Word problem Math

Question

Marie has a rectangular board of 18 inches by 23 inches around which she wants to put a uniform border of shells. If she has enough shells for a border whose area is 86 square inches, determine the width of the border. 

 

Answer 1

Make sure you draw a sketch to help you set up this problem. 

The border will extend beyond the rectangle above, below, left, and right by the same amount, x.

Consider the border to be four thin rectangles along the sides and four little squares outside the four corners of the rectangle. The trick is to make sure you don't over or under count the portions that extend beyond the rectangles.

 

Each of those little squares are x by x for an area of x². There are 4 of them, so that makes 4x².

The thin rectangles along the width of the board will be 18 by x, for an area of 18x. There are 2 of those making 36x.

The thin rectangles along the length of the board will be 23 by x, for an area of 23x. There are 2 of those making 46x.

 

So, 4x²+36x+46x=86 in²

 

Combine like terms to get 4x²+82x = 86 in²

 

Now turn this into a quadratic equation equal to zero: 4x² + 82x - 86 = 0

 

Factor to get (2x - 2)(2x + 43) = 0

 

x = 1 or x = -21.5        reject the negative value since that is an impossible border width

 

Therefore the width of the border is 1 inch.

 

 

 

Answer 2

Hi Theresa;

Great question!  Thank you for asking!

 

First, we must understand that (18)(23)=414.  This means that the area of the board is 414 square inches.  Whereas the area at issue is 86.  This means that the shells will not be around the exterior, but rather within the interior.

 

(18 - x)(23 - x)=86

 

Are you familiar with FOIL?

 

FIRST...(18)(23)=414

OUTER...-18x

INNER...-23x

LAST...x²

 

The equation is now...

 

414 - 18x - 23x + x² = 86

 

Let's subtract 86 from both sides, while we combine 18x and 23x...

 

-86 + 414 - 41x + x² = -86 + 86

 

328 - 41x + x² = 0

 

As with any other FOIL equation, the x cannot be isolated.  We now must play a guessing game.

The answer is approximately 11.

 

328 - (41(11)) + (11)²= -2 which almost = 0

 

 

 

 

 

 

Answer 3

The area A of the border = (area of border plus board) minus (area of board)

 

If x is the width of the border, then the area of the border and the board is (18+2x)(23+2x). The area of the board is 18*23.

 

Therefore,

 

A= (18+2x)(23+2x)-18(23) = 82x +4x²

 

Set this equal to 86 and solve:

 

4x²+82x=86

4x²+82x-86=0

 

Use the quadratic formula to solve this quadratic equation and get the two solutions x = -21.5 and x =1. Since the width x must be positive, the answer is x = 1 inch.