To analyze the piecewise function you're describing, we need to identify the domain and the range based on the points provided:
- Domain: The domain represents the set of possible input values (in this case, time in minutes, denoted as x).
The time values provided are:
- Start at \( x = 0 \) (initial time)
- First drain occurs until \( x = 20 \) minutes
- Second drain valves opened at \( x = 40 \) minutes
- Break at \( x = 50 \) minutes
- Finally drained to \( x = 70 \) minutes
Thus, the time spans from 0 minutes to 70 minutes. Therefore, the domain is:
\[ 0 \leq x \leq 70 \]
- Range: The range represents the set of possible output values (in this case, water in gallons, denoted as y).
The water values at the key time points are:
- \( y = 500 \) gallons at \( x = 0 \) minutes,
- \( y = 400 \) gallons at \( x = 20 \) minutes (after draining 100 gallons),
- \( y = 200 \) gallons at \( x = 40 \) minutes (after another 200 gallons have been drained),
- \( y = 200 \) gallons remains at \( x = 50 \) minutes (during break),
- Finally, \( y = 0 \) gallons at \( x = 70 \) minutes (when the water heater is completely drained).
Thus, the water values span from 0 gallons to 500 gallons. Therefore, the range is:
\[ 0 \leq y \leq 500 \]
In summary, the domain and the range of this relation are:
- Domain: \( 0 \leq x \leq 70 \)
- Range: \( 0 \leq y \leq 500 \)