Question : If the volume of two circular cones is in the ratio 4 : 1 and their diameter is in the ratio 5 : 4, then the ratio of their height is:
Option 1: 25 : 16
Option 2: 25 : 64
Option 3: 64 : 25
Option 4: 16 : 25
Correct Answer: 64 : 25
Solution :
Given that the volume of the two cones is in the ratio 4 : 1.
$⇒\frac{V_1}{V_2} = \frac {4} {1}$
Also, given that the diameter of the cones is in the ratio 5 : 4.
$⇒\frac{d_1}{d_2}= \frac{5}{4}$
$⇒\frac{r_1}{r_2} =\frac{\frac{d_1}{2}}{\frac{d_2}{2}}= \frac{5}{4}$ where $r_1$ and $r_2$ denotes the radii of the two cones.
Substituting these ratios into the formula,
$⇒\frac{V_1}{V_2} = \frac{\frac{1}{3}\pi r_1^2h_1}{\frac{1}{3}\pi r_2^2h_2} = \frac{r_1^2h_1}{r_2^2h_2} $
$⇒\frac {4} {1}=(\frac{5}{4})^2×\frac{h_1}{h_2}$
$⇒\frac{h_1}{h_2}=\frac {64}{25} $
Hence, the correct answer is 64 : 25.
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