Question : The heights of two cones are in the ratio 7 : 5 and their diameters are in the ratio 10 : 21.
What is the ratio of their volumes? (Where $\pi=\frac{22}{7}$)
Option 1: 26 : 47
Option 2: 14 : 19
Option 3: 20 : 63
Option 4: 17 : 21
Correct Answer: 20 : 63
Solution :
Ratio of heights, $h_1:h_2 =\frac{h_1}{h_2} = \frac{7}{5}$
Ratio of diameters, $d_1 : d_2 = \frac{d_1}{d_2} = \frac{10}{21}$
⇒ Ratio of radii, $r_1 : r_2 = \frac{r_1}{r_2} = \frac{10}{21}$
Ratio of volumes of cones, $V_1 : V_2 = \frac{\frac{1}{3} \pi r_1^2 h_1}{\frac{1}{3} \pi r_2^2 h_2}= \frac{r_1^2 h_1}{r_2^2 h_2} = \frac{10^2 ×7}{21^2 × 5} = \frac{20}{63} = 20 :63$
Hence, the correct answer is 20 : 63.
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