The ratio of the base radii of a cylinder and a cone is 3:4, and their heights are in the ratio 2:3. Then the ratio of their volumes is:

Question

Accepted Answer

Let:  
- $$ x_{cy} $$ = radius of the cylinder, $$ x_{co} $$ = radius of the cone.  
- $$ h_{cy} $$ = height of the cylinder, $$ h_{co} $$ = height of the cone.

### Step 1: Ratios provided in the question  
- The ratio of radii is $$ \frac{x_{cy}}{x_{co}} = \frac{3}{4} $$.  
- The ratio of heights is $$ \frac{h_{cy}}{h_{co}} = \frac{2}{3} $$.

### Step 2: Formula for volumes  
The volume of a cylinder is given by:  
$$
V_{	ext{cylinder}} = \pi x_{cy}^2 h_{cy}.
$$  
The volume of a cone is given by:  
$$
V_{	ext{cone}} = \frac{1}{3} \pi x_{co}^2 h_{co}.
$$

The ratio of their volumes is:  
$$
\frac{V_{	ext{cylinder}}}{V_{	ext{cone}}} = \frac{\pi x_{cy}^2 h_{cy}}{\frac{1}{3} \pi x_{co}^2 h_{co}} = 3 \cdot \frac{x_{cy}^2}{x_{co}^2} \cdot \frac{h_{cy}}{h_{co}}.
$$

### Step 3: Substituting values from ratios  
Substitute $$ \frac{x_{cy}}{x_{co}} = \frac{3}{4} $$:  
$$
\frac{x_{cy}^2}{x_{co}^2} = \left(\frac{3}{4}ight)^2 = \frac{9}{16}.
$$  
Substitute $$ \frac{h_{cy}}{h_{co}} = \frac{2}{3} $$:  
$$
\frac{V_{	ext{cylinder}}}{V_{	ext{cone}}} = 3 \cdot \frac{9}{16} \cdot \frac{2}{3}.
$$

Simplify:  
$$
\frac{V_{	ext{cylinder}}}{V_{	ext{cone}}} = \frac{9}{16} \cdot 2 = \frac{18}{16} = \frac{9}{8}.
$$

### Final Answer:  
The ratio of their volumes is **9:8**, corresponding to option **(b)**.

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