🧠:<think>Okay, so I've got this problem about Alex planning a multi-purpose commercial building. It's divided into two parts: space optimization and material cost minimization. Let me try to tackle them step by step.Starting with the first part, the space optimization problem. The plot is 150 meters by 200 meters. So, first, I should calculate the total available land area. That's straightforward: length times width. So, 150 * 200 = 30,000 square meters. Got that.Now, the total floor area of the building shouldn't exceed 60% of this land area. So, 60% of 30,000 is 0.6 * 30,000 = 18,000 square meters. So, the building can't be more than 18,000 m² in total.Next, the building has three parts: retail, office, and storage. The retail space needs at least 30% of the total building area, and the office space needs at least 50%. Hmm, wait, that seems like a problem because 30% + 50% is 80%, which is more than the total building area. That can't be right. Maybe I misread it.Wait, let me check again. It says retail requires at least 30% of the total building floor area, and office requires at least 50%. So, if the total building area is, say, T, then retail is at least 0.3T and office is at least 0.5T. But 0.3T + 0.5T = 0.8T, which is less than T, so storage can take the remaining 0.2T. But wait, is that the case?Wait, no, because the total floor area is 18,000 m². So, if retail is at least 30% of 18,000, which is 5,400 m², and office is at least 50% of 18,000, which is 9,000 m². So, 5,400 + 9,000 = 14,400 m². That leaves 18,000 - 14,400 = 3,600 m² for storage. So, the maximum storage area would be 3,600 m².But wait, the problem says "determine the maximum possible area allocated for the storage space." So, to maximize storage, we need to minimize the areas allocated to retail and office, right? Because if we allocate the minimum required to retail and office, the remaining will be the maximum for storage.So, the constraints are:1. Retail ≥ 30% of total building area (T)2. Office ≥ 50% of total building area (T)3. Total building area ≤ 18,000 m²But wait, if T is the total building area, which is ≤18,000, then:Retail ≥ 0.3TOffice ≥ 0.5TStorage = T - Retail - OfficeTo maximize storage, we need to set Retail and Office to their minimums.So, Retail = 0.3TOffice = 0.5TThen, Storage = T - 0.3T - 0.5T = 0.2TBut T is ≤18,000, so Storage ≤0.2*18,000=3,600 m².Wait, but is there a way to have T less than 18,000? Because if T is less, then storage could be less, but we want to maximize storage, so we should set T to its maximum, which is 18,000. So, Storage would be 0.2*18,000=3,600 m².So, the maximum storage area is 3,600 m².But let me set this up as a linear programming problem to confirm.Let me define variables:Let R = retail areaO = office areaS = storage areaTotal building area T = R + O + SConstraints:1. R ≥ 0.3T2. O ≥ 0.5T3. T ≤ 18,000We need to maximize S.Express S in terms of T:S = T - R - OBut from constraints 1 and 2:R ≥ 0.3TO ≥ 0.5TSo, to maximize S, set R = 0.3T and O = 0.5T. Then S = T - 0.3T - 0.5T = 0.2T.Since T can be up to 18,000, then S = 0.2*18,000=3,600.So, the maximum storage area is 3,600 m².Now, moving on to the second part, the material cost minimization problem.The costs are 5 per m² for retail, 8 for office, and 2 for storage. We need to formulate the total annual maintenance cost as a function of R, O, S, and then find the critical points to minimize it.But wait, from the first part, we have fixed areas: R=5,400, O=9,000, S=3,600. So, the total cost would be 5*5,400 + 8*9,000 + 2*3,600.But the problem says "given the areas found from the optimization problem above," so we have fixed areas. But then it says "formulate a multi-variable function for the total annual maintenance cost. Determine the critical points and identify the conditions under which the maintenance cost is minimized."Wait, but if the areas are fixed, then the cost is fixed as well. So, maybe I'm misunderstanding.Alternatively, perhaps the areas are variables, and we need to minimize the cost subject to the constraints from the first part.Wait, let me read again."Given the areas found from the optimization problem above, formulate a multi-variable function for the total annual maintenance cost. Determine the critical points and identify the conditions under which the maintenance cost is minimized."Hmm, so maybe the areas are fixed as R=5,400, O=9,000, S=3,600, so the cost is fixed. But that doesn't make sense for a multi-variable function. Alternatively, perhaps it's a separate problem where the areas are variables, but subject to the same constraints as the first part, and we need to minimize the cost.Wait, perhaps the first part was just to find the maximum storage, but for the second part, we can vary the areas within the constraints to minimize the cost.Let me think.In the first part, we maximized storage by setting R and O to their minimums. For the second part, perhaps we can adjust R and O within their constraints to minimize the total cost.So, the total cost function would be C = 5R + 8O + 2S.But since S = T - R - O, and T is fixed at 18,000, we can write C in terms of R and O.But wait, T is fixed at 18,000, so S = 18,000 - R - O.So, C = 5R + 8O + 2(18,000 - R - O) = 5R + 8O + 36,000 - 2R - 2O = (5R - 2R) + (8O - 2O) + 36,000 = 3R + 6O + 36,000.Now, we need to minimize C = 3R + 6O + 36,000.Subject to the constraints:1. R ≥ 0.3T = 0.3*18,000 = 5,4002. O ≥ 0.5T = 0.5*18,000 = 9,0003. R + O ≤ 18,000 (since S ≥0)But wait, R and O are already set to their minimums in the first part, so if we try to minimize C, which is 3R + 6O, we should set R and O as small as possible, which is exactly what we did in the first part. So, the minimum cost would be when R=5,400 and O=9,000, giving C=3*5,400 + 6*9,000 + 36,000.Calculating that: 3*5,400=16,200; 6*9,000=54,000; 16,200+54,000=70,200; 70,200+36,000=106,200.So, the total cost is 106,200 per year.But the problem says "formulate a multi-variable function for the total annual maintenance cost. Determine the critical points and identify the conditions under which the maintenance cost is minimized."Wait, but in this case, the function is linear, so the minimum occurs at the vertices of the feasible region. The feasible region is defined by R ≥5,400, O ≥9,000, and R + O ≤18,000.So, the vertices are:1. R=5,400, O=9,000: this is the point where both R and O are at their minimums.2. R=5,400, O=18,000 -5,400=12,600: but wait, O can't exceed 18,000 - R, but O's minimum is 9,000, so this point is R=5,400, O=12,600.3. R=18,000 - O, with O=9,000: R=9,000, O=9,000.Wait, but R can't exceed 18,000 - O, but R's minimum is 5,400. So, the feasible region is a polygon with vertices at (5,400,9,000), (5,400,12,600), and (9,000,9,000).Wait, let me check:If R is at minimum 5,400, then O can vary from 9,000 up to 18,000 -5,400=12,600.Similarly, if O is at minimum 9,000, then R can vary from 5,400 up to 18,000 -9,000=9,000.So, the feasible region is a rectangle with vertices at (5,400,9,000), (5,400,12,600), (9,000,9,000), and (9,000,12,600). Wait, no, because R + O ≤18,000, so when R=9,000, O can be at most 9,000, but O's minimum is 9,000, so that point is (9,000,9,000). Similarly, when O=12,600, R=5,400.So, the feasible region is actually a line segment from (5,400,9,000) to (5,400,12,600) and another from (5,400,9,000) to (9,000,9,000). Wait, no, because R and O can vary within their constraints as long as R + O ≤18,000.Wait, perhaps it's better to plot it, but since I can't, I'll think through.The constraints are:1. R ≥5,4002. O ≥9,0003. R + O ≤18,000So, the feasible region is the set of points where R is between 5,400 and 18,000 - O, and O is between 9,000 and 18,000 - R.But since R and O are both bounded below, the feasible region is a polygon with vertices at (5,400,9,000), (5,400,12,600), and (9,000,9,000). Because when R=5,400, O can go up to 12,600, and when O=9,000, R can go up to 9,000.So, the vertices are:A: (5,400,9,000)B: (5,400,12,600)C: (9,000,9,000)Now, to find the minimum of C=3R +6O +36,000, we evaluate C at each vertex.At A: C=3*5,400 +6*9,000 +36,000=16,200 +54,000 +36,000=106,200At B: C=3*5,400 +6*12,600 +36,000=16,200 +75,600 +36,000=127,800At C: C=3*9,000 +6*9,000 +36,000=27,000 +54,000 +36,000=117,000So, the minimum is at point A: (5,400,9,000) with C=106,200.Therefore, the minimum cost is achieved when R=5,400 and O=9,000, which is exactly the same as the first part where we maximized storage. So, this makes sense because storage has the lowest maintenance cost, so to minimize total cost, we should maximize storage, which requires minimizing R and O to their minimums.So, the critical point is at R=5,400, O=9,000, and S=3,600.Therefore, the conditions for minimum maintenance cost are when retail and office spaces are at their minimum required areas, allowing storage to take up the maximum possible area.</think>

🧠:<think>Alright, so I've got these two sub-problems to solve related to the spaceship illustration. Let me tackle them one by one. I'll start with Sub-problem 1.Sub-problem 1:The main body of the spaceship is modeled by the paraboloid equation ( z = x^2 + y^2 ). The height from the base to the highest point is 20 meters. I need to find the radius of the base.Hmm, okay. So, a paraboloid is a three-dimensional shape that's like a parabola rotated around its axis. In this case, the equation is ( z = x^2 + y^2 ). That means for any point (x, y) on the base, the height z is determined by the sum of the squares of x and y. The highest point of the spaceship would be the vertex of the paraboloid. But wait, in the equation ( z = x^2 + y^2 ), the vertex is at the origin (0,0,0). So, if the height is 20 meters from the base to the highest point, does that mean the vertex is 20 meters above the base? Or is the base at z=0, and the highest point is at z=20?Wait, the problem says the height is exactly 20 meters from its base to its highest point. So, the base is probably at some lower z-value, and the highest point is 20 meters above the base. But in the equation given, z is a function of x and y, so the base would be where z is at its minimum, which is at the origin. So, if the base is at z=0, then the highest point is at z=20. That makes sense.So, the equation is ( z = x^2 + y^2 ). At the highest point, which is the vertex, z is 20. But wait, in the equation, the vertex is at (0,0,0). So, maybe the equation is scaled differently. Maybe it's ( z = a(x^2 + y^2) ), where a is a scaling factor.Wait, the given equation is ( z = x^2 + y^2 ). So, if I plug in x=0 and y=0, z=0. So, the vertex is at the origin. If the height is 20 meters, that must mean that the maximum z-value is 20. So, the paraboloid extends from z=0 up to z=20. Therefore, the base is at z=0, and the highest point is at z=20.But then, the radius of the base would be the radius at z=0, which is zero. That doesn't make sense because the base can't have zero radius. Maybe I misunderstood the problem.Wait, perhaps the base is not at z=0. Maybe the base is at some z-value, and the height is 20 meters from the base to the highest point. So, the total height is 20 meters, meaning the distance between the base and the vertex is 20 meters.But in the equation ( z = x^2 + y^2 ), the vertex is at (0,0,0). So, if the base is at z = -20, then the height from the base to the vertex would be 20 meters. But then, the radius of the base would be the radius at z = -20. Let's see.Wait, but in the equation ( z = x^2 + y^2 ), z is always non-negative because it's the sum of squares. So, z cannot be negative. Therefore, the base must be at z=0, and the highest point is at z=20. So, the radius of the base would be the radius at z=0, which is zero. That can't be right because the base must have some positive radius.Wait, maybe the equation is different. Maybe it's ( z = a(x^2 + y^2) ), and we need to find a such that when z=20, the radius is the base radius. But the problem says the equation is ( z = x^2 + y^2 ). So, perhaps the base is at z=20, and the vertex is at z=0. But then, the height would be 20 meters from the base to the vertex. But in that case, the base is at z=20, and the vertex is at z=0, so the height is 20 meters.Wait, but the equation is ( z = x^2 + y^2 ). So, if the base is at z=20, then the radius at the base would be sqrt(20), because z = x^2 + y^2 = 20, so x^2 + y^2 = 20, so radius r = sqrt(20). But sqrt(20) is approximately 4.472 meters. But the problem says the height is 20 meters from the base to the highest point. If the base is at z=20, then the highest point is at z=0, which is 20 meters below. But that would mean the spaceship is upside down, which is possible, but the problem doesn't specify that.Alternatively, maybe the equation is ( z = -x^2 - y^2 ), which would open downward, so the vertex is at (0,0,0), and the base is at z=-20, with radius sqrt(20). But again, the problem states the equation is ( z = x^2 + y^2 ), so it opens upward.Wait, perhaps the height is measured from the vertex to the base. So, if the vertex is at z=0, and the base is at z=20, then the radius at the base is sqrt(20). But that would make the spaceship a paraboloid opening upward, with the vertex at the bottom and the base at the top. That might make sense.But the problem says the height is from the base to the highest point. So, if the base is at z=20, and the highest point is at z=0, that would be a height of 20 meters, but it's upside down. Alternatively, if the base is at z=0, and the highest point is at z=20, then the radius at the base is zero, which doesn't make sense.Wait, maybe I'm overcomplicating this. Let's think differently. The paraboloid equation is ( z = x^2 + y^2 ). The height from the base to the highest point is 20 meters. So, the base is a circle at some z-value, and the highest point is 20 meters above the base.But in the equation, z is a function of x and y. So, if the base is at z = h, then the highest point is at z = h + 20. But the equation is ( z = x^2 + y^2 ), so the highest point would be where x and y are maximum. Wait, no, because as x and y increase, z increases without bound. So, unless we have a finite size.Wait, maybe the spaceship is a finite paraboloid, meaning it has a certain height and a certain base radius. So, the equation is ( z = x^2 + y^2 ), and the height from the base (z=0) to the highest point is 20 meters. So, the highest point is at z=20, which occurs when x^2 + y^2 = 20. Therefore, the radius at the base (z=0) is zero, but that can't be.Wait, maybe the base is not at z=0. Maybe the base is at z=20, and the vertex is at z=0, so the height is 20 meters. Then, the radius at the base (z=20) would be sqrt(20). But then, the equation would be ( z = x^2 + y^2 ), so at z=20, x^2 + y^2 = 20, so radius r = sqrt(20) ≈ 4.472 meters.But the problem says the height is from the base to the highest point. If the base is at z=20, and the highest point is at z=0, that's a height of 20 meters, but the spaceship would be upside down. Alternatively, if the base is at z=0, and the highest point is at z=20, then the radius at the base is zero, which is not possible.Wait, maybe the equation is scaled differently. Maybe it's ( z = (x^2 + y^2)/a ), so that when z=20, x^2 + y^2 = a*20. Then, the radius at z=20 would be sqrt(a*20). But the problem states the equation is ( z = x^2 + y^2 ), so a=1.I'm getting confused. Let me try to visualize this. A paraboloid opening upward has its vertex at the bottom. So, the base would be a circle at some height z=h, and the vertex is at z=0. The height from the base to the vertex is h. So, if h=20, then the base is at z=20, and the radius at the base is sqrt(20). Therefore, the radius is sqrt(20) meters.But the problem says the height is 20 meters from the base to the highest point. If the base is at z=20, and the highest point is at z=0, that's a height of 20 meters. So, the radius of the base is sqrt(20). Simplifying sqrt(20) is 2*sqrt(5), which is approximately 4.472 meters.Wait, but the problem says the height is from the base to the highest point, which is 20 meters. So, if the base is at z=20, and the highest point is at z=0, that's a height of 20 meters. Therefore, the radius of the base is sqrt(20). So, the answer is sqrt(20) meters, which can be simplified to 2*sqrt(5) meters.But let me double-check. If the equation is ( z = x^2 + y^2 ), then at the base (z=20), x^2 + y^2 = 20, so the radius is sqrt(20). Yes, that makes sense.So, the radius of the base is 2*sqrt(5) meters.Sub-problem 2:The antenna array consists of 5 concentric circular rings, each with radii increasing by 2 meters from the innermost to the outermost ring. The innermost ring has a radius of 1 meter. I need to calculate the total area covered by all 5 rings combined.Wait, concentric circular rings. So, each ring is an annulus, meaning the area between two circles. So, the innermost ring has an inner radius of 0 and an outer radius of 1 meter. Then, each subsequent ring increases the outer radius by 2 meters.Wait, but the problem says "each with radii increasing by 2 meters from the innermost to the outermost ring." So, the innermost ring has a radius of 1 meter. Then, the next one has a radius of 1 + 2 = 3 meters, then 5, 7, 9 meters. So, the outermost ring has a radius of 9 meters.But wait, if they are concentric rings, each ring is an annulus. So, the first ring (innermost) is from 0 to 1 meters. The second ring is from 1 to 3 meters. The third is from 3 to 5 meters. The fourth is from 5 to 7 meters. The fifth is from 7 to 9 meters.So, each ring's area is the area of the outer circle minus the area of the inner circle.So, the areas would be:1st ring: π*(1)^2 - π*(0)^2 = π*1 = π m²2nd ring: π*(3)^2 - π*(1)^2 = π*(9 - 1) = 8π m²3rd ring: π*(5)^2 - π*(3)^2 = π*(25 - 9) = 16π m²4th ring: π*(7)^2 - π*(5)^2 = π*(49 - 25) = 24π m²5th ring: π*(9)^2 - π*(7)^2 = π*(81 - 49) = 32π m²Now, adding all these areas together:π + 8π + 16π + 24π + 32π = (1 + 8 + 16 + 24 + 32)π = 81π m²So, the total area is 81π square meters.Wait, let me verify. The radii are 1, 3, 5, 7, 9. So, the areas of the annuli are:Between 0-1: π(1² - 0²) = πBetween 1-3: π(3² - 1²) = π(9 - 1) = 8πBetween 3-5: π(25 - 9) = 16πBetween 5-7: π(49 - 25) = 24πBetween 7-9: π(81 - 49) = 32πAdding them up: π + 8π + 16π + 24π + 32π = (1 + 8 + 16 + 24 + 32)π = 81πYes, that seems correct.Alternatively, another way to think about it is that the total area covered by all rings is the area of the largest circle minus the area of the innermost circle. But wait, no, because each ring is an annulus, and the total area is the sum of all annuli, which is the same as the area of the largest circle (radius 9) minus the area of the innermost circle (radius 1). But wait, that would be π*(9)^2 - π*(1)^2 = 81π - π = 80π. But that contradicts the previous result of 81π.Wait, that's a problem. Which one is correct?Wait, no. Because if you have 5 rings, each with increasing radii, the total area covered is the area of the largest circle (radius 9) minus the area of the smallest circle (radius 1). So, that would be π*(9)^2 - π*(1)^2 = 81π - π = 80π.But when I calculated each annulus and summed them, I got 81π. There's a discrepancy here.Wait, let's see. The first ring is from 0 to 1: area π.The second ring is from 1 to 3: area 8π.Third: 16πFourth: 24πFifth: 32πTotal: π + 8π + 16π + 24π + 32π = 81π.But if I consider the total area as the largest circle minus the smallest, it's 81π - π = 80π.So, which is correct?Wait, the issue is whether the innermost ring includes the area from 0 to 1, which is a full circle, not an annulus. So, the first ring is a full circle, and the subsequent rings are annuli. Therefore, the total area is the sum of the first ring (a circle) plus the areas of the four annuli.Wait, but the problem says "5 concentric circular rings". So, each ring is an annulus. So, the innermost ring would be from 0 to 1, which is a circle, but in reality, an annulus with inner radius 0 and outer radius 1. So, it's still an annulus, just with inner radius 0.Therefore, the total area is the sum of all five annuli, which is 81π.But when I subtract the smallest circle from the largest, I get 80π. So, why the difference?Because the innermost annulus (from 0 to 1) is a full circle, and the rest are annuli. So, the total area is the area of the largest circle (radius 9) minus the area of the innermost circle (radius 1). Wait, no, that would be 81π - π = 80π. But according to the annulus sum, it's 81π.Wait, perhaps I made a mistake in the annulus areas. Let me recalculate.First ring (0-1): π*(1)^2 - π*(0)^2 = πSecond ring (1-3): π*(3)^2 - π*(1)^2 = 9π - π = 8πThird ring (3-5): π*(5)^2 - π*(3)^2 = 25π - 9π = 16πFourth ring (5-7): π*(7)^2 - π*(5)^2 = 49π - 25π = 24πFifth ring (7-9): π*(9)^2 - π*(7)^2 = 81π - 49π = 32πTotal: π + 8π + 16π + 24π + 32π = 81πBut the area of the largest circle is 81π, and the area of the innermost circle is π. So, 81π - π = 80π. But according to the annulus sum, it's 81π. So, there's a discrepancy of π.Wait, perhaps the innermost ring is not including the area from 0 to 1, but just the ring itself. But that doesn't make sense because a ring from 0 to 1 is just a circle. So, if we consider that the innermost ring is a circle, then the total area is 81π. But if we consider that the innermost ring is an annulus with inner radius 0, then it's still a circle, so the total area is 81π.Wait, maybe the problem is considering the innermost ring as a circle, and the other four as annuli. So, the total area is π + 8π + 16π + 24π + 32π = 81π.Alternatively, if the innermost ring is considered as an annulus with inner radius 0, then the total area is 81π.But if we consider the total area as the largest circle minus the innermost circle, it's 80π. So, which is correct?Wait, let's think about it. The total area covered by all five rings is the area of the largest circle (radius 9) minus the area of the innermost circle (radius 1). Because each ring is an annulus, and the innermost ring is from 0 to 1, which is a circle, but the subsequent rings are annuli. So, the total area is the area of the largest circle minus the area of the innermost circle.Wait, no. Because the innermost ring is from 0 to 1, which is a circle, and the next ring is from 1 to 3, which is an annulus, and so on. So, the total area is the sum of all these, which includes the innermost circle. So, the total area is 81π.But the area of the largest circle is 81π, and the innermost circle is π. So, the area between 1 and 9 is 80π, but the total area including the innermost circle is 81π.So, the problem says "the total area covered by all 5 rings combined." So, if each ring is an annulus, including the innermost one, which is a circle, then the total area is 81π.Alternatively, if the innermost ring is considered as an annulus with inner radius 0, then it's a circle, and the total area is 81π.Therefore, the answer should be 81π square meters.But let me check again. The innermost ring has a radius of 1 meter. So, it's a circle with radius 1. The next ring has a radius of 3 meters, so it's an annulus from 1 to 3. The third ring is from 3 to 5, and so on.So, the areas are:1st ring: π*(1)^2 = π2nd ring: π*(3)^2 - π*(1)^2 = 9π - π = 8π3rd ring: π*(5)^2 - π*(3)^2 = 25π - 9π = 16π4th ring: π*(7)^2 - π*(5)^2 = 49π - 25π = 24π5th ring: π*(9)^2 - π*(7)^2 = 81π - 49π = 32πTotal: π + 8π + 16π + 24π + 32π = 81πYes, that's correct.Alternatively, if I think of it as the area of the largest circle (radius 9) minus the area of the innermost circle (radius 1), that would be 81π - π = 80π. But that would exclude the innermost ring. But since the innermost ring is included, the total area is 81π.Therefore, the total area covered by all five rings is 81π square meters.Wait, but let me think again. If the innermost ring is a circle, and the other four are annuli, then the total area is the sum of the innermost circle plus the four annuli. So, that would be π + (8π + 16π + 24π + 32π) = π + 80π = 81π.Yes, that's correct.So, the total area is 81π square meters.Final AnswerSub-problem 1: The radius of the base is boxed{2sqrt{5}} meters.Sub-problem 2: The total area covered by all 5 rings is boxed{81pi} square meters.</think>

🧠:<think>Okay, so I have this problem where a pharmacist and a computer engineer are working together to develop a new drug formulation. The efficacy of the drug is modeled by this function E(x) = (a x³ + b x² + c x + d)/(x² + 1). I need to solve two sub-problems related to this function.Starting with Sub-problem 1: They say that the maximum efficacy occurs at x = 4 mg. I need to find the relationship between the constants a, b, c, and d by taking the derivative of E(x) and setting it to zero at x = 4.Alright, so to find the maximum, I know I need to take the derivative of E(x) with respect to x, set it equal to zero, and plug in x = 4. That should give me an equation involving a, b, c, and d.Let me recall how to take the derivative of a quotient. The derivative of E(x) = numerator/denominator is (num’ * denom - num * denom’) / denom². So, let me define numerator = a x³ + b x² + c x + d, and denominator = x² + 1.First, compute the derivatives of numerator and denominator:Numerator derivative: num’ = 3a x² + 2b x + cDenominator derivative: denom’ = 2xSo, E’(x) = [ (3a x² + 2b x + c)(x² + 1) - (a x³ + b x² + c x + d)(2x) ] / (x² + 1)²Now, since the maximum occurs at x = 4, E’(4) = 0. So, the numerator must be zero at x = 4.Let me compute the numerator at x = 4:Numerator = (3a*(4)^2 + 2b*4 + c)*(4^2 + 1) - (a*(4)^3 + b*(4)^2 + c*4 + d)*(2*4)Simplify step by step:First, compute each part:3a*(16) = 48a2b*4 = 8bSo, the first part of the first term is 48a + 8b + c.Then, 4^2 + 1 = 16 + 1 = 17.So, the first term is (48a + 8b + c)*17.Now, the second part:a*(64) = 64ab*(16) = 16bc*4 = 4cSo, the numerator of the second term is 64a + 16b + 4c + d.Multiply this by 2*4 = 8:So, the second term is (64a + 16b + 4c + d)*8.Putting it all together, the numerator is:(48a + 8b + c)*17 - (64a + 16b + 4c + d)*8 = 0Let me compute each multiplication:First term: (48a + 8b + c)*17Compute 48a*17: 48*17 = 816, so 816a8b*17 = 136bc*17 = 17cSo, first term is 816a + 136b + 17cSecond term: (64a + 16b + 4c + d)*8Compute 64a*8 = 512a16b*8 = 128b4c*8 = 32cd*8 = 8dSo, second term is 512a + 128b + 32c + 8dNow, subtract the second term from the first term:(816a + 136b + 17c) - (512a + 128b + 32c + 8d) = 0Compute each coefficient:816a - 512a = 304a136b - 128b = 8b17c - 32c = -15c-8dSo, the equation becomes:304a + 8b - 15c - 8d = 0Hmm, so that's the relationship between a, b, c, d.Wait, let me double-check my calculations to make sure I didn't make a mistake.First term: (48a + 8b + c)*1748*17: 40*17=680, 8*17=136, so 680+136=816a. Correct.8b*17=136b. Correct.c*17=17c. Correct.Second term: (64a + 16b + 4c + d)*864*8=512a. Correct.16*8=128b. Correct.4*8=32c. Correct.d*8=8d. Correct.Subtracting:816a - 512a = 304a136b - 128b = 8b17c - 32c = -15c-8dYes, that seems correct.So, the equation is 304a + 8b -15c -8d = 0.I can write this as 304a + 8b -15c -8d = 0.Alternatively, I can factor out some common factors. Let's see:304a: 16*19a8b: 8b-15c: -15c-8d: -8dHmm, not sure if there's a common factor. Maybe 8 is a common factor for some terms:304a = 8*38a8b = 8*1b-15c remains as is-8d = 8*(-1)dSo, factor out 8:8*(38a + b - d) -15c = 0But that might not be particularly helpful. Alternatively, maybe write the equation as:304a + 8b -15c = 8dSo, d = (304a + 8b -15c)/8Which simplifies to:d = 38a + b - (15/8)cHmm, that could be a way to express d in terms of a, b, c.Alternatively, leave it as 304a + 8b -15c -8d = 0.Either way, that's the relationship.So, for Sub-problem 1, the relationship is 304a + 8b -15c -8d = 0.Moving on to Sub-problem 2: The computer engineer wants the efficacy E(x) > 0.75 for dosages between 3 mg and 5 mg. So, E(x) > 0.75 for 3 ≤ x ≤ 5. I need to determine the range of values for a, b, c, d that satisfy this condition.Hmm, okay. So, E(x) = (a x³ + b x² + c x + d)/(x² + 1) > 0.75 for all x in [3,5].So, that inequality must hold for all x between 3 and 5.Let me rewrite the inequality:(a x³ + b x² + c x + d)/(x² + 1) > 0.75Multiply both sides by (x² + 1), which is always positive, so the inequality sign doesn't change:a x³ + b x² + c x + d > 0.75(x² + 1)So, a x³ + b x² + c x + d - 0.75x² - 0.75 > 0Simplify:a x³ + (b - 0.75)x² + c x + (d - 0.75) > 0So, define a new function F(x) = a x³ + (b - 0.75)x² + c x + (d - 0.75)We need F(x) > 0 for all x in [3,5].So, F(x) is a cubic polynomial. We need it to be positive on the interval [3,5].To ensure that F(x) > 0 on [3,5], we need to make sure that the minimum of F(x) on [3,5] is greater than 0.So, perhaps we can find the minimum of F(x) on [3,5] and set that greater than 0.Alternatively, since F(x) is a cubic, it can have up to two turning points. So, we need to check the critical points within [3,5] and also the endpoints.So, let's compute F'(x):F'(x) = 3a x² + 2(b - 0.75)x + cSet F'(x) = 0 to find critical points:3a x² + 2(b - 0.75)x + c = 0This is a quadratic equation. Let me denote:A = 3aB = 2(b - 0.75)C = cSo, quadratic equation: A x² + B x + C = 0Solutions are x = [-B ± sqrt(B² - 4AC)]/(2A)We need to check whether these critical points lie within [3,5].If they do, then we need to evaluate F(x) at those points and ensure they are positive. Also, evaluate F(x) at x=3 and x=5, and ensure they are positive.Alternatively, if the critical points are outside [3,5], then the extrema on [3,5] are at the endpoints.But since F(x) is a cubic, which can have two critical points, it's possible that one or both are in [3,5]. So, we need to consider both possibilities.But this seems a bit involved because it's a general case with variables a, b, c, d.Alternatively, perhaps we can express this condition as F(x) > 0 for x in [3,5], which is equivalent to:For all x in [3,5], a x³ + (b - 0.75)x² + c x + (d - 0.75) > 0But without knowing specific values for a, b, c, d, it's hard to pin down exact ranges. However, maybe we can express this condition in terms of inequalities that a, b, c, d must satisfy.Wait, but we also have the relationship from Sub-problem 1: 304a + 8b -15c -8d = 0. So, perhaps we can use that to express one variable in terms of the others, say d in terms of a, b, c, and then substitute into the inequality.From Sub-problem 1, we have:d = (304a + 8b -15c)/8So, let me substitute d into F(x):F(x) = a x³ + (b - 0.75)x² + c x + (d - 0.75)Substitute d:F(x) = a x³ + (b - 0.75)x² + c x + [(304a + 8b -15c)/8 - 0.75]Simplify the constant term:(304a + 8b -15c)/8 - 0.75 = (304a + 8b -15c)/8 - 3/4Convert 3/4 to eighths: 3/4 = 6/8So, (304a + 8b -15c - 6)/8So, F(x) = a x³ + (b - 0.75)x² + c x + (304a + 8b -15c - 6)/8Hmm, that might not be the most straightforward way. Maybe instead of substituting d, I can keep F(x) as it is and use the relationship to reduce the number of variables.Alternatively, perhaps it's better to consider that since F(x) must be positive on [3,5], we can write inequalities for F(3) > 0 and F(5) > 0, and also ensure that any minima within [3,5] are positive.But since F(x) is a cubic, it's possible that it could dip below zero between 3 and 5 even if the endpoints are positive, depending on the coefficients.So, perhaps a safer approach is to ensure that F(x) is positive at x=3, x=5, and also that the minimum value of F(x) on [3,5] is positive.To find the minimum, we can find the critical points by solving F'(x)=0 and then evaluate F(x) at those points.But since F'(x) is quadratic, we can have 0, 1, or 2 real roots. Depending on the coefficients, the critical points may or may not lie within [3,5].This is getting a bit complicated, but let me try to outline the steps:1. Compute F(3) > 0 and F(5) > 0.2. Find the critical points by solving F'(x)=0.3. For each critical point x_c in [3,5], compute F(x_c) > 0.4. If all these conditions are satisfied, then F(x) > 0 on [3,5].So, let's start with computing F(3) and F(5).First, F(3):F(3) = a*(27) + (b - 0.75)*(9) + c*(3) + (d - 0.75) > 0Simplify:27a + 9b - 6.75 + 3c + d - 0.75 > 0Combine constants: -6.75 - 0.75 = -7.5So, F(3) = 27a + 9b + 3c + d - 7.5 > 0Similarly, F(5):F(5) = a*(125) + (b - 0.75)*(25) + c*(5) + (d - 0.75) > 0Simplify:125a + 25b - 18.75 + 5c + d - 0.75 > 0Combine constants: -18.75 - 0.75 = -19.5So, F(5) = 125a + 25b + 5c + d - 19.5 > 0Now, from Sub-problem 1, we have the relationship:304a + 8b -15c -8d = 0We can use this to express d in terms of a, b, c:From 304a + 8b -15c -8d = 0,8d = 304a + 8b -15cSo, d = (304a + 8b -15c)/8Let me substitute this into F(3) and F(5):First, F(3):27a + 9b + 3c + d - 7.5 > 0Substitute d:27a + 9b + 3c + (304a + 8b -15c)/8 - 7.5 > 0Multiply through by 8 to eliminate the denominator:8*(27a) + 8*(9b) + 8*(3c) + 304a + 8b -15c - 8*7.5 > 0Compute each term:8*27a = 216a8*9b = 72b8*3c = 24c304a remains8b remains-15c remains-8*7.5 = -60So, combining all terms:216a + 72b + 24c + 304a + 8b -15c -60 > 0Combine like terms:a: 216a + 304a = 520ab: 72b + 8b = 80bc: 24c -15c = 9cConstants: -60So, 520a + 80b + 9c -60 > 0We can write this as:520a + 80b + 9c > 60Similarly, let's do the same for F(5):F(5) = 125a + 25b + 5c + d - 19.5 > 0Substitute d:125a + 25b + 5c + (304a + 8b -15c)/8 - 19.5 > 0Multiply through by 8:8*125a + 8*25b + 8*5c + 304a + 8b -15c - 8*19.5 > 0Compute each term:8*125a = 1000a8*25b = 200b8*5c = 40c304a remains8b remains-15c remains-8*19.5 = -156Combine all terms:1000a + 200b + 40c + 304a + 8b -15c -156 > 0Combine like terms:a: 1000a + 304a = 1304ab: 200b + 8b = 208bc: 40c -15c = 25cConstants: -156So, 1304a + 208b + 25c -156 > 0Which can be written as:1304a + 208b + 25c > 156So, now we have two inequalities:1. 520a + 80b + 9c > 602. 1304a + 208b + 25c > 156Additionally, we need to ensure that any critical points within [3,5] result in F(x) > 0.But this is getting quite involved. Maybe instead of trying to handle the critical points, we can consider that if F(x) is positive at x=3 and x=5, and the function is increasing or decreasing appropriately, but it's not straightforward.Alternatively, perhaps we can consider that since F(x) is a cubic, and we have the derivative F'(x) = 3a x² + 2(b - 0.75)x + c.If we can ensure that F'(x) does not have any roots in [3,5], then F(x) is either always increasing or always decreasing on [3,5]. If it's always increasing, then the minimum is at x=3, and if it's always decreasing, the minimum is at x=5. So, if we can ensure that F'(x) doesn't cross zero in [3,5], then we just need to check F(3) and F(5).But if F'(x) does have roots in [3,5], then we need to check F(x) at those critical points as well.So, perhaps we can consider two cases:Case 1: F'(x) has no real roots in [3,5]. Then, F(x) is monotonic on [3,5], so we just need F(3) > 0 and F(5) > 0.Case 2: F'(x) has one or two real roots in [3,5]. Then, we need to ensure that F(x) at those roots is also > 0.But without specific values, it's hard to determine. Maybe we can find conditions on a, b, c such that F'(x) has no roots in [3,5].Alternatively, perhaps we can find the discriminant of F'(x) and see if it's negative, meaning no real roots. But if the discriminant is positive, we have to check if the roots lie within [3,5].The discriminant D of F'(x) = 3a x² + 2(b - 0.75)x + c is:D = [2(b - 0.75)]² - 4*3a*c = 4(b - 0.75)^2 - 12a cIf D < 0, then F'(x) has no real roots, so F(x) is monotonic.If D ≥ 0, then F'(x) has real roots, which may or may not lie in [3,5].So, let's consider D:D = 4(b - 0.75)^2 - 12a cIf D < 0, then F'(x) has no real roots, so F(x) is either always increasing or always decreasing on [3,5]. Then, as long as F(3) > 0 and F(5) > 0, we're good.If D ≥ 0, then we have critical points, and we need to check if they lie in [3,5], and if so, ensure F(x) at those points is > 0.But this is getting quite complex, and without specific values, it's hard to proceed. Maybe the problem expects us to just write the inequalities from F(3) > 0 and F(5) > 0, along with the relationship from Sub-problem 1.So, summarizing:From Sub-problem 1: 304a + 8b -15c -8d = 0From Sub-problem 2:F(3) = 27a + 9b + 3c + d - 7.5 > 0F(5) = 125a + 25b + 5c + d - 19.5 > 0We can substitute d from Sub-problem 1 into these inequalities to get:For F(3):520a + 80b + 9c > 60For F(5):1304a + 208b + 25c > 156Additionally, we might need to consider the critical points, but perhaps for the purpose of this problem, these two inequalities along with the relationship from Sub-problem 1 are sufficient.Alternatively, if we consider that F(x) must be positive on [3,5], and given that F(x) is a cubic, it's possible that even if F(3) and F(5) are positive, F(x) could dip below zero in between. So, to be thorough, we might need to ensure that the minimum of F(x) on [3,5] is positive.But without knowing the exact behavior, it's difficult. Maybe the problem expects us to just provide the inequalities from F(3) and F(5) after substituting d.So, in conclusion, the range of values for a, b, c, d must satisfy:1. 304a + 8b -15c -8d = 02. 520a + 80b + 9c > 603. 1304a + 208b + 25c > 156These are the necessary conditions for E(x) > 0.75 on [3,5].Alternatively, if we express d in terms of a, b, c, we can write d = (304a + 8b -15c)/8, and then the inequalities become:520a + 80b + 9c > 601304a + 208b + 25c > 156Which can be simplified by dividing by common factors if possible.Looking at the first inequality: 520a + 80b + 9c > 60Divide all terms by, say, 5:104a + 16b + (9/5)c > 12Not particularly helpful.Similarly, the second inequality: 1304a + 208b + 25c > 156Divide by 4:326a + 52b + (25/4)c > 39Still not very clean.Alternatively, perhaps we can express these inequalities in terms of a, b, c, and d with the relationship from Sub-problem 1.But I think the key takeaway is that we have three conditions:1. 304a + 8b -15c -8d = 02. 520a + 80b + 9c > 603. 1304a + 208b + 25c > 156These inequalities define the range of a, b, c, d.Alternatively, we can express these inequalities in terms of a, b, c by substituting d.So, from 304a + 8b -15c -8d = 0, we have d = (304a + 8b -15c)/8Substitute into F(3) and F(5):F(3) = 27a + 9b + 3c + d - 7.5 > 0Substitute d:27a + 9b + 3c + (304a + 8b -15c)/8 - 7.5 > 0Multiply through by 8:216a + 72b + 24c + 304a + 8b -15c -60 > 0Combine like terms:(216a + 304a) = 520a(72b + 8b) = 80b(24c -15c) = 9cSo, 520a + 80b + 9c -60 > 0Similarly for F(5):125a + 25b + 5c + d - 19.5 > 0Substitute d:125a + 25b + 5c + (304a + 8b -15c)/8 - 19.5 > 0Multiply through by 8:1000a + 200b + 40c + 304a + 8b -15c -156 > 0Combine like terms:(1000a + 304a) = 1304a(200b + 8b) = 208b(40c -15c) = 25cSo, 1304a + 208b + 25c -156 > 0Therefore, the conditions are:1. 304a + 8b -15c -8d = 02. 520a + 80b + 9c > 603. 1304a + 208b + 25c > 156These are the necessary conditions for the efficacy to be greater than 75% between 3 mg and 5 mg.I think that's as far as I can go without more specific information. So, the range of a, b, c, d is defined by these three inequalities.</think>

🧠:<think>Alright, so Jeremy is this guy who's been planting trees for 30 years, right? He started 30 years ago and every year after that, he plants double the number of trees he did the previous year. That sounds like a geometric progression to me. Let me try to wrap my head around this.First, the problem is asking how many trees Jeremy has planted in total by the end of the 30th year. Okay, so we're dealing with a geometric series here because each year he's doubling the number of trees. That means the number of trees each year forms a geometric sequence where each term is twice the previous one.Let me recall the formula for the sum of a geometric series. The sum ( S_n ) of the first ( n ) terms of a geometric series where the first term is ( a ) and the common ratio is ( r ) is given by:[S_n = a times frac{r^n - 1}{r - 1}]In this case, Jeremy started planting trees 30 years ago, so that's 30 terms. The common ratio ( r ) is 2 because he doubles the number each year. But wait, the problem doesn't specify how many trees he planted in the first year. Hmm, that might be an issue. Let me check the second question.Oh, the second question actually gives us that the total number of trees ( S_{30} ) is ( 2^{30} - 1 ). It also mentions that the initial number of trees is ( T_0 ) and the common ratio is 2. So maybe the first question is just asking for the same thing but without knowing ( T_0 )?Wait, no. The first question says Jeremy planted the first tree 30 years ago. So does that mean he planted 1 tree in the first year? That would make sense because it says "the first tree." So if he planted 1 tree in the first year, then each subsequent year he plants double the previous year. So the number of trees each year would be 1, 2, 4, 8, ..., up to the 30th year.So in that case, the first term ( a ) is 1, the common ratio ( r ) is 2, and the number of terms ( n ) is 30. Plugging into the formula:[S_{30} = 1 times frac{2^{30} - 1}{2 - 1} = 2^{30} - 1]So that would be the total number of trees. But wait, the second question is also about ( S_{30} = 2^{30} - 1 ) and asks for ( T_0 ). So if we use the same formula, if ( S_{30} = 2^{30} - 1 ), then:[S_{30} = T_0 times frac{2^{30} - 1}{2 - 1} = T_0 times (2^{30} - 1)]So if ( S_{30} = 2^{30} - 1 ), then:[T_0 times (2^{30} - 1) = 2^{30} - 1]Which implies that ( T_0 = 1 ). So that makes sense because he planted 1 tree in the first year, and the rest follows.But wait, in the first question, it just says Jeremy planted the first tree 30 years ago and every subsequent year he plants double the number. So does that mean he planted 1 tree in the first year, and then 2, 4, etc.? If that's the case, then the total number of trees is indeed ( 2^{30} - 1 ).But let me double-check. If he plants 1 tree in year 1, then 2 in year 2, 4 in year 3, ..., up to year 30. The total is the sum of a geometric series with 30 terms, starting at 1, ratio 2.Yes, so the sum is ( 2^{30} - 1 ). So both questions are connected. The first question is essentially asking for the total, which is ( 2^{30} - 1 ), and the second question is confirming that if the total is ( 2^{30} - 1 ), then the initial term ( T_0 ) must be 1.So, to answer the first question: Jeremy has planted a total of ( 2^{30} - 1 ) trees by the end of the 30th year.For the second question, since the total is given as ( 2^{30} - 1 ), we can solve for ( T_0 ) and find that ( T_0 = 1 ).Wait, but let me make sure I didn't mix up the questions. The first question is just asking for the total, and the second is giving the total and asking for ( T_0 ). So in the first question, since he planted the first tree 30 years ago, that implies ( T_0 = 1 ), so the total is ( 2^{30} - 1 ). In the second question, they give the total as ( 2^{30} - 1 ) and ask for ( T_0 ), which is 1.So both questions are consistent and lead to the same conclusion.Just to be thorough, let me calculate ( 2^{30} ). I know that ( 2^{10} = 1024 ), so ( 2^{20} = 1,048,576 ), and ( 2^{30} = 1,073,741,824 ). Therefore, ( 2^{30} - 1 = 1,073,741,823 ). So Jeremy has planted over a billion trees? That seems like a lot, but considering he's been doubling every year for 30 years, it adds up exponentially.Yeah, that makes sense. So the total number of trees is ( 2^{30} - 1 ), which is 1,073,741,823 trees. And the initial number of trees ( T_0 ) is 1.Final Answer1. The total number of trees Jeremy has planted is boxed{2^{30} - 1}.2. The initial number of trees ( T_0 ) is boxed{1}.</think>