🧠:<think>Okay, so I have this problem about Elara and her memorial garden. It's a parabolic garden, and I need to figure out the equation of the parabola and then find its focus. Let me take it step by step.First, part 1: I need to find the values of a, b, and c for the quadratic equation y = ax² + bx + c. They gave me some information: the garden intersects the x-axis at x = -2 and x = 4, and it passes through the point (1, 6). Hmm, okay.Since the parabola intersects the x-axis at x = -2 and x = 4, that means these are the roots of the quadratic equation. So, I can express the equation in its factored form first. The factored form of a quadratic with roots at x = -2 and x = 4 is y = a(x + 2)(x - 4). That makes sense because when x is -2 or 4, y becomes zero, which are the x-intercepts.Now, I need to find the value of 'a'. They also told me that the parabola passes through the point (1, 6). So, I can plug x = 1 and y = 6 into the equation to solve for 'a'.Let me write that out:6 = a(1 + 2)(1 - 4)Simplify the terms inside the parentheses:1 + 2 is 3, and 1 - 4 is -3.So, 6 = a * 3 * (-3)Multiply 3 and -3: 3 * (-3) = -9So, 6 = -9aTo solve for 'a', divide both sides by -9:a = 6 / (-9) = -2/3Okay, so a is -2/3. Now, I can write the equation in factored form:y = (-2/3)(x + 2)(x - 4)But I need to express it in standard form, which is y = ax² + bx + c. So, I have to expand the factored form.Let me do that step by step.First, multiply (x + 2)(x - 4):(x + 2)(x - 4) = x*(x - 4) + 2*(x - 4) = x² - 4x + 2x - 8 = x² - 2x - 8So, now the equation is y = (-2/3)(x² - 2x - 8)Now, distribute the -2/3:y = (-2/3)x² + (4/3)x + (16/3)So, in standard form, that's:y = (-2/3)x² + (4/3)x + 16/3Therefore, a = -2/3, b = 4/3, and c = 16/3.Wait, let me double-check my multiplication:(x + 2)(x - 4) = x² - 4x + 2x - 8 = x² - 2x - 8. That's correct.Then, multiplying by -2/3:-2/3 * x² = (-2/3)x²-2/3 * (-2x) = (4/3)x-2/3 * (-8) = 16/3Yes, that's correct. So, part 1 is done.Now, part 2: Finding the focus of the parabola defined by y = (-2/3)x² + (4/3)x + 16/3.Hmm, I remember that the standard form of a parabola is y = ax² + bx + c, and the focus can be found using the formula. But I need to recall the exact formula.I think the vertex form of a parabola is y = a(x - h)² + k, where (h, k) is the vertex. The focus is located at (h, k + 1/(4a)) if the parabola opens upwards or downwards.Since our parabola has a negative coefficient for x², it opens downward. So, the focus will be below the vertex.First, I need to find the vertex of the parabola. To do that, I can use the formula for the x-coordinate of the vertex, which is x = -b/(2a).From our equation, a = -2/3, b = 4/3.So, x = -(4/3)/(2*(-2/3)) = -(4/3)/(-4/3) = (4/3)/(4/3) = 1.So, the x-coordinate of the vertex is 1. Now, to find the y-coordinate, plug x = 1 back into the equation.Wait, but we already know that when x = 1, y = 6 because the parabola passes through (1, 6). So, the vertex is at (1, 6).Wait, is that correct? Let me verify.If x = 1, then y = (-2/3)(1)² + (4/3)(1) + 16/3 = (-2/3) + (4/3) + (16/3) = (-2 + 4 + 16)/3 = 18/3 = 6. Yes, that's correct. So, the vertex is indeed at (1, 6).Now, since the parabola is in the form y = a(x - h)² + k, with vertex (h, k) = (1, 6), and a = -2/3.The formula for the focus is (h, k + 1/(4a)). Let me compute that.First, compute 1/(4a):a = -2/3, so 4a = 4*(-2/3) = -8/3Therefore, 1/(4a) = 1/(-8/3) = -3/8So, the focus is at (h, k + 1/(4a)) = (1, 6 + (-3/8)) = (1, 6 - 3/8)Convert 6 to eighths: 6 = 48/8, so 48/8 - 3/8 = 45/8So, 45/8 is equal to 5.625.Therefore, the focus is at (1, 45/8) or (1, 5.625).Wait, let me make sure I didn't make a mistake in the calculation.1/(4a) where a = -2/3:1/(4*(-2/3)) = 1/(-8/3) = -3/8. That's correct.So, adding that to k (which is 6):6 + (-3/8) = 6 - 3/8 = 45/8. Yes, that's correct.So, the focus is at (1, 45/8).Alternatively, 45 divided by 8 is 5.625, so (1, 5.625). Both are correct, but probably better to leave it as a fraction, 45/8.Let me just recap:1. Found the quadratic equation by using the roots and the given point.2. Expanded it to standard form.3. Found the vertex using x = -b/(2a), which turned out to be (1, 6).4. Used the formula for the focus, which is (h, k + 1/(4a)), calculated 1/(4a) as -3/8, so the focus is at (1, 45/8).I think that's all. Let me just make sure I didn't make any arithmetic errors.Calculating 1/(4a):a = -2/3, so 4a = -8/3, reciprocal is -3/8. Correct.Adding to k: 6 is 48/8, 48/8 - 3/8 = 45/8. Correct.So, the focus is at (1, 45/8). Yep, that seems right.Final Answer1. The values are ( a = -dfrac{2}{3} ), ( b = dfrac{4}{3} ), and ( c = dfrac{16}{3} ). 2. The coordinates of the focus are ( boxed{left(1, dfrac{45}{8}right)} ).</think>

🧠:<think>First, I need to calculate the average speed for the first 250 miles of the race. The driver completes this distance in 1 hour and 45 minutes. I'll convert the time into hours to make the calculation easier. 45 minutes is 0.75 hours, so the total time is 1.75 hours. By dividing the distance by the time, I can find the average speed.Next, I'll determine how many full pit stops the driver needs to make during the 500-mile race. The car's fuel tank holds 22 gallons, and it consumes fuel at a rate of 4 miles per gallon. I'll calculate the total fuel required for the race and then see how many full refuels are needed based on the tank capacity. Since the driver starts with a full tank, I'll subtract that initial fuel from the total requirement to find out how many additional pit stops are necessary.</think>

🧠:<think>Okay, so I have this problem about a graphic designer creating a film poster inspired by French cinema, which uses symmetry and the golden ratio. The problem has two parts, and I need to solve both. Let me take it step by step.Starting with part 1: The designer has a rectangle inscribed in a golden spiral, and the rectangle follows the golden ratio. The golden ratio, φ, is approximately 1.618. The shorter side of the rectangle is given as 'a', and I need to express the area of the rectangle in terms of 'a' and φ. Then, compute the area when a = 10 cm.Alright, so first, let's recall what the golden ratio is. The golden ratio φ is defined such that the ratio of the longer side to the shorter side of a rectangle is φ. So, if the shorter side is 'a', the longer side must be φ times 'a'. That is, longer side = φ * a.Therefore, the rectangle has sides of length 'a' and 'φa'. The area of a rectangle is calculated by multiplying its length and width. So, the area should be a * φa.Let me write that down:Area = a * (φ * a) = φ * a².So, that's the area in terms of 'a' and φ. Now, when a = 10 cm, I can substitute that into the formula.Area = φ * (10 cm)² = φ * 100 cm².Since φ is approximately 1.618, let me compute that:Area ≈ 1.618 * 100 cm² = 161.8 cm².Hmm, that seems straightforward. Let me just make sure I didn't miss anything. The rectangle is inscribed in a golden spiral, but since the rectangle itself follows the golden ratio, the sides are just 'a' and 'φa', so the area is indeed φa². Yeah, that makes sense.Moving on to part 2: The designer wants to place a circular element within the rectangle. The circle is tangent to the longer side of the rectangle and fits perfectly within the shorter side. I need to calculate the radius of this circle and its area in terms of 'a', then find the radius and area when a = 10 cm.Alright, let me visualize this. The rectangle has a shorter side 'a' and a longer side 'φa'. The circle is tangent to the longer side and fits perfectly within the shorter side. Hmm, so the circle must fit within the shorter side, which is 'a', and be tangent to the longer side.Wait, if the circle is tangent to the longer side, that means the diameter of the circle is equal to the shorter side of the rectangle. Because if the circle is tangent to the longer side, the distance from the center of the circle to the longer side is equal to the radius. But since the circle also fits perfectly within the shorter side, the diameter must be equal to the shorter side.Let me think again. The shorter side is 'a', so if the circle is fitting perfectly within the shorter side, the diameter of the circle is equal to 'a'. Therefore, the radius would be a/2.But wait, is that correct? If the circle is tangent to the longer side, then the distance from the center of the circle to the longer side is equal to the radius. But the shorter side is 'a', so the circle must fit within that. So, the diameter can't exceed 'a', so the maximum diameter is 'a', making the radius a/2.Alternatively, maybe the circle is inscribed such that it touches both the shorter and longer sides? Wait, no, the problem says it's tangent to the longer side and fits perfectly within the shorter side. So, perhaps the diameter is equal to the shorter side, meaning radius is a/2.Let me try to draw this mentally. Imagine the rectangle with shorter side vertical and longer side horizontal. The circle is placed such that it touches the longer side (the right side, say) and fits within the shorter side (the top and bottom). So, the diameter of the circle must be equal to the shorter side 'a', so radius is a/2.But wait, if the circle is tangent to the longer side, the center of the circle must be at a distance of radius from the longer side. So, the center is at (φa - r, something). But since the circle also fits within the shorter side, the vertical position of the center must be such that the circle doesn't exceed the shorter side.Wait, maybe I'm overcomplicating it. If the circle is tangent to the longer side, which is of length φa, and fits within the shorter side of length 'a', then the diameter of the circle must be equal to 'a', so the radius is a/2. Because if the circle is tangent to the longer side, the radius is the distance from the center to the longer side, and since the circle also must fit vertically, the diameter can't exceed 'a'.Alternatively, perhaps the circle is inscribed such that it touches the longer side and the top and bottom sides. In that case, the diameter would be equal to the shorter side, so radius is a/2. That seems to make sense.So, radius r = a/2.Then, the area of the circle is πr² = π(a/2)² = πa²/4.So, in terms of 'a', the radius is a/2 and the area is πa²/4.Now, when a = 10 cm, the radius is 10/2 = 5 cm, and the area is π*(10)²/4 = π*100/4 = 25π cm², which is approximately 78.54 cm².Wait, let me double-check. If the circle is tangent to the longer side, which is φa, and fits within the shorter side, which is 'a', then the diameter is 'a', so radius is a/2. That seems correct.Alternatively, if the circle were inscribed in the rectangle such that it touches all four sides, it would have to be a square, but in this case, the rectangle isn't a square, it's a golden rectangle. So, the circle can only touch the longer side and the top and bottom, or left and right. But the problem says it's tangent to the longer side and fits perfectly within the shorter side, so I think the diameter is equal to the shorter side, hence radius a/2.Yes, that seems consistent.So, summarizing:1. Area of the rectangle is φa², which is approximately 161.8 cm² when a = 10 cm.2. Radius of the circle is a/2, so 5 cm when a = 10 cm, and the area is πa²/4, which is 25π cm² or approximately 78.54 cm².I think that's it. Let me just make sure I didn't make any calculation errors.For part 1:Area = φ * a² = 1.618 * 10² = 1.618 * 100 = 161.8 cm². Correct.For part 2:Radius = a/2 = 10/2 = 5 cm. Area = π*(5)² = 25π ≈ 78.54 cm². Correct.Yes, everything checks out.</think>

🧠:<think>First, I need to determine how many breaks Alex takes in a day. Since Alex studies for 8 hours and takes a 10-minute break every hour, there are a total of 8 breaks.Next, I'll figure out how many jogging breaks and jumping jacks breaks there are. Since the breaks alternate starting with jogging, there will be 4 jogging breaks and 4 jumping jacks breaks.For the jogging breaks, each break is 10 minutes long, and Alex runs at a speed of 6 km/h. I'll convert the jogging time into hours to calculate the distance. Each jogging break is 10 minutes, which is 1/6 of an hour. So, the distance for one jogging break is 6 km/h multiplied by 1/6 hour, which equals 1 km. Over 4 jogging breaks, the total distance jogged is 4 km.For the jumping jacks, each break is 10 minutes, and Alex does 30 jumping jacks per minute. I'll calculate the number of jumping jacks per break by multiplying 30 by 10, which equals 300 jumping jacks per break. Over 4 jumping jacks breaks, the total number of jumping jacks is 1200.</think>