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Math problem set Grade 9 Engineering & Design

Harrowmede Reservoir: A Dry-Year Water Budget in Acre-Feet

A grade 9 set where students build a dry-year water budget for a fictional reservoir, converting MGD to acre-feet and solving for the years left before dead pool.

Objectives

  1. Students will convert a steady daily demand into an annual volume expressed in acre-feet.
  2. Students will assemble inflow, demand, and evaporation into a single annual water balance.
  3. Students will write and solve a linear equation for the number of dry years a reservoir can supply.

Vocabulary

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.
Water budget
A water budget is a way of tracking all the water coming into an area, like rain and snowmelt, against all the water leaving it, like evaporation and streamflow. Scientists use water budgets to see whether a watershed or aquifer is gaining or losing water over time.
Dead pool
A reservoir level so low that water can no longer flow out through the dam on its own. It is the worst-case marker in a long drought.
Evaporation
Evaporation happens when liquid water, often warmed by the sun, turns into an invisible gas called water vapor that rises into the air. It is how water leaves oceans, lakes, and puddles.

Also: inflow

Teacher overview

Skelbrook is a fictional town whose only supply is Harrowmede Reservoir, a shallow impoundment on a small creek. Planners describe a reservoir with a water budget, which adds up what comes in, what people take out, and what the sun removes from the surface, then reports the change in storage. This set has students build one dry year of that budget in acre-feet, write the storage equation, and solve for the year in which the reservoir would reach the level below which it cannot deliver water. The error to watch for is treating evaporation as a small correction, because on a wide, shallow reservoir the surface loss can rival a whole season of town demand.

Harrowmede Reservoir: A Dry-Year Water Budget in Acre-Feet — Grade 9 Engineering & Design
Name: ____________________________ Date: ______________

Problems

The town of Skelbrook and Harrowmede Reservoir are fictional, and every figure below belongs to this scenario.

Harrowmede Reservoir holds 9,600 acre-feet when full and covers 320 acres of surface. The lowest 1,500 acre-feet cannot be delivered, because the outlet works sit above that level. The reservoir begins the dry year 78 percent full. Skelbrook uses 2.8 MGD, steady all year. Dry-year inflow to the reservoir totals 2,400 acre-feet, and dry-year evaporation removes 3.5 feet of depth from the whole surface. Use 1 acre-foot = 325,851 gallons and a 365-day year.

  1. Skelbrook uses 2.8 MGD every day of a 365-day year. How many gallons does the town use in that year?

  2. Convert the yearly demand from problem 1 into acre-feet. Use 1 acre-foot = 325,851 gallons and round to the nearest tenth.

  3. Evaporation removes 3.5 feet of depth from the reservoir's 320-acre surface. How many acre-feet does that remove?

  4. The reservoir gains 2,400 acre-feet of inflow and loses the town demand from problem 2 and the evaporation from problem 3. How many acre-feet of storage does the reservoir lose over the dry year? Round to the nearest tenth.

  5. The reservoir begins the dry year 78 percent full out of 9,600 acre-feet. How many acre-feet is that?

  6. Let S stand for storage in acre-feet and n stand for the number of dry years. Write an equation that starts at your answer to problem 5 and subtracts your answer to problem 4 for each year. Then solve it for the value of n at which S reaches the 1,500 acre-feet that cannot be delivered. Round to the nearest hundredth of a year.

  7. Skelbrook adopts restrictions that cut town demand by 15 percent. Using your answer from problem 2 as the starting demand, what is the new yearly demand in acre-feet? Round to the nearest tenth.

  8. Recalculate the yearly loss using the reduced demand from problem 7, the same 2,400 acre-feet of inflow, and the same evaporation from problem 3. How many acre-feet does the reservoir now lose each dry year? Round to the nearest tenth.

  9. Using the reduced yearly loss from problem 8 and the same starting storage from problem 5, how many dry years pass before storage reaches 1,500 acre-feet? Round to the nearest hundredth of a year.

  10. A resident says a 15 percent cut in demand should buy 15 percent more time. Use your rounded answers from problems 6 and 9. By what percent does the cut actually extend the supply? Round to the nearest tenth of a percent.

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: 1,022,000,000 gallons.
    2.8 million gallons per day x 365 days = 1,022 million gallons, written out as 1,022,000,000 gallons.

  2. Answer: 3,136.4 acre-feet.
    1,022,000,000 gallons divided by 325,851 gallons per acre-foot = 3,136.40 acre-feet, which rounds to 3,136.4 acre-feet.

  3. Answer: 1,120 acre-feet.
    An acre-foot is one acre covered one foot deep, so depth times area gives the volume directly: 3.5 feet x 320 acres = 1,120 acre-feet.

  4. Answer: 1,856.4 acre-feet.
    Inflow minus the two losses: 2,400 minus 3,136.4 minus 1,120 = negative 1,856.4, so storage falls by 1,856.4 acre-feet over the year. Evaporation alone removes 1,120 acre-feet, more than a third as much as the town's entire yearly demand.

  5. Answer: 7,488 acre-feet.
    9,600 acre-feet x 0.78 = 7,488 acre-feet in storage at the start of the dry year.

  6. Answer: 3.23 years.
    The equation is S = 7,488 minus 1,856.4n. Set S = 1,500: 7,488 minus 1,856.4n = 1,500. Subtract to get 1,856.4n = 5,988, then divide: n = 3.2256, which rounds to 3.23 years.

  7. Answer: 2,665.9 acre-feet.
    A 15 percent cut leaves 85 percent of the demand. 3,136.4 x 0.85 = 2,665.94 acre-feet, which rounds to 2,665.9 acre-feet.

  8. Answer: 1,385.9 acre-feet.
    2,400 minus 2,665.9 minus 1,120 = negative 1,385.9, so the reservoir now loses 1,385.9 acre-feet each dry year.

  9. Answer: 4.32 years.
    The equation becomes S = 7,488 minus 1,385.9n. Set S = 1,500: 1,385.9n = 5,988, so n = 4.3207, which rounds to 4.32 years.

  10. Answer: 33.7 percent.
    The gain is 4.32 minus 3.23 = 1.09 years. Then 1.09 divided by 3.23 = 0.3375, or 33.7 percent. Accept answers between 33.5 and 34.0 percent, since students who keep unrounded values get 33.9 percent. The gain beats 15 percent because the cut applies only to demand, while inflow and evaporation stay fixed, so the yearly deficit shrinks by about a quarter rather than by a seventh.

Differentiation

Support

Give these students a blank budget frame with four labeled rows, inflow, town demand, evaporation, and change in storage, so problem 4 becomes a matter of filling boxes rather than tracking signs. Print the definition of an acre-foot beside a sketch of one acre covered one foot deep for problem 3, and supply the equation frame for problem 6 with only the two numbers missing.

Extension

Ask students to find the reduction in demand, as a percent, that would stretch the fictional Skelbrook supply to exactly six dry years. Then have them explain what that percent would mean for households in practice and why a utility usually pairs restrictions with a second source rather than relying on restrictions alone.

ELL supports

Pre-teach inflow, demand, evaporation, and dead pool with a labeled cross-section of a reservoir showing arrows in, arrows out, and a marked line for the level that cannot be delivered. Provide the sentence frame "The reservoir loses ___ acre-feet each year because ___ comes in and ___ goes out" for problem 4. Allow problems 6 and 9 to be answered with the equation and the final number only.

Extensions

Have students graph storage against years for both cases in this fictional scenario, plotting S = 7,488 minus 1,856.4n and S = 7,488 minus 1,385.9n on the same axes with a horizontal line at 1,500 acre-feet. Ask them to label where each line crosses that horizontal line and to explain what the difference in slopes represents. Students can then describe in writing how the graph would change if a wet year restored 3,000 acre-feet partway through.