Two right circular cylinders of equal volume have their heights in the ratio 1:2. The ratio of their radii is :

Options

A. `sqrt(2) :1`
B. `2:1`
C. `1:2`
D. `1:4`

Answer

To solve the problem of finding the ratio of the radii of two right circular cylinders with equal volumes and heights in the ratio of 1:2, we can follow these steps:

### Step 1: Understand the Volume Formula
The volume \( V \) of a right circular 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: Define the Heights and Radii
Let the height of the first cylinder be \( h \) and the height of the second cylinder be \( 2h \) (since the ratio of their heights is 1:2). Let the radius of the first cylinder be \( r_1 \) and the radius of the second cylinder be \( r_2 \).

### Step 3: Write the Volume Equations
Using the volume formula, we can express the volumes of both cylinders:
- Volume of the first cylinder:
\[ V_1 = \pi r_1^2 h \]
- Volume of the second cylinder:
\[ V_2 = \pi r_2^2 (2h) \]

### Step 4: Set the Volumes Equal
Since the volumes of the two cylinders are equal, we can set the equations equal to each other:
\[ \pi r_1^2 h = \pi r_2^2 (2h) \]

### Step 5: Simplify the Equation
We can cancel \( \pi \) and \( h \) from both sides (assuming \( h \neq 0 \)):
\[ r_1^2 = 2 r_2^2 \]

### Step 6: Solve for the Ratio of the Radii
Taking the square root of both sides gives:
\[ \frac{r_1}{r_2} = \sqrt{2} \]
This can be expressed as:
\[ r_1 : r_2 = \sqrt{2} : 1 \]

### Conclusion
Thus, the ratio of the radii \( r_1 \) to \( r_2 \) is \( \sqrt{2} : 1 \).

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