SOLUTION: Cynthia Besch wants to buy a rug for a room that is 21ft wide and 28ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 494 square feet of ca

Algebra ->  Quadratic Equations and Parabolas -> SOLUTION: Cynthia Besch wants to buy a rug for a room that is 21ft wide and 28ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 494 square feet of ca      Log On


   

Question 1195712: Cynthia Besch wants to buy a rug for a room that is 21ft wide and 28ft long. She wants to leave a uniform strip of floor around the rug. She can afford to buy 494 square feet of carpeting. What dimensions should the rug​ have?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
Cynthia Besch wants to buy a rug for a room that is 21ft wide and 28ft long.
She wants to leave a uniform strip of floor around the rug.
She can afford to buy 494 square feet of carpeting.
What dimensions should the rug​ have?
~~~~~~~~~~~~~~~~~~~~~~~~~ 

Let "x" be the uniform width of the strip. 

Then the dimensions of the rug are (21-2x) and (28-2x). 

Thus the area of the rug is (21-2x)*(28-2x).


According to the condition, it must be equal to 494.  It gives you an equation

(21-2x)*(28-2x) = 494


588+-+42x+-+56x+%2B+4x%5E2 = 494

4x%5E2+-+98x+%2B+94 = 0

2x%5E2+-+49x+%2B+47 = 0

The discriminant   d = 49%5E2+-+4%2A2%2A47 = 2025.

x%5B1%2C2%5D = %2849+%2B-+sqrt%282025%29%29%2F%282%2A2%29 = %2849+%2B-+45%29%2F4.


There are two roots.


x%5B1%5D = %2849+%2B+45%29%2F4 = 23.5  is, obviously, too big (more than the wide of the room) 
                                       and, therefore, does not work as the solution to the problem.


x%5B2%5D = %2849-45%29%2F4 = 1  perfectly suits as the solution.


Check.  (21-2*1)*(28-2*1) = 19*26 = 494,  the area of the rug


Answer.  The rug should be 19 by 26 feet.

Solved.



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


The response from the other tutor shows a good typical formal algebraic method for solving the problem.

But you can also get good experience in problem solving by using logical reasoning and a bit of simple arithmetic.

If the dimensions of the room are 21 by 28 feet and the strip surrounding the rug is of uniform width, then the dimensions of the rug are 21-x and 28-x feet. The numbers given in the problem are whole numbers, so the solution will almost certainly involve whole numbers. So subtract whole numbers from each dimension of the room to find a pair of numbers whose product is 494.

Subtracting 1 from each dimension gives us 27 by 20 feet; that product is obviously not 494.

Subtracting 2 from each dimension gives us 26 by 19 feet. The units digit of that product is 4 (9 times 6 is 54), so that could be our answer. And, multiplying, we find that indeed 26*19 is 494. So the dimensions of the rug are 26 by 19 feet.

ANSWER: 26 feet by 19 feet