what were the original dimensions of th cardboard??

Question

A box is created from a sheet of cardboard with equal squares cut out of each corner. The square cut from the corner were 3 inches on each side in the volume of the box is 24 in.³ If the length of the original piece of cardboard is 2 inches more than the width, what were the original dimensions of the cardboard?

Answer 1

Hi Maddie, let w represent the original width of the cardboard sheet. The the original length would be w + 2.

 

The volume of the box formed with a height of 3 inches is given by: V = l * w * h

 

To form the box, 3 inch squares are removed from each corner. This tells us the width of the box formed by removing the 3 inch squares is equal to w - 6 inches.

 

The length of the box formed by removing the 3 inch squares is w + 2 - 6, or w - 4.

 

Substituting into the equation for the volume of the box gives:

 

(w - 4) * (w - 6) * 3 = 24

 

Divide both sides by 3:

 

(w - 4) * (w - 6) = 8

 

w² - 6w - 4w + 24 = 8

 

Combine terms and subtract 8 from both sides:

 

w² - 10w + 16 = 0

 

Factoring:

 

(w - 2) * (w - 8) = 0

 

Using the zero property rule:

 

w - 2 = 0

 

w = 2

 

w - 8 = 0

 

w = 8

 

Solution:

 

Discard the solution, w = 2, because using it would give a negative width to the cardboard sheet.

 

The original dimensions to the cardboard sheet are: width = 8 inches, length = width + 2, or 10 inches.

 

Check:

 

(w - 6) * (w - 4) * 3 = 24

 

Substituting:

 

(8 - 6) * (8 - 4) * 3 = 24

 

4 * 2 * 3 = 24

 

24 = 24

 

Values check.

 

Questions?