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 \(\frac{r_4}{r_1}.\)
+26367 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:
\(\text{Let $A_{\circ_{1} }= \pi r_1^2$ } \\ \text{Let $A_{\circ_{4} } = \pi r_4^2$ } \)
\(\begin{array}{|llcll|} \hline & 1*A_1 &=& A_{\circ_{1}} \\ & 2*A_1 = A_2 &=& A_{\circ_{2}}- A_{\circ_{1}} \\ & 3*A_1 = A_3 &=& A_{\circ_{3}}- A_{\circ_{2}} \\ & 4*A_1 = A_4 &=& A_{\circ_{4}}- A_{\circ_{3}} \\ \hline \text{sum}: & 10A_1&=& A_{\circ_{1}}+A_{\circ_{2}} - A_{\circ_{1}}+A_{\circ_{3}} - A_{\circ_{2}}+A_{\circ_{4}} - A_{\circ_{3}} \\ & 10A_1&=& A_{\circ_{4}} \quad | \quad A_1 = A_{\circ_{1}} \\ & 10A_{\circ_{1}}&=& A_{\circ_{4}} \\ & 10\pi r_1^2&=& \pi r_4^2 \\ & 10r_1^2 &=& r_4^2 \\\\ & \dfrac{r_4^2}{r_1^2} &=& 10 \\\\ & \mathbf{\dfrac{r_4}{r_1}} &=& \mathbf{\sqrt{10}} \\ \hline \end{array} \)
