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Math problem set Grade 11 Math

Hollis Junction Water Rates: Linear Versus Exponential Growth

A grade 11 problem set contrasting a fixed-percent water rate increase with a steady pipe leak, using function models and average rate of change.

Objectives

  1. Students will distinguish situations modeled by linear functions from those modeled by exponential functions.
  2. Students will construct a linear and an exponential function from a described relationship.
  3. Students will compute and interpret average rate of change over an interval.

Vocabulary

Key words: linear, exponential, average rate of change, compound increase

Teacher overview

This set follows the town of Hollis Junction (a fictional town) as it looks at two quantities that both grow over time but grow in different ways. One is a water bill that rises by a fixed percent each year, and the other is water lost from a steady pipe leak that adds the same amount every week. Students build a function for each, then use average rate of change to see why one curve bends upward while the other stays straight. A common misconception this set corrects is treating a percent increase as if it added the same number of dollars each year, which confuses exponential growth with linear growth.

Hollis Junction Water Rates: Linear Versus Exponential Growth — Grade 11 Math
Name: ____________________________ Date: ______________

Problems

Hollis Junction's base water bill is 40 dollars per month, and the utility plans to raise it by 6 percent each year, compounding. Separately, a home in town has a steady pipe leak that wastes 180 gallons of water every week.

  1. Classify each situation as linear or exponential, and give the reason:
    (a) a bill that increases by 3 dollars every year;
    (b) a bill that increases by 6 percent every year.
  2. Write an exponential function B(t) for the monthly bill after t years, starting from 40 dollars and growing 6 percent per year.
  3. Find the monthly bill after 5 years. Round to the nearest cent.
  4. Find the average rate of change of the bill from year 0 to year 5, in dollars per year. Round to the nearest cent.
  5. Find the average rate of change of the bill from year 5 to year 10, in dollars per year, and state whether it is larger or smaller than your answer to problem 4. Use B(10) = 40 × 1.06 raised to the 10th power.
  6. Write a linear function L(w) for the total gallons lost to the leak after w weeks, at 180 gallons per week.
  7. Find the average rate of change of L(w) over weeks 0 to 10 and over weeks 10 to 20, and state what you notice.
  8. Find the total gallons lost to the leak after 52 weeks.
  9. Using your average-rate-of-change answers, explain in two or three sentences why the bill and the leak are fundamentally different kinds of growth.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key
  1. Answer: (a) linear; (b) exponential. A fixed 3-dollar increase adds equal differences each year, which is linear; a fixed 6-percent increase multiplies by the same factor of 1.06 each year, which is exponential.
  2. Answer: B(t) = 40 × 1.06^t. Starting value 40 dollars, multiplied by the growth factor 1.06 once for each year t.
  3. Answer: 53.53 dollars. B(5) = 40 × 1.06^5 = 40 × 1.338226 = 53.529, which rounds to 53.53 dollars.
  4. Answer: 2.71 dollars per year. Average rate = (B(5) − B(0)) ÷ (5 − 0) = (53.53 − 40.00) ÷ 5 = 13.53 ÷ 5 = 2.706, about 2.71 dollars per year.
  5. Answer: 3.62 dollars per year. B(10) = 40 × 1.06^10 = 40 × 1.790847 = 71.63; average rate = (71.63 − 53.53) ÷ 5 = 18.10 ÷ 5 = 3.62 dollars per year, which is larger than the 2.71 from problem 4.
  6. Answer: L(w) = 180w. Each week adds 180 gallons, so total loss is 180 times the number of weeks w.
  7. Answer: 180 gallons per week. Weeks 0 to 10: (1,800 − 0) ÷ 10 = 180; weeks 10 to 20: (3,600 − 1,800) ÷ 10 = 180. The rate is the same over both intervals, which is the signature of a linear function.
  8. Answer: 9,360 gallons. L(52) = 180 × 52 = 9,360 gallons, close to the commonly cited 9,400 gallons a year for a household leak.
  9. Answer: The leak's average rate of change is constant at 180 gallons per week, so its graph is a straight line. The bill's average rate of change grows from 2.71 to 3.62 dollars per year across equal five-year spans, so its graph bends upward. Constant rate of change means linear; a rate of change that keeps growing because the bill is multiplied by the same factor, 1.06, each year means exponential.

Differentiation

Support

Provide the powers of 1.06 in a small table (1.06^1 through 1.06^10) so students can focus on setting up the average-rate-of-change quotient rather than on exponent arithmetic. Work problems 2 and 3 together before releasing the rest.

Extension

Ask students to find the first whole year in which the monthly bill passes 60 dollars, and to explain why an exponential model reaches that level sooner than a linear model that adds a flat 2.71 dollars each year would.

ELL supports

Provide a two-column organizer labeled "same amount added" and "same factor multiplied," and have students sort the bill and the leak into the correct column before writing functions. Pair "average rate of change" with a simple slope picture.

Extensions

Have students pull their own household's water rate schedule, if the local utility publishes one, and decide whether recent increases have looked more linear or more exponential across the last several years, supporting the claim with an average-rate-of-change calculation.