The radii of the bases of a right circular cylinder and a right circular cone are equal. Their vertical heights are 7 cm and 14 cm respectively. Find the ratio of their volumes.

Answer

To find the ratio of the volumes of a right circular cylinder and a right circular cone with equal radii and different heights, we can follow these steps:

### Step 1: Identify the given values
- Let the radius of both the cylinder and the cone be \( r \) (the exact value is not needed since they are equal).
- Height of the cylinder \( h_c = 7 \) cm.
- Height of the cone \( h_o = 14 \) cm.

### Step 2: Write the formulas for the volumes
- The volume \( V_c \) of the cylinder is given by the formula:
\[ V_c = \pi r^2 h_c \]
- The volume \( V_o \) of the cone is given by the formula:
\[ V_o = \frac{1}{3} \pi r^2 h_o \]

### Step 3: Find the ratio of the volumes
To find the ratio of the volumes \( \frac{V_c}{V_o} \):
\[ \frac{V_c}{V_o} = \frac{\pi r^2 h_c}{\frac{1}{3} \pi r^2 h_o} \]

### Step 4: Simplify the ratio
- The \( \pi r^2 \) terms cancel out:
\[ \frac{V_c}{V_o} = \frac{h_c}{\frac{1}{3} h_o} \]
- This can be rewritten as:
\[ \frac{V_c}{V_o} = \frac{h_c \cdot 3}{h_o} \]

### Step 5: Substitute the heights
Substituting the heights into the equation:
\[ \frac{V_c}{V_o} = \frac{7 \cdot 3}{14} \]

### Step 6: Calculate the ratio
\[ \frac{V_c}{V_o} = \frac{21}{14} = \frac{3}{2} \]

### Conclusion
Thus, the ratio of the volumes of the cylinder to the cone is:
\[ \text{Ratio} = 3:2 \]

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