SOLUTION: Cynthia Besch wants to buy a rug for a room that is 19 ft wide and 31 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 253 square feet of

Algebra ->  Quadratic Equations and Parabolas  -> Quadratic Equations Lessons -> SOLUTION: Cynthia Besch wants to buy a rug for a room that is 19 ft wide and 31 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 253 square feet of      Log On


   

Question 1197474: Cynthia Besch wants to buy a rug for a room that is 19 ft wide and 31 ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 253 square feet of carpeting. What dimensions should the rug​ have?
Found 2 solutions by ikleyn, math_tutor2020:
Answer by ikleyn(52776) About Me  (Show Source):
You can put this solution on YOUR website!
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About this legendary personage Cynthia Besch and relevant problems on her rug see the lesson
    - Cynthia Besch wants to buy a rug for a room
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Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

x = width of the uniform strip (i.e. the gap between the rug edge and the wall)

Diagram

The larger rectangle is 19 by 31
The smaller rectangle inside is (19-2x) by (31-2x)

The smaller rectangle area is 253 sq ft
length*width = area
(19-2x)*(31-2x) = 253
19*(31-2x) - 2x*(31-2x) = 253
589 - 38x - 62x + 4x^2 = 253
589 - 100x + 4x^2 = 253
589 - 100x + 4x^2 - 253 = 0
4x^2 - 100x + 336 = 0

Compare this to ax^2+bx+c = 0
a = 4
b = -100
c = 336

Use the quadratic formula to isolate x.
x+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

x+=+%28-%28-100%29%2B-sqrt%28%28-100%29%5E2-4%284%29%28336%29%29%29%2F%282%284%29%29

x+=+%28100%2B-sqrt%2810000-5376%29%29%2F%288%29

x+=+%28100%2B-sqrt%284624%29%29%2F%288%29

x+=+%28100%2B-++68%29%2F%288%29

x+=+%28100%2B68%29%2F%288%29 or x+=+%28100-68%29%2F%288%29

x+=+%28168%29%2F%288%29 or x+=+%2832%29%2F%288%29

x+=+21 or x+=+4

If x = 21, then 19-2x = 19-2*21 = 19-42 = -23 which isn't possible.
A negative length or width makes no sense.
We'll ignore x = 21.

The only practical solution is x = 4.
If x = 4, then 19-2x = 19-2*4 = 19-8 = 11
If x = 4, then 31-2x = 31-2*4 = 31-8 = 23
Both results are positive.
The carpet is 11 by 23 to get an area of 11*23 = 253 sq ft. This verifies the answer.

Answer: 11 ft by 23 ft