Discussion
Ajay has 2 cones with him. 1st cone has radius twice the other and height half the other. What is volume ratio of 1st and 2nd cones?
Answer: A
Let the second cone have radius \( r \) and height \( h \). The first then has radius \( 2r \) and height \( \tfrac{h}{2} \).
\[ V_{1} = \frac{1}{3}\pi (2r)^{2}\left(\frac{h}{2}\right) = \frac{1}{3}\pi \cdot 4r^{2} \cdot \frac{h}{2} = \frac{2}{3}\pi r^{2}h \]
\[ V_{2} = \frac{1}{3}\pi r^{2} h \]
\[ \frac{V_{1}}{V_{2}} = \frac{2/3}{1/3} = 2 \quad\Longrightarrow\quad 2 : 1 \]
Doubling the radius quadruples the volume while halving the height only halves it, so the net effect is a doubling - radius counts twice over, height only once.
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