Question

1. circles a large circle has an area of 400 cm² and a small circle has an area of 200 cm². the radius of the large circle is \\(sqrt{\frac{400}{pi}}\\) cm. the radius of the small circle is \\(sqrt{\frac{200}{pi}}\\) cm. find the difference of the radius of the large circle and the radius of the small circle.

Explanation:

Step1: Recall the formula for the area of a circle, \( A = \pi r^2 \), so \( r=\sqrt{\frac{A}{\pi}} \). The radius of the large circle \( R=\sqrt{\frac{400}{\pi}} \) and the radius of the small circle \( r = \sqrt{\frac{200}{\pi}} \). We need to find \( R - r \).

Step2: Simplify \( \sqrt{\frac{400}{\pi}}-\sqrt{\frac{200}{\pi}} \). Factor out \( \frac{1}{\sqrt{\pi}} \): \( \frac{\sqrt{400}-\sqrt{200}}{\sqrt{\pi}} \).

Step3: Simplify the square roots: \( \sqrt{400} = 20 \), \( \sqrt{200}=\sqrt{100\times2}=10\sqrt{2} \). So we have \( \frac{20 - 10\sqrt{2}}{\sqrt{\pi}} \). Or, rationalizing the denominator (optional), but we can also factor out 10: \( 10\frac{2-\sqrt{2}}{\sqrt{\pi}} \). Alternatively, if we want to write it as \( \frac{20}{\sqrt{\pi}}-\frac{10\sqrt{2}}{\sqrt{\pi}} \), but a more simplified form by factoring: \( \frac{10(2 - \sqrt{2})}{\sqrt{\pi}} \) or rationalizing \( \frac{10\sqrt{\pi}(2 - \sqrt{2})}{\pi} \). Wait, actually, let's check the initial approach. Wait, the problem says "the radius of the large circle is \( \sqrt{\frac{400}{\pi}} \) cm" and "the radius of the small circle is \( \sqrt{\frac{200}{\pi}} \) cm". So the difference is \( \sqrt{\frac{400}{\pi}}-\sqrt{\frac{200}{\pi}} \). Let's factor out \( \frac{1}{\sqrt{\pi}} \): \( \frac{\sqrt{400}-\sqrt{200}}{\sqrt{\pi}}=\frac{20 - 10\sqrt{2}}{\sqrt{\pi}} \). We can also factor 10: \( 10\frac{2 - \sqrt{2}}{\sqrt{\pi}} \). If we rationalize the denominator: multiply numerator and denominator by \( \sqrt{\pi} \), we get \( \frac{10(2 - \sqrt{2})\sqrt{\pi}}{\pi} \). But maybe a simpler way: \( \sqrt{\frac{400}{\pi}}=\frac{20}{\sqrt{\pi}} \), \( \sqrt{\frac{200}{\pi}}=\frac{10\sqrt{2}}{\sqrt{\pi}} \), so the difference is \( \frac{20 - 10\sqrt{2}}{\sqrt{\pi}} = \frac{10(2 - \sqrt{2})}{\sqrt{\pi}} \) cm. Alternatively, we can write it as \( \frac{10\sqrt{\pi}(2 - \sqrt{2})}{\pi} \) after rationalizing. But let's check the calculation again. Wait, the area of the large circle is 400 cm², so \( R=\sqrt{\frac{400}{\pi}}=\frac{20}{\sqrt{\pi}} \), small circle area 200 cm², so \( r = \sqrt{\frac{200}{\pi}}=\frac{10\sqrt{2}}{\sqrt{\pi}} \). Then \( R - r=\frac{20 - 10\sqrt{2}}{\sqrt{\pi}} \). We can factor 10: \( 10\frac{2 - \sqrt{2}}{\sqrt{\pi}} \). If we approximate, but the problem probably expects an exact form. Let's simplify \( \sqrt{\frac{400}{\pi}}-\sqrt{\frac{200}{\pi}}=\sqrt{\frac{200\times2}{\pi}}-\sqrt{\frac{200}{\pi}}=\sqrt{\frac{200}{\pi}}\times\sqrt{2}-\sqrt{\frac{200}{\pi}}=\sqrt{\frac{200}{\pi}}(\sqrt{2}-1) \). Since \( \sqrt{\frac{200}{\pi}}=\frac{10\sqrt{2}}{\sqrt{\pi}} \), so \( \frac{10\sqrt{2}}{\sqrt{\pi}}(\sqrt{2}-1)=\frac{10(2 - \sqrt{2})}{\sqrt{\pi}} \).

Answer:

The difference in radii is \( \boldsymbol{\frac{10(2 - \sqrt{2})}{\sqrt{\pi}}} \) cm (or equivalent forms like \( \frac{20}{\sqrt{\pi}}-\frac{10\sqrt{2}}{\sqrt{\pi}} \) or \( \frac{10\sqrt{\pi}(2 - \sqrt{2})}{\pi} \) after rationalizing). If we want a numerical approximation, using \( \pi\approx3.14 \), \( \sqrt{2}\approx1.414 \): \( \frac{10(2 - 1.414)}{\sqrt{3.14}}\approx\frac{10(0.586)}{\sqrt{3.14}}\approx\frac{5.86}{1.772}\approx3.31 \) cm, but the exact form is better. However, the problem might have a simpler approach. Wait, wait, maybe I made a mistake. Wait, the area of the large circle is 400, small is 200. So \( R = \sqrt{\frac{400}{\pi}}=\frac{20}{\sqrt{\pi}} \), \( r=\sqrt{\frac{200}{\pi}}=\frac{10\sqrt{2}}{\sqrt{\pi}} \). Then \( R - r=\frac{20 - 10\sqrt{2}}{\sqrt{\pi}} = 10\frac{2 - \sqrt{2}}{\sqrt{\pi}} \). Alternatively, factor out \( \sqrt{\frac{100}{\pi}} \): \( \sqrt{\frac{100}{\pi}}(\sqrt{4}-\sqrt{2})=\frac{10}{\sqrt{\pi}}(2 - \sqrt{2}) \), which is the same as before. So the exact difference is \( \frac{10(2 - \sqrt{2})}{\sqrt{\pi}} \) centimeters.