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Area in between two concentric circles.

I am working on a problem and a bit stuck.

The problem is that we have two concentric circles, with the area inclosed by the inner circle equal to $7000\pi$ square meters . We are then given that the ratio of the circumference of the outer to the inner circle is equal to $8:7$ .

I realize that in order to solve this we need to understand the formulas for the area of a circle and the circumference of a circle.

Thus, we recognize that $C = 2\pi r$ and $A = \pi r^2$

We then conclude that $7000\pi = \pi r^2$ and find $r = \sqrt{7000}$

Now that we have this value, we need to find it in terms of the ratio $8:7$ in order to find the radius of the larger circle so that we can calculate its area and subtract that of the smaller circle. This is where I am stuck however, how do we convert the radius $\sqrt{7000}$ in to the appropriate ratio?

Do we just simply divide by seven and then multiply this value by eight?

Thanks.

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    $\begingroup$ Yes, you just multiply the radius by $\frac 87$. $\endgroup$ – Ross Millikan Aug 29 '20 at 2:55
  • $\begingroup$ @RossMillikan Thanks! $\endgroup$ – Ethan Aug 29 '20 at 2:56
  • $\begingroup$ If the linear measure is in ratio $8:7$ then the area measure is in ratio $64:49$ ($8^2:7^2$). Which means that the ratio of the area between them to the area of the smaller circle is $15:49$. $\endgroup$ – CogitoErgoCogitoSum Aug 29 '20 at 3:05
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$$7C_{outer}=8C_{inner}$$

$$7(2\pi r_{outer})=8(2\pi r_{inner})$$

$$14\pi r_{outer}=16\pi r_{inner}$$

$$r_{outer}=\frac{8}{7} \ r_{inner}$$

$$r_{outer}=\frac{8}{7}\sqrt{7000}$$

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  • $\begingroup$ Your first equation is incorrect . It should be $8C_{inner}=7C_{outer}$. This flows all the way through. The outer radius should be larger than the inner radius, while you have it $\frac 78$ the size. $\endgroup$ – Ross Millikan Aug 29 '20 at 2:57
  • $\begingroup$ @RossMillikan Sorry about that. I realized once I read it over. Thanks for letting me know. $\endgroup$ – Kman3 Aug 29 '20 at 2:58
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If you need to find the area between those two circles , use the hint given below :

$ \frac {Area_{outer}}{Area_ {inner}} =\frac{ R²_{1}} {R²_{2}}. \,$

$\frac{Area_{outer}} {Area_ {inner}} =\frac{ 64}{49}. $

$\frac{(Area_{outer} - Area_ {inner})} { Area_{inner}} = \frac{15} {49}. $

${Area_{outer} - Area_ {inner}} = Area_{remaining} $

$ =\frac {15}{49}× Area_{ inner} = \frac{15}{49} × 7000π = \frac{15000π}{7} sq. unit$

Hope it helps.

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  • $\begingroup$ @Ethan was that helpful ?? $\endgroup$ – A Student 4ever Aug 29 '20 at 5:57

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