Application of Derivatives and Optimization of Volume

Question

Using Calculus, determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 384 square centimeters.

Can someone explain using derivative rules

Thank you so much!

Answer 1

Set base length to x and height to y; we can get Volume = x^2 * y and surface area = 2x^2+4xy = 384.

Rewrite y in terms of x based on the surface area formula and we get y = (384-2x^2)/4x, and we can plug this term back to the volume function and get Volume = x^2 * ((384-2x^2)/4x).

In order to find the max volume, we need to determine the maximum point on the volume graph, and to do that, we need to set the derivative of the volume function to zero and solve for x to get a potential minimum or maximum. So V' = (384/4) - (3x^2 / 2) = 0, solve for x and get x1 = 8, x2 = -8. Obviously, length can't be negative so x = 8. Then, we can find y which is also 8.

Therefore, the dimension of the rectangular solid is 8cm by 8cm by 8cm which is a cube.

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