SOLUTION: Two circular regions are tangent to each other, one being larger than the other. The distance between the centers is 10 feet. Find the radius of each circle if the combined area

Algebra ->  Circles -> SOLUTION: Two circular regions are tangent to each other, one being larger than the other. The distance between the centers is 10 feet. Find the radius of each circle if the combined area       Log On


   

Question 70334: Two circular regions are tangent to each other, one being larger than the other. The distance between the centers is 10 feet.
Find the radius of each circle if the combined area is 52pi square feet.
Answer by bucky(2189) About Me  (Show Source):
You can put this solution on YOUR website!
Without going into the geometric detail, if you draw a diagram of the two tangent circles
mark their centers, and draw a line connecting the two centers, the line will pass through
the point of tangency. Therefore, the line joining the two centers will be comprised
of the two radii. Call one of the radii x feet. Since the two radii will add together to be 10 feet,
the second radius will be 10-x feet.
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It might not be a bad idea to make a sketch of the two circles with the line joining
their centers so you can understand what is happening.
.
You know that the Area of a circle is given by the equation:
.
A+=+pi%2AR%5E2
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where A represents the Area and R is the radius of the circle. So for this problem we
can compute the combined Area of the two circles by substituting x for one radius
and 10-x for the other and add the resulting two Areas. In equation form in which we
use "At" for total Area is:
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At+=+%28pi%2Ax%5E2%29+%2B+%28pi%2A%2810-x%29%5E2%29
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Factor a pi out of each of the two terms on the right side:
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At+=+pi%2A%28x%5E2+%2B+%2810-x%29%5E2%29
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Square the (10-x) term to make the equation:
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At+=+pi%2A%28x%5E2+%2B+100+-+20x+%2B+x%5E2%29
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Simplify the expression in the parentheses and rearrange it in descending powers of x:
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At+=+pi%2A%282x%5E2+-+20x+%2B+100%29
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At this point we use the fact that the combined Area of the two circles is:52%2Api square feet.
by substituting this for At in the equation. Our equation then becomes:
.
52%2Api+=++pi%2A%282x%5E2+-+20x+%2B+100%29
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Cancel the pi multipliers on both sides by dividing both sides by pi to get:
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52+=+2x%5E2+-+20x+%2B+100
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Subtract 52 from both sides and then transpose the equation:
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2x%5E2+-+20x+%2B+48+=+0
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Divide both sides by 2 to simplify the equation to:
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x%5E2+-+10x+%2B+24+=+0
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This equation factors to:
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%28x-6%29%2A%28x-4%29+=+0
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And this equation will be true if either of the factors equals zero. Therefore, to solve
for x, set each factor equal to zero.
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If x-6=0 then x=6 and if x-4=0 then x=4. If x=6 then the
other radius is 10-6+=+4 and if x=4 then the other radius is 10+-+4=6}.
From this, it is obvious that one radius must be 6 feet long and the other radius
must be 4 feet long.
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Hope this helps to clarify the problem for you.