Math problem set Grade 10 Engineering & Design

Vireo Landing Filtration Plant: Sizing Basins and Filters

A grade 10 engineering math set that sizes a fictional plant's sedimentation basin and filters, checking detention time and filter loading against standard design ranges.

Grade: Grade 10
Subject: Engineering & Design
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: Items 1 through 4 carry consistent units through a basin-volume, gallon-conversion, and detention-time chain, the unit-guided multi-step problem solving this standard describes.

All materials for HSN-Q.A.1 →

HSG-MG.A.3 Common Core Math

Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost).

How this material aligns: Items 5 through 9 apply geometric measures of a basin and filters to check whether the design meets detention-time and loading-rate constraints, the design-under-constraint work this standard names.

All materials for HSG-MG.A.3 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-17.

Objectives

Students will calculate a rectangular sedimentation basin's volume in cubic feet and gallons.
Students will calculate detention time and filter loading rate and compare them to standard design ranges.
Students will explain how detention time and loading rate connect to a plant's turbidity performance.

Vocabulary

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.
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.

Also: sedimentation basin, filter loading rate

Teacher overview

The fictional Vireo Landing Water Treatment Plant treats river water in two design stages that this set examines: a sedimentation basin where heavy particles settle out, and sand filters that trap the finer particles that stay suspended. Engineers size both stages so water moves slowly enough to be treated but the plant still keeps up with demand. Students calculate the basin's volume and detention time, then the filters' combined surface area and loading rate, and check each result against the ranges plants normally target. A common misconception in this math is thinking a bigger basin is always better: too little detention time means particles do not settle, but a basin far larger than needed costs more to build and takes more land, so the design has to land inside a target range.

Vireo Landing Filtration Plant: Sizing Basins and Filters — Grade 10 Engineering & Design

Name: ____________________________ Date: ______________

Problems

The Vireo Landing plant (a fictional facility built for this set) runs a design flow of 2.0 MGD. Its rectangular sedimentation basin is 80 feet long, 25 feet wide, and holds water 12 feet deep. The plant has two identical sand filters, each 20 feet long and 15 feet wide.

Find the sedimentation basin's volume in cubic feet, using Volume = Length × Width × Depth.
Convert the basin volume to gallons, using the conversion that 1 cubic foot holds 7.48 gallons.
Convert the plant's design flow of 2.0 MGD to gallons per minute (gpm), using the conversion that 1 MGD equals 694 gpm.
Calculate the basin's detention time in hours. Convert the flow to gallons per day, divide the basin volume in gallons by it to get days, then multiply by 24 hours per day.
Water typically spends about 2 to 4 hours in a sedimentation basin. Does the Vireo Landing basin fall within that typical range at 2.0 MGD?
Find the total sand-filter surface area in square feet, adding the surface area of both filters.
Calculate the filter loading rate in gallons per minute per square foot, using loading rate = flow (gpm) ÷ filter surface area (sq ft), with the flow from item 3.
Conventional filters usually run at 2 to 6 gpm per square foot. Is the loading rate from item 7 within that range?
When one filter is taken offline for backwashing, all of the plant's flow passes through the single remaining filter. Recalculate the loading rate on that one filter, and state whether it is still inside the 2-to-6 range.
Explain how keeping detention time and filter loading inside their design ranges helps Vireo Landing meet the finished-water turbidity limit of 0.3 NTU that conventional plants must satisfy in most of their monthly samples.

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: 24,000 cubic feet.
Volume = 80 ft × 25 ft × 12 ft = 24,000 cubic feet.
Answer: 179,520 gallons.
24,000 cubic feet × 7.48 gallons/cubic foot = 179,520 gallons.
Answer: 1,388 gpm.
2.0 MGD × 694 gpm/MGD = 1,388 gpm.
Answer: about 2.15 hours.
Flow of 2.0 MGD is 2,000,000 gallons per day. Detention time = 179,520 ÷ 2,000,000 = 0.08976 days. Converting to hours: 0.08976 × 24 = 2.154, which rounds to about 2.15 hours.
Answer: Yes.
A detention time of about 2.15 hours falls inside the typical 2-to-4-hour range. The Ten States Standards and several state design rules call for at least 4 hours in a conventional basin, though, so a designer would check the state's rule before approving this basin.
Answer: 600 square feet.
Each filter is 20 ft × 15 ft = 300 square feet, and the two together are 2 × 300 = 600 square feet.
Answer: about 2.31 gpm per square foot.
Loading rate = 1,388 gpm ÷ 600 sq ft = 2.313, which rounds to about 2.31 gpm per square foot.
Answer: Yes.
A loading rate of about 2.31 gpm per square foot is inside the standard 2-to-6 gpm per square foot range, so the filters are not being overloaded at the design flow.
Answer: about 4.63 gpm per square foot.
With one filter offline, all 1,388 gpm passes through the remaining 300 square feet: 1,388 ÷ 300 = 4.627, which rounds to about 4.63 gpm per square foot. That is still inside the 2-to-6 range, so the plant can keep filtering during a backwash.
Answer: Accept any answer explaining that adequate detention time lets heavier particles settle out in the basin, and a loading rate inside the design range keeps water moving slowly enough for the sand to trap the finer particles instead of pushing them through. Both steps together lower the cloudiness of the finished water, helping the plant stay at or below the 0.3 NTU turbidity limit.

Differentiation

Support (below grade level)

Provide the basin-volume, detention-time, and loading-rate formulas on a reference card with the two constants (7.48 gallons per cubic foot and 694 gpm per MGD) written in, and work items 1 and 2 together so students carry the volume forward correctly before finding detention time.

Extension (above grade level)

Ask students to find the highest flow, in MGD, at which the Vireo Landing basin would still provide at least 2 hours of detention time, working backward from the detention-time formula, then explain what that upper limit means for the plant on a high-demand day.

ELL supports

Provide a labeled top-view diagram of the basin and the two filters with length, width, and depth marked, and a small glossary card for sedimentation, detention time, loading rate, and backwash. Allow students to present items 1 through 9 as numbered calculations and answer item 10 with a short sentence and a labeled sketch.