A piece of cardboard measuring 9 inches by 14 inches is formed into an open-top box by cutting squares with side length xx from each corner and folding up the sides.

Question

A piece of cardboard measuring 9 inches by 14 inches is formed into an open-top box by cutting squares with side length x from each corner and folding up the sides.

Find a formula for the volume of the box in terms of x

V(X)=

Find the value for x that will maximize the volume of the box

x=

Answer 1

V(x) = (14 - 2x)(9 - 2x)(x) = 126x - 46x² + 4x³ for 0 <= x <= 9/2

V'(x) = 12x² - 92x + 126 = 2(6x²-46x+63)

x = 23/6 ± √151/6

Adding gives a relative minimum (where x is not in the valid range); subtracting gives the x value at the relative maximum, which is what we are seeking.

23/6 - √151/6