Question

Sabrina is making an open box from a piece of cardboard that has a width of 12 inches and a length of 18 inches. She'll form the box by making cuts at the corners and folding up the sides. If she wants the box to have a volume of 224 in^3 how long should she make the cuts?

Answer 1

Remember that

The volume of the box is given by

V=L*W*H

where

V=224 in3

L=(18-2x) in

W=(12-2x) in

H=x in

substitute in the formula

224=(18-2x)(12-2x)(x)

224=[(18)(12)-18(2x)-2x(12)+(2x)^2]x

224=[216-36x-24x+4x^2]x

224=[216-60x+4x^2]x

224=216x-60x^2+4x^3

4x^3-60x^2+216x-224=0

Solve the cubic equation by graphing

The values of x are

x=2

x=2.725

x=10.275 -----> is not a solution

Remember that

The domain for x is the interval

2x<12

x<6 in

interval (0,6)

Verify both values of x

For x=2 in

V=(18-2x)(12-2x)(x)

V=(18-2(2))(12-2(2))(2)

V=(14)(8)(2)=224 in3 -----> is ok

For x=2.725 in

V=(18-2(2.725))(12-2(2.725))(2.725)

V=(12.55)(6.55)(2.725)=224 in3

therefore

The answer is

The value of x can be x=2 in or x=2.725 in