A cylinder has 'r' as the radius of the base and 'h' as the height. The radius of base of another cylinder, having double the volume but the same height as that of the first cylinder must be equal to

Options

A. `(r)/(sqrt(2))`
B. 2r
C. `r sqrt(2)`
D. `sqrt(2r)`

Answer

To solve the problem, we need to find the radius of the base of a second cylinder that has double the volume of the first cylinder but the same height. Let's break it down step by step.

### Step 1: Write the formula for the volume of the first cylinder.
The volume \( V_1 \) of a cylinder is given by the formula:
\[ V_1 = \pi r^2 h \]
where \( r \) is the radius of the base and \( h \) is the height.

### Step 2: Write the formula for the volume of the second cylinder.
Let the radius of the second cylinder be \( R \). The volume \( V_2 \) of the second cylinder is:
\[ V_2 = \pi R^2 h \]

### Step 3: Set up the equation for the volumes.
According to the problem, the volume of the second cylinder is double that of the first cylinder:
\[ V_2 = 2V_1 \]
Substituting the volume formulas we have:
\[ \pi R^2 h = 2(\pi r^2 h) \]

### Step 4: Simplify the equation.
We can cancel \( \pi \) and \( h \) from both sides (assuming \( h \neq 0 \)):
\[ R^2 = 2r^2 \]

### Step 5: Solve for \( R \).
To find \( R \), we take the square root of both sides:
\[ R = \sqrt{2r^2} \]
This simplifies to:
\[ R = r\sqrt{2} \]

### Conclusion:
The radius of the base of the second cylinder is \( R = r\sqrt{2} \).

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