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Math problem set Grade 12 Math

Fallowfield Water Utility: Revenue and an Inclining Block Rate

A grade 12 problem set on utility finance, covering a fictional utility's revenue need with a fixed charge and a three-tier inclining block rate.

Objectives

  1. Students will split a utility's revenue requirement between fixed charges and a tiered volumetric rate.
  2. Students will write bill equations for each rate tier and test a household bill against an affordability threshold.

Vocabulary

Revenue requirement
The total amount of money a utility must collect from customers each year to run the water system safely. Rates are set so that all the bills together add up to this amount.
Fixed charge
The part of a water bill that stays the same every month no matter how much water a family uses. It helps pay for the pipes, meters, and workers the utility needs even when no water is flowing.

Also: inclining block rate, affordability

Teacher overview

Fallowfield Water Utility is a fictional utility that must set rates to cover a fixed yearly revenue requirement while encouraging conservation. It uses a fixed monthly charge plus an inclining block rate, where each additional block of water costs more per gallon than the last. This set has students split the revenue need between fixed and volumetric charges, write a bill equation for each tier, and check a household's bill against the affordability threshold. A common error with tiered rates is applying the highest tier's price to every gallon used; each tier's price applies only to the gallons that fall inside that tier.

Fallowfield Water Utility: Revenue and an Inclining Block Rate — Grade 12 Math
Name: ____________________________ Date: ______________

Problems

Fallowfield Water Utility (a fictional utility) needs 4,200,000 dollars in revenue this year. It serves 9,000 accounts and charges each a fixed 18.00 dollars per month. Its inclining block rate, per 1,000 gallons, is 5.00 dollars for the first 4,000 gallons, 7.50 dollars for gallons from 4,001 to 10,000, and 11.00 dollars for anything above 10,000 gallons. Let g stand for a household's monthly use in thousands of gallons.

  1. Find the utility's total yearly revenue from the fixed monthly charge across all 9,000 accounts.
  2. Subtract the fixed revenue from the 4,200,000-dollar requirement to find how much must come from volumetric charges.
  3. A sample household uses 8,000 gallons in a month. Find its volumetric charge, applying each tier's price only to the gallons inside that tier.
  4. Find that household's total monthly bill, including the fixed charge.
  5. For a household that stays within the first tier (g at most 4), write an equation for the monthly bill B in dollars, then use it to find the bill at g = 3.
  6. For usage in the second tier (g between 4 and 10), the bill is B = 38 + 7.50(g - 4). Use it to check the bill at g = 8 against your answer from problem 4.
  7. A household earns 52,000 dollars a year. The affordability threshold is about 4.5 percent of household income. Write and solve for the yearly water-and-sewer dollar amount at that threshold.
  8. The sample household's water bill is 68 dollars a month. If its sewer bill equals its water bill, find the combined yearly cost and state whether it is above or below the threshold from problem 7.
  9. Explain why an inclining block rate can both raise the needed revenue and encourage households to conserve, compared with a single flat price per gallon.

From Water for Schools (waterforschools.org). This material is in the public domain (CC0) and may be copied, edited, and shared freely.

Answer Key

  1. Answer: 1,944,000 dollars.
    Fixed revenue = 18.00 dollars x 12 months x 9,000 accounts = 216 x 9,000 = 1,944,000 dollars per year.

  2. Answer: 2,256,000 dollars.
    Volumetric revenue needed = 4,200,000 - 1,944,000 = 2,256,000 dollars.

  3. Answer: 50.00 dollars.
    First tier: 4,000 gallons at 5.00 dollars per 1,000 = 20.00 dollars. Second tier: the remaining 4,000 gallons at 7.50 dollars per 1,000 = 30.00 dollars. Total volumetric charge = 20.00 + 30.00 = 50.00 dollars.

  4. Answer: 68.00 dollars.
    Total bill = fixed charge + volumetric charge = 18.00 + 50.00 = 68.00 dollars for the month.

  5. Answer: 33.00 dollars.
    Within the first tier the bill is B = 18 + 5g. At g = 3, B = 18 + 5(3) = 18 + 15 = 33.00 dollars.

  6. Answer: 68.00 dollars.
    B = 38 + 7.50(g - 4). At g = 8, B = 38 + 7.50(4) = 38 + 30 = 68.00 dollars, which matches the household bill from problem 4.

  7. Answer: 2,340 dollars.
    Let T be the threshold amount. T = 0.045 x 52,000 = 2,340 dollars per year of combined water and sewer cost.

  8. Answer: Below the threshold.
    Water is 68 dollars x 12 = 816 dollars a year. With sewer equal to water, the combined cost is 1,632 dollars a year, which is below the 2,340-dollar threshold.

  9. Answer: Higher blocks raise revenue while pricing heavy use up.
    Because each added block costs more per gallon, households that use a lot pay a rising price, which both brings in the revenue the utility needs and gives every household a reason to cut back on the most expensive gallons. A single flat price would collect the same amount only by charging light users more and would give no extra signal to reduce heavy use.

Differentiation

Support

Provide a filled-in tier table showing the price and gallon range for each block, and pre-compute problem 1 so students can begin the revenue split from a known fixed-revenue figure. Allow calculators throughout.

Extension

Ask students to find the average monthly use per account that would exactly close the volumetric revenue gap from problem 2 if every gallon were billed at the first-tier price of 5.00 dollars per 1,000 gallons, then to compare the result with the 4,000-gallon limit of the first tier and discuss why real utilities cannot count on first-tier sales alone.

ELL supports

Provide a labeled bar diagram showing the three tiers as stacked blocks of increasing price, glossing "fixed charge," "volumetric," "tier," and "affordability." Offer a sentence frame for problem 9: "An inclining block rate raises revenue because ___ and encourages conservation because ___."

Extensions

Have students find their own utility's rate sheet, identify whether it uses a flat or a tiered volumetric rate, and compute a sample monthly bill for a household using 8,000 gallons under that real structure.