Question

A conical cavity of maximum volume is carved out from a wooden solid hemisphere of radius 10 cm. Curved surface area of the cavity carved out is (use \(\pi = 3.14\))

• A. \(314 \sqrt{2}\) \(\text{cm}^{2}\)
• B. \(314\) \(\text{cm}^{2}\)
• C. \(\frac{3140}{3}\) \(\text{cm}^{2}\)
• D. \(3140 \sqrt{2}\) \(\text{cm}^{2}\)

Hint:

For any cone inscribed in a hemisphere to have maximum volume, its height must be equal to its base radius, which are both equal to the radius of the hemisphere. The slant height will always be \(r\sqrt{2}\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To carve out a cone of maximum volume from a hemisphere, the base of the cone must coincide with the base of the hemisphere, and the vertex of the cone must be at the center of the hemisphere's base or the top of the dome. In this case, the radius of the cone (\(r\)) will be equal to the radius of the hemisphere (\(R\)), and the height of the cone (\(h\)) will also be equal to the radius of the hemisphere (\(R\)).
Step 2: Key Formula or Approach:
The curved surface area (CSA) of a cone is given by:
\[ \text{CSA} = \pi r l \]
where \(l\) is the slant height, calculated as \(l = \sqrt{r^2 + h^2}\).
Step 3: Detailed Explanation:
Given: Radius of hemisphere \(R = 10\) cm.
For maximum volume, for the cone:
Radius \(r = R = 10\) cm
Height \(h = R = 10\) cm
First, we find the slant height (\(l\)):
\[ l = \sqrt{10^2 + 10^2} = \sqrt{100 + 100} = \sqrt{200} = 10\sqrt{2} \text{ cm} \]
Now, calculate the Curved Surface Area:
\[ \text{CSA} = \pi \times 10 \times 10\sqrt{2} \]
\[ \text{CSA} = 3.14 \times 100\sqrt{2} \]
\[ \text{CSA} = 314\sqrt{2} \text{ cm}^2 \]
Step 4: Final Answer:
The curved surface area of the cavity is \(314 \sqrt{2}\) \(\text{cm}^{2}\).