Math problem set Grade 8 Engineering & Design
Two Treatment Plant Bids for Thistledown: A Systems of Equations Set
A grade 8 problem set where students write and solve a system of linear equations to compare two water treatment plant bids and test them against a cost criterion.
Objectives
- Students will write a linear cost equation for each of two competing treatment plant proposals.
- Students will solve a system of two linear equations to find the treated-water volume where both proposals cost the same.
- Students will use the solved system to evaluate which proposal better meets a stated engineering cost criterion at different usage levels.
Vocabulary
Key words: system of equations, break-even point, capital cost, operating cost, criterion
Teacher overview
The fictional town of Thistledown needs a new drinking-water treatment plant and has received two construction bids from competing engineering firms. Riverbend Engineering proposes a plant with a higher construction cost but a lower cost per gallon treated afterward; Bluepoint Engineering proposes a cheaper plant to build but a higher cost for every gallon it treats. This problem set has students write each proposal as a linear cost equation and solve the resulting system to find the exact point where the two bids cost the same, the same reasoning an engineering review board uses when comparing competing infrastructure proposals.
A common misconception in this math is assuming the proposal with the lower construction cost is automatically the cheaper long-term choice. Watch for students who answer "Bluepoint" for every question in this set simply because its construction cost is lower; the system of equations shows that which bid is cheaper depends entirely on how much water the plant actually treats each year.
Every figure in this scenario is fictional and chosen to keep the arithmetic manageable.
Problems
Thistledown's review board is comparing two fictional construction bids for its new treatment plant. Use these figures, stated in the bids, for all problems. Let x represent the number of thousands of gallons treated in a year.
- Riverbend Engineering: a yearly construction-loan payment of $180,000, plus a treatment cost of $2.50 for every 1,000 gallons treated.
- Bluepoint Engineering: a yearly construction-loan payment of $120,000, plus a treatment cost of $4.00 for every 1,000 gallons treated.
- Write an equation for Riverbend's total annual cost, C(x), in terms of x.
- Write an equation for Bluepoint's total annual cost, C(x), in terms of x.
- Solve the system of equations algebraically to find the value of x where the two bids cost the same amount.
- State the break-even point as an ordered pair (thousands of gallons treated, total cost in dollars), and explain in one sentence what this point means for Thistledown's decision.
- Thistledown currently treats about 25,000 thousand gallons of water per year. Calculate each bid's total cost at this volume and state which bid costs less.
- Town planners project that Thistledown's treated volume will grow to 90,000 thousand gallons per year once new housing is built. Calculate each bid's total cost at this volume, state which bid costs less, and by how much.
- The review board's written criterion requires the chosen bid to cost no more than $250,000 per year at a projected volume of 30,000 thousand gallons. Calculate each bid's cost at this volume and state which bid, if either, meets the criterion.
- Challenge. Thistledown's treated volume is expected to keep growing every year toward the 90,000 thousand-gallon projection in problem 6. Using your answers from problems 5 through 7, explain in two or three sentences why the review board should not choose a bid based only on which one is cheaper today.
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: C(x) = 180,000 + 2.5x.
The $180,000 yearly construction payment is the fixed starting value, and $2.50 is charged for every one of the x thousands of gallons treated.Answer: C(x) = 120,000 + 4x.
The $120,000 yearly construction payment is the fixed starting value, and $4.00 is charged for every one of the x thousands of gallons treated.Answer: 40,000 thousand gallons.
Set the two equations equal: 180,000 + 2.5x = 120,000 + 4x. Subtract 2.5x from both sides: 180,000 = 120,000 + 1.5x. Subtract 120,000 from both sides: 60,000 = 1.5x. Divide both sides by 1.5: x = 40,000.Answer: (40,000, $280,000).
At x = 40,000: Riverbend's cost is 180,000 + 2.5(40,000) = 180,000 + 100,000 = 280,000. Bluepoint's cost is 120,000 + 4(40,000) = 120,000 + 160,000 = 280,000. This point means that at 40,000 thousand gallons treated per year, both bids cost exactly $280,000, so below that volume Bluepoint is cheaper and above it Riverbend is cheaper.Answer: Bluepoint is cheaper, by $22,500.
Riverbend: 180,000 + 2.5(25,000) = 180,000 + 62,500 = 242,500. Bluepoint: 120,000 + 4(25,000) = 120,000 + 100,000 = 220,000. 242,500 − 220,000 = 22,500.Answer: Riverbend is cheaper, by $75,000.
Riverbend: 180,000 + 2.5(90,000) = 180,000 + 225,000 = 405,000. Bluepoint: 120,000 + 4(90,000) = 120,000 + 360,000 = 480,000. 480,000 − 405,000 = 75,000.Answer: Only Bluepoint meets the criterion.
Riverbend: 180,000 + 2.5(30,000) = 180,000 + 75,000 = 255,000, which exceeds $250,000. Bluepoint: 120,000 + 4(30,000) = 120,000 + 120,000 = 240,000, which is under $250,000.Answer: Accept any well-reasoned response that references the break-even point. A strong answer notes that Bluepoint looks cheaper at today's volume and even meets the board's $250,000 criterion at 30,000 thousand gallons, but the system of equations shows the two bids cross at 40,000 thousand gallons. Since Thistledown's volume is projected to grow past that point toward 90,000 thousand gallons, choosing Bluepoint for its lower cost today would commit the town to a bid that becomes the more expensive option, by $75,000 a year, once the town grows as expected.
Differentiation
Support
Provide both cost equations already written on the page, and give students a table with the columns "x," "Riverbend cost," and "Bluepoint cost" pre-labeled so they only need to substitute and compute for problems 5 through 7. Allow a calculator for every problem.
Extension
Ask students to introduce a third, hypothetical bid from a fictional firm called Cedarline Engineering with a yearly construction-loan payment of $150,000 and a treatment cost of $3.00 per 1,000 gallons, then determine the volume range where Cedarline's bid is the cheapest of all three options.
ELL supports
Provide a labeled diagram showing "construction cost" as a starting bar and "treatment cost" as a repeating stack added on top, so the fixed-cost-plus-rate structure of each equation is visible before students write the equations symbolically. Allow the explanation in problems 4 and 8 to be written as short bullet points rather than full paragraphs.
Extensions
Have students calculate how the break-even volume would change if Riverbend lowered its treatment cost from $2.50 to $2.00 per 1,000 gallons, and have them explain in writing whether that change would make Riverbend the better choice for Thistledown sooner or later than at 40,000 thousand gallons.