Question

A conical cavity of maximum volume is carved out from a wooden solid hemisphere of radius $10\text{ cm}$. 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 of maximum volume carved from a hemisphere of radius $R$, the slant height is always $R\sqrt{2}$.
This yields a direct shortcut formula for the curved surface area: $\text{CSA} = \sqrt{2}\pi R^2$.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
A wooden solid hemisphere has a radius of $R = 10\text{ cm}$.
A conical cavity of the maximum possible volume is carved out from this hemisphere.
We need to find the curved surface area (CSA) of this carved-out conical cavity.

Step 2: Key Formula or Approach:
For a cone carved out of a hemisphere to have the maximum possible volume:
- The circular base of the cone must lie on the flat circular base of the hemisphere. Thus, the radius of the cone's base is equal to the radius of the hemisphere: $r = R = 10\text{ cm}$.
- The height of the cone must be equal to the radius of the hemisphere: $h = R = 10\text{ cm}$.
The slant height $l$ of the cone is given by:
\[ l = \sqrt{r^2 + h^2} \]
The curved surface area (CSA) of a cone is calculated as:
\[ \text{CSA} = \pi r l \]

Step 3: Detailed Explanation:

• Determine the radius $r$ and height $h$ of the maximum volume cone:
\[ r = 10\text{ cm} \]
\[ h = 10\text{ cm} \]

• Compute the slant height $l$ of the cone:
\[ l = \sqrt{10^2 + 10^2} \]
\[ l = \sqrt{100 + 100} = \sqrt{200} \]
\[ l = 10\sqrt{2}\text{ cm} \]

• Write down the formula for the curved surface area of the cone:
\[ \text{CSA} = \pi r l \]

• Substitute the values of $r$, $l$, and $\pi = 3.14$:
\[ \text{CSA} = 3.14 \times 10 \times 10\sqrt{2} \]
\[ \text{CSA} = 314\sqrt{2}\text{ cm}^2 \]


Step 4: Final Answer:
The curved surface area of the carved-out cavity is $314\sqrt{2}\text{ cm}^2$.