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Math problem set Grade 11 Math

Ansel Gorge Reservoir: Drawdown Functions and Storage Equations

A grade 11 problem set modeling a fictional reservoir's drought drawdown and refill with linear functions, graph features, and a two-variable storage equation.

Objectives

  1. Students will write linear equations relating reservoir elevation to elapsed time.
  2. Students will interpret intercepts, intervals of increase and decrease, and a relative minimum on a reservoir graph.
  3. Students will build and solve a two-variable equation relating stored gallons to days of release.

Vocabulary

Drawdown
Drawdown is how far the water level in a well drops when a pump pulls water out. The harder and longer you pump, the bigger the drawdown gets.
Acre-foot
The amount of water it takes to cover one acre of land, about the size of a football field, one foot deep. Western states use acre-feet to measure large amounts of water.

Also: elevation, relative minimum

Teacher overview

This set follows Ansel Gorge Reservoir (a fictional reservoir) behind Ansel Dam through a drought and its recovery. Students first model the water-surface elevation as a straight line while the level falls, then as another line while rain refills it, and read the key features of the combined graph. A separate storage question has students convert the reservoir's full volume into gallons and write an equation for how fast a release empties it. A common misconception this set corrects is reading a graph's relative minimum as its starting value, and another is comparing a storage in acre-feet directly against a release measured in gallons without converting first.

Ansel Gorge Reservoir: Drawdown Functions and Storage Equations — Grade 11 Math
Name: ____________________________ Date: ______________

Problems

Ansel Gorge Reservoir starts a drought with its water surface at an elevation of 240 feet. For 10 weeks the level falls a steady 2 feet per week. Then rain refills it a steady 4 feet per week for 5 weeks. When full, the reservoir holds 12,000 acre-feet. Use one acre-foot equals 325,851 gallons.

  1. Write a linear equation for the elevation E, in feet, as a function of the week t during the drought, for weeks 0 through 10.
  2. State the E-intercept of your drought equation and explain what it means for the reservoir.
  3. Find the elevation at week 10.
  4. During the refill, the level rises 4 feet per week from the week-10 elevation. Write an equation for E as a function of t for weeks 10 through 15, and find the elevation at week 15.
  5. Looking at the full 15-week graph, name the interval where the level is decreasing, the week and value of the relative minimum, and the interval where the level is increasing.
  6. The refill raised the level by the same amount the drought lowered it. How many weeks of refill at 4 feet per week were needed to recover the drop from problem 3?
  7. Convert the reservoir's full storage of 12,000 acre-feet into gallons.
  8. The dam releases water at 30 million gallons per day with no inflow. Let d be the number of days of release. Write a two-variable equation for the stored gallons G remaining, starting from the full volume in problem 7.
  9. Using your equation from problem 8, find how many days of release it takes for storage to fall to half full, which is 6,000 acre-feet. Round to the nearest day.

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: E = 240 − 2t. The elevation starts at 240 feet and drops 2 feet for each week t during the drought.
  2. Answer: 240 ft. The E-intercept is E = 240 when t = 0, meaning the water surface stood at 240 feet at the start of the drought.
  3. Answer: 220 ft. E = 240 − 2(10) = 240 − 20 = 220 feet at week 10.
  4. Answer: 240 ft. The refill equation is E = 220 + 4(t − 10); at t = 15, E = 220 + 4(5) = 220 + 20 = 240 feet, back to the starting level.
  5. Answer: minimum 220 ft at week 10. The level decreases over weeks 0 to 10, reaches a relative minimum of 220 feet at week 10, and increases over weeks 10 to 15.
  6. Answer: 5 weeks. The drought lowered the level by 240 − 220 = 20 feet, and 20 ÷ 4 = 5 weeks of refill at 4 feet per week were needed.
  7. Answer: 3,910,212,000 gal. 12,000 acre-feet × 325,851 gallons per acre-foot = 3,910,212,000 gallons.
  8. Answer: G = 3,910,212,000 − 30,000,000d. The release of 30 million gallons per day is 30,000,000 gallons each day, subtracted from the full volume for each day d.
  9. Answer: 65 days. Half full is 6,000 × 325,851 = 1,955,106,000 gallons; solving 3,910,212,000 − 30,000,000d = 1,955,106,000 gives 30,000,000d = 1,955,106,000, so d = 65.17, about 65 days.

Differentiation

Support

Provide pre-labeled axes (weeks on the horizontal axis from 0 to 15, elevation on the vertical axis from 210 to 250 feet) so students can plot the two line segments and read the relative minimum directly before writing equations.

Extension

Ask students to write a single piecewise function that gives the elevation for any week from 0 to 15, then to find the average rate of change of elevation across the whole 15-week span and explain why it is zero.

ELL supports

Pair "drawdown," "refill," and "relative minimum" with a simple sketch of a reservoir cross-section showing the water surface falling and then rising. Provide the phrase "increasing interval" and "decreasing interval" with arrows on a sample graph.