Math help please

Question

A rectangular piece of metal is 25 in longer than it is wide. Squares with sides 5 in long are cut from the four corners and the flaps are folded upward to form an open box. If the volume of the box is 1750 in^3?, what were the original dimensions of the piece of? metal?

 

 

original width______

 

original length______

 

Answer 1

Hi Amber.

 

This problem requires you to know what the formula for the volume of a box is:

 

V = L * W * H

 

where

 

L = length

W = width

H = height

 

We start off with a sheet of metal, which is just a length and a width (no height). Since we don't know what the length and the width are explicitly, let's start by calling some variables. I'm going to call the width of this sheet of metal x.

 

We're told the length of the sheet is 25 inches longer than the width. The means:

 

L = x + 25

 

Then we're told that squares 5 inches long are cut from all four of the corners and folded up to make the box. You know, then, that the height of your box will be 5 inches.

 

H = 5

 

Since 5 in is removed from all four corners, both the length of the box and the width of the box will lose 10 inches from the original dimensions of the sheet metal.

 

W = x - 10

L = (x + 25) - 10 = x+15

 

We're told the volume of the box is 1750 in^3. So,

 

V = L*W*H

 

1750 = (x+15)(x-10)(5)

 

(x+15)(x-10) = 350

 

x^2 + 5x - 150 = 350

 

x^2 + 5x - 500 = 0

 

(x+25)(x-20) = 0

 

x = -25, 20

 

Since we're dealing with length here, it can't be negative. Disregard the -25.

 

x = 20

 

That tells us our original width of the sheet metal is 20 in. Since the length is 25 more than the width, the original length of the box was 20+25 = 45 in.

 

Hope this helps.

Answer 2

Volume = width X length X height = 1750 cubic inches

 

We are given that height = 5 inches

 

5WL = 1750

 

or

 

WL = 350

 

But L = W + 25

 

So

 

W (W+25) = 350

 

W^2 + 25W -350 = 0

 

Use quadratic formula to solve Must use the plus sign to get sensible answer):

 

W = -12.5 + 0.5 SQRT(225 + 4(350))

 

W = -12.5 + 0.5(45) = 10

 

Therefore, Width = 10 inches and Length = Width + 25 = 35 inches

 

Check answer: 10 X 35 X 5 = 1750

 

Update:

I calculated the NEW length and width and mistakenly forgot to add 10 (twice 5!) to the results.

 

Width = 20 and Length = 45

 

Answer 3

Alright, Amber, let's do this!

 

The rectangular piece of metal has a width of w and a length of w + 25, right?

 

If we cut 5-inch squares out of each corner, we lose 10 inches from both the width and the length, so our width is now w - 10 and the length is w + 15.

 

Folding up the flaps gives us a height of 5 inches.

 

How do we find volume? Length x width x height...

 

(w + 15)(w - 10)5 = 1750

5(w² + 5w - 150) = 1750

w² + 5w - 150 = 350

w² + 5w - 500 = 0

(w + 25)(w - 20) = 0

 

w + 25 = 0

w = -25

Length is always measured positively, so this answer is out.

 

w - 20 = 0

w = 20

 

Soooo, the original width is 20 inches and the original length is 20 + 25 = 45 inches.

 

Hope this helps!