Worksheet Grade 12 Math

Elderton Filtration Plant: Loading, Backwash, Detention

A grade 12 worksheet computing filter loading rates as a density, sedimentation-basin detention time, and a turbidity check for a fictional water treatment plant.

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

Standards

HSG-MG.A.2 Common Core Math

Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

How this material aligns: Items 3, 5, and 9 apply a density concept, gallons per minute per square foot of filter surface, to judge whether the plant's filters are loaded within the accepted operating range.

All materials for HSG-MG.A.2 →

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: Items 1, 6, and 7 carry units through a multi-step conversion from million gallons per day to gallons per minute and on to a detention time in minutes and hours.

All materials for HSN-Q.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 a rapid filter's loading rate in gpm per square foot and compare it to the operating range.
Students will calculate sedimentation-basin detention time and check a finished-water turbidity result against the rule.

Vocabulary

Backwash: Backwash is when a treatment plant reverses the flow of water through a filter to flush out the dirt and particles it has trapped. It is like rinsing out a dirty sponge so the filter can keep working well.
Detention time: Detention time is how long water stays inside a tank or basin before flowing out. Water must stay long enough for each treatment step, like settling or disinfecting, to finish its job.
Turbidity: Turbidity is a measure of how cloudy or murky water looks because of tiny particles floating in it. Clear water has low turbidity, and muddy water has high turbidity.

Also: filter loading rate

Teacher overview

Elderton Filtration Plant is a fictional conventional treatment plant that runs several rapid sand filters and one sedimentation basin. Operators judge a filter by its loading rate, the flow it treats for each square foot of surface, and they judge a basin by its detention time, the average time water spends settling. This worksheet has students compute both, then check a finished-water turbidity result against the drinking-water rule. A frequent error is dividing flow by the number of filters instead of by the total filter area; the loading rate is a density per square foot, not a share per filter.

Elderton Filtration Plant: Loading, Backwash, Detention — Grade 12 Math

Name: ____________________________ Date: ______________

Student Worksheet

Elderton Filtration Plant (a fictional plant) treats 4.0 million gallons per day (MGD) through 4 identical rapid filters, each 20 feet by 25 feet. Its rectangular sedimentation basin measures 100 feet long, 30 feet wide, and 15 feet deep. Use these relationships:

1 MGD = 694 gallons per minute (gpm).
1 cubic foot = 7.48 gallons.
Filter loading rate = flow (gpm) / total filter surface area (sq ft). Conventional filters typically run 2 to 6 gpm per square foot.
Detention time (minutes) = basin volume (gallons) / flow (gpm). Sedimentation basins are typically designed for 2 to 4 hours.

Convert the plant flow of 4.0 MGD to gallons per minute.
Find the total filter surface area, adding up all 4 filters.
Find the filter loading rate in gpm per square foot, using your answers from items 1 and 2.
Compare your loading rate to the typical range of 2 to 6 gpm per square foot. Is it within, below, or above that range?
One filter is taken offline for backwash. Recompute the loading rate carried by the 3 filters still in service.
Find the sedimentation basin's volume in gallons, using its length, width, and depth.
Find the basin's detention time in minutes, then convert it to hours. Is it within the typical 2-to-4-hour design range?
A finished-water sample from Elderton reads 0.18 NTU. The rule requires turbidity at or below 0.3 NTU in most samples and never above 1 NTU. Does this sample comply?
During peak demand the plant flow rises to 6.0 MGD. Recompute the loading rate with all 4 filters in service.
Explain why taking a filter offline for backwash raises the loading rate on the remaining filters, and why operators avoid backwashing during peak demand.

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: 2,776 gpm.
4.0 MGD x 694 gpm per MGD = 2,776 gallons per minute.
Answer: 2,000 sq ft.
Each filter is 20 ft x 25 ft = 500 square feet. Four filters give 4 x 500 = 2,000 square feet.
Answer: 1.388 gpm per sq ft.
Loading rate = 2,776 gpm / 2,000 sq ft = 1.388 gallons per minute per square foot.
Answer: Below the range.
1.388 is less than the low end of the typical 2-to-6 range, so Elderton's filters are lightly loaded at average flow.
Answer: 1.85 gpm per sq ft.
With one filter offline, the surface area is 3 x 500 = 1,500 sq ft. Loading rate = 2,776 gpm / 1,500 sq ft = 1.85 gallons per minute per square foot.
Answer: 336,600 gallons.
Volume = 100 ft x 30 ft x 15 ft = 45,000 cubic feet. In gallons, 45,000 x 7.48 = 336,600 gallons.
Answer: 121 minutes.
Detention time = 336,600 gallons / 2,776 gpm = 121.3 minutes. Dividing by 60 gives about 2.0 hours, which is within the typical 2-to-4-hour range.
Answer: Yes, it complies.
0.18 NTU is below the 0.3 NTU limit that applies to most samples and well below the 1 NTU ceiling, so the sample meets the rule.
Answer: 2.08 gpm per sq ft.
At 6.0 MGD, flow = 6.0 x 694 = 4,164 gpm. Loading rate = 4,164 gpm / 2,000 sq ft = 2.08 gallons per minute per square foot, now just inside the typical range.
Answer: Fewer filters must carry the same total flow.
The plant still treats the full flow when one filter is backwashing, so that flow is divided over less surface area, which raises the loading rate on each remaining filter. During peak demand the flow is already high, so backwashing then would push the loading rate higher still, which is why operators schedule backwash for low-demand hours.

Differentiation

Support

Provide the loading-rate and detention-time formulas already labeled with their units, and pre-compute the plant flow in gpm so students can start from a known value. Give the filter dimensions as a small table rather than in sentence form.

Extension

Ask students to find the plant flow, in MGD, at which all four filters would reach the top of the typical loading range, and to explain what that flow would mean for how the plant is operated during a hot, dry week.

ELL supports

Provide a labeled plan-view diagram of the four filters and the sedimentation basin, glossing "loading rate," "backwash," "detention time," and "turbidity." Offer a sentence frame for item 10: "When one filter is offline, the same flow goes through ___ area, so the loading rate ___."

Extensions

Have students contact a local water utility or read its annual water-quality report, find the reported average turbidity, and compare it with the rule's limits and with Elderton's sample result.