Question

The water flow rate through an 8-inch pipe is 1,600 gallons per minute. The water flow rate through a 4-inch pipe is 260 gallons per minute. A 6-inch pipe has a water flow rate that is one-half of the rate of an 8-inch pipe.

Part A

Complete the equation to relate the flow of water from the 8-inch pipe, e, to the flow of water from the 4-inch pipe, f.

80
e =
f



Part B

Complete the equation to relate the flow of water from the 6-inch pipe, s, to the flow of water from the 8-inch pipe, e.



s =
e

Answer 1

To complete the equations based on the provided information, we can start by identifying the flow rates.

### Part A
- The flow rate through the 8-inch pipe (e) is 1,600 gallons per minute.
- The flow rate through the 4-inch pipe (f) is 260 gallons per minute.

The relationship between the flow rates is determined by the equation:
\[
e = k \cdot f
\]
where \( k \) is a constant that describes the relationship between the two flow rates.

To find \( k \):
\[
k = \frac{e}{f} = \frac{1600 \, \text{gpm}}{260 \, \text{gpm}} \approx 6.15
\]

Thus, the equation becomes:
\[
e = \frac{1600}{260} f \quad \text{or approximately} \quad e \approx 6.15f
\]

### Part B
- The flow rate through the 6-inch pipe (s) is stated to be one-half of the rate of the 8-inch pipe (e). Thus, we can express this relationship as:
\[
s = \frac{1}{2} e
\]

We can substitute the flow rate of the 8-inch pipe into the equation to express it in terms of e:
\[
s = \frac{1}{2} \cdot 1600 = 800 \, \text{gpm}
\]

So you can fill in the final equation as:
\[
s = \frac{1}{2} e
\]

### Final Answers:
- **Part A:** \( e = \frac{1600}{260} f \)
- **Part B:** \( s = \frac{1}{2} e \)