Question

You have a piece of cardboard that has a length of 9 feet and a width of 16 feet. You are going to use this piece of cardboard to make an open top box by cutting equal size squares out of each corner and folding up the sides to construct the box.

a. if X represents the side lengths of the square cut outs, write an equation to represent the volume of the box
b. sketch the graph below. What is the maximum volume for the open top box? What are the dimensions for the box with the maximum volume?

Answer 1

A piece of cardboard that has a length of 9 feet and a width of 16 feet.

We are going to use this piece of cardboard to make an open top box by cutting equal size squares out of each corner and folding up the sides to construct the box.

a)Let X represents the side lengths of the square cut outs

sides reduced by x

16 feet becomes 16 - 2x

9 becomes 9-2x

volume , V of the cuboid or rectangular box

V=length x breadth x height

V = (16 - 2x)(9-2x)x = 4x^3 -50x^2+144x

b) The graph is attached with this answer

Maximum value of Volume at x=2

Length = 12, breadth = 5 and height = 2

Volume . V = 12 x 5 x 2 = 120 cube units

Dimensions are, 12,5 and 2