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: Build the CSA formula symbolically before plugging in numbers.
For the largest cone that fits inside a hemisphere of radius $R$, the cone's base radius and height both equal $R$, so slant height $l=\sqrt{r^2+h^2}=\sqrt{R^2+R^2}=R\sqrt{2}$.
Step 2: Write the CSA purely in terms of R. \[ \text{CSA} = \pi r l = \pi (R)(R\sqrt{2}) = \sqrt{2}\,\pi R^2 \] This single formula now works for any radius without repeating the Pythagoras step.
Step 3: Substitute R = 10 cm and pi = 3.14. \[ \text{CSA} = \sqrt{2} \times 3.14 \times 100 = 314\sqrt{2}\text{ cm}^2 \]
\[ \boxed{314\sqrt{2}\text{ cm}^2} \]