A man goes to a shop and buys a cylindrical jar with a capacity of $83259 ext{ cm}^3$. He also purchases a conical vessel whose capacity is $\frac{1}{27}$ of the cylindrical jar. The height of the conical vessel is $14 ext{ cm}$, which is four times the height of the cylindrical jar. What is the ratio of the radius of the cylindrical jar to the radius of the conical vessel?

Question

A man goes to a shop and buys a cylindrical jar with a capacity of $83259 	ext{ cm}^3$. He also purchases a conical vessel whose capacity is $\frac{1}{27}$ of the cylindrical jar. The height of the conical vessel is $14 	ext{ cm}$, which is four times the height of the cylindrical jar. What is the ratio of the radius of the cylindrical jar to the radius of the conical vessel?

Accepted Answer

**Solution:**

1. Let the volume of the cylindrical jar be $V$.

2. The volume of the conical vessel is given as $\frac{1}{27}$ of the cylindrical jar, so its volume is $\frac{V}{27}$.

3. The volume formulas are:
   - Cylinder: $V = \pi r^2 h$
   - Cone: $V_{cone} = \frac{1}{3} \pi R^2 H$

4. Given that the height of the conical vessel $H$ is four times the height of the cylinder $h$, so:
   $$H = 4h$$

5. Substitute the values into the volume ratio:
   $$\frac{V}{V/27} = \frac{\pi r^2 h}{\frac{1}{3} \pi R^2 (4h)} = 27$$

6. Simplify the expression:
   $$\frac{\pi r^2 h}{\frac{4}{3} \pi R^2 h} = 27$$
   $$\frac{r^2}{\frac{4}{3} R^2} = 27$$
   $$r^2 	imes \frac{3}{4 R^2} = 27$$
   $$\frac{3 r^2}{4 R^2} = 27$$

7. Multiply both sides by $\frac{4 R^2}{3}$:
   $$r^2 = 27 	imes \frac{4 R^2}{3} = 36 R^2$$

8. Taking square roots:
   $$\frac{r}{R} = 6 : 1$$

**Answer:** $6 : 1$

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