Read the following information carefully to answer the questions.
A hemispherical bowl is filled with hot water to the brim. The contents of the bowl are transferred into a cylindrical vessel whose radius is `50%` more than its height.
The hemispherical bowl is joined at one end of the cylindrical vessel and the solid obtained by combining these figure is filled completely with water. If a drain pipe is connected to it then pipe will empty 539 `cm^3` of water in one minute. If radius of the solid is 21 cm, then what is the time in which whole of the water will be emptied from the solid? (Given that diameter of cylinder and hemisphere are same.)

Options

A. 72 minutes
B. 90 minutes
C. 60 minutes
D. 75 minutes

Answer

To solve the problem step by step, we need to find the total volume of the solid formed by the hemispherical bowl and the cylindrical vessel, and then calculate the time it takes to drain that volume using the given drain rate.

### Step 1: Understand the dimensions
1. Let the height of the cylindrical vessel be \( h \).
2. The radius of the cylinder is \( r \), which is 50% more than its height. Therefore, we can express the radius as:
\[ r = 1.5h \]

### Step 2: Relate the radius of the cylinder to the hemispherical bowl
Since the diameter of the cylinder and the hemisphere are the same, we have:
\[ \text{Diameter of hemisphere} = 2r = 2r \]
Thus, the radius of the hemisphere is also \( r \).

### Step 3: Calculate the volume of the cylinder
The volume \( V_c \) of the cylinder is given by the formula:
\[ V_c = \pi r^2 h \]

### Step 4: Calculate the volume of the hemisphere
The volume \( V_h \) of the hemisphere is given by the formula:
\[ V_h = \frac{2}{3} \pi r^3 \]

### Step 5: Substitute the radius in terms of height
Since \( r = 1.5h \), we can substitute this into the volume formulas.

**Volume of the cylinder:**
\[ V_c = \pi (1.5h)^2 h = \pi (2.25h^2) h = 2.25\pi h^3 \]

**Volume of the hemisphere:**
\[ V_h = \frac{2}{3} \pi (1.5h)^3 = \frac{2}{3} \pi (3.375h^3) = 2.25\pi h^3 \]

### Step 6: Total volume of the solid
The total volume \( V \) of the solid formed by the cylinder and hemisphere is:
\[ V = V_c + V_h = 2.25\pi h^3 + 2.25\pi h^3 = 4.5\pi h^3 \]

### Step 7: Find the height using the given radius
We are given that the radius of the solid is 21 cm. Since \( r = 1.5h \):
\[ 1.5h = 21 \implies h = \frac{21}{1.5} = 14 \text{ cm} \]

### Step 8: Calculate the total volume using the height
Now substitute \( h = 14 \) cm into the volume formula:
\[ V = 4.5\pi (14)^3 \]
Calculating \( 14^3 \):
\[ 14^3 = 2744 \]
Thus,
\[ V = 4.5\pi \times 2744 = 12348\pi \text{ cm}^3 \]

### Step 9: Calculate the volume in numerical form
Using \( \pi \approx 3.14 \):
\[ V \approx 12348 \times 3.14 \approx 38767.32 \text{ cm}^3 \]

### Step 10: Calculate the time to drain the volume
The drain pipe empties 539 cm³ of water per minute. Therefore, the time \( T \) to drain the total volume is given by:
\[ T = \frac{V}{\text{drain rate}} = \frac{38767.32}{539} \]
Calculating this gives:
\[ T \approx 71.94 \text{ minutes} \approx 72 \text{ minutes} \]

### Final Answer
The total time to empty the whole of the water will be approximately **72 minutes**.