Question 815574: The sum of diameters of two circles is 112 cm and the sum of their areas is 5236 cm^2. Find the radii of the two circles.
Answer by TimothyLamb(4379)   (Show Source):
You can put this solution on YOUR website! a = pi*r^2
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m = radius of smaller circle
n = radius of larger circle
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2m + 2n = 112
2m = 112 - 2n
m = 56 - n
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pi*m^2 + pi*n^2 = 5236
m^2 + n^2 = 5236/pi
m^2 + n^2 = 1666.6706
mm + nn = 1666.6706
(56 - n)(56 - n) + nn = 1666.6706
3136 - 112n + nn + nn = 1666.6706
3136 - 112n + 2nn = 1666.6706
2nn - 112n + 1469.3294 = 0
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the above quadratic equation is in standard form, with a=2, b=-112, and c=1469.3294
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to solve the quadratic equation, plug this:
2 -112 1469.3294
into this: https://sooeet.com/math/quadratic-equation-solver.php
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the roots of the quadratic are:
35.0239092
20.9760908
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since both roots are positive, they both make sense as the radii of circles, so the two roots are in fact the radii of the two circles.
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Answer:
n = 35.0239092 cm
m = 20.9760908 cm
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you can easily confirm this by setting n equal to the first root, and solving for m:
m = 56 - n
m = 56 - 35.0239092
m = 20.976091
in the m result above, note the slight rounding of the trailing ...08 to ...1
this is due to my handheld calculator automatically rounding to the number of digits in its display, whereas the online quadratic solver has a slightly higher number of significant digits in its ROOTS output.
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