SOLUTION: A rectangular rug has a decorative interior with a 1/2 foot border of uniform width around the outside. The length of the decorative area is 3 feet more than the width. If the area

Question 1096124: A rectangular rug has a decorative interior with a 1/2 foot border of uniform width around the outside. The length of the decorative area is 3 feet more than the width. If the area of the rug (including the border) is 108 ft2, find the dimensions of the rug (including the border). Show all work. Answer with a complete sentence.
Found 2 solutions by ankor@dixie-net.com, greenestamps:
Answer by ankor@dixie-net.com(22740) (Show Source): You can put this solution on YOUR website!
A rectangular rug has a decorative interior with a 1/2 foot border of uniform width around the outside.
The length of the decorative area is 3 feet more than the width.
If the area of the rug (including the border) is 108 ft2, find the dimensions of the rug (including the border).
:
let L = the length of the decorated area
let w = the width
Also "length of the decorative area is 3 feet more than the width"
L = w+3
:
Twice the width of the border (1 ft), is added to the decorated dimensions, therefore
(L+1)(w+1) = 108
Replace L with (w+3)
((w+3)+1)(w+1) = 108
(w+4)(w+1) = 108
FOIL
w^2 + w + 4w + 4 = 108
w^2 + 5w + 4 - 108 = 0
w^2 + 5w - 104 = 0
You can use the quadratic formula, a=1, b=5, c=-104;but this will factor
(x+13)(x-8) =
the positive solution is all we want here
x = 8ft is the width of the decorated area
then, obviously 11 ft is the length
Find the overall dimensions, add 1 ft
12 ft by 9 ft is the overall dimensions of the rug
:
;
Check by finding the area with these dimensions: 12 * 9 = 108

Answer by greenestamps(13200) (Show Source): You can put this solution on YOUR website!

If the length of the decorative area is 3 more than the width, and the border is uniform width, then the length of the whole rug will still be 3 more than the width. So the problem is easier to solve if you just solve the equation

x = -12 or x = 9; obviously the positive answer is the one that makes sense.

Since the problem asks for the area of the rug including the border, we know the width is 9 and the length is x+3 = 12.

Note that solving the problem algebraically involves solving a quadratic equation; to do that by factoring, we need to find two numbers whose difference is 3 and whose product is 108.

But that is exactly what the problem asks us to do, WITHOUT doing any algebra: find a length and a width whose product is 108, with the length 3 more than the width.

So using formal algebra to solve this problem is a waste of time -- except as practice in solving problems using algebra.
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