A box with a square base and open top must have a volume of 296352 cm³. A(x) We wish to find the dimensions of the box that minimize the amount of material used. Step 1: Find a formula for the surface area of the box in terms of only x, the length of one side of the square base. Hint: use the volume formula to express the height, y of the box in terms of x, and then substitute this into the surface area formula. Simplify your formula as much as possible. = x //////// Step 2: Find the derivative, A'(x). A'(x) = /////// x Find the critical values, that is solve A'(x) = 0 for x. Hint: Multiply both sides for ² The critical value(s) are x = Y We next have to make sure that this value of a gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x). A"(x) = and then x = y = Evaluate A"(x) at the x-value you gave above. The dimensions of the box that minimizes the volume is Part 3 of 3

A box with a square base and open top must have a volume of 296352 cm³. A(x) We wish to find the dimensions of the box that minimize the amount of material used. Step 1: Find a formula for the surface area of the box in terms of only x, the length of one side of the square base. Hint: use the volume formula to express the height, y of the box in terms of x, and then substitute this into the surface area formula. Simplify your formula as much as possible. = x //////// Step 2: Find the derivative, A'(x). A'(x) = /////// x Find the critical values, that is solve A'(x) = 0 for x. Hint: Multiply both sides for ² The critical value(s) are x = Y We next have to make sure that this value of a gives a minimum value for the surface area. Let's use the second derivative test. Find A"(x). A"(x) = and then x = y = Evaluate A"(x) at the x-value you gave above. The dimensions of the box that minimizes the volume is Part 3 of 3

Mathematics For Machine Technology

8th Edition

ISBN:9781337798310

Author:Peterson, John.

Publisher:Peterson, John.

Chapter61: Areas Of Circles, Sectors, And Segments

Section: Chapter Questions

Problem 50A

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