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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
 #1
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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:

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

 

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

 

laugh

 May 25, 2020
 #2
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Area of circle       A1 = pi            area of  A4 = 10 pi

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

 May 25, 2020

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