Question

Assertion (A): The area of canvas cloth required to just cover a heap of rice in the form of a cone of diameter 14 m and height 24 m is $175\pi$ sq.m.
Reason (R): The curved surface area of a cone of radius $r$ and slant height $l$ is $\pi r l$.

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of Assertion (A)
• C. Assertion (A) is true, but Reason (R) is false.
• D. Assertion (A) is false, but Reason (R) is true.

Answer 1

The Correct Option is A

Step 1: Analyze the assertion (A):
We are given that the heap of rice is in the form of a cone with a diameter of 14 m and a height of 24 m. The area of canvas cloth required to cover the heap is the curved surface area of the cone.
The diameter of the cone is 14 m, so the radius \( r = \frac{14}{2} = 7 \, \text{m} \).
The height of the cone is given as 24 m. We need to find the slant height \( l \) of the cone, which can be found using the Pythagorean theorem.
The slant height \( l \) is given by: \[ l = \sqrt{r^2 + h^2} \] Substitute \( r = 7 \) m and \( h = 24 \) m: \[ l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \, \text{m} \] Now, use the formula for the curved surface area of a cone: \[ \text{Curved Surface Area} = \pi r l \] Substitute \( r = 7 \) m and \( l = 25 \) m: \[ \text{Curved Surface Area} = \pi \times 7 \times 25 = 175\pi \, \text{sq.m} \] Thus, the area of the canvas cloth required to cover the heap is indeed \( 175\pi \) square meters. Therefore, the assertion is correct.

Step 2: Analyze the reason (R):
The reason (R) states that the curved surface area of a cone with radius \( r \) and slant height \( l \) is given by the formula: \[ \text{Curved Surface Area} = \pi r l \] This is the correct formula for the curved surface area of a cone, so the reason is also true.

Step 3: Conclusion:
Both the assertion (A) and the reason (R) are true, and the reason correctly explains the assertion. Therefore, the correct answer is:
\[ \boxed{\text{Both (A) and (R) are true, and (R) is the correct explanation of (A)}} \]