Solve this Word Problem

Question

You are making an open box from a rectangular sheet of cardboard by cutting squares 2 inches in length from each corner and folding up the sides. The length of the cardboard is 8 inches more than the width.

 

1.) Write a polynomial the represents the whole volume of the box.

 

2.) What is the volume of the box if the width of sheet of cardboard is 6 inches.

Answer 1

Hi Nathalie!

 

Since we are given information in terms of, and ultimately given a value for, the width, let's set up the problem in terms of the width.  Without a diagram to follow, this may be a little tricky to follow at times, but we can always go back over parts, as needed.

 

Both the lend and width of the original cardboard piece are reduced by 4 inches (from the two-inch squares being cut off both ends of each side).  These modified lengths become the length and width of the box once it is folded.  So we have (all lengths in inches):

 

w = width of original cardboard piece

l = length of original cardboard piece = (w+8)

w_b = width of box = w - 4

l_b = length of box = l - 4 = (w+8) - 4 = (w+4)

h_b = height of box = 2  (since the height is as tall as the sides of the squares cut out of each corner)

 

So the volume of the box, v_b, in cubic inches (in³) is given by:

 

v_b = l_b · w_b · h_b = (w-4)(w+4)(2) = 2(w²-16).  [note: this uses the difference of two squares formula, that a² - b² = (a-b)(a+b)].  This is a polynomial for the volume in terms of the width of the original cardboard

 

Putting in a value of w = 6 inches for the original cardboard gives a value of v_b = 40 in³

 

I hope this helps!  Just let me know if you have more questions about this.