The ratio of the radii of two cylinder is 1:2 and the ratio of their heigths is 2:1. Find the ratio of their volumes.

Answer

To find the ratio of the volumes of two cylinders given the ratio of their radii and heights, we can follow these steps:

### Step 1: Define the variables
Let the radius of the first cylinder be \( r \) and the radius of the second cylinder be \( r_1 \). According to the problem, the ratio of the radii is given as:
\[ \frac{r}{r_1} = \frac{1}{2} \]
This means:
\[ r_1 = 2r \]

### Step 2: Define the heights
Let the height of the first cylinder be \( h \) and the height of the second cylinder be \( h_1 \). The ratio of the heights is given as:
\[ \frac{h}{h_1} = \frac{2}{1} \]
This means:
\[ h = 2h_1 \]

### Step 3: Write the volume formulas
The volume \( V \) of a cylinder is given by the formula:
\[ V = \pi r^2 h \]
So, the volume of the first cylinder \( V_1 \) is:
\[ V_1 = \pi r^2 h \]
And the volume of the second cylinder \( V_2 \) is:
\[ V_2 = \pi r_1^2 h_1 \]

### Step 4: Substitute the values
Now, substitute \( r_1 \) and \( h \) in terms of \( r \) and \( h_1 \):
\[ V_2 = \pi (2r)^2 h_1 \]
\[ V_2 = \pi (4r^2) h_1 \]

### Step 5: Express \( h_1 \) in terms of \( h \)
From the earlier step, we know \( h = 2h_1 \), which gives:
\[ h_1 = \frac{h}{2} \]

### Step 6: Substitute \( h_1 \) into \( V_2 \)
Now substitute \( h_1 \) into the volume of the second cylinder:
\[ V_2 = \pi (4r^2) \left(\frac{h}{2}\right) \]
\[ V_2 = 2\pi r^2 h \]

### Step 7: Find the ratio of the volumes
Now we can find the ratio of the volumes \( \frac{V_1}{V_2} \):
\[ \frac{V_1}{V_2} = \frac{\pi r^2 h}{2\pi r^2 h} \]
The \( \pi \) and \( r^2 h \) terms cancel out:
\[ \frac{V_1}{V_2} = \frac{1}{2} \]

### Step 8: Express the ratio in simplest form
Thus, the ratio of the volumes of the two cylinders is:
\[ V_1 : V_2 = 1 : 2 \]

### Final Answer
The ratio of the volumes of the two cylinders is \( 1 : 2 \).

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