Question

In the figure given below, a cylinder is inserted into a cone, and the vertical height of the cone is 30 cm. The diameter of the cylinder is 8 cm. What is the volume of the cone? The base of the cylinder and the base of the cone are on the same plane.

• A. \(3000\pi \text{ cm}^3\)
• B. \(4860\pi \text{ cm}^3\)
• C. \(2800\pi \text{ cm}^3\)
• D. Cannot be determined

Answer 1

The Correct Option is A

Given:
Height of the cone, \(AD = 30\) cm  
Diameter of the cylinder = 8 cm  
Radius of the cylinder, \(r = \frac{8}{2} = 4\) cm  
Since the base of the cylinder and the base of the cone are on the same plane, the height of the cylinder and the height of the cone are equal.
In triangle \(ACD\):
\[\tan \angle ACD = \frac{AD}{DC} = \sqrt{3} \quad (\text{since } \angle ACD = 60^\circ)\]
\[DC = \frac{AD}{\sqrt{3}} = \frac{30}{\sqrt{3}} = 10\sqrt{3} \text{ cm}\]
Therefore, the radius of the cone is \(DC = 10\sqrt{3}\) cm.
The volume of the cone is given by:
\[\text{Volume of the cone} = \frac{1}{3} \pi r^2 h = \frac{1}{3} \pi (10\sqrt{3})^2 (30) = \frac{1}{3} \pi (300) (30) = 3000\pi \text{ cm}^3\]
Therefore, the volume of the cone is \(3000\pi \text{ cm}^3\).
Thus, the correct answer is A.

Answer 2

From the figure, the base half-angle of the cone (angle ACD) is 60°, with the height \(AD=30\) cm. Instead of finding the base radius directly from the tangent ratio, find the slant length \(AC\) first using the sine ratio, then obtain the radius from the cosine ratio.

\[\sin60^{\circ}=\frac{AD}{AC} \implies AC=\frac{30}{\sin60^{\circ}}=\frac{30}{\frac{\sqrt3}{2}}=\frac{60}{\sqrt3}=20\sqrt3\text{ cm}\]

\[\cos60^{\circ}=\frac{DC}{AC} \implies DC=AC\times\cos60^{\circ}=20\sqrt3\times\frac12=10\sqrt3\text{ cm}\]

So the base radius of the cone is \(10\sqrt3\) cm, and its volume is:
\[V=\frac13\pi r^{2}h=\frac13\pi(10\sqrt3)^{2}(30)=\frac13\pi(300)(30)=3000\pi\text{ cm}^{3}\]

1. Option A \(3000\pi\): Matches exactly.
2. Option B \(4860\pi\): Does not match.
3. Option C \(2800\pi\): Does not match.
4. Option D (Cannot be determined): Ruled out since the volume can be found uniquely from the given figure.

The volume of the cone is \(3000\pi\text{ cm}^{3}\).

Hence, the correct answer is Option A: \(3000\pi \text{ cm}^3\).