Quiz & test questions Grade 8 Engineering & Design

Amberlake's New Elevated Tank: A Grade 8 Volume and Cost Assessment

A grade 8 assessment on a fictional water authority's elevated tank, testing cylinder volume, scientific notation, and a systems-of-equations cost comparison.

Grade: Grade 8
Subject: Engineering & Design
Material: Quiz & test questions
Time: 45 min
Prep: No prep
Materials: calculator

Standards

8.G.C.9 Common Core Math

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

How this material aligns: Items 2 through 5 have students apply the cylinder volume formula to find the elevated tank's capacity in cubic feet and gallons and explain the formula's constant, the real-world volume application this standard targets.

All materials for 8.G.C.9 →

8.EE.C.8 Common Core Math

Analyze and solve pairs of simultaneous linear equations, understanding that solutions correspond to points of intersection, and solving systems by graphing and algebraically in the context of real-world problems.

How this material aligns: Items 6 through 9 have students write and solve a system of linear equations comparing two construction bids and evaluate the results against a cost criterion, the flagship task this standard describes.

All materials for 8.EE.C.8 →

MS-ETS1-2 NGSS

Evaluate competing design solutions using a systematic process to determine how well they meet the criteria and constraints of the problem.

How this material aligns: Items 6 through 9 compare the steel and concrete bids at the break-even capacity and at 440,000 gallons, then judge each against Amberlake's stated $900,000 cost criterion, the systematic evaluation of competing solutions this standard describes; the sizing items 1 through 5 are not part of this alignment.

All materials for MS-ETS1-2 →

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This material is in the public domain (CC0) and may be copied, edited, and shared freely, without attribution. Updated 2026-07-16.

Objectives

Students will calculate the volume and gallon capacity of a cylindrical elevated tank.
Students will express a large gallon capacity in scientific notation.
Students will write and solve a system of linear equations to compare two construction bid options and evaluate them against a cost criterion.

Vocabulary

Key words: cylinder volume, scientific notation, system of equations, break-even point, criterion

Teacher overview

This 45-minute assessment (26 points total) checks whether students can apply cylinder volume, scientific notation, and systems of linear equations to a single engineering scenario: a new elevated water tank proposed for the fictional Amberlake Regional Water Authority. It mixes two multiple-choice items, two short-answer items, and five computation items, moving from sizing the tank itself to comparing two competing construction bids against a stated cost criterion.

Amberlake's New Elevated Tank: A Grade 8 Volume and Cost Assessment — Grade 8 Engineering & Design

Name: ____________________________ Date: ______________

Assessment

The fictional Amberlake Regional Water Authority is planning a new cylindrical elevated tank with a diameter of 50 feet and a depth of 30 feet.

(2 points) Which expression correctly represents the cylinder volume formula used for this tank, where D is the diameter and depth is the tank's height?
- A) 0.785 × D × depth
- B) 0.785 × D² × depth
- C) D² × depth
- D) (4/3) × 3.14 × D³
(4 points) Using Volume (cu ft) = 0.785 × Diameter² × depth, calculate the tank's volume in cubic feet.
(3 points) Convert your answer from item 2 to gallons, using the conversion 1 cubic foot = 7.48 gallons.
(3 points) Express the tank's capacity in gallons, from item 3, in scientific notation, rounded to three significant figures.
(2 points) Short answer: In your own words, explain what the constant 0.785 represents in the cylinder volume formula.
(4 points) Two construction bids are being compared for the new tank. The steel-tank bid has a base cost of $310,000 plus $1.20 for every gallon of capacity. The concrete-tank bid has a base cost of $250,000 plus $1.50 for every gallon of capacity. Set up a system of equations and solve for the tank capacity, in gallons, at which both bids cost the same amount.
(3 points) Using a rounded tank capacity of 440,000 gallons, calculate the total cost of the steel bid and the total cost of the concrete bid, and state which bid costs less.
(2 points) Which statement is best supported by the cost data in items 6 and 7?
- A) The concrete bid is always the cheaper choice, no matter the tank's capacity.
- B) Above 200,000 gallons of capacity, the steel bid becomes the cheaper choice.
- C) The two bids never cost the same amount at any capacity.
- D) Tank capacity does not affect either bid's total cost.
(3 points) Short answer: Amberlake's engineering criteria require the chosen design to cost no more than $900,000 for a 440,000-gallon tank. Using your answers from items 6 and 7, which bid, if either, meets this criterion? Explain using the numbers you calculated.

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: B) 0.785 × D² × depth. This is the cylinder volume formula used throughout this assessment, where the diameter is squared and multiplied by 0.785 (an approximation of pi divided by 4) and then by the tank's depth.
Answer: 58,875 cubic feet. 0.785 × 50² × 30 = 0.785 × 2,500 × 30 = 0.785 × 75,000 = 58,875 cubic feet.
Answer: 440,385 gallons. 58,875 × 7.48 = 440,385 gallons.
Answer: 4.40 × 10^5 gallons. 440,385 rounds to three significant figures as 440,000, which is written in scientific notation as 4.40 × 10^5.
Answer: Accept any answer that identifies 0.785 as an approximation of pi divided by 4 (π ÷ 4), which converts a diameter-squared measurement into the same result that using the radius in the standard formula πr² would give.
Answer: 200,000 gallons. 310,000 + 1.2x = 250,000 + 1.5x. Subtract 1.2x from both sides: 310,000 = 250,000 + 0.3x. Subtract 250,000 from both sides: 60,000 = 0.3x. Divide both sides by 0.3: x = 200,000.
Answer: The steel bid is cheaper, by $72,000. Steel: 310,000 + 1.2(440,000) = 310,000 + 528,000 = 838,000. Concrete: 250,000 + 1.5(440,000) = 250,000 + 660,000 = 910,000. 910,000 − 838,000 = 72,000.
Answer: B) Above 200,000 gallons of capacity, the steel bid becomes the cheaper choice. The break-even point found in item 6 is 200,000 gallons, and at the higher 440,000-gallon capacity tested in item 7, the steel bid was cheaper, confirming this statement.
Answer: Only the steel bid meets the criterion. The steel bid's total cost of $838,000 (from item 7) is under the $900,000 limit, while the concrete bid's total cost of $910,000 exceeds it.

Differentiation

Support

Provide the cylinder volume formula and the gallon conversion factor printed on a reference card taped to the desk, and give students the system of equations from item 6 already written so they only need to solve it. Allow a calculator for every item.

Extension

Ask students to determine what base cost the concrete bid would need, keeping its $1.50 per-gallon rate the same, for it to break even with the steel bid at exactly 300,000 gallons of capacity instead of 200,000.

ELL supports

Provide a labeled diagram of the cylindrical tank showing diameter and depth as distinct measurements, and a simple two-column chart showing "base cost" and "cost per gallon" for each bid before item 6. Allow item 5's and item 9's short answers to be written as a single sentence using a provided sentence frame.

Extensions

Have students calculate, using only the figures given in this assessment, how the break-even capacity from item 6 would change if the steel bid's per-gallon rate dropped from $1.20 to $1.00, and explain in writing whether that change makes the steel bid the better choice at a smaller or larger range of tank capacities than before.