Lesson Cynthia Besch wants to buy a rug for a room

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Cynthia Besch wants to buy a rug for a room


This problem came to the forum so many times and proved itself to be so popular that I decided to make the lesson
(with the goal to provide,  at least,  a permanent link).

Problem 1

Cynthia Besch wants to buy a rug for a room that is  8 ft  wide and  19 ft  long.
She wants to leave a uniform strip of floor around the rug.  She can afford to buy
102 square feet of carpeting.  What dimensions should the rug have?

Solution 

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

Then the dimensions of the rug are (8-2x) and (19-2x). 

Thus the area of the rug is (8-2x)*(19-2x).


According to the condition, it must be equal to 102.  It gives you an equation

(8-2x)*(19-2x) = 102


152+-+38x+-+16x+%2B+4x%5E2 = 102

4x%5E2+-+54x+%2B+50 = 0

2x%5E2+-+27x+%2B+25 = 0

The discriminant   d = 27%5E2+-+4%2A2%2A25 = 529.

x%5B1%2C2%5D = %2827+%2B-+sqrt%28529%29%29%2F%282%2A2%29 = %2827+%2B-+23%29%2F4.


There are two roots.


x%5B1%5D = %2827+%2B+23%29%2F4 = 12.5  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 = %2827-23%29%2F4 = 1  perfectly suits as the solution.


Check.  (8-2*1)*(19-2*1) = 6*17 = 102.


Answer.  The rug should be 6 by 17 feets.

Solved.


It is classic algebraic solution.

You can solve the problem MENTALLY, too. 

The difference between the room dimensions is 19-8 = 11 ft.

Since the strip is of uniform width, the dimensions of the rug must have the same difference of 11 ft.


So, you will solve the problem, if you find the decomposition of the number of 102 into the product of two factors,

whose difference is 11.


There are, actually, only one such a possibility 102 = 6*17, which you can find in a minute (if not in 3 seconds).

At this point your mental solution is completed.

Thank you for you patience reading my solution from the beginning to the end.

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You can find many other similar solved problems in the lessons
    - Problems on the area and the dimensions of a rectangle surrounded by a strip
    - Problems on a circular pool and a walkway around it
    - OVERVIEW of lessons on dimensions and the area of rectangles and circles and their elements
in this site.

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