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Sample Question 7:
Step 1: Let the side length of the biggest square be \(x\). Hence, the diameter of the first circle is also \(x\).
Step 2: The square inscribed in the aforementioned circle will have a diagonal of length \(x\).
Step 3: Now, the side length of this smaller square can be calculated as \(x/\sqrt{2}\) or \(x\sqrt{2}/2\).
Step 4: The smaller circle inscribed in this square will have a diameter equal to the side of the square, which is \(x\sqrt{2}/2\). So, its radius is \(x\sqrt{2}/4\).
Step 5: Now, we can calculate the area of the smaller circle, which equals \(\pi (x\sqrt{2}/4)^2\), that simplifies to \(\pi x^2/8\).
Step 6: The area of the largest square is \(x^2\).
Step 7: Therefore, the ratio of the area of the smaller circle to the area of the larger square is \((\pi x^2/8)/x^2\).
Step 8: Simplifying this, we get \(\pi/8\). Hence, the ratio of the areas of the smaller circle to the larger square is \(\pi/8\). This corresponds to option \(\textbf{(B)} \frac{\pi}{8}\).
A circle is inscribed in a square, then a square is inscribed in this circle, and finally, a circle is inscribed in this square. What is the ratio of the area of the smaller circle to the area of the larger square?
\(\mathrm{(A)} \frac{\pi}{16} \qquad \mathrm{(B)} \frac{\pi}{8} \qquad \mathrm{(C)} \frac{3\pi}{16} \qquad \mathrm{(D)} \frac{\pi}{4} \qquad \mathrm{(E)} \frac{\pi}{2}\)
Answer Keys
Question 7: BSolutions
Question 7Step 1: Let the side length of the biggest square be \(x\). Hence, the diameter of the first circle is also \(x\).
Step 2: The square inscribed in the aforementioned circle will have a diagonal of length \(x\).
Step 3: Now, the side length of this smaller square can be calculated as \(x/\sqrt{2}\) or \(x\sqrt{2}/2\).
Step 4: The smaller circle inscribed in this square will have a diameter equal to the side of the square, which is \(x\sqrt{2}/2\). So, its radius is \(x\sqrt{2}/4\).
Step 5: Now, we can calculate the area of the smaller circle, which equals \(\pi (x\sqrt{2}/4)^2\), that simplifies to \(\pi x^2/8\).
Step 6: The area of the largest square is \(x^2\).
Step 7: Therefore, the ratio of the area of the smaller circle to the area of the larger square is \((\pi x^2/8)/x^2\).
Step 8: Simplifying this, we get \(\pi/8\). Hence, the ratio of the areas of the smaller circle to the larger square is \(\pi/8\). This corresponds to option \(\textbf{(B)} \frac{\pi}{8}\).