Question : The ratio of the curved surface area of two cones is 1 : 4 and the ratio of slant height of the two cones is 2 : 1. What is the ratio of the radius of the two cones?
Option 1: 1 : 2
Option 2: 1 : 4
Option 3: 1 : 8
Option 4: 1 : 1
Correct Answer: 1 : 8
Solution :
The curved surface area of a cone $ = \pi rl$ where \(r\) is the radius and \(l\) is the slant height of the cone.
Given that the ratio of the curved surface areas of two cones = 1 : 4
$⇒\frac{\pi r_1 l_1}{\pi r_2 l_2} = \frac{1}{4}$
Also, given that the ratio of the slant heights of the two cones = 2 : 1
$⇒\frac{l_1}{l_2} = \frac{2}{1}$
$⇒\frac{r_1}{r_2} = \frac{1}{4} \times \frac{1}{2} = \frac{1}{8}$
Hence, the correct answer is 1 : 8.
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