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Two concentric circles $P$ and $Q$ have radii $r$ and $R$, $r<R$. If a
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18 Jan 2025, 00:40
18 Jan 2025, 00:40
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Two concentric circles $P$ and $Q$ have radii $r$ and $R$, $r<R$. If area of the smaller circle is equal to the difference in the areas of the two circles, which of the following expresses r in terms of $\(R\)$ ?
(A) $\(\frac{R}{4}\)$
(B) $\(\frac{R}{3}\)$
(C) $\(\frac{R}{2}\)$
(D) $\(\frac{R}{\sqrt{2}}\)$
(E) $\(\quad R \sqrt{2}\)$
(A) $\(\frac{R}{4}\)$
(B) $\(\frac{R}{3}\)$
(C) $\(\frac{R}{2}\)$
(D) $\(\frac{R}{\sqrt{2}}\)$
(E) $\(\quad R \sqrt{2}\)$
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Two concentric circles $P$ and $Q$ have radii $r$ and $R$, $r<R$. If a
[#permalink]
26 Feb 2025, 12:56
26 Feb 2025, 12:56
Expert Reply
$\(\mathrm{P} \& \mathrm{Q}\)$ are two concentric circle having radii $\(\mathrm{r} \& \mathrm{R}\)$ respectively, where $\(\mathrm{r}<\mathrm{R}\)$.
The area of the smaller circle i.e. circle $\(P=\pi r^2 \&\)$ the area of the bigger circle i.e. circle $\(Q=\pi R^2\)$
It is given that the area of the smaller circle i.e. circle $\(P\)$ is equal to the difference in the areas of the two circles, so we get $\(\pi r^2=\pi R^2-\pi r^2\)$ i.e. $\(2 \pi r^2=\pi R^2 \Rightarrow r=\frac{R}{\sqrt{2}}\)$
Hence the value of $\(r\)$ in terms of $\(R\)$ is $\(r=\frac{R}{\sqrt{2\)$, so the answer is (D).
The area of the smaller circle i.e. circle $\(P=\pi r^2 \&\)$ the area of the bigger circle i.e. circle $\(Q=\pi R^2\)$
It is given that the area of the smaller circle i.e. circle $\(P\)$ is equal to the difference in the areas of the two circles, so we get $\(\pi r^2=\pi R^2-\pi r^2\)$ i.e. $\(2 \pi r^2=\pi R^2 \Rightarrow r=\frac{R}{\sqrt{2}}\)$
Hence the value of $\(r\)$ in terms of $\(R\)$ is $\(r=\frac{R}{\sqrt{2\)$, so the answer is (D).
gmatclubot
Two concentric circles $P$ and $Q$ have radii $r$ and $R$, $r<R$. If a [#permalink]
26 Feb 2025, 12:56
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