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 input data.
A conical heap of rice is described by:
- Diameter = 14 m, implying radius $r = \frac{14}{2} = 7$ m.
- Height $h = 24$ m.
The objective is to determine the area of canvas needed for coverage, which corresponds to the cone's curved surface area.

Step 2: Identify the relevant formula.
The curved surface area (CSA) of a cone is calculated using:
\[ \text{CSA} = \pi r l \] where $r$ is the radius and $l$ is the slant height.

Step 3: Calculate the slant height.
The slant height $l$ can be found using the Pythagorean theorem, given $r$ and $h$. It forms the hypotenuse of a right triangle with $r$ and $h$ as legs:
\[ l = \sqrt{r^2 + h^2} \] Substituting $r = 7$ m and $h = 24$ m:
\[ l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \text{ m} \]

Step 4: Compute the curved surface area.
With the slant height $l = 25$ m, the CSA is:
\[ \text{CSA} = \pi \times 7 \times 25 = 175\pi \text{ sq.m.} \] The required canvas area is $175\pi$ sq.m., confirming the statement.

Step 5: Verify the provided reason.
The formula for the curved surface area of a cone, $\text{CSA} = \pi r l$, is correctly stated.

Step 6: Final assessment.
Both the assertion and the reason are true. The statement is therefore valid.