(b) Find the values of x for which the volume is greater than 200in^3 and find largest volume

Question

A box with an open top is to be constructed from a rectangular piece of cardboard with dimensions 12 in. by 30 in. by cutting out equal squares of side x at each corner and then folding up the sides.

(a) Find a function that models the volume V of the box. 

(b) Find the values of x for which the volume is greater than 200 in³. (Round your answers to three decimal places. Enter your answer using interval notation.) 

(c) Find the largest volume that such a box can have. (Round your answer to three decimal places.) 

Answer 1

(b) x: (0.652,5)

Answer 2

V = (30 - 2x)(12 - 2x)x = 4x(90 - 21x + x²)

dV/dx = 4(90 - 21x + 3x²) which by quadratic formula has roots at approximately 11.35 and 2.641

The 11.35 root is extraneous to the problem and when x = 2.641 the volume is maximized at approximately 438.553

The value of x at which the volume is 200 is approximately .655