Water flows into a tank which is 200 m long and 15 m wide, through a pipe of cross-section `0.3 m xx 0.2 m` at 20 km/hour. Then the time (in hours) for the water level in the tank to reach 8 m is

Options

A. a)50
B. b)120
C. c)150
D. d)20

Answer

To solve the problem, we will follow these steps:

### Step 1: Calculate the volume of water needed to fill the tank to a height of 8 m.
The volume \( V \) of the tank can be calculated using the formula for the volume of a rectangular prism:
\[ V = \text{length} \times \text{width} \times \text{height} \]
Given:
- Length \( l = 200 \, m \)
- Width \( w = 15 \, m \)
- Height \( h = 8 \, m \)

Substituting the values:
\[ V = 200 \, m \times 15 \, m \times 8 \, m = 24000 \, m^3 \]

### Step 2: Calculate the cross-sectional area of the pipe.
The cross-sectional area \( A \) of the pipe can be calculated using the formula:
\[ A = \text{width} \times \text{height} \]
Given:
- Width of the pipe = 0.3 m
- Height of the pipe = 0.2 m

Substituting the values:
\[ A = 0.3 \, m \times 0.2 \, m = 0.06 \, m^2 \]

### Step 3: Calculate the speed of water flow in meters per second.
The speed of water flow is given as 20 km/h. We need to convert this speed into meters per second (m/s):
\[ \text{Speed in m/s} = \frac{20 \times 1000}{3600} = \frac{20000}{3600} \approx 5.56 \, m/s \]

### Step 4: Calculate the volume of water flowing through the pipe in one second.
The volume of water \( V_f \) flowing through the pipe in one second can be calculated using the formula:
\[ V_f = A \times \text{speed} \]
Substituting the values:
\[ V_f = 0.06 \, m^2 \times 5.56 \, m/s \approx 0.3336 \, m^3/s \]

### Step 5: Calculate the time taken to fill the tank.
To find the time \( t \) taken to fill the tank, we can use the formula:
\[ t = \frac{\text{Volume of tank}}{\text{Volume flow rate}} \]
Substituting the values:
\[ t = \frac{24000 \, m^3}{0.3336 \, m^3/s} \approx 71826.79 \, s \]

### Step 6: Convert the time from seconds to hours.
To convert seconds to hours:
\[ t_{\text{hours}} = \frac{t_{\text{seconds}}}{3600} \]
Substituting the value:
\[ t_{\text{hours}} = \frac{71826.79 \, s}{3600} \approx 19.95 \, hours \]

### Final Answer:
The time taken for the water level in the tank to reach 8 m is approximately **20 hours**.

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