A company decides to drain the water heater to flush out sediments. The water heater has a capacity of 500 gallons. It drains 100 gallons in 20 minutes. After 20 minutes, they open another drain valve and it drains 200 gallons in the next 20 minutes. The drain valves are closed for 10 minutes, while the workers take a break and then the water heater is drained until the water heater is completely empty.

Question

What are the domain and the range of this relation?

The piecewise function represents time in minutes on the x-axis and water in gallons on the y-axis. The function starts from (0, 500), (20, 400), (40, 200), (50, 200), and (70, 0).
x ≥ 0
y ≥ 0
0 ≤ x ≤ 500
0 ≤ x ≤ 70
0 ≤ y ≤ 500
0 ≤ y ≤ 70

Answer 1

To determine the domain and range of the piecewise function representing the draining process of the water heater, we first consider the points given and the context of the problem.

Domain

The domain represents the set of all possible values for the independent variable \( x \) (time in minutes). From the information provided, we have the following key points:

1. The process starts at \( t = 0 \) minutes.
2. The water heater is completely drained at \( t = 70 \) minutes.

Given this, the domain is all the time values from the start to the end of the draining process.

Therefore, the domain is: \[ \text{Domain: } [0, 70] \]

Range

The range represents the set of all possible values for the dependent variable \( y \) (water in gallons). The draining process occurs from the full capacity of the water heater down to empty.

Initially, when \( t = 0 \), the water heater is full:

• At \( t = 0 \), \( y = 500 \) gallons.
• After draining, at \( t = 70 \) minutes, \( y = 0 \) gallons (the water heater is empty).

Thus, the range is the set of water amounts from full to empty, inclusive of both endpoints.

Therefore, the range is: \[ \text{Range: } [0, 500] \]

Summary

In conclusion:

• Domain: \( [0, 70] \) (time in minutes)
• Range: \( [0, 500] \) (water in gallons)