calculus question

Question

Determine the dimensions of a rectangular solid (with a square base) with maximum volume if its surface area is 73.5 square centimeters.

 ? = cm (smallest value)

 ? = cm

? =  cm (largest value)

Answer 1

Assign variables: You have 2 equal side---> call these x

The third side can be y

For surface area: You have 2 sides with an area of x^2, and 4 sides with an area of x*y.

Add these up and set equal to 73.5 :

2x^2 + 4xy = 73.5 <<equation 1>>

Next, write the equation for the volume: V = x^2 * y <<equation 2>>

Solve equation 1 to get y = f(x)

4xy = 73.5 - 2x^2

y = 73.5/4x - x/2

use this to replace y in equation 2:

V = (x^2) * (73.5/4x - x/2)

V= 73.5x/4 - (x^3) /2

Finally, take the 1st derivative (dV/dx) , set equal to zero, and solve for x