SOLUTION: A garden 7 m by 12 m will be expanded by planting a border of flowers. The border will be of the same width around the entire garden and has an area of 92 im

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Question 1207920: A garden 7 m by 12 m will be expanded by planting a border of flowers. The border will be of the same width around the entire garden and has an area of 92 im
Found 3 solutions by ikleyn, greenestamps, math_tutor2020:
Answer by ikleyn(52786)   (Show Source): You can put this solution on YOUR website!
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A garden 7 m by 12 m will be expanded by planting a border of flowers.
The border will be of the same width around the entire garden and has an area of 92 square meters.
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        The question absents in this post,  which is inaccuracy
        and disrespect to tutors,  both in the same time. 


The garden is a rectangle 7 m by 12 m. Its area is 7*12 = 84 m^2.

Let w be the uniform width of the border of flowers around the garden.


The extended garden is a rectangle of dimensions (7+2x) by (12+2x) meters.
The area of the extended garden is (7+2x)*(12+2x) m^2.


The difference of these two areas is the area of the border around the garden.


So, we write this equation

    (7+2x)*(12+2x) - 84 = 92  square meters.    (1)


At this point, the setup is complete. 
Now our goal is to find x from this equation.    
Simplify it

    84 + 24x + 14x + 4x^2 - 84 = 92.

    4x^2 + 38x - 92 = 0,

    2x^2 + 19x - 46 = 0.


Use the quadratic formula

     =  =  = .


One root is positive, the other root is negative.

Naturally, we discard the negative root and accept the positive root

    x =  =  = 2 meters.


ANSWER.  The uniform width of the border around the garden is 2 meters.


CHECK.    We check if equation (1) is correct.

          Its left side is (7+2*2)*(12+2*2) - 84 = use your calculator = 92 m^2,  

                                                   precisely as the given area of the border.

Solved.

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

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
    - Problems on a circular pool and a walkway around it
in this site.

Learn the subject from there.



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


The other tutor showed a typical formal algebraic solution.

If formal algebra is not required, and if the speed of obtaining the answer is important -- as on a timed competitive exam -- then the problem can be solved easily and quickly with logical reasoning and a little mental arithmetic.

The area of the original garden is 7*12 = 84 square meters; with the added border of flowers of uniform width, the area is 84+92 = 176 square meters.

The difference between the width and length is 5 meters; since the border of flowers is of uniform width, the difference between the length and width of the expanded garden is again 5 meters.

So to find the solution, you need only find two numbers that differ by 5 and have a product of 176. A bit of playing with numbers shows those two numbers to be 11 and 16.

The garden without the border is 7 by 12 meters; with the border it is 11 by 16 meters. Since the border is uniform width on all four sides, the width of the border is 2 meters.
Answer by math_tutor2020(3817)   (Show Source): You can put this solution on YOUR website!

Some time ago I solved a similar problem involving a 12 by 10 rectangle

The 12 by 10 rectangle enlarges to a (12+2x) by (10+2x) rectangle.

Using this template idea, your 7 by 12 rectangle enlarges to a (7+2x) by (12+2x) rectangle.
A = Large rectangle area = (7+2x)*(12+2x) = 4x^2+38x+84
B = small rectangle area = 7*12 = 84 square meters
A-B = (4x^2+38x+84) - (84) = 4x^2+38x

A-B = area of the flower border = 92

4x^2+38x = 92
4x^2+38x-92 = 0
Use of the quadratic formula will yield the two roots x = -11.5 and x = 2
A negative border width makes no sense, so we ignore it.


Answer: Border width is 2 meters
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