If 30,000 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
Question
If 30,000 cm2 of material is available to make a box with a square base and an open top, find the largest possible volume of the box.
Solution
Sure, here's how to solve this problem:
Step 1: Let's denote the side of the square base as x (in cm) and the height of the box as h (in cm). The surface area of the box (which is given as 30,000 cm^2) can be calculated as the area of the base plus the area of the four sides. In mathematical terms, this is x^2 + 4xh.
Step 2: We know that this surface area is 30,000 cm^2, so we can set up the equation x^2 + 4xh = 30,000.
Step 3: We want to maximize the volume of the box, which is given by the formula V = x^2h. To do this, we need to express h in terms of x using the equation from step 2. Solving for h, we get h = (30,000 - x^2) / (4x).
Step 4: Substitute h in the volume formula, we get V = x^2 * (30,000 - x^2) / (4x) = x(30,000 - x^2) / 4.
Step 5: To find the maximum volume, we need to take the derivative of V with respect to x, set it equal to zero, and solve for x. The derivative of V is (30,000 - 3*x^2) / 4.
Step 6: Setting the derivative equal to zero gives us 30,000 - 3*x^2 = 0. Solving for x, we get x = sqrt(10,000) = 100 cm.
Step 7: Substitute x = 100 cm back into the equation from step 3 to find h. We get h = (30,000 - 100^2) / (4*100) = 50 cm.
Step 8: The maximum volume of the box is therefore V = 100^2 * 50 = 500,000 cm^3.
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