SOLUTION: The Bob's family room is 13 ft by 16 ft, and they want to carpet it, except for a border of uniform width. What would be the width of the border if they can afford only 108 sq ft o

Click here to see ALL problems on Miscellaneous Word Problems

Question 1124151: The Bob's family room is 13 ft by 16 ft, and they want to carpet it, except for a border of uniform width. What would be the width of the border if they can afford only 108 sq ft of carpet?
Found 2 solutions by ikleyn, Theo:
Answer by ikleyn(52788)   (Show Source):
You can put this solution on YOUR website!
.
1.   The easy way to solve it MENTALLY

The difference the length minus the width is 16 - 13 = 3 ft. If the non-carpeted border is of uniform length, then the difference between the length and the width of the carpet must be the same 3 ft. So we need to find a decomposition of the number 108 (108 sq.ft., the area) into the product of two numbers with the difference of 3 between them. As soon as you re-formulated the problem in this way, you can guess the answer MENTALLY and MOMENTARILY: the numbers are 12 and 9. Answer. The dimensions of the carpet are 12 and 9 feet. The uniform width of the border is 2 ft. ( = = . )

2.   Formal algebra solution

Let x be the uniform border width. Then the dimensions of the carpet are (16-2x) and (13-2x) feet. So the area of the carpet is (16-2x)*(13-2x) = 108. It is your equation to find the unknown x. Simplify it; write as a quadratic equation in standard form and solve by using the quadratic formula or factoring.

-------------------

If you want to see many other similar solved problems, look into the lessons
    - Problems on the area and the dimensions of a rectangle surrounded by a strip
    - Cynthia Besch wants to buy a rug for a room
in this site.

Also, you have this free of charge online textbook in ALGEBRA-I in this site
    - ALGEBRA-I - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic
"Dimensions and the area of rectangles and circles and their elements".

Save the link to this online textbook together with its description

Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson

to your archive and use it when it is needed.

Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
13 by 16 = 208 square feet.
that's the area of the room.
let x equal the width of the border all around.

the width of the carpet will be 13 - 2x
the length of the carpet will be 16 - 2x.

the area of the carpet is 108 square feet.

therefore (13 - 2x) * (16 - 2x) = 108

simplify to get 13 * 16 - 13 * 2x - 2x * 16 - 2x * -2x

combine like terms to get 208 - 58x + 4x^2 = 108

subtract 108 from both sides of the equation to get 100 - 58x + 4x^2 = 0

reorder the terms in descending order of degree to get 4x^2 - 58x + 100 = 0

factor this quadratic equation to get x = 12.5 or x = 2.

x can't be 12.5 because 13 - 2x would be negative.

therefore x has to be equal to 2 or not at all.

when x = 2, you get (13 - 4) * ( 16 - 4) becomes 9 * 12 = 108.

the area of the rug is 108 square feet.

the area of the room is (9+4) * (12+4) = 13 * 16 = 208.

the width of the border all around will be 2 feet.

the area of the border all around should be 100 square feet.

along the length you get 16 * 2 = 32 * 2 = 64 square feet.

along the width you get 13 * 2 = 26 * 2 = 52 square feet.

total them up and you get 116 square feet.

the 4 corners of the room have been double counted.

therefore you have to subtract 4 * (2 * 2) = 4 * 4 = 16 square feet.

116 square feet minus 16 square feet = 100 square feet for the area of the border all around.

it all checks out.

your solution is that the width of the border is 2 feet.

here's my diagram.