SOLUTION: How do I find the perimeter of a rectangle that's in a circle when all I know is the diameter (6 inches) and the area of the square (15). I've been at this for a week please help

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Question 1190163: How do I find the perimeter of a rectangle that's in a circle when all I know is the diameter (6 inches) and the area of the square (15). I've been at this for a week please help
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52788)   (Show Source): You can put this solution on YOUR website!
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How do I find the perimeter of a rectangle that's in a circle when all I know is the diameter (6 inches) and the area of the square (15).
I've been at this for a week please help
~~~~~~~~~~~~~~~~~


After reading your post,  I have a question:

        does the problem talks about a  RECTANGLE  or about a  SQUARE ?


////////////////


When formulated in a correct way,  this problem has a nice, short, simple and elegant  " exact "  solution. 

    +-----------------------------------------------------------------------+
    |    Find the perimeter of a rectangle inscribed in a circle of the     |
    |    diameter 6 inches, if the area of the rectangle is 15 sq. inches.  | 
    +-----------------------------------------------------------------------+


Let x and y be the dimensions of the rectangle.

Then you have these two equations


    x^2 + y^2 = 36      (1)    (The Pythagoras, applied to the legs and the hypotenuse.
                                which is the diameter of the circle)

    xy        = 15      (2)    (area equation).



Multiply equation (2) by 2 (both sides) and add it to equation (1).  You will get

    x^2 + 2xy + y^2 = 36 + 2*15,

or

    (x + y)^2 = 66,   which implies  x + y = .      (3)



The perimeter of the rectangle is  2x + 2y = ( from equation (3) ) =  = 16.248  inches (approximately).


Thus you have BOTH "exact" solution and approximate (rounded) numerical value.

Solved.




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


A square of area 15 can't be inscribed in a circle of diameter 6; so I will assume that the area of 15 is of a rectangle -- as suggested in the first phrase of your post -- instead of the area of a square.

Let x be one dimension of the rectangle
Then 15/x is the other dimension

The diameter of the circle is the diagonal of the rectangle:






The equation does not factor over the integers, so use a numerical method or a graphing calculator to find the possible values of x. I leave that to you.

Note the equation is an even function. It has four zeros -- two positive values and the opposites of those two.

Obviously the negative solutions make no sense in the problem.

The two positive solutions are the two dimensions of the rectangle.

Use those dimensions to find the perimeter.
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