Worksheet Grade 9 Math

Draymoor Transmission Main: A Pipe Velocity and Flow Worksheet

A grade 9 worksheet using the continuity equation Q = A x V to find pipe velocity and flow in a fictional water district's transmission mains.

Grade: Grade 9
Subject: Math
Material: Worksheet
Time: 40 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: Students carry units through the multi-step conversions from cubic feet per second to gallons per minute and million gallons per day in items 4 and 5, choosing consistent units in the continuity formula.

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: Items 7 and 9 ask students to write a one-variable equation from Q = A x V and solve it for an unknown flow or velocity, the core create-and-solve skill this standard names.

All materials for HSA-CED.A.1 →

Download Word file

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 use the continuity equation Q = A x V to find velocity, flow, or pipe area.
Students will convert a flow between cubic feet per second, gallons per minute, and million gallons per day.

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 equation, velocity, cross-sectional area

Teacher overview

The Draymoor Water District (a fictional utility used for this worksheet) moves treated water through large transmission mains before it reaches neighborhood pipes. Operators watch the velocity in those mains, because water that moves too slowly lets sediment settle, while water that moves too fast wears the pipe walls. This worksheet uses one formula, the continuity equation Q = A x V, to connect pipe size, flow, and velocity. Watch for the common error of using diameter in place of the cross-sectional area; the area must be found first with A = 0.785 x D squared.

Draymoor Transmission Main: A Pipe Velocity and Flow Worksheet — Grade 9 Math

Name: ____________________________ Date: ______________

Student Worksheet

Draymoor runs a main transmission line that is 2.0 feet in diameter. Use these rules for every problem. Round to two decimal places unless told otherwise.

Cross-sectional area of a round pipe: A = 0.785 x D squared (D in feet, A in square feet).
Continuity equation: Q = A x V (Q in cubic feet per second, A in square feet, V in feet per second).
1 cubic foot per second (cfs) = 448.8 gallons per minute (gpm).
1 million gallons per day (MGD) = 694 gpm.
Operators want velocity between 2 and 5 feet per second in this main.

Find the cross-sectional area of the 2.0-foot main using A = 0.785 x D squared.
The district measures a flow of 12 cfs in this main. Find the velocity in feet per second.
Does the velocity from item 2 fall inside the 2-to-5 feet-per-second range operators prefer? Explain in one sentence.
Convert the 12 cfs flow to gallons per minute.
Convert the flow from item 4 to million gallons per day (MGD). Round to two decimals.
A parallel main is 1.5 feet in diameter. Find its cross-sectional area.
Operators want at least the 2 feet-per-second minimum velocity in the 1.5-foot main. Write the equation Q = A x V for this pipe, then solve for the smallest flow in cfs that reaches 2 feet per second.
Suppose the flow stays the same but the pipe diameter is doubled. Explain what happens to the velocity, and why.
Write an equation for velocity V as a function of flow Q in the 2.0-foot main, using your area from item 1. Then find V when Q = 8 cfs.
During peak demand the district raises the flow in the 2.0-foot main to 18 cfs. Find the new velocity, and state whether it stays under the 5 feet-per-second 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 (2.0) squared = 0.785 x 4 = 3.14 square feet.
Answer: 3.82 ft/s. Rearrange Q = A x V to V = Q / A = 12 / 3.14 = 3.82 feet per second.
Answer: Yes. A velocity of 3.82 feet per second sits between the 2 and 5 feet-per-second limits, so the main is neither too slow (sediment settling) nor too fast (pipe wear).
Answer: 5,385.6 gpm. 12 cfs x 448.8 gpm per cfs = 5,385.6 gallons per minute.
Answer: 7.76 MGD. 5,385.6 gpm / 694 gpm per MGD = 7.76 million gallons per day. (Checking with 12 cfs / 1.547 cfs per MGD also gives 7.76 MGD.)
Answer: 1.77 sq ft. A = 0.785 x (1.5) squared = 0.785 x 2.25 = 1.766, which rounds to 1.77 square feet.
Answer: 3.53 cfs. Q = A x V = 1.766 x 2 = 3.53 cubic feet per second is the smallest flow that reaches the 2 feet-per-second minimum. (Using the rounded area of 1.77 square feet gives 3.54 cfs, which is also acceptable.)
Answer: The velocity drops to one-fourth. Area depends on the square of the diameter, so doubling the diameter multiplies the area by four. With the same flow spread over four times the area, velocity falls to one-fourth of its old value.
Answer: 2.55 ft/s. The equation is V = Q / 3.14. At Q = 8 cfs, V = 8 / 3.14 = 2.55 feet per second.
Answer: 5.73 ft/s. V = 18 / 3.14 = 5.73 feet per second, which is above the 5 feet-per-second limit, so the flow is too fast for this main.

Differentiation

Support

Give students the area value for the 2.0-foot main (3.14 square feet) already computed, and provide a printed reference card that shows the three rearranged forms of Q = A x V (solve for Q, for V, and for A). Allow unrounded calculator answers.

Extension

Ask students to find the diameter (in feet) that would carry the peak flow from item 10 at exactly the 5 feet-per-second limit, working backward from A = Q / V and then D = square root of (A / 0.785).

ELL supports

Provide a labeled cross-section diagram of a round pipe showing diameter, radius, and area as separate ideas. Pre-teach the words "velocity," "flow," and "area" with a simple picture of water moving through a pipe, and post the unit abbreviations (cfs, gpm, MGD) with their full names beside the worksheet.