the vertical. So, if the normal vector is (a,b,c), then the z-component c is related to the other components.Earlier, I found that a² + b² = c². So, c = sqrt(a² + b²). Let me choose a and b such that this holds.But I also need another condition. The plane passes through (2,2,0), so 2a + 2b + d = 0. So, d = -2a - 2b.So, the equation of the plane is ax + by + cz - 2a - 2b = 0.But I need another condition to determine a, b, c.Wait, maybe the mirror plane is such that it's symmetric with respect to the line y=x? Or maybe it's aligned in some way.Alternatively, perhaps the mirror plane is such that it's rising at 45° from the point (2,2,0). So, the direction of the mirror's slope is 45°, which could mean that the normal vector has a certain relation.Wait, maybe I can consider the direction of the mirror. If it's inclined at 45°, then the angle between the normal vector and the vertical is 45°, so the normal vector makes 45° with (0,0,1). So, the angle between n and (0,0,1) is 45°, which is consistent with what I had before.But I still need another condition to find a, b, c.Wait, maybe the mirror plane is such that it's orthogonal to some direction. Alternatively, perhaps the mirror is placed such that it's symmetric with respect to the line y=x. So, maybe the normal vector lies in the plane y=x, z=0? Hmm, not sure.Alternatively, perhaps the mirror plane is such that it contains the point (2,2,0) and has a normal vector making 45° with the vertical. So, let me assume that the normal vector is (a, a, c), since it's symmetric with respect to x and y because of the line y=x.Wait, that might make sense because the mirror intersects the line y=x, so maybe it's symmetric across that line. So, if I set a = b, then the normal vector is (a, a, c). Then, from earlier, a² + a² = c² => 2a² = c² => c = a√2.So, the normal vector is (a, a, a√2). Let me choose a=1 for simplicity, so normal vector is (1,1,√2). Then, the equation of the plane is:1x + 1y + √2 z + d = 0Since it passes through (2,2,0):2 + 2 + 0 + d = 0 => d = -4So, the equation is x + y + √2 z - 4 = 0.Wait, let me check if this satisfies the angle condition. The normal vector is (1,1,√2), and the floor's normal is (0,0,1). The angle between them is:cosθ = (0 + 0 + √2 * 1) / (sqrt(1+1+2) * sqrt(0+0+1)) = √2 / (2 * 1) = √2/2, which is 45°. Perfect, that works.So, the equation of the mirror plane is x + y + √2 z = 4.Wait, let me write it as x + y + √2 z - 4 = 0.So, that should be the answer for part 1.Now, moving on to part 2: Alex wants to place a light source at (4,4,2) and determine the path of the light reflected off the mirror. I need to find where the light hits the mirror and the direction of the reflected light.First, I need to find the point where the light from (4,4,2) hits the mirror plane x + y + √2 z = 4.To find this point, I can parametrize the line from the light source to the mirror. Since the light travels in a straight line, I can write parametric equations for the line.Let me denote the light source as point P(4,4,2). The direction vector of the light towards the mirror is unknown, but since it's a straight line, I can assume it's moving towards the mirror plane. Wait, actually, the light is going from P to the mirror, so the direction vector is towards the mirror.But I don't know the direction yet. Alternatively, since the mirror is a plane, the light will hit the mirror at some point Q, and then reflect according to the reflection law: angle of incidence equals angle of reflection.But to find Q, I need to find where the line from P intersects the mirror plane.Wait, but the light is going from P(4,4,2) to the mirror. So, the line from P to Q lies on the mirror plane? No, the line from P to Q is just a straight line that intersects the mirror plane at Q.So, parametrize the line from P(4,4,2) in some direction. But since we don't know the direction, maybe we can express it in terms of a parameter t.Wait, but we can write the parametric equations of the line as:x = 4 + aty = 4 + btz = 2 + ctWhere (a,b,c) is the direction vector. But we don't know a, b, c. However, since the line intersects the mirror plane at Q, we can substitute x, y, z into the plane equation and solve for t.But without knowing a, b, c, this might not be straightforward. Alternatively, perhaps we can assume that the light travels towards the mirror, so the direction vector is towards the mirror. But without knowing where it's going, it's tricky.Wait, maybe I can use the fact that the reflection law will hold. So, if I can find Q, then I can find the reflected direction. But since I don't know Q yet, maybe I need another approach.Alternatively, perhaps I can use the formula for reflection across a plane. The reflection of a point across a plane can be found, and then the reflected light direction would be the line from Q to the reflection of P.Wait, that might work. Let me recall that the reflection of a point across a plane can be calculated, and the reflected light would appear to come from that reflection.So, if I reflect the light source P across the mirror plane, the reflected light would travel in a straight line from the reflection to the point where it was reflected. So, the reflected light direction would be the line from Q to the reflection of P.But actually, in this case, the light is going from P to Q, and then reflects off the mirror. So, the reflected direction can be found using the reflection formula.Alternatively, perhaps I can find Q by solving the intersection of the line from P to the mirror plane.Wait, but I don't know the direction of the line. Hmm.Wait, perhaps I can parametrize the line from P(4,4,2) towards the mirror plane. Let me denote the direction vector as (a,b,c). Then, the parametric equations are:x = 4 + aty = 4 + btz = 2 + ctThis line intersects the mirror plane x + y + √2 z = 4. So, substituting:(4 + at) + (4 + bt) + √2(2 + ct) = 4Simplify:4 + at + 4 + bt + 2√2 + √2 ct = 4Combine like terms:(4 + 4 + 2√2) + (a + b + √2 c)t = 4So,(8 + 2√2) + (a + b + √2 c)t = 4Then,(a + b + √2 c)t = 4 - 8 - 2√2(a + b + √2 c)t = -4 - 2√2But I don't know a, b, c. Hmm, this seems like I'm stuck.Wait, maybe I need to consider that the direction vector (a,b,c) is such that the line from P to Q is the incident ray, and the reflection follows the law of reflection. But without knowing Q, it's difficult.Alternatively, perhaps I can use vector projections. Let me recall that the reflection of a vector over a plane can be found using the formula:R = V - 2(V • n / ||n||²) nWhere V is the incident vector, n is the normal vector of the plane.But in this case, the incident vector is from Q to P, and the reflected vector is from Q to the reflection point.Wait, maybe I can find Q by solving the system where the line from P intersects the mirror plane.But I need another condition. Wait, perhaps the line from P to Q is such that Q lies on the mirror plane, and the reflection direction can be found using the normal vector.Alternatively, maybe I can use the fact that the point Q is the intersection point, so I can write the parametric line from P and solve for t when it hits the plane.But I don't know the direction vector. Hmm.Wait, perhaps I can assume that the light travels towards the mirror in some direction, but without loss of generality, maybe I can set the direction vector as (a,b,c) and solve for t.Wait, but I have one equation from the plane intersection, but three variables a, b, c. That's not enough.Wait, perhaps I can express the direction vector in terms of the normal vector. Let me think.Wait, the direction vector of the incident ray (from P to Q) can be expressed as some vector, and the reflected direction can be found using the normal vector.But maybe I need to find Q such that the angle between the incident vector and the normal equals the angle between the reflected vector and the normal.Alternatively, perhaps it's easier to use the formula for reflection across a plane.Let me recall that the reflection of a point across a plane can be found using the formula:If you have a point P(x0,y0,z0) and a plane ax + by + cz + d = 0, then the reflection P' of P across the plane is given by:P' = P - 2 * ( (ax0 + by0 + cz0 + d) / (a² + b² + c²) ) * (a, b, c)So, in our case, the mirror plane is x + y + √2 z - 4 = 0, so a=1, b=1, c=√2, d=-4.So, the reflection of P(4,4,2) across the mirror plane is:First, compute the numerator: (1*4 + 1*4 + √2*2 - 4) = (4 + 4 + 2√2 - 4) = (4 + 2√2)Denominator: a² + b² + c² = 1 + 1 + 2 = 4So, the scalar factor is 2 * (4 + 2√2)/4 = 2*(1 + (√2)/2) = 2 + √2Wait, no, wait. The formula is:P' = P - 2 * ( (ax0 + by0 + cz0 + d) / (a² + b² + c²) ) * (a, b, c)So, compute (ax0 + by0 + cz0 + d):= 1*4 + 1*4 + √2*2 - 4= 4 + 4 + 2√2 - 4= 4 + 2√2So, the scalar is 2 * (4 + 2√2) / (1 + 1 + 2) = 2*(4 + 2√2)/4 = (4 + 2√2)/2 = 2 + √2So, the reflection point P' is:P' = (4,4,2) - (2 + √2)*(1,1,√2)Compute each component:x: 4 - (2 + √2)*1 = 4 - 2 - √2 = 2 - √2y: 4 - (2 + √2)*1 = 4 - 2 - √2 = 2 - √2z: 2 - (2 + √2)*√2 = 2 - (2√2 + 2) = 2 - 2√2 - 2 = -2√2So, P' is (2 - √2, 2 - √2, -2√2)Now, the reflected light would appear to come from P', so the reflected ray is the line from Q to P'. But since the light is going from P to Q and then reflects to some point, the direction of the reflected light is towards P'.But actually, the reflected direction is the direction from Q to P'. So, if I can find Q, then I can find the direction vector as P' - Q.But wait, Q is the intersection point of the line from P to the mirror plane. But since P' is the reflection, the line from P to Q is the same as the line from Q to P' in the reflected path.Wait, maybe I can find Q by finding the intersection of the line from P to P' with the mirror plane.Wait, that makes sense because the reflection path would be such that the incident path from P to Q and the reflected path from Q to P' are symmetric with respect to the mirror plane.So, the line from P to P' passes through Q. So, let me parametrize the line from P(4,4,2) to P'(2 - √2, 2 - √2, -2√2).The direction vector from P to P' is:(2 - √2 - 4, 2 - √2 - 4, -2√2 - 2) = (-2 - √2, -2 - √2, -2 - 2√2)So, parametric equations:x = 4 + t*(-2 - √2)y = 4 + t*(-2 - √2)z = 2 + t*(-2 - 2√2)We need to find t where this line intersects the mirror plane x + y + √2 z = 4.Substitute x, y, z into the plane equation:[4 + t*(-2 - √2)] + [4 + t*(-2 - √2)] + √2 [2 + t*(-2 - 2√2)] = 4Simplify:4 - (2 + √2)t + 4 - (2 + √2)t + √2*2 + √2*(-2 - 2√2)t = 4Combine like terms:(4 + 4 + 2√2) + [ - (2 + √2) - (2 + √2) - (2√2 + 2*2) ] t = 4Wait, let me compute each part step by step.First, expand each term:First term: 4 + t*(-2 - √2)Second term: 4 + t*(-2 - √2)Third term: √2*2 + √2*t*(-2 - 2√2) = 2√2 + t*(-2√2 - 2*(√2)^2) = 2√2 + t*(-2√2 - 4)Now, add all three terms:[4 + 4 + 2√2] + [ (-2 - √2)t + (-2 - √2)t + (-2√2 - 4)t ] = 4Simplify constants:4 + 4 + 2√2 = 8 + 2√2Now, the coefficients of t:(-2 - √2) + (-2 - √2) + (-2√2 - 4) = (-2 - √2 -2 - √2 -2√2 -4) = (-2 -2 -4) + (-√2 -√2 -2√2) = (-8) + (-4√2)So, the equation becomes:8 + 2√2 + (-8 -4√2)t = 4Bring constants to one side:(-8 -4√2)t = 4 - 8 - 2√2(-8 -4√2)t = -4 - 2√2Multiply both sides by -1:(8 + 4√2)t = 4 + 2√2Factor numerator and denominator:Left side: 4*(2 + √2) tRight side: 2*(2 + √2)So,4*(2 + √2) t = 2*(2 + √2)Divide both sides by (2 + √2):4t = 2So, t = 2/4 = 1/2Now, plug t=1/2 into the parametric equations to find Q:x = 4 + (1/2)*(-2 - √2) = 4 -1 - (√2)/2 = 3 - (√2)/2y = 4 + (1/2)*(-2 - √2) = 4 -1 - (√2)/2 = 3 - (√2)/2z = 2 + (1/2)*(-2 - 2√2) = 2 -1 - √2 = 1 - √2So, the point Q is (3 - (√2)/2, 3 - (√2)/2, 1 - √2)Now, to find the direction of the reflected light, we can take the direction from Q to P', which is the same as the direction from Q to the reflection point.But since we have the parametric line from P to P', and we found Q at t=1/2, the reflected direction is the direction from Q to P', which is the same as the direction vector from Q to P'.But since the direction vector from P to P' is (-2 - √2, -2 - √2, -2 - 2√2), and Q is at t=1/2, the direction from Q to P' is the same as the direction vector, because it's a straight line.Alternatively, the reflected direction can be found using the reflection formula.But perhaps it's easier to compute the direction vector from Q to P':P' is (2 - √2, 2 - √2, -2√2)Q is (3 - √2/2, 3 - √2/2, 1 - √2)So, the direction vector is:(2 - √2 - (3 - √2/2), 2 - √2 - (3 - √2/2), -2√2 - (1 - √2))Simplify each component:x: 2 - √2 -3 + √2/2 = (-1) + (-√2 + √2/2) = -1 - (√2)/2y: same as x: -1 - (√2)/2z: -2√2 -1 + √2 = (-1) + (-2√2 + √2) = -1 - √2So, the direction vector is (-1 - √2/2, -1 - √2/2, -1 - √2)But we can factor out -1:= - (1 + √2/2, 1 + √2/2, 1 + √2)Alternatively, we can write it as (- (2 + √2)/2, - (2 + √2)/2, - (2 + 2√2)/2 )Wait, let me compute each component:x: -1 - √2/2 = -(1 + √2/2) = -(2 + √2)/2Similarly for y: same as xz: -1 - √2 = -(1 + √2)So, the direction vector is (-(2 + √2)/2, -(2 + √2)/2, -(1 + √2))We can write this as:(- (2 + √2)/2, - (2 + √2)/2, - (1 + √2))Alternatively, we can factor out -1:= - [ (2 + √2)/2, (2 + √2)/2, (1 + √2) ]But direction vectors can be scaled, so we can write it as:(2 + √2, 2 + √2, 2(1 + √2)) multiplied by -1/2Wait, let me see:If I factor out - (2 + √2)/2 from x and y, and - (1 + √2) from z, it's a bit messy. Maybe it's better to leave it as is.Alternatively, we can write the direction vector as:( - (2 + √2)/2, - (2 + √2)/2, - (1 + √2) )But to make it simpler, we can multiply numerator and denominator to rationalize or make it look cleaner, but perhaps it's acceptable as is.So, the direction of the reflected light is given by the vector (- (2 + √2)/2, - (2 + √2)/2, - (1 + √2)).Alternatively, we can write it as:( - (2 + √2), - (2 + √2), - 2(1 + √2) ) scaled by 1/2, but I think the first form is fine.So, to summarize:The point where the light hits the mirror is Q(3 - √2/2, 3 - √2/2, 1 - √2), and the direction of the reflected light is (- (2 + √2)/2, - (2 + √2)/2, - (1 + √2)).Alternatively, we can write the direction vector as:( - (2 + √2), - (2 + √2), - 2(1 + √2) ) scaled by 1/2, but I think the first form is acceptable.Wait, let me double-check the calculations for Q.We had t=1/2, so plugging into the parametric equations:x = 4 + (1/2)*(-2 - √2) = 4 -1 - (√2)/2 = 3 - √2/2Similarly for y.z = 2 + (1/2)*(-2 - 2√2) = 2 -1 - √2 = 1 - √2Yes, that's correct.And the direction vector from Q to P' is:P' - Q = (2 - √2 - (3 - √2/2), 2 - √2 - (3 - √2/2), -2√2 - (1 - √2))Compute each component:x: 2 - √2 -3 + √2/2 = (-1) + (-√2 + √2/2) = -1 - √2/2Similarly for y.z: -2√2 -1 + √2 = (-1) + (-2√2 + √2) = -1 - √2Yes, that's correct.So, the direction vector is (-1 - √2/2, -1 - √2/2, -1 - √2)Alternatively, we can factor out -1:= - (1 + √2/2, 1 + √2/2, 1 + √2)But to make it look neater, perhaps we can write it as:= - ( (2 + √2)/2, (2 + √2)/2, (2 + 2√2)/2 )Wait, because 1 + √2/2 = (2 + √2)/2, and 1 + √2 = (2 + 2√2)/2.So, yes, the direction vector can be written as:- ( (2 + √2)/2, (2 + √2)/2, (2 + 2√2)/2 )Which simplifies to:( - (2 + √2)/2, - (2 + √2)/2, - (2 + 2√2)/2 )Alternatively, we can factor out - (2 + √2)/2 from x and y, and - (2 + 2√2)/2 from z, but it's essentially the same.So, the direction vector is:( - (2 + √2)/2, - (2 + √2)/2, - (2 + 2√2)/2 )We can also write this as:( - (2 + √2), - (2 + √2), - (2 + 2√2) ) multiplied by 1/2.But in any case, that's the direction vector.So, to recap:1. The equation of the mirror plane is x + y + √2 z = 4.2. The light hits the mirror at Q(3 - √2/2, 3 - √2/2, 1 - √2), and the reflected direction is given by the vector (- (2 + √2)/2, - (2 + √2)/2, - (2 + 2√2)/2 ).I think that's it.</think>