Question

The radius of the base of a right circular cone is 7 cm and its curved surface area is 550 $\text{cm}^2$. The volume of the cone is: (use $\pi$ = 22/7)

• A. 1223 $\text{cm}^3$
• B. 1233 $\text{cm}^3$
• C. 1322 $\text{cm}^3$
• D. 1232 $\text{cm}^3$

Hint:

Recognizing the standard Pythagorean triplet $(7, 24, 25)$ saves precious calculation time when finding the height of the cone.
If radius is 7 and slant height is 25, the vertical height is immediately 24.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
The goal of this question is to calculate the volume of a right circular cone given its base radius and its curved surface area.

Step 2: Key Formula or Approach:
The curved surface area of a cone is:
\[ \text{CSA} = \pi r l \]
The relation between slant height $l$, radius $r$, and vertical height $h$ is:
\[ l^2 = r^2 + h^2 \]
The volume $V$ of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]

Step 3: Detailed Explanation:
Let us execute the calculations step-by-step:

•

Step 1: Find the slant height ($l$):
Given radius $r = 7\text{ cm}$ and $\text{CSA} = 550\text{ cm}^2$.
Using $\pi r l = 550$:
\[ \frac{22}{7} \times 7 \times l = 550 \implies 22 \times l = 550 \implies l = 25\text{ cm} \]

•

Step 2: Find the vertical height ($h$):
Using the Pythagorean relationship:
\[ h^2 = l^2 - r^2 = 25^2 - 7^2 = 625 - 49 = 576 \]
\[ h = \sqrt{576} = 24\text{ cm} \]

•

Step 3: Calculate the volume ($V$):
Substitute $r = 7\text{ cm}$ and $h = 24\text{ cm}$ into the volume formula:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 7 \times 7 \times 24 \]
\[ V = \frac{1}{3} \times 22 \times 7 \times 24 \]
\[ V = 22 \times 7 \times 8 = 1232\text{ cm}^3 \]

Step 4: Final Answer:
The volume of the cone is 1232 $\text{cm}^3$, matching option (D).