A rectangular tank is 225 m by 162 m at the base. With what speed must the water flow into it through an aperture 60 cm by 45 cm that the level may be raised 20 cm in 5 hours.

Options

A. 5000 m/hr
B. 5200 m/hr
C. 5400 m/hr
D. 5600 m/hr

Answer

To solve the problem step by step, we need to find the speed at which water must flow into the rectangular tank to raise the water level by 20 cm in 5 hours.

### Step 1: Determine the volume of water needed to raise the water level.
The dimensions of the tank are given as:
- Length (L) = 225 m
- Breadth (B) = 162 m
- Height increase (h) = 20 cm = 0.2 m (since 1 cm = 0.01 m)

The volume of water needed (V) can be calculated using the formula for the volume of a cuboid:
\[ V = L \times B \times h \]

Substituting the values:
\[ V = 225 \, \text{m} \times 162 \, \text{m} \times 0.2 \, \text{m} \]
\[ V = 225 \times 162 \times 0.2 \]
\[ V = 7290 \, \text{m}^3 \]

### Step 2: Calculate the area of the aperture.
The dimensions of the aperture are given as:
- Length = 60 cm = 0.6 m
- Breadth = 45 cm = 0.45 m

The area (A) of the aperture can be calculated as:
\[ A = \text{Length} \times \text{Breadth} \]
\[ A = 0.6 \, \text{m} \times 0.45 \, \text{m} \]
\[ A = 0.27 \, \text{m}^2 \]

### Step 3: Determine the total volume of water that flows through the aperture in 5 hours.
Let the speed of water flowing through the aperture be \( x \) m/h. The volume of water flowing through the aperture in 5 hours can be calculated as:
\[ \text{Volume} = \text{Area} \times \text{Speed} \times \text{Time} \]
\[ \text{Volume} = A \times x \times \text{Time} \]
\[ \text{Volume} = 0.27 \, \text{m}^2 \times x \, \text{m/h} \times 5 \, \text{h} \]
\[ \text{Volume} = 1.35x \, \text{m}^3 \]

### Step 4: Set up the equation to find the speed \( x \).
Since the volume of water needed to raise the level is equal to the volume flowing through the aperture:
\[ 1.35x = 7290 \]
Now, solve for \( x \):
\[ x = \frac{7290}{1.35} \]
\[ x = 5400 \, \text{m/h} \]

### Final Answer:
The speed at which the water must flow into the tank is **5400 m/h**.