Two cylindrical cans has the base of the same size. The diameter of each is 14cm. The height of one can is 10cm and another 20cm. Find the ratio of their volumes.

Options

A. `3:2`
B. `2:3`
C. `1:2`
D. `2:1`

Answer

To find the ratio of the volumes of two cylindrical cans with the same base size, we can follow these steps:

### Step 1: Understand the formula for the volume of a cylinder
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
where \( r \) is the radius and \( h \) is the height of the cylinder.

### Step 2: Identify the dimensions of the cylinders
Given:
- Diameter of each can = 14 cm
- Therefore, the radius \( r \) of each can = \( \frac{14}{2} = 7 \) cm
- Height of the first can \( h_1 = 10 \) cm
- Height of the second can \( h_2 = 20 \) cm

### Step 3: Write the volumes of both cylinders
Using the formula for volume:
- Volume of the first can \( V_1 = \pi (7^2)(10) \)
- Volume of the second can \( V_2 = \pi (7^2)(20) \)

### Step 4: Set up the ratio of the volumes
To find the ratio of the volumes \( \frac{V_1}{V_2} \):
\[ \frac{V_1}{V_2} = \frac{\pi (7^2)(10)}{\pi (7^2)(20)} \]

### Step 5: Simplify the ratio
The \( \pi \) and \( 7^2 \) terms cancel out:
\[ \frac{V_1}{V_2} = \frac{10}{20} = \frac{1}{2} \]

### Step 6: Write the final ratio
Thus, the ratio of the volumes of the two cans is:
\[ 1 : 2 \]