A rectangular solid with a square base has a surface area of 37.5 square centimeters. (Let x represent the length of the sides of the square base and let y represent the height.)

Question

also, (a) Determine the dimensions that yield the maximum volume.

Answer 1

Write equations based on the information supplied....

For starters, we are told that V (volume) = x^2 * y <------equation 1

We have 2 unknowns, so we need one more equation. That will come from this statement:

"has a surface area of 37.5 square centimeters"

2 square faces, each with an area of x^2.

4 sides, each with an area of x*y

Thus A = 2x^2 + 4xy = 37.5 <------equation 2

We need an equation for V, so:

solve equation 2 for y:

2x^2 / 4x + y = 37.5 / x

y = 37.5 / x - 2x^2 / 4x

y = 37.5/x - x/2

Substitute into equation 1:

V = x^2 * ( 37.5/x - x/2 )

V = 37.5x - x^3 / 2

To find the maximum V, take the 1st derivative of V, and set + to 0:

V' = 37.5 - 3/2 * x^2 = 0

3/2 * x^2 = 37.5

x^2 = 37.5 * 2/3 = 25

x = 5

Use equation 2 to find y

Answer 2

2x² + 4xy = 37.5

4xy = 37.5 - 2x²

y = (37.5 - 2x²) / 4x

V(x) = x²y

V(x) = x²(37.5 - 2x²) / 4x

V(x) = 9.375x - 0.5x²

V(x) = -0.5(x² - 18.75x + (18.75/2)²) + 0.5(18.75/2)²

V(x) = -0.5(x - 9.375)² + 43.9453125