Maximum volume of a cube

Question

What is the maximum volume of a cube with a side length of 24

The cube dimensions are as follows:
L = 24-2x, W = 24-2x and H = x. The max volume is where the volume equation is the greatest.
determine:
1. The volume equation.
2. The maximum volume.
3. The value of x that maximizes the volume.

Answer 1

The volume of a cube is L*W*H.
So with L = 24-2x, W = 24-2x and H = x you get:

V(x)=(24-2x)*(24-2x)*x=((24-2x)^2)*x=(5…

The max volume is found when the derivative of V(x) equals zero.

V'(x)=576-192x+12x^2=0

Solving that we get two points: x=12 and x=4

We try both solutions in the Volume equation:

V(4)=1024
V(12)=0

So the correct answer is 1024 when x=4.