A flask in the shape of a right circular cone of height 24 cm is filled with water. The water is poured in right circular cylindrical flask whose radius is `(1)/(3)`rd of radius of the base of the circular cone. Then the height of the water in the cylindrical flask is

Options

A. a)32 cm
B. b)24 cm
C. c)48 cm
D. d)72 cm

Answer

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

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

Given:
- Height of the cone \( h = 24 \) cm
- Let the radius of the base of the cone be \( r \).

So, the volume of the cone becomes:
\[ V_{\text{cone}} = \frac{1}{3} \pi r^2 \times 24 \]
\[ V_{\text{cone}} = 8 \pi r^2 \]

### Step 2: Determine the Radius of the Cylindrical Flask
The radius of the cylindrical flask is given as \( \frac{1}{3} \) of the radius of the cone. Therefore, the radius of the cylindrical flask \( r_c \) is:
\[ r_c = \frac{1}{3} r \]

### Step 3: Calculate the Volume of the Cylinder
The volume \( V_c \) of a right circular cylinder is given by the formula:
\[ V_c = \pi r_c^2 h_c \]
where \( h_c \) is the height of the water in the cylindrical flask.

Substituting \( r_c \):
\[ V_c = \pi \left(\frac{1}{3} r\right)^2 h_c \]
\[ V_c = \pi \left(\frac{1}{9} r^2\right) h_c \]
\[ V_c = \frac{1}{9} \pi r^2 h_c \]

### Step 4: Set the Volumes Equal
Since the volume of water remains the same when poured from the cone to the cylinder, we can set the volumes equal to each other:
\[ 8 \pi r^2 = \frac{1}{9} \pi r^2 h_c \]

### Step 5: Simplify and Solve for \( h_c \)
Dividing both sides by \( \pi r^2 \) (assuming \( r \neq 0 \)):
\[ 8 = \frac{1}{9} h_c \]

Now, multiplying both sides by 9:
\[ h_c = 8 \times 9 \]
\[ h_c = 72 \text{ cm} \]

### Conclusion
The height of the water in the cylindrical flask is \( 72 \) cm.

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