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Lesson plan Grade 11 Math

Sable Crossing's Growth: Timing a Treatment Plant Expansion

A grade 11 lesson where students model a fictional town's maximum-day water demand exponentially and use logarithms to time a treatment plant expansion.

Objectives

  1. Students will construct an exponential function for a town's maximum-day water demand from a growth rate, a per-resident production rate and a maximum-day factor, carrying units through to MGD.
  2. Students will solve exponential equations with logarithms to find when demand reaches a planning trigger and the plant's rated capacity.
  3. Students will use a construction lead time, a linear-growth comparison and a conservation scenario to decide when an expansion must begin and how large it should be.

Vocabulary

Key words: exponential growth, growth factor, average-day demand, maximum-day factor, treatment capacity, MGD, lead time, logarithm

Teacher overview

Sable Crossing, a fictional town used for this lesson, has a 4.0 million-gallon-per-day (MGD) treatment plant and a population growing about 3 percent a year, and its utility board has to decide when to start designing an expansion that takes six years to build. Students build an exponential model of the town's maximum-day demand from its population, its production per resident and its maximum-day factor, then use logarithms to find when demand reaches the board's 80 percent design trigger and the plant's full capacity. Treatment plants are generally designed to meet maximum-day demand, while storage tanks cover the shorter swings within a single day, so the maximum day rather than the average day is the figure that matters here. Two common misconceptions are addressed directly: comparing average-day demand to rated capacity, which makes the plant look about half used, and treating a percentage growth rate as a fixed number of new residents each year, which delays the date the plant runs short.

Procedure

Hook (5 min): Hand out the Sable Crossing student handout and read the fictional council member's statement at the top aloud: "Our town uses only about half of what the plant can treat, and at 3 percent growth it takes more than 20 years for demand to double, so we have decades before we need to think about this." Ask students to write one reason the statement might be wrong before they calculate anything, and collect two or three reasons on the board for later.

Direct instruction (15 min): Work Questions 1 to 4 with the class. Write P(t) = 18,000(1.03)^t and point out that 1.03 is the growth factor, so each year's population is 103 percent of the year before. Build the demand chain with units written at every step: people × gallons per person per day = gallons per day, then × 1.5 for the maximum day, then ÷ 1,000,000 for MGD. Show that the constants combine into a single starting value, so M(t) = 2.97(1.03)^t in MGD. Return to the hook: the 1.98 MGD average day is about half of 4.0 MGD, yet the 2.97 MGD maximum day is already about three quarters of it. Then model the solving method students will use in every remaining question: isolate the power, take the logarithm of both sides, and divide.

Guided practice (15 min): In pairs, students complete Questions 5 to 7. They solve for the year maximum-day demand reaches capacity and check the answer by evaluating M(10) and M(11), then find when the board's 80 percent trigger is reached and the latest date design can begin given the six-year lead time. As a formative check, circulate during Question 5 and ask each pair to show the line where they took the logarithm; any pair that computed (4.0 ÷ 2.97 − 1) ÷ 0.03 has treated the growth as linear and will get 11.56 rather than 10.07, so redirect them to the growth-factor form before they move on. Before independent practice, ask the class whether the board's 80 percent policy would have started design in time and take a show of hands; at least three quarters of the room should say yes and cite the 2.52-year and 4.07-year results.

Independent practice (10 min): Each student completes Questions 8 to 10, which compare the council member's constant-increase assumption with the exponential model, test how much time a conservation program would buy, and size the expansion for a 20-year horizon.

Closure / exit ticket (5 min): Students answer Question 11 on the handout for the neighboring fictional town of Ferrin Ridge and turn it in. Sort the tickets into three piles: correct function and correct subtraction of the lead time, correct function but lead time missing or added, and a linear or average-day approach. Use the second and third piles to plan a short opening review for the next class.

Sable Crossing's Growth: Timing a Treatment Plant Expansion — Grade 11 Math
Name: ____________________________ Date: ______________

Student Handout

The town of Sable Crossing, its plant, its records and its neighbor Ferrin Ridge are fictional. Use these values for every question. Time t is measured in years after January 1, 2026, so t = 0 is January 1, 2026 and t = 10 is January 1, 2036.

Sable Crossing planning data Value
Population on January 1, 2026 18,000 people
Planning growth rate 3 percent per year, every year
Average-day production per resident 110 gallons per person per day
Maximum-day factor (highest day ÷ average day, from the town's records) 1.5
Treatment plant rated capacity 4.0 MGD
Board policy: begin designing an expansion when maximum-day demand reaches 80 percent of rated capacity
Time to design and build an expansion 6 years

The USGS estimates that a person uses about 80 to 100 gallons of water a day at home. Sable Crossing's figure of 110 gallons per resident is larger because the plant also supplies businesses, schools and parks, and some treated water is lost from leaking pipes. One MGD is 1,000,000 gallons per day.

A council member says: "Our town uses only about half of what the plant can treat, and at 3 percent growth it takes more than 20 years for demand to double, so we have decades before we need to think about this."

  1. Write a function P(t) for Sable Crossing's population t years after January 1, 2026. Use it to find the population on January 1, 2036, to the nearest whole person.
  2. Find Sable Crossing's average-day demand on January 1, 2026, in MGD.
  3. Find Sable Crossing's maximum-day demand on January 1, 2026, in MGD, and state it as a percent of the plant's 4.0 MGD rated capacity.
  4. Write a function M(t) for maximum-day demand in MGD. Combine all of the constants into a single starting value, and show the units that cancel.
  5. Solve M(t) = 4.0 with logarithms. Give t to the nearest hundredth of a year and the calendar year it falls in. Check your answer by finding M(10) and M(11) to the nearest hundredth.
  6. Find when maximum-day demand reaches the board's 80 percent design trigger. Give t to the nearest hundredth of a year.
  7. Using your answer to Question 5 and the 6-year design and construction time, find the latest t at which design can begin and still have the expansion running before demand reaches capacity. How much earlier than this does the 80 percent trigger in Question 6 occur?
  8. The council member assumes the town will add 540 people every year, since 540 is 3 percent of 18,000. Write the linear population function this assumption gives, and find when maximum-day demand would reach 4.0 MGD under it, to the nearest hundredth of a year. How many years later is this than your answer to Question 5? Explain why the linear assumption gives a later date.
  9. Suppose a conservation program lowered average-day production to 100 gallons per resident starting January 1, 2026, with the growth rate and maximum-day factor unchanged. Find when maximum-day demand would reach 4.0 MGD, to the nearest hundredth of a year, and how many years the program would gain compared with Question 5, to the nearest tenth.
  10. The board wants the expansion to cover maximum-day demand through January 1, 2046 (t = 20). Find M(20) to the nearest hundredth of an MGD, then find the added capacity the expansion needs, rounded up to the next tenth of an MGD.
  11. Exit ticket. The fictional town of Ferrin Ridge had a maximum-day demand of 2.4 MGD on January 1, 2026, growing 2 percent per year. Its plant is rated at 3.0 MGD, and an expansion takes 5 years to design and build. Write Ferrin Ridge's demand function, find when it reaches 3.0 MGD, and find the latest t at which Ferrin Ridge can begin design, each to the nearest hundredth of a year.

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

Answer Key

All values below come from the fictional Sable Crossing and Ferrin Ridge data on the handout. Logarithms are base 10; natural logarithms give the same quotients.

  1. Answer: 24,190 people.
    P(t) = 18,000(1.03)^t. On January 1, 2036, t = 10, and (1.03)^10 = 1.343916, so P(10) = 18,000 × 1.343916 = 24,190.49, which rounds to 24,190 people.

  2. Answer: 1.98 MGD.
    18,000 people × 110 gallons per person per day = 1,980,000 gallons per day, and 1,980,000 ÷ 1,000,000 = 1.98 MGD. This is 49.5 percent of the 4.0 MGD capacity, which is the "about half" the council member describes.

  3. Answer: 2.97 MGD, which is 74.25 percent of capacity.
    The maximum day is 1.98 MGD × 1.5 = 2.97 MGD, and 2.97 ÷ 4.0 = 0.7425, or 74.25 percent. On the day that matters for the plant, it is already about three quarters used.

  4. Answer: M(t) = 2.97(1.03)^t MGD.
    M(t) = 18,000(1.03)^t people × 110 gallons per person per day × 1.5 ÷ 1,000,000 gallons per million gallons. The "people" units cancel, leaving gallons per day, and dividing by 1,000,000 gallons per million gallons leaves million gallons per day. The constants combine as 18,000 × 110 × 1.5 ÷ 1,000,000 = 2.97.

  5. Answer: about 10.07 years, early in 2036.
    Setting 2.97(1.03)^t = 4.0 gives (1.03)^t = 4.0 ÷ 2.97 = 1.346801. Taking the logarithm of both sides, t = log(1.346801) ÷ log(1.03) = 0.129304 ÷ 0.012837 = 10.07 years after January 1, 2026, which falls in early 2036. Check: M(10) = 2.97 × 1.343916 = 3.99 MGD, just under capacity, and M(11) = 2.97 × 1.384234 = 4.11 MGD, over capacity.

  6. Answer: about 2.52 years, in mid-2028.
    80 percent of 4.0 MGD is 3.2 MGD. 2.97(1.03)^t = 3.2 gives (1.03)^t = 3.2 ÷ 2.97 = 1.077441, so t = log(1.077441) ÷ log(1.03) = 0.032394 ÷ 0.012837 = 2.52 years. Check: M(2) = 3.15 MGD and M(3) = 3.25 MGD, so the trigger is crossed between them.

  7. Answer: about 4.07 years, early in 2030; the trigger comes about 1.55 years earlier.
    Demand reaches capacity at t = 10.07, so design must begin no later than 10.07 − 6 = 4.07 years after January 1, 2026. The 80 percent trigger at t = 2.52 comes 4.07 − 2.52 = 1.55 years before that deadline, so following the board's policy leaves about a year and a half of margin for delays in design, funding or construction.

  8. Answer: about 11.56 years, about 1.49 years later than the exponential model.
    The linear assumption gives L(t) = 18,000 + 540t. Maximum-day demand is 110 × 1.5 = 165 gallons per resident on the peak day, so capacity is reached when 165(18,000 + 540t) = 4,000,000. Then 18,000 + 540t = 24,242.42, so 540t = 6,242.42 and t = 11.56 years, in mid-2037. The difference is 11.56 − 10.07 = 1.49 years. The linear model adds 3 percent of the 2026 population every year, while the exponential model adds 3 percent of each year's larger population. By the tenth year the exponential model adds about 705 people (0.03 × 18,000 × 1.03^9 = 704.58), well above 540, so the linear model falls further behind each year and reports a later date than the town can safely plan for.

  9. Answer: about 13.30 years, a gain of about 3.2 years.
    The new starting value is 18,000 × 100 × 1.5 ÷ 1,000,000 = 2.7 MGD, so 2.7(1.03)^t = 4.0 gives (1.03)^t = 1.481481, and t = log(1.481481) ÷ log(1.03) = 0.170696 ÷ 0.012837 = 13.30 years, in spring 2039. The gain is 13.30 − 10.07 = 3.23 years using the rounded answers, or 3.22 years using unrounded values, and both round to 3.2 years. Conservation delays the date but does not remove the need to expand, because the population keeps growing.

  10. Answer: 1.4 MGD of added capacity, for a 5.4 MGD plant.
    M(20) = 2.97(1.03)^20 = 2.97 × 1.806111 = 5.36 MGD. The expansion must add at least 5.36 − 4.0 = 1.36 MGD, and rounding up to the next tenth gives 1.4 MGD. Rounding down to 1.3 MGD would leave the plant short before 2046.

  11. Answer: F(t) = 2.4(1.02)^t MGD; demand reaches 3.0 MGD at about t = 11.27, and design must begin by about t = 6.27, in spring 2032.
    The starting value is 2.4 MGD and the growth factor is 1.02. Setting 2.4(1.02)^t = 3.0 gives (1.02)^t = 1.25, so t = log(1.25) ÷ log(1.02) = 0.096910 ÷ 0.008600 = 11.27 years. Subtracting the 5-year lead time, Ferrin Ridge must begin design by t = 11.27 − 5 = 6.27 years after January 1, 2026.

Differentiation

Support

Work Questions 1 to 4 as a class and leave the finished M(t) = 2.97(1.03)^t on the board for the rest of the lesson. Give students a printed card with the solving steps for a(b)^t = C in order: divide both sides by a, take the logarithm of both sides, divide log(C ÷ a) by log(b). Allow graphing calculators so students can confirm each logarithm answer by graphing y = 2.97(1.03)^x and y = 4.0 and finding the intersection. For Question 8, provide the equation 165(18,000 + 540t) = 4,000,000 already set up so the student's work is solving it and explaining the comparison.

Extension

Ask students to suppose the 1.4 MGD expansion from Question 10 comes online exactly at t = 10.07, giving the fictional Sable Crossing plant 5.4 MGD of capacity. Using M(t) = 2.97(1.03)^t, they find when maximum-day demand reaches 80 percent of the new capacity (4.32 MGD), which is about t = 12.68, in late summer 2038, and then write a short memo to the board explaining what this result implies about planning only one expansion at a time when growth is exponential. Stronger students can also derive a general formula for the latest design start, t = log(C ÷ D₀) ÷ log(1 + r) − L, where C is capacity, D₀ is the starting maximum-day demand, r is the growth rate and L is the lead time, and use it to check Questions 7 and 11.

ELL supports

Pre-teach "growth factor," "average day," "maximum day," "capacity," and "lead time" with a labeled timeline drawn on the board: the 2026 starting point, the 80 percent trigger, the latest design start and the date capacity is reached, each marked with its t value as the class finds it. Provide the units chain as a fill-in frame (people × ___ gallons per person per day × ___ ÷ 1,000,000 = ___ MGD) and sentence starters for Question 8: "The linear model gives a later date because ___."