To solve the problem step by step, let's break it down into manageable parts:
### Step 1: Calculate the Volume of Water Required to Raise the Tank Level
The dimensions of the tank are given as:
- Length = 7 m
- Width = 4 m
- Height increase = 4.5 m
To find the volume of water required to raise the water level by 4.5 m, we use the formula for the volume of a rectangular prism:
\[
\text{Volume} = \text{Length} \times \text{Width} \times \text{Height}
\]
Substituting the values:
\[
\text{Volume} = 7 \, \text{m} \times 4 \, \text{m} \times 4.5 \, \text{m} = 126 \, \text{m}^3
\]
### Step 2: Convert the Volume to Cubic Centimeters
Since the dimensions of the pipe are given in centimeters, we need to convert the volume from cubic meters to cubic centimeters.
1 cubic meter = 1,000,000 cubic centimeters.
So,
\[
126 \, \text{m}^3 = 126 \times 1,000,000 \, \text{cm}^3 = 126,000,000 \, \text{cm}^3
\]
### Step 3: Calculate the Time in Minutes
The total time given is 6 hours and 18 minutes. We need to convert this time into minutes.
\[
\text{Total time} = 6 \times 60 + 18 = 378 \, \text{minutes}
\]
### Step 4: Calculate the Volume of Water Flowing Through the Pipe
The cross-sectional area of the pipe is given as:
- Width = 5 cm
- Depth = 4 cm
The area \( A \) of the pipe is:
\[
A = \text{Width} \times \text{Depth} = 5 \, \text{cm} \times 4 \, \text{cm} = 20 \, \text{cm}^2
\]
If water flows through the pipe at a rate of \( x \) cm/min, the volume of water flowing through the pipe in 1 minute is:
\[
\text{Volume per minute} = A \times x = 20 \, \text{cm}^2 \times x \, \text{cm/min} = 20x \, \text{cm}^3/\text{min}
\]
### Step 5: Calculate the Total Volume of Water Flowing in 378 Minutes
The total volume of water that flows through the pipe in 378 minutes is:
\[
\text{Total Volume} = 20x \times 378 = 7560x \, \text{cm}^3
\]
### Step 6: Set Up the Equation
According to the problem, the total volume of water flowing through the pipe must equal the volume required to raise the tank level:
\[
7560x = 126,000,000
\]
### Step 7: Solve for \( x \)
Now, we can solve for \( x \):
\[
x = \frac{126,000,000}{7560} = 16666.67 \, \text{cm/min}
\]
### Step 8: Convert \( x \) to m/s
To find the speed in meters per second, we convert from centimeters per minute to meters per second:
\[
x = \frac{16666.67 \, \text{cm/min}}{100} \times \frac{1}{60} = 2.77778 \, \text{m/s}
\]
### Final Answer
The speed at which water should run through the pipe is approximately **2.78 m/s**.
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