When squares of side x inches are cut from the corners of a 35-inch-square piece of cardboard, an open-top box can be formed by folding up the sides. The volume of this box is given by , find the rate of change of volume with respect to the size of the squares.

Question

When squares of side x inches are cut from the corners of a 35-inch-square piece of cardboard, an open-top box can be formed by folding up the sides. The volume of this box is given by  , find the rate of change of volume with respect to the size of the squares. ​

Quizplus · Accepted Answer

Let the length of each side of the square cut from each corner be $x$. Then the length of each side of the base of the box is $35-2x$, and the height of the box is $x$.
The volume of the box is $(35-2x)^2x=4x^3-140x^2+1225x$. Taking the derivative with respect to $x$ gives $\frac{dV}{dx}=12x^2-280x+1225$.
Thus, the rate of change of volume with respect to the size of the squares is $\frac{dV}{dx}\biggvert_{x=5} = 12(5)^2-280(5)+1225= \boxed{	extbf{(C) }215}$.

When Squares of Side X Inches Are Cut from the Corners