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 x be the side of each cut square. Then the height of the box is x, and the length and width are both 35-2x. Therefore, the volume of the box is given by V(x) = x(35-2x)^2. 
Using the power rule and chain rule, we differentiate V(x) with respect to x: 
V'(x) = (35-2x)^2 - 2x(35-2x)(-4) 
       = (35-2x)(74-6x) 
To find the rate of change of volume with respect to the size of the squares, we evaluate V'(x) at x = 0 (corresponding to no squares cut) and simplify: 
V'(0) = (35-0)(74-6(0)) = 2590 
Therefore, the rate of change of volume with respect to the size of the squares is 2590 cubic inches per inch, which is equivalent to the answer choice C.

When Squares of Side X Inches Are Cut from the Corners