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

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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(52790)   (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.

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    - Problems on the area and the dimensions of a rectangle surrounded by a strip
    - Cynthia Besch wants to buy a rug for a room
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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.