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Worksheet Grade 11 Math

Brenhollow's Center Pivots: Circles, Acre-Feet, and Pump Hours

A grade 11 worksheet on a fictional farm district's quarter-mile center pivot, moving from circle area to acre-feet, gallons, pump hours, and an aquifer cap.

Objectives

  1. Students will compute the irrigated area of a center pivot and compare it against the square field it sits in.
  2. Students will convert an application depth into acre-inches, acre-feet, and gallons.
  3. Students will test a season irrigation plan against a stated annual groundwater allowance.

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.

Also: center pivot, acre-inch, application depth, allowance

Teacher overview

A center pivot waters a circle inside a square field, which is why irrigated acreage and field acreage are never the same number, and why a grower planning a season has to start from the circle rather than the property line. This worksheet follows one quarter-mile pivot in the fictional Brenhollow farm district, which draws from a shared aquifer with an annual allowance per acre. Students carry a single inch of applied water from acres through acre-inches, acre-feet, and gallons, then scale it to a full season and test the result against the allowance. The common error is to treat an acre-inch and an acre-foot as interchangeable, which throws every later figure off by a factor of twelve.

Brenhollow's Center Pivots: Circles, Acre-Feet, and Pump Hours — Grade 11 Math
Name: ____________________________ Date: ______________

Student Worksheet

Brenhollow, its pivot, and its aquifer allowance are fictional. Use these values for every item, and keep full calculator precision until the final rounding in each item.

  • The pivot lateral is 1,320 ft long, so the irrigated circle has a radius of 1,320 ft.
  • The pivot sits at the center of a square field measuring 2,640 ft on each side.
  • The pump supplies the pivot at a steady 900 gpm.
  • The crop needs 22 inches of applied irrigation water across the season.
  • The aquifer allowance for this field is 1.7 acre-feet per acre per year.
  • Conversions: 1 acre = 43,560 square feet; 1 acre-foot = 325,851 gallons; 12 acre-inches = 1 acre-foot.
  • Once you have found the irrigated acreage in item 2, use that rounded value for every later item.
  1. Find the area of the irrigated circle in square feet. Round to the nearest square foot.
  2. Convert that area to acres. Round to one decimal place.
  3. Find the area of the square field in acres.
  4. What percent of the square field does the pivot water? Round to one decimal place.
  5. How many acres sit in the corners the pivot never reaches? Round to one decimal place.
  6. How many acre-inches of water does it take to apply 1.00 inch across the irrigated acres?
  7. Convert that volume to acre-feet. Round to three decimal places.
  8. Convert that volume to gallons. Round to the nearest gallon.
  9. How many hours must the pump run at 900 gpm to deliver 1.00 inch? Round to one decimal place.
  10. Find the season total of 22 inches in acre-feet. Round to two decimal places.
  11. Find the season total in gallons. Round to the nearest gallon.
  12. Find the season pump time in hours. Round to one decimal place.
  13. Find this field's annual allowance in acre-feet. Round to two decimal places.
  14. How many inches of water may the grower apply and stay inside the allowance? Round to one decimal place.
  15. Compare items 10 and 13, then explain two changes the grower could make so the season plan fits the allowance without leaving the crop short of the water it needs.

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: 5,473,911 sq ft.
    Area = pi times the radius squared. The radius squared is 1,320 x 1,320 = 1,742,400 square feet, and 1,742,400 x 3.14159265 = 5,473,911 square feet.
  2. Answer: 125.7 acres.
    5,473,911 divided by 43,560 square feet per acre = 125.664 acres, which rounds to 125.7 acres.
  3. Answer: 160 acres.
    The square field covers 2,640 x 2,640 = 6,969,600 square feet, and 6,969,600 divided by 43,560 = 160 acres.
  4. Answer: 78.6 percent.
    125.7 divided by 160 = 0.7856, or 78.6 percent. Students who keep the unrounded 125.664 acres get 78.5 percent, which matches the exact ratio of pi divided by 4; accept either.
  5. Answer: 34.3 acres.
    160 - 125.7 = 34.3 acres lie in the four corners outside the circle.
  6. Answer: 125.7 acre-inches.
    One inch applied over 125.7 acres is 125.7 acre-inches by definition of the unit.
  7. Answer: 10.475 acre-ft.
    125.7 acre-inches divided by 12 acre-inches per acre-foot = 10.475 acre-feet.
  8. Answer: 3,413,289 gal.
    10.475 acre-feet x 325,851 gallons per acre-foot = 3,413,289.2 gallons.
  9. Answer: 63.2 hr.
    3,413,289 gallons divided by 900 gallons per minute = 3,792.5 minutes, and 3,792.5 divided by 60 = 63.21 hours.
  10. Answer: 230.45 acre-ft.
    Each inch takes 10.475 acre-feet, so 22 inches take 10.475 x 22 = 230.45 acre-feet.
  11. Answer: 75,092,363 gal.
    230.45 acre-feet x 325,851 gallons per acre-foot = 75,092,363 gallons. Multiplying the item 8 result by 22 gives the same figure.
  12. Answer: 1,390.6 hr.
    75,092,363 divided by 900 = 83,436 minutes, and 83,436 divided by 60 = 1,390.6 hours. That is about 58 days of continuous pumping.
  13. Answer: 213.69 acre-ft.
    125.7 acres x 1.7 acre-feet per acre = 213.69 acre-feet.
  14. Answer: 20.4 in.
    213.69 acre-feet divided by 10.475 acre-feet per inch = 20.40 inches.
  15. Answer: The plan exceeds the allowance by 16.76 acre-feet, so the grower must cut applied water or draw less from the aquifer.
    The season plan calls for 230.45 acre-feet while the allowance is 213.69 acre-feet, an overage of 16.76 acre-feet, or about 1.6 inches of application. Reasonable changes include switching to a crop or variety with a lower season requirement, replacing the pivot sprinkler package with low-pressure drop nozzles that lose less water to evaporation and drift, irrigating on a soil-moisture schedule so no pass is wasted after rain, or buying or leasing allowance from another landowner in the district. Simply pumping more is not available, because the allowance is the limit.

Differentiation

Support

Give students a diagram of the circle inscribed in the square with the radius, the side length, and the four corners already labeled, so items 1 through 5 become substitution rather than interpretation. Provide a filled-in unit ladder showing acres, acre-inches, acre-feet, and gallons in order, and work items 6 and 7 together before students continue alone.

Extension

Ask students to find the radius a pivot would need to reach 90 percent of the square field, then explain why no circle inscribed in a square can reach 100 percent of it. Students who finish early can compute the cost of the season pumping if electricity for the pump runs at a rate the class chooses, and compare that against the value of the corner acres left dry.

ELL supports

Pre-teach "pivot," "lateral," "application depth," "allowance," and "acre-foot" with a labeled overhead sketch of a circular field inside a square one. Provide a two-column glossary card pairing each unit with a plain description, such as "acre-inch: enough water to cover one acre one inch deep," and allow item 15 to be answered as a labeled list rather than in full sentences.

Extensions

Have students look up the irrigation water requirement for a crop grown in their own state, using a university extension bulletin, then rework items 10 through 14 with that requirement in place of the 22 inches used here. Comparing a humid-region crop with an arid-region crop shows quickly why groundwater allowances are argued over so heavily in some states and barely noticed in others.