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# Can someone explain how to solve this problem?

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Four circles are drawn. Let $$A_1, A_2, A_3, A_4$$ be the areas of the regions, so $$A_1$$ is the area inside the smallest circle, $$A_2$$ is the area outside the smallest circle and inside the second-smallest circle, and so on. The areas satisfy

$$A_1 = \frac{A_2}{2} = \frac{A_3}{3} = \frac{A_4}{4}$$

Let $$r_1$$ denote the radius of the smallest circle, and let $$r_1$$ denote the radius of the largest circle. Find $$\frac{r_4}{r_1}.$$

May 25, 2020

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Four circles are drawn.
Let $$A_1, A_2, A_3, A_4$$ be the areas of the regions, so
$$A_1$$is the area inside the smallest circle,
$$A_2$$ is the area outside the smallest circle and inside the second-smallest circle, and so on.
The areas satisfy
$$A_1 = \frac{A_2}{2} = \frac{A_3}{3} = \frac{A_4}{4}$$
Let $$r_1$$ denote the radius of the smallest circle, and let $$r_4$$ denote the radius of the largest circle. Find $$\frac{r_4}{r_1}$$.

My attempt:

$$\text{Let A_{\circ_{1} }= \pi r_1^2 } \\ \text{Let A_{\circ_{4} } = \pi r_4^2 }$$

$$\begin{array}{|llcll|} \hline & 1*A_1 &=& A_{\circ_{1}} \\ & 2*A_1 = A_2 &=& A_{\circ_{2}}- A_{\circ_{1}} \\ & 3*A_1 = A_3 &=& A_{\circ_{3}}- A_{\circ_{2}} \\ & 4*A_1 = A_4 &=& A_{\circ_{4}}- A_{\circ_{3}} \\ \hline \text{sum}: & 10A_1&=& A_{\circ_{1}}+A_{\circ_{2}} - A_{\circ_{1}}+A_{\circ_{3}} - A_{\circ_{2}}+A_{\circ_{4}} - A_{\circ_{3}} \\ & 10A_1&=& A_{\circ_{4}} \quad | \quad A_1 = A_{\circ_{1}} \\ & 10A_{\circ_{1}}&=& A_{\circ_{4}} \\ & 10\pi r_1^2&=& \pi r_4^2 \\ & 10r_1^2 &=& r_4^2 \\\\ & \dfrac{r_4^2}{r_1^2} &=& 10 \\\\ & \mathbf{\dfrac{r_4}{r_1}} &=& \mathbf{\sqrt{10}} \\ \hline \end{array}$$

May 25, 2020
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Area of circle       A1 = pi            area of  A4 = 10 pi

r1 =  1                           r4 = sqrt( 10pi / pi )

May 25, 2020