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The Question: A central angle of two concentric circles is $\frac{3\pi}{14}$. The area of the large sector is twice the area of the small sector. What is the ratio of the lengths of the radii of the two circles?

Answer: 0.71:1

The answer I end up with is $\sqrt{2}$:$1$, instead of $\frac{\sqrt{2}}{2}$: 1 (which I'm assuming is how they got 0.71). My book says the angle measure is superfluous, and that "areas of similar figures are proportional to the squares of linear measures associated with those figures". Using the similar figures property, this was my answer:

if $r_2$ = radius of larger circle and $r_1$ = radius of smaller circle and likewise for $a_2$ and $a_1$, then ($\frac{r_2}{r_1}$)$^2$ = $\frac{a_2}{a_1}$. Since $\frac{a_2}{a_1}$ = 2, taking the root of both sides yields $\sqrt{2}$ = $\frac{r_2}{r_1}$. Any explanation would be appreciated, thank you very much!

ak_27
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  • What are you actually asking? – Xetrov Apr 10 '17 at 20:11
  • $\frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}$ - maybe this is where you are going wrong. Plus the order of the ratio. – Chinny84 Apr 10 '17 at 20:12
  • @simplest_mathematics Sorry if it wasn't clear, I just didn't know why my answer was wrong/how they got their answer. – ak_27 Apr 10 '17 at 20:12
  • They have written as the answer the ratio of the smaller to the larger, and you have found the ratio of the larger to the smaller. – David Quinn Apr 10 '17 at 20:13
  • @DavidQuinn My answer was sqrt(2):1, and theirs was 0.71:1. – ak_27 Apr 10 '17 at 20:16
  • Explanation of what? Everything you said is perfectly correct. The ratio of the long to short radius is root (2) to 1. The ratio of the short radius to long is 1/root (2) $\approx$ .71 to 1 (and SHAME on your book for thinking that was a acceptable answer, especially at the teaching level). And shame on your book for not specifying whether it wanted the ratio of the long to short, or the short to long. – fleablood Apr 10 '17 at 20:16
  • @DavidQuinn Sorry, got super confused there! You're right, thanks! – ak_27 Apr 10 '17 at 20:17
  • @fleablood Thanks! This had me so confused, didn't even think about the possibility that they might have done it the other way! – ak_27 Apr 10 '17 at 20:18
  • r2/r1= root (2). r1/r2 = 1/root (2) =.71. No harm no foul. But I'm not going to be taking your book to the prom after that sort of shenanigans. – fleablood Apr 10 '17 at 20:19

2 Answers2

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The question does not specify which radius comes first in the proportion. The book answer gives $r_1:r_2=\frac {\sqrt 2}2:1$ while you are saying $r_2:r_1=\sqrt 2:1$. You agree with the book but present the data in the opposite order. I also object strongly to saying $\frac {\sqrt 2}2=0.71$ but that is another issue.

Ross Millikan
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let r = radius of the smaller circle and R = radius of the larger circle. We will get R:r= √2: 1. Since the order in taking which radius first we can take r:R =1:√2 or r/R=1/√2= 0.707 i.e., r:R=0.707:1

            r:R=0.71:1
Niranjana P V
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