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Math problem set Grade 9 Math

Rennick Valley Center Pivots: Run Times and Canal Capacity

A grade 9 problem set on scheduling six center pivots under a canal limit using acre-inches, run times, inequalities, and wheat water use per pound.

Objectives

  1. Students will convert acre-feet to acre-inches and gallons and use a pump rate to find how long a center pivot must run.
  2. Students will write and solve inequalities that represent a canal's delivery limit and interpret which solutions are viable.
  3. Students will compare irrigation water per pound of wheat with a published total water footprint and explain why the two figures measure different quantities.

Vocabulary

Flow rate
Flow rate measures how much water moves past a point in a certain amount of time, like gallons per minute through a garden hose. Bigger pipes and stronger pushes give bigger flow rates.
Water footprint
All the water it takes to make the things we use and eat, counting water used far away. A hamburger's water footprint includes the water that grew the cattle's feed.

Also: acre-inch, allocation, center pivot, constraint

Teacher overview

The fictional Rennick Valley farm district feeds six center-pivot sprinklers from one canal, the north lateral, which can carry only so much water at a time. Growers have to decide how many pivots may run at once, how long each must run to put a given depth of water on its field, and how far the season's allocation will stretch. This set works through those scheduling questions with acre-inches and pump rates, uses inequalities to describe the canal's limit, and ends by comparing the district's irrigation water per pound of wheat with published water-footprint figures. A common misconception arises in problem 4. Students who solve 750p ≤ 4,000 and get p ≤ 5.33 often round up to 6 pivots, which would draw 4,500 gpm and exceed the limit, because they do not yet treat the number of running pivots as a whole number that can be no greater than 5.33.

Rennick Valley Center Pivots: Run Times and Canal Capacity — Grade 9 Math
Name: ____________________________ Date: ______________

Problems

Rennick Valley (a fictional farm district) delivers irrigation water through its north lateral canal to six center-pivot sprinklers. Use these facts for every problem.

  • Each pivot irrigates a circle of 125 acres, so the six pivots cover 750 acres in all.
  • Each pivot pumps 750 gallons per minute (gpm) while it runs.
  • The north lateral can deliver at most 4,000 gpm at any moment.
  • The north lateral's allocation for the season is 1,150 acre-feet.
  • 1 acre-foot = 325,851 gallons, and 1 foot = 12 inches.
  • One week has 168 hours.
  1. An acre-inch is the volume of water that covers 1 acre to a depth of 1 inch. Find the number of gallons in 1 acre-inch. Keep two decimal places.
  2. How many gallons does it take to apply 1 inch of water to one 125-acre pivot circle?
  3. How many hours must one pivot run at 750 gpm to apply 1 inch of water to its circle? Round to the nearest tenth of an hour.
  4. Let p be the number of pivots running at the same time. Write an inequality that keeps the running pivots within the canal's 4,000 gpm limit, solve it, and state the greatest number of pivots that can run at once. Explain why the exact solution of the inequality is not itself a possible answer.
  5. During the hottest week of the summer, growers ask for 2 inches of water on every one of the six circles. Using your answer to problem 4, find the most water the pivots can deliver in one week, in gallons. Decide whether the request can be met, and if it cannot, find the shortfall in gallons.
  6. The district decides to give all six circles the same depth d, in inches, during that week. Write an inequality comparing the water needed for depth d on six circles with the weekly delivery limit from problem 5, solve it, and round d down to the nearest hundredth of an inch.
  7. For the whole season, the growers plan to grow wheat on all 750 acres and apply 14 inches of irrigation water. How many acre-feet does this plan use, and how many acre-feet of the 1,150 acre-foot allocation are left over?
  8. Irrigated wheat in Rennick Valley yields about 4,800 pounds per acre. Find the total wheat harvest from 750 acres, convert the plan's water use from problem 7 to gallons, and find the irrigation water used per pound of wheat. Round to the nearest tenth of a gallon per pound.
  9. Water Footprint Network global averages put the total water used to produce 1 pound of beef at about 1,800 gallons and 1 pound of wheat at about 220 gallons. Most of the beef figure is rainwater used by the pasture and feed crops cattle eat, not irrigation or tap water. How many pounds of wheat take the same total water as 1 pound of beef? Round to the nearest tenth.
  10. A classmate converts the full 1,150 acre-foot allocation to gallons, divides by 220 gallons per pound, and says the north lateral can grow that many pounds of wheat. Find the classmate's number, rounded to the nearest whole pound. Then use the units and the meaning of each figure to explain why it does not match the harvest you found in problem 8.

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: 27,154.25 gallons. An acre-inch is one twelfth of an acre-foot, so 325,851 gallons per acre-foot / 12 inches per foot = 27,154.25 gallons per acre-inch.
  2. Answer: 3,394,281.25 gallons. 125 acres x 1 inch = 125 acre-inches; 125 acre-inches x 27,154.25 gallons per acre-inch = 3,394,281.25 gallons.
  3. Answer: 75.4 hours. 3,394,281.25 gallons / 750 gallons per minute = 4,525.71 minutes; 4,525.71 minutes / 60 minutes per hour = 75.43 hours, about 75.4 hours, or roughly 3.1 days of continuous running for one inch.
  4. Answer: 5 pivots. The running pivots draw 750p gpm, so the constraint is 750p ≤ 4,000. Dividing both sides by 750 gives p ≤ 5.33. The number of pivots must be a whole number, so 5.33 is not a possible answer, and rounding up to 6 would draw 6 x 750 = 4,500 gpm, more than the canal can deliver. The greatest viable value is p = 5, which draws 5 x 750 = 3,750 gpm.
  5. Answer: No; the shortfall is 2,931,375 gallons. Five pivots deliver 5 x 750 = 3,750 gpm. One week is 168 hours x 60 minutes per hour = 10,080 minutes, so the weekly limit is 3,750 gpm x 10,080 minutes = 37,800,000 gallons. The request is 6 circles x 2 inches = 12 circle-inches, and 12 x 3,394,281.25 gallons = 40,731,375 gallons. Since 40,731,375 is greater than 37,800,000, the request cannot be met, and the shortfall is 40,731,375 - 37,800,000 = 2,931,375 gallons. The same conclusion follows in pivot-hours: the request needs 12 x 75.43 = about 905 pivot-hours, while only 5 pivots x 168 hours = 840 pivot-hours are available.
  6. Answer: 1.85 inches. Depth d on one circle takes 3,394,281.25d gallons, so six circles take 6 x 3,394,281.25d = 20,365,687.5d gallons. The constraint is 20,365,687.5d ≤ 37,800,000, so d ≤ 37,800,000 / 20,365,687.5 = 1.8561. Rounding down keeps the schedule within the limit, so d = 1.85 inches, even though ordinary rounding would give 1.86. Check: 20,365,687.5 x 1.85 = 37,676,521.875 gallons, which is less than 37,800,000 gallons, while 1.86 inches would need 37,880,178.75 gallons, which is too much.
  7. Answer: 875 acre-feet used, with 275 acre-feet left over. 14 inches = 14/12 feet. 750 acres x 14/12 feet = 10,500/12 = 875 acre-feet. 1,150 acre-feet - 875 acre-feet = 275 acre-feet left over.
  8. Answer: 79.2 gallons per pound. Harvest: 750 acres x 4,800 pounds per acre = 3,600,000 pounds. Water: 875 acre-feet x 325,851 gallons per acre-foot = 285,119,625 gallons. Water per pound: 285,119,625 gallons / 3,600,000 pounds = 79.1999, about 79.2 gallons per pound. Working per acre gives the same result: 14 acre-inches x 27,154.25 gallons per acre-inch = 380,159.5 gallons per acre, and 380,159.5 / 4,800 = 79.2 gallons per pound.
  9. Answer: 8.2 pounds. 1,800 gallons / 220 gallons per pound = 8.18 pounds, about 8.2 pounds of wheat.
  10. Answer: 1,703,312 pounds, which is not a valid harvest estimate. The classmate's work is 1,150 acre-feet x 325,851 gallons per acre-foot = 374,728,650 gallons, and 374,728,650 / 220 = 1,703,312.05, about 1,703,312 pounds. The number is less than half of the 3,600,000 pounds that problem 8 shows the district can grow with only 875 of its 1,150 acre-feet. The two figures look like they share a unit, gallons per pound, yet they count different water. The 220 gallons per pound is a global average of all the water a pound of wheat uses, much of it rain and soil moisture, while the allocation counts only canal water, and Rennick Valley's irrigation water per pound is about 79.2 gallons. Dividing irrigation gallons by total-water gallons per pound mixes the two quantities. The harvest is also limited by land: 750 acres at 4,800 pounds per acre is 3,600,000 pounds no matter how much allocation is left over.

Differentiation

Support

Give students a card with "1 acre-foot = 325,851 gallons," "1 foot = 12 inches," and "1 hour = 60 minutes," and set up problem 1 together. For problems 5 and 6, provide a table with columns for pivots running, gpm, minutes in a week, and gallons delivered, so students fill in one row at a time before comparing totals. Allow a calculator throughout.

Extension

Suppose the fictional district installs a new turnout that raises the north lateral's limit to 4,500 gpm. Ask students to redo problems 4 through 6 with the new limit and decide whether the 2-inch peak-week request can now be met. (All six pivots can then run, delivering 6 x 750 = 4,500 gpm and 4,500 x 10,080 = 45,360,000 gallons in a week, which covers the 40,731,375-gallon request.) Students can also find how many inches per day one pivot applies when it runs around the clock, which is 24 / 75.43 = about 0.32 inch per day.

ELL supports

Pre-teach "acre-inch," "center pivot," "constraint," and "allocation" with a sketch of a pivot circle and a one-acre square covered one inch deep. For problem 10, offer the sentence frame "The figure of 220 gallons per pound counts ___, while the allocation counts only ___," and allow students to answer the inequality problems with a labeled number line instead of full sentences.