You are planning to make an open rectangular box from an 8-inch by 15-inch piece of cardboard by cutting congruent squares from the corners and folding up the sides. what is the largest volume you can make from a box this way in cubic inches?

Let us say that x is the cut that we will make on the sides to make a box, therefore the new dimensions are:

l = 15 - 2x

w = 8 - 2x

It is 2x since we cut on two sides.

We know that volume is:

V = l w x

V = (15 - 2x) (8 - 2x) x

V = 120x - 30x^2 - 16x^2 + 4x^3

V = 120x - 46x^2 + 4x^3

Taking the 1st derivative:

dV/dx = 120 - 92x + 12x^2

Set dV/dx = 0 to get maxima:

120 - 92x + 12x^2 = 0

Divide by 12:

x^2 - (92/12) x + 10 = 0

(x - (92/24)) ^2 = - 10 + (92/24) ^2

x - 92/24 = ±2.17

x = 1.66, 6

We cannot have x = 6 because that will make our w negative, so:

x = 1.66 inches

So the largest volume is:

V = 120x - 46x^2 + 4x^3

V = 120 (1.66) - 46 (1.66) ^2 + 4 (1.66) ^3

V = 90.74 cubic inches