A conical flask is full of water. The flask has base radius r and height h. This water is poured into a cylindrical flask of base radius mr. The height of water in the cylindrical flask is

Options

A. `(m)/(2h)`
B. `(h)/(2) m^(2)`
C. `(2h)/(m)`
D. `(h)/(3m^(2))`

Answer

To find the height of the water in the cylindrical flask after pouring the water from the conical flask, we can follow these steps:

### Step 1: Calculate the volume of the conical flask
The volume \( V \) of a cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
where \( r \) is the base radius and \( h \) is the height of the cone.

### Step 2: Write the volume of the cylindrical flask
The volume \( V' \) of a cylinder is given by the formula:
\[ V' = \pi (mr)^2 h' \]
where \( mr \) is the base radius of the cylinder and \( h' \) is the height of the water in the cylinder that we need to find.

### Step 3: Set the volumes equal
Since the water from the conical flask is poured into the cylindrical flask, the volumes must be equal:
\[ \frac{1}{3} \pi r^2 h = \pi (mr)^2 h' \]

### Step 4: Simplify the equation
We can cancel \( \pi \) from both sides:
\[ \frac{1}{3} r^2 h = (mr)^2 h' \]
This simplifies to:
\[ \frac{1}{3} r^2 h = m^2 r^2 h' \]

### Step 5: Divide both sides by \( r^2 \)
Assuming \( r \neq 0 \), we can divide both sides by \( r^2 \):
\[ \frac{1}{3} h = m^2 h' \]

### Step 6: Solve for \( h' \)
Now, we can solve for \( h' \):
\[ h' = \frac{h}{3m^2} \]

### Final Result
The height of the water in the cylindrical flask is:
\[ h' = \frac{h}{3m^2} \]