Math problem set Grade 12 Math

Deepwell Booster Station: Dynamic Head and Energy Cost

A grade 12 problem set computing total dynamic head, water and brake horsepower, and daily pumping energy cost for a fictional booster pump station.

Grade: Grade 12
Subject: Math
Material: Math problem set
Time: 45 min
Prep: No prep
Materials: Calculator

Standards

HSN-Q.A.1 Common Core Math

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

How this material aligns: Problems 1 through 5 carry units through the horsepower formula, converting a pressure loss in psi to feet of head and horsepower to kilowatts before any cost is found.

All materials for HSN-Q.A.1 →

HSA-CED.A.1 Common Core Math

Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

How this material aligns: Problem 8 has students write and solve a one-variable equation for the daily run hours that keep the pump's energy use under a stated kilowatt-hour budget.

All materials for HSA-CED.A.1 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-23.

Objectives

Students will compute total dynamic head and convert pump horsepower into a daily energy use in kilowatt-hours.
Students will write and solve an equation for the run hours allowed by an energy budget.

Vocabulary

Total dynamic head (TDH): Total dynamic head is the total amount of lift a pump has to provide to move water where it needs to go, adding up the height it must lift the water, the friction it must overcome in the pipes, and the pressure needed at the far end. Engineers calculate it to pick a pump that is powerful enough for the job.

Also: water horsepower, brake horsepower, pump efficiency

Teacher overview

Deepwell Booster Station is a fictional station that lifts drinking water uphill to a higher-pressure zone. To size its pump and estimate its electric bill, operators find the total dynamic head, convert it into horsepower, and then into energy at the local electricity rate. This set builds that chain one step at a time. A common error is to skip the efficiency step and treat water horsepower as the power the motor actually draws; the brake horsepower is always larger than the water horsepower because no pump is perfectly efficient.

Deepwell Booster Station: Dynamic Head and Energy Cost — Grade 12 Math

Name: ____________________________ Date: ______________

Problems

Deepwell Booster Station (a fictional station) pumps 1,200 gallons per minute (gpm) against a static lift of 180 feet. A gauge shows 25 psi of friction and fitting losses. The pump efficiency is 0.75. Use these relationships:

1 psi = 2.31 feet of head.
Water horsepower = (flow in gpm x total dynamic head in ft) / 3,960.
Brake horsepower = water horsepower / pump efficiency.
1 horsepower = 0.746 kilowatts.
Energy (kWh) = power (kW) x run time (hours). The electricity rate is a fictional 0.14 dollars per kilowatt-hour.

Convert the 25 psi friction loss to feet of head.
Find the total dynamic head, adding the static lift of 180 feet to the friction head from problem 1.
Find the water horsepower, using flow 1,200 gpm and your total dynamic head from problem 2.
Find the brake horsepower, using the pump efficiency of 0.75.
Convert the brake horsepower to kilowatts.
If the pump runs 18 hours per day, find the daily energy use in kilowatt-hours.
Find the daily electricity cost at the fictional rate of 0.14 dollars per kilowatt-hour.
The utility caps this pump at 1,000 kilowatt-hours per day. Write and solve an equation for the greatest number of hours per day the pump can run under that cap.
A new high-efficiency pump would raise the efficiency from 0.75 to 0.85 at the same flow and head. Find its brake horsepower, its kilowatts, and its daily cost at 18 hours, then state the daily savings compared with problem 7.

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

Answer Key

Answer: 57.75 ft.
25 psi x 2.31 ft per psi = 57.75 feet of head.
Answer: 237.75 ft.
Total dynamic head = static lift + friction head = 180 + 57.75 = 237.75 feet.
Answer: 72.05 hp.
Water horsepower = (1,200 gpm x 237.75 ft) / 3,960 = 285,300 / 3,960 = 72.05 water horsepower.
Answer: 96.06 hp.
Brake horsepower = 72.05 / 0.75 = 96.06 horsepower. The motor must supply more than the water horsepower because the pump is 75 percent efficient.
Answer: 71.66 kW.
96.06 hp x 0.746 kW per hp = 71.66 kilowatts.
Answer: 1,290 kWh.
Energy = 71.66 kW x 18 hours = 1,289.9, which rounds to about 1,290 kilowatt-hours.
Answer: 180.59 dollars.
Cost = 1,289.9 kWh x 0.14 dollars per kWh = 180.59 dollars per day.
Answer: 13.9 hours.
Let h be the run hours. Energy = 71.66 x h, and the cap gives 71.66h = 1,000. Solving, h = 1,000 / 71.66 = 13.955, which must be rounded down to 13.9 hours per day so that energy use stays at or under the 1,000 kWh cap.
Answer: 21.24 dollars saved per day.
Brake horsepower = 72.05 / 0.85 = 84.76 hp. In kilowatts, 84.76 x 0.746 = 63.23 kW. Daily energy = 63.23 x 18 = 1,138.2 kWh, and cost = 1,138.2 x 0.14 = 159.35 dollars. The savings are 180.59 - 159.35 = 21.24 dollars per day, because a more efficient pump turns less electricity into the same lift.

Differentiation

Support

Provide the four relationships on a labeled reference card and pre-compute the friction head in problem 1 so students can begin the horsepower chain from a known total dynamic head. Allow calculator use and accept answers rounded to one decimal place.

Extension

Ask students to find the electricity rate at which the high-efficiency pump would pay back a purchase price of 9,400 dollars within one year of 18-hour days, using the daily savings they found in problem 9.

ELL supports

Provide a cutaway diagram of a booster station labeling the static lift, the friction losses, the pump, and the motor. Offer a sentence frame for the horsepower steps: "Water horsepower measures ___, while brake horsepower measures ___, so brake horsepower is ___."

Extensions

Have students graph daily cost against pump efficiency for efficiencies from 0.60 to 0.90 at the fixed flow and head, and describe how the curve explains why utilities invest in efficient pumps even when they cost more to buy.