NTSE Stage 2 SAT 2016 Question Paper

Question : 51 of 100

Marks: +1, -0

A circular metallic sheet is divided into two parts in such a way that each part can be folded into a cone. If the ratio of their curved surface areas is 1 : 2, the the ratio of their volumes is :

1:8
$1:√16$
$1:√10$
2:3

Validate

Solution:

{\Area \of \sector \ADB}/{\text "Area of sector ACD"}={\θ/360°×\π\r^2}/{{360°-\θ}/360°×\π\r^2

⇒

1/2=\θ/{360°-\θ}

⇒

\q=120°

∴

\ov {\ADB}=\θ/360°×2\π\r= {2\π\r}/3

⇒

\ov{\ACB}={4\π\r}/3

\ov {\ADB}=\circumference \of \base=2\π\r_1

{2\π\r}/3=2\π\r_1

⇒

\r_1=\r/3

Similarly

\r_={2\r}/3

\h_1=√{\r^2-\r^2_1}

\h_1=√{\r^2-\r^2/9}

={2√2\r}/3

Similarly,

\h_2={√5\r}/3

\V_1/\V_2={1/2\π\r^2_1\h_1}/{1/3\π\r^2_2\h_2}

=(\r_1/\r_1)^2(\h_1/\h_2)^2

=1/4×{2√2}/√5

=1/√10

So the ratio of their volume is 1 :

√10

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