College Algebra Word Problem

Question

A concrete walk of uniform width is to be laid around a rectangular flower bed that is 60 feet long and 30 feet wide. If the area of the walk is to be 279 square feet, what is the width of the walk. 

Answer 1

 Let x be the extra width and length of the walk. (Drawing a picture may help.)  You should create four small squares in the corner and two long/two short.  Calculating the area of each square/rectangle,  you should get x², 60x, x², 30x, x², 60x, x², and 30x. 

Combine like terms to get: 4x² + 180x = 279 or 4x² + 180x - 279 =0

Use your quadratic formula to find the positive answer (a negative width wouldn't work). Hint:it's a friendly decimal between 1 & 2.

This is the extra width of the walk! 

Answer 2

Look at the difference in area of the outside and the given inside.

Outside = (60 + 2W)(30 + 2W)

Inside = 60 * 30

Difference = 279

Gives: 4Wˆ2 + 180W - 279 = zero.

Apply the quadratic solution giving -22.5 +/- 24  

A negative answer is silly. So, W = 1.5

Check it. It works - giving 279 as the walking area!

Answer 3

Another way to find it

 

 

The interior area is 60x30= 1800 and the concrete walk area is 279

 

That means if you draw the exterior rectangle (including the walk), the total area is 1800+279= 2079

 

Let's say the extra space between rectangles is X , and the area of a rectangle is L x W

 

Then one side is 60+2X and the other is 30+2X, then

.

(60+2X) x (30+2X) = 2079

 

1800 + 120X + 60X + 4X² = 2079

 

4X² +180X +1800 = 2079

 

4X² +180 -279 = 0

 

Applying quadratic formula in equation aX2 +bx +c = 0   =>       X=  -b +/- √(b² - 4ac)

                                                                                                                2a

 

Two possible solutions: X= -180+192=   1.5    or       X= -180-192= -46.5

                                              8                                        8

 

 

There cannot be negative solutions because we are talking about length.

 

X= 1.5 feet