The volume of a right circular cone is given as $3234 ext{ cm}^3$. The diameter of the cone is $42 ext{ cm}$, and the height of the cone is equal to the height of a circular cylinder. The radius of the circular cylinder is $\frac{5}{7}$ of the radius of the cone. Find the volume of the circular cylinder.

Question

The volume of a right circular cone is given as $3234 		ext{ cm}^3$. The diameter of the cone is $42 		ext{ cm}$, and the height of the cone is equal to the height of a circular cylinder. The radius of the circular cylinder is $\frac{5}{7}$ of the radius of the cone. Find the volume of the circular cylinder.

Accepted Answer

**Solution:**

1. Given the volume of the cone is $3234 	ext{ cm}^3$ and the diameter of the cone is $42 	ext{ cm}$. Therefore, the radius of the cone, $r_c = \frac{42}{2} = 21 	ext{ cm}$.

2. The formula for the volume of a cone is:
   $$V_{cone} = \frac{1}{3} \pi r_c^2 h$$

3. Substitute the known values:
   $$3234 = \frac{1}{3} 	imes \pi 	imes 21^2 	imes h$$

4. Calculate $21^2 = 441$:
   $$3234 = \frac{1}{3} 	imes \pi 	imes 441 	imes h$$

5. Multiply both sides by 3:
   $$9702 = \pi 	imes 441 	imes h$$

6. Solve for $h$:
   $$h = \frac{9702}{\pi 	imes 441}$$

7. Using $\pi \approx 3.14$:
   $$h = \frac{9702}{3.14 	imes 441} = \frac{9702}{1384.74} \approx 7 	ext{ cm}$$

8. The height of the cylinder is the same as the height of the cone, so $h_{cyl} = 7 	ext{ cm}$.

9. The radius of the cylinder is $\frac{5}{7}$ of the radius of the cone:
   $$r_{cyl} = \frac{5}{7} 	imes 21 = 15 	ext{ cm}$$

10. The volume of the cylinder is given by:
    $$V_{cyl} = \pi r_{cyl}^2 h_{cyl} = \pi 	imes 15^2 	imes 7$$

11. Calculate $15^2 = 225$:
    $$V_{cyl} = 3.14 	imes 225 	imes 7 = 3.14 	imes 1575 = 4945.5 \approx 4950 	ext{ cm}^3$$

**Answer:** $4950 	ext{ cm}^3$

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