Question : The ratio of the height and the diameter of a right circular cone is 6: 5 and its volume is $\frac{2200}{7} \mathrm{~cm}^3$. What is its slant height?
(Take $\pi=\frac{22}{7}$ )
Option 1: 26 cm
Option 2: 13 cm
Option 3: 25 cm
Option 4: 5 cm
Correct Answer: 13 cm
Solution :
The height and diameter of a right circular cone is in the ratio 6 : 5.
The volume of the right circular cone = $\frac{2200}{7}\ \text{cm}^3$
The volume of a right circular cone = $\frac{1}{3}\pi r^2 h$, where $r$ is the radius and $h$ is the height.
Let the height and diameter of the cone be 6x and 5x respectively
Then, Volume $=\frac{1}{3}\pi r^2 h$
$⇒ (\frac{1}{3})\pi \ (\frac{5\text{x}}{2})^2 \ 6\text{x}=\frac{2200}{7}$
$⇒ \text{x}^3 = \frac{(2200 × 3 × 7 × 4)}{(7 × 25 × 22 × 6)}$
$⇒ \text{x}^3 = 8$
$⇒ \text{x}= 2$
So, the height will be 12 cm, and the radius will be 5 cm.
Slant height = $\sqrt{\text{height}^2 + \text{radius}^2}$
⇒ Slant height = $\sqrt{12^2 + 5^2}= 13\ \text{cm}$
Hence, the correct answer is 13 cm.
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