Worksheet Grade 10 Math

Cliffmere Transmission Main: Flow, Velocity, and Continuity

A grade 10 worksheet applying Q = A x V to a fictional transmission main: circle area, velocity checks against a design limit, and cfs, gpm, and MGD conversions.

Grade: Grade 10
Subject: Math
Material: Worksheet
Time: 40 min
Prep: No prep
Materials: Calculator

Standards

HSG-MG.A.1 Common Core Math

Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

How this material aligns: Items 1 and 7 model the pipe's cross-section as a circle and use its area, the geometric-shape modeling of a real object this standard describes.

All materials for HSG-MG.A.1 →

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 4 and 5 convert one flow across cfs, gpm, and MGD, carrying units through each step exactly as this standard's multi-step reasoning demands.

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 apply Q = A x V to relate flow, pipe area, and velocity.
Students will convert a single flow rate among cfs, gpm, and MGD.

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.

Also: continuity, velocity, cross-section

Teacher overview

Water moving through a pipe follows the continuity rule Q = A x V, where Q is flow, A is the pipe's cross-section area, and V is the water's velocity. Operators use this rule to check that a big main is not running too fast, because high velocity can wear pipes and cause pressure surges. This worksheet uses the fictional city of Cliffmere and its transmission main. Teaching note: keep flow in cubic feet per second and area in square feet so the velocity comes out in feet per second without a hidden conversion.

Cliffmere Transmission Main: Flow, Velocity, and Continuity — Grade 10 Math

Name: ____________________________ Date: ______________

Student Worksheet

The city of Cliffmere has one large transmission main. This scenario is fictional, and its numbers are set for this worksheet. Use Q = A x V and the circle-area rule A = 0.785 x D squared, with D in feet.

Main data:

Diameter: 24 inches, which is 2.0 feet
Design velocity limit: 5 feet per second
Conversions: 1 cfs = 448.8 gpm, and 1 MGD = 1.547 cfs

Find the cross-section area of the main in square feet.
The main carries 12.0 cfs. Find the water velocity in feet per second.
Is the velocity from item 2 within the 5 feet per second limit? By how much?
Convert 12.0 cfs to gpm.
Convert 12.0 cfs to MGD.
At peak hour the flow rises to 18.0 cfs. Find the new velocity, then state whether it passes the 5 feet per second limit.
To keep velocity at or below 5 feet per second while carrying 18.0 cfs, find the smallest area needed and the matching diameter in feet.
Explain why an operator would rather add a second pipe than let one main run well above the velocity limit.

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: 3.14 sq ft.
A = 0.785 x D squared = 0.785 x (2.0) squared = 0.785 x 4 = 3.14 square feet.
Answer: 3.82 ft/s.
V = Q / A = 12.0 cfs / 3.14 sq ft = 3.82 feet per second.
Answer: Yes, by 1.18 ft/s.
The velocity 3.82 is below the 5 feet per second limit, with room of 5 - 3.82 = 1.18 feet per second.
Answer: 5,386 gpm.
12.0 cfs x 448.8 gpm per cfs = 5,385.6, which rounds to 5,386 gpm.
Answer: 7.76 MGD.
12.0 cfs divided by 1.547 cfs per MGD = 7.757, which rounds to 7.76 MGD.
Answer: 5.73 ft/s.
V = 18.0 / 3.14 = 5.73 feet per second. That is above the 5 feet per second limit by 0.73, so it does not pass.
Answer: 2.14 ft.
Smallest area is A = Q / V = 18.0 / 5 = 3.6 square feet. Then D = square root of (3.6 / 0.785) = square root of 4.586 = 2.14 feet, about 26 inches.
Answer: A second pipe splits the flow, so each main stays below the velocity limit.
Running one main far above the limit speeds up pipe wear and raises the risk of pressure surges, while two pipes share the flow and keep velocity safe.

Differentiation

Support

Provide the circle-area rule A = 0.785 x D squared already worked for the 2.0-foot diameter, so students can focus on the Q = A x V steps. Give a conversion card listing 1 cfs = 448.8 gpm and 1 MGD = 1.547 cfs.

Extension

Have students find what diameter would hold velocity at or below 4 feet per second at the peak flow from item 6, then explain why a lower velocity target leads to a larger, more expensive pipe.

ELL supports

Pre-teach "velocity," "cross-section," and "flow rate" with a labeled drawing of a pipe end showing the circle area and an arrow for velocity. Provide a sentence frame for item 8: "A second pipe helps because ___."