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: Understanding the given information:
We are given a heap of rice in the form of a cone with:
- Diameter = 14 m, so radius $r = \frac{14}{2} = 7$ m.
- Height $h = 24$ m.
We need to find the area of canvas cloth required to cover the heap, which is the curved surface area of the cone.

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

Step 3: Finding the slant height ($l$):
We can find the slant height using the Pythagorean theorem since we are given the radius and height of the cone. The slant height $l$ is the hypotenuse of the right-angled triangle formed by the radius, height, and slant height, so: \[ l = \sqrt{r^2 + h^2} \] Substituting the values of $r = 7$ m and $h = 24$ m: \[ l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \text{ m} \]

Step 4: Calculating the curved surface area:
Now that we have the slant height $l = 25$ m, we can calculate the curved surface area: \[ \text{CSA} = \pi \times 7 \times 25 = 175\pi \text{ sq.m.} \] Thus, the area of canvas cloth required to cover the heap is $175\pi$ sq.m., which matches the assertion.

Step 5: Checking the reason (R):
The formula for the curved surface area of a cone is correctly stated as: \[ \text{CSA} = \pi r l \] Therefore, the reason (R) is true.

Step 6: Conclusion:
Since both the assertion (A) and reason (R) are true, the statement is correct.