Question 201686
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And I will be both pleased and thankful if you stop typing your messages in ALL CAPS.  All caps is the electronic equivalent of shouting and is therefore both annoying and rude.


Furthermore, I suspect that the sum of the areas of the two circles in your problem is actually *[tex \Large 130\pi\text{ cm^2}] instead of *[tex \Large 130\text{ cm^2}] as stated.


Going on that presumption, if the centers, or centres if you insist, are separated by a distance of 14 cm, then the point of tangency where the two circles touch is also the end point of a radius for each of the circles.  Hence the sum of the measures of the radii for the two circles is 14 cm.


Let *[tex \Large r] represent the measure of the radius of one of the circles, then *[tex \Large 14 - r] must represent the radius of the other circle.  Furthermore, the area of the first circle is *[tex \Large \pi r^2], the area of the second circle is *[tex \Large \pi(14-r)^2], and we are given that the sum of these areas is *[tex \Large 130\pi\text{ cm^2}].  Hence:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \pi r^2 + \pi(14-r)^2 = 130\pi]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ r^2 + (14-r)^2 = 130]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ 2r^2 - 28r + 66 = 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ r^2 - 14r + 33 = 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ (r - 3)(r - 11) = 0]


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ r = 3] or *[tex \LARGE r = 11]


One radius is 3 and the other is 11.


Now, if my assumption about the given area was wrong, then you would end up with the following computational horror:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ r = \frac{14 \pm sqrt{196 - 4\left(\frac{98\pi-130}{\pi}\right)}}{2} ]


Which, by the way, is a conjugate pair of complex numbers which tells me that the solution is impossible if the combined area is only *[tex \Large 130\text{ cm^2}] instead of a little over 3 times that value.  In fact, the maximum possible sum of the radii of two circles with a combined area of *[tex \Large 130\text{ cm^2}] is *[tex \Large \sqrt{2\cdot\frac{130}{\pi}}\ \approx\ 9.1]



John
*[tex \LARGE e^{i\pi} + 1 = 0]
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