If the diameter of the base of a cone is $56 ext{ cm}$ and its curved surface area is $3080 ext{ cm}^2$, then what will be its volume (in $ ext{cm}^3$)? (Use $\pi = \frac{22}{7}$)

Question

If the diameter of the base of a cone is $56 	ext{ cm}$ and its curved surface area is $3080 	ext{ cm}^2$, then what will be its volume (in $	ext{cm}^3$)? (Use $\pi = \frac{22}{7}$)

Accepted Answer

**Solution:**

1. Given:
   - Diameter $d = 56 	ext{ cm}$
   - Radius $r = \frac{d}{2} = 28 	ext{ cm}$
   - Curved Surface Area (CSA) $= 3080 	ext{ cm}^2$
   - $\pi = \frac{22}{7}$

2. Use the formula for curved surface area of a cone:
   $$	ext{CSA} = \pi r l$$
   where $l$ is the slant height.

3. Substitute the values:
   $$3080 = \frac{22}{7} 	imes 28 	imes l$$

4. Simplify to find $l$:
   $$l = \frac{3080 	imes 7}{22 	imes 28} = \frac{21560}{616} = 35 	ext{ cm}$$

5. Use Pythagorean theorem to find height $h$:
   $$l^2 = r^2 + h^2$$
   $$35^2 = 28^2 + h^2$$
   $$1225 = 784 + h^2$$
   $$h^2 = 1225 - 784 = 441$$
   $$h = \sqrt{441} = 21 	ext{ cm}$$

6. Calculate the volume $V$ of the cone:
   $$V = \frac{1}{3} \pi r^2 h$$
   $$V = \frac{1}{3} 	imes \frac{22}{7} 	imes 28^2 	imes 21$$

7. Calculate stepwise:
   - $28^2 = 784$
   - $$V = \frac{1}{3} 	imes \frac{22}{7} 	imes 784 	imes 21$$
   - Simplify $\frac{22}{7} 	imes 784 = 22 	imes 112 = 2464$
   - $$V = \frac{1}{3} 	imes 2464 	imes 21 = \frac{1}{3} 	imes 51744 = 17248 	ext{ cm}^3$$

**Answer:** $17248 	ext{ cm}^3$

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