Question

A box without a top is to be made from a rectangular piece of cardboard, with dimensions 12 in. by 16 in., by cutting out square corners with side length x and folding up the sides.

A) Write an equation for the volume V of the box in terms of x.
B) Use technology to estimate the value of x, to the nearest tenth, that gives the greatest volume.
Please explain how you go your answer

Answer 1

Explanation:

Length (a) = 12-2x

Width (b) = 16-2x

Height (h) = x

Volume = a*b*h = x(12-2x)(16-2x) = 192x-56x^2+4x^3

Use derivative to find max.

dV/dx = 192-112x+12x^2 = 0

x = 7.07, 2.26

You can use second derivative or take two closest points and find gradient of them to find max x.

Max(x) = 2.26