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The lengths of arcs of two sectors are in the ratio $2: 1$ and their a
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02 May 2025, 01:12
02 May 2025, 01:12
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The lengths of arcs of two sectors are in the ratio $2: 1$ and their areas are in the ratio $1: 2$. What is the ratio of the diameters of the two sectors? (Indicate all such answers).
(A) $1: 5$
(B) $1: 4$
(C) $1: 3$
(D) $1: 2$
(E) $1: 1$
(A) $1: 5$
(B) $1: 4$
(C) $1: 3$
(D) $1: 2$
(E) $1: 1$
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Re: The lengths of arcs of two sectors are in the ratio $2: 1$ and their a
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07 May 2025, 05:00
07 May 2025, 05:00
Expert Reply
\(\frac{l_1}{l_2}=\frac{2}{1}\)
GRe angles geometry.png [ 18.37 KiB | Viewed 49 times ]
\(\begin{aligned}
&\begin{aligned}
& l_1=\frac{\theta_1}{360} \times 2 \pi r_1 \quad l_2=\frac{\theta_2}{360} \times 2 \pi r_2 \Rightarrow \frac{l_1}{l_2}=\frac{2}{1}=\frac{\theta_1}{\theta_2} \frac{r_1}{r_2} \\
& \Rightarrow A_1=\frac{\theta_1}{360} \times \pi r_1^2 \quad \text { and } A_2=\frac{\theta_2}{360} \times \pi r_2^2 \\
& \Rightarrow \frac{A_1}{A_2}=\frac{\theta_1}{\theta_2} \times \frac{r_1^2}{r_2^2}=\frac{1}{2} \\
& \Rightarrow \frac{1}{2}=\frac{\theta_1 r_1}{\theta_2 r_2} \times \frac{r_1}{r_2} \Rightarrow \frac{1}{2}=2 \times \frac{r_1}{r_2} \Rightarrow \frac{1}{2 \times 2}=\frac{2 r_1}{2 r_2} \Rightarrow \frac{d_1}{d_2}=\frac{1}{4}=1: 4 .
\end{aligned}\\
&\text { Thus, the correct answer is } B \text {. }
\end{aligned}\)
Attachment:
GRe angles geometry.png [ 18.37 KiB | Viewed 49 times ]
\(\begin{aligned}
&\begin{aligned}
& l_1=\frac{\theta_1}{360} \times 2 \pi r_1 \quad l_2=\frac{\theta_2}{360} \times 2 \pi r_2 \Rightarrow \frac{l_1}{l_2}=\frac{2}{1}=\frac{\theta_1}{\theta_2} \frac{r_1}{r_2} \\
& \Rightarrow A_1=\frac{\theta_1}{360} \times \pi r_1^2 \quad \text { and } A_2=\frac{\theta_2}{360} \times \pi r_2^2 \\
& \Rightarrow \frac{A_1}{A_2}=\frac{\theta_1}{\theta_2} \times \frac{r_1^2}{r_2^2}=\frac{1}{2} \\
& \Rightarrow \frac{1}{2}=\frac{\theta_1 r_1}{\theta_2 r_2} \times \frac{r_1}{r_2} \Rightarrow \frac{1}{2}=2 \times \frac{r_1}{r_2} \Rightarrow \frac{1}{2 \times 2}=\frac{2 r_1}{2 r_2} \Rightarrow \frac{d_1}{d_2}=\frac{1}{4}=1: 4 .
\end{aligned}\\
&\text { Thus, the correct answer is } B \text {. }
\end{aligned}\)
gmatclubot
Re: The lengths of arcs of two sectors are in the ratio $2: 1$ and their a [#permalink]
07 May 2025, 05:00
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