Question

A piece of cardboard is 2.9 times as long as it is wide. It is to be made into a box with an open top by cutting 3-inch squares from each corner and folding up the sides. Let x represent the width (in inches) of the original piece of cardboard.

a) Represent the length of the original piece of cardboard in terms of x
b) Give the restrictions on x. What will be the dimensions of the bottom rectangular base of the box?
c) Determine the function V that represents the volume of the box in terms of x
d) For what dimensions of the bottom of the box will the volume be 520 in^3
e) Find the values of x if such a box is to have a volume between 600 and 800in^3. Between which 2 values must x be in order to produce this range of volumes?

Answer 1

To find the dimensions of the box, we need to consider the original piece of cardboard and the cuts made to create the box. Let's break down the problem step by step.

1. Let's assume the width of the original piece of cardboard is x inches.

2. According to the given information, the length of the cardboard is 2.9 times its width. Therefore, the length can be expressed as 2.9x inches.

3. To create the box, we need to cut 3-inch squares from each corner of the cardboard. Since there are four corners, a total of 4 * 3 = 12 inches will be removed from both the length and width.

4. After cutting out the squares, the new dimensions of the cardboard will be (x - 6) inches for the width and (2.9x - 6) inches for the length.

5. The height of the box will be equal to the height of each folded side, which is 3 inches.

Now, let's calculate the dimensions of the box:

Width: The width of the box will be equal to (x - 6) inches.

Length: The length of the box will be equal to (2.9x - 6) inches.

Height: The height of the box will be equal to 3 inches.

Therefore, we have determined that after cutting out squares from each corner and folding up the sides, the dimensions of the resulting box will be (x - 6) inches for width, (2.9x - 6) inches for length, and 3 inches for height.

Explanation: