Question

Find the height of the right circular cone of maximum volume, which is inscribed in a sphere of radius 12cm.

Hint:

For a cone of maximum volume inscribed in a sphere of radius \( R \), the height is always \( \frac{4}{3}R \) and the volume is \( \frac{8}{27} \) of the sphere's volume.

Answer 1

Step 1: Understanding the Concept:
This is an optimization problem. We must express the volume of the cone in terms of a single variable and use calculus to find the height for maximum volume.
Step 2: Detailed Explanation:
Let \( R = 12 \) be the radius of the sphere.
Let \( h \) be the height and \( r \) be the radius of the cone.
If \( x \) is the distance from the center of the sphere to the base of the cone, then \( h = R + x \) and \( r^2 = R^2 - x^2 \).
Volume of cone \( V = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (R^2 - x^2)(R + x) \).
Differentiate \( V \) with respect to \( x \):
\[ \frac{dV}{dx} = \frac{\pi}{3} [ (R^2 - x^2)(1) + (R + x)(-2x) ] \] \[ \frac{dV}{dx} = \frac{\pi}{3} [ R^2 - x^2 - 2Rx - 2x^2 ] = \frac{\pi}{3} (R^2 - 2Rx - 3x^2) \].
For maxima, \( \frac{dV}{dx} = 0 \):
\( 3x^2 + 2Rx - R^2 = 0 \implies (3x - R)(x + R) = 0 \).
Since \( x>0 \), \( x = R/3 \).
Height \( h = R + R/3 = \frac{4R}{3} = \frac{4(12)}{3} = 16 \text{ cm} \).
Step 3: Final Answer:
The height is 16 cm.