Comparing Three Rain Barrels Modeled as Linear Functions
A greenhouse at Lincoln Middle School collects rainwater in three identical 60-liter barrels. Students recorded the volume every hour after a storm began. Barrel A's data are in the table below. Barrel B is modeled by y = 3.5x + 18, where y is liters and x is hours since the storm began. Barrel C's model is a line that passes through (0, 10) and (4, 30). The barrels started the storm with different amounts left over from the previous day, and the inflow rates stayed constant during the storm. The class must decide which barrel to use first for watering seedlings and when to stop trusting the models. A student says, 'Barrel C will always have the most water because its line is steepest.' The teacher wants a written recommendation that states the time interval over which each model is valid.
| x (hours) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| y (liters) | 12 | 16.5 | 21 | 25.5 |
1. Find the rate of change and initial value for each barrel. Rank the barrels by filling rate and by starting volume, and explain how the table, equation, and two points each support the result.
Read the worked answer
Barrel A: from the table, (16.5 − 12) ÷ (1 − 0) = 4.5 L/h, and the value at x = 0 is 12 L. Each hour adds 4.5 L, which is constant, so the model is y = 4.5x + 12.
Barrel B: y = 3.5x + 18, so the rate is 3.5 L/h and the initial volume is 18 L.
Barrel C: slope = (30 − 10) ÷ (4 − 0) = 20 ÷ 4 = 5 L/h; the y-intercept is 10 L, so y = 5x + 10.
Filling rates from greatest to least: C (5) > A (4.5) > B (3.5). Starting volumes from greatest to least: B (18) > A (12) > C (10). A steeper line means a faster rate, not a larger amount, so the student is conflating slope with height.
2. At what time do barrels A and B contain the same volume? What is that volume? At that same time, how much is in barrel C?
Read the worked answer
Set the models equal: 4.5x + 12 = 3.5x + 18. Subtract 3.5x: x + 12 = 18, so x = 6 hours.
Volume: 4.5(6) + 12 = 27 + 12 = 39 L. Check with B: 3.5(6) + 18 = 21 + 18 = 39 L. So A and B both hold 39 L at 6 hours.
Barrel C at 6 hours: 5(6) + 10 = 40 L. So C is 1 L above the other two at that moment. This is the point where C overtakes B and A: before 6 hours B is highest, and after 6 hours C is highest (until a barrel fills).
3. Evaluate the student's claim that barrel C will always have the most water. Use the 60-L capacity and state the valid domain for each model.
Read the worked answer
The claim is false. At x = 0, C has 10 L while B has 18 L, so B has more. Solving 5x + 10 = 3.5x + 18 gives 1.5x = 8, so x ≈ 5.33 hours, the time C catches B; before that, B is highest. C is highest only from about 5.33 hours until a barrel fills.
Capacity limits: C reaches 60 L at (60 − 10) ÷ 5 = 10 hours. A reaches 60 L at (60 − 12) ÷ 4.5 ≈ 10.67 hours. B reaches 60 L at (60 − 18) ÷ 3.5 = 12 hours. After a barrel reaches 60 L, its linear model no longer applies because the volume cannot keep increasing; the graph would be horizontal, not continued upward.
So each model is valid only for x ≥ 0 and 0 ≤ y ≤ 60, and C is not always highest. Between 10 and 10.67 hours, C is full and A is still rising, so A is highest among the barrels still filling.