A square-based rectangular prism has a total surface area of 2400cm^2. What are the dimensions of the prism if its volume is a maximum?
Let x represent the sides of the base of the square-based rectangular prism, and let h be the height of the prism.
We are told that the surface area is 2400 cm^2.
The equation we will use to define surface area will be
Solving for h, we get
Since , we can now substitute the expression we came up with into h:
Differentiate and then set equal to zero.
We take the positive value, since length can't be negative.
Now this implies that
Thus, the dimensions of the prism are
This prism is actually a cube.
Does this make sense?
--Chris