The diameter of the base of a cylinder is $28 ext{ cm}$ and its curved surface area is $704 ext{ cm}^2$. A cone has a height that is $25\%$ greater than the height of the cylinder, and the radius of the cone is equal to the radius of the cylinder. What is the volume of the cone?

Question

The diameter of the base of a cylinder is $28 	ext{ cm}$ and its curved surface area is $704 	ext{ cm}^2$. A cone has a height that is $25\%$ greater than the height of the cylinder, and the radius of the cone is equal to the radius of the cylinder. What is the volume of the cone?

Accepted Answer

**Solution:**

1. Given the diameter of the cylinder's base is $28 	ext{ cm}$, so the radius $r = \frac{28}{2} = 14 	ext{ cm}$.

2. The curved surface area (CSA) of the cylinder is given by:
   $$	ext{CSA} = 2 \pi r h = 704$$

3. Substitute $r = 14$ and $\pi = \frac{22}{7}$:
   $$2 	imes \frac{22}{7} 	imes 14 	imes h = 704$$

4. Simplify:
   $$2 	imes 22 	imes 2 	imes h = 704$$
   $$88h = 704$$

5. Solve for $h$:
   $$h = \frac{704}{88} = 8 	ext{ cm}$$

6. The height of the cone is $25\%$ more than the cylinder's height:
   $$h_{cone} = 8 + 0.25 	imes 8 = 8 + 2 = 10 	ext{ cm}$$

7. The radius of the cone is the same as the cylinder's radius, $r = 14 	ext{ cm}$.

8. Volume of the cone is:
   $$V = \frac{1}{3} \pi r^2 h = \frac{1}{3} 	imes \frac{22}{7} 	imes 14^2 	imes 10$$

9. Calculate stepwise:
   $$14^2 = 196$$
   $$V = \frac{1}{3} 	imes \frac{22}{7} 	imes 196 	imes 10$$

10. Simplify:
    $$\frac{22}{7} 	imes 196 = 22 	imes 28 = 616$$

11. So,
    $$V = \frac{1}{3} 	imes 616 	imes 10 = \frac{6160}{3} 	ext{ cm}^3$$

**Answer:** $\frac{6160}{3} 	ext{ cm}^3$

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