Compare Rate and Output

These supplied continuous linear models give total water for 0 to 6 minutes. The graph points are also listed so you can read the quantities without relying on color.

1 Guided comparison

Use A’s points at 2 and 6 minutes to calculate its rate and recover its initial value. B begins with 9 liters and gains 1 liter per minute. Write both rules. Which has the greater rate? Which has greater total at 0, 2 and 6 minutes?

The rules for the graph are A = 2.5t + 3 and B = t + 9. Equality requires 2.5t + 3 = t + 9. Remove t from each side, then 3 from each side: 1.5t = 6. Thus t = 4 minutes, and each total is 13 liters. Before 4 minutes B has more; after 4 A has more, within the stated 0-to-6-minute domain.

A larger change rate describes how output changes as input changes. Greater output is a comparison at a particular common input. Initial values also affect that output. Outside the supplied domain, an extended expression is a conditional prediction, not a guaranteed reading.

2 Your comparison

For 0 to 8 minutes, supplied linear model C begins with 2 liters and gains 2 liters per minute. Model D is given by a table below. Find D’s rate and initial value, write both rules, and find when totals are equal. State which has more before and after that time, within the domain.

3 Two observations and a proposed rule

A record gives (1, 6) and (3, 12), in minutes and liters, but supplies no full model. One possible process is 3t + 3 liters throughout 0 to 6 minutes. Another is 3t + 3 through minute 3, then t + 9 after minute 3 through minute 6.

Check that both possibilities fit both observations. What does each give at five minutes? If a full linear model over 0 to 6 were explicitly assumed, which expression would those two points determine? Does the record alone establish that assumption?

Two distinct points determine a line under a linear assumption. They do not by themselves establish that every later physical reading belongs to that line.

4 Equal rates, different starts

On 0 to 6 minutes, E = 2t + 5 and F = 2t + 1 give total liters. Do they ever have equal totals? Explain how equal rates affect the difference between their outputs.