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Math problem set Grade 12 Math

Halbrook Water: Uniform Versus Inclining Block Rate Break-Evens

A grade 12 set comparing a uniform water rate with a three-block piecewise rate: two break-evens, interval answers, and a year of seasonal bills.

Objectives

  1. Students will model an inclining block water rate as a piecewise linear function and solve it against a uniform rate piece by piece, rejecting solutions that fall outside a piece's domain.
  2. Students will state, in interval notation, the usage ranges over which each plan is cheaper and confirm them on a graph.
  3. Students will show with a year of monthly usage data why a block-rate bill evaluated at average usage misstates the annual cost.

Vocabulary

Key words: inclining block rate, piecewise function, break-even, marginal price, interval notation

Teacher overview

The fictional Halbrook Municipal Water utility offers residential customers a choice of two rate plans. Plan A charges a small base fee and one uniform price for every 1,000 gallons, while Plan B, the Conservation Block plan, charges a larger base fee and three prices that rise as monthly use passes 8,000 and then 15,000 gallons. Because Plan B is a piecewise function, its cost line can cross Plan A's line more than once, so students solve the system piece by piece, keep only the solutions that fall inside each piece's domain, and report the answer as intervals. The set ends with one household's twelve months of usage, which shows why the choice between the plans cannot be made from average use alone. A common error with block rates is evaluating the bill once at average monthly use and multiplying by 12; because the higher block prices fall on the heaviest months, that shortcut understates the annual cost of the block plan.

Halbrook Water: Uniform Versus Inclining Block Rate Break-Evens — Grade 12 Math
Name: ____________________________ Date: ______________

Problems

Let x be a household's monthly use in thousands of gallons, and let A(x) and B(x) be the monthly bills in dollars. Use these two fictional Halbrook rate plans for every problem.

Plan Base fee per month Price per 1,000 gallons
Plan A, Uniform Rate $12.00 $6.00 for every 1,000 gallons
Plan B, Conservation Block $27.00 $3.50 for the first 8,000 gallons; $7.00 for use above 8,000 up to 15,000 gallons; $11.00 for use above 15,000 gallons

Each Plan B price applies only to the gallons that fall inside its block.

  1. Write A(x). Then write B(x) as a piecewise function with three pieces, each in the form mx + b, with its domain stated. Show that the pieces agree at x = 8 and at x = 15.

  2. Find each plan's monthly bill at 4,000, 10,000, and 18,000 gallons, and state which plan is cheaper at each usage and by how much.

  3. Solve A(x) = B(x) separately on each of Plan B's three pieces. For each solution, decide whether it lies inside that piece's domain, reject any that do not, and give every valid break-even usage with its bill.

  4. Let D(x) = B(x) − A(x). Write D(x) as a piecewise function, then use it to state, in interval notation, the usages for which Plan B is cheaper and the usages for which Plan A is cheaper.

  5. Graph both plans on one set of axes for 0 ≤ x ≤ 20, with labeled and scaled axes. Mark the two corners of Plan B's graph and every crossing point, and explain how the graph confirms your intervals from problem 4.

  6. The marginal price is what the next 1,000 gallons adds to the bill. Give each plan's marginal price for a household already using 4,000 gallons, 10,000 gallons, and 18,000 gallons. At which of these usages does Plan B send a stronger signal to conserve than Plan A?

One Halbrook household recorded its use for a full year.

Month Jan Feb Mar Apr May Jun Jul Aug Sep Oct Nov Dec
Use (thousands of gallons) 6 5 6 7 9 14 19 20 15 8 6 5

  1. Find the household's average monthly use. A neighbor estimates each plan's yearly cost by finding the bill at that average use and multiplying by 12. What yearly cost does this shortcut give for each plan, and which plan does it recommend?

  2. Find the household's actual yearly cost under each plan by billing each month separately. Which plan is actually cheaper for this household, and by how much?

  3. Explain why the shortcut in problem 7 gives the correct yearly cost for Plan A but not for Plan B. Use the monthly bills and the slopes of Plan B's pieces in your explanation.

  4. The Halbrook board wants Plan B to stop being cheaper than Plan A at 12,000 gallons instead of 13,000 gallons, and it will change only the price of the second block (use above 8,000 up to 15,000 gallons). Find the new second-block price, then check that the change does not create a valid break-even on the third piece.

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: A(x) = 12 + 6x; B(x) = 27 + 3.5x for 0 ≤ x ≤ 8, B(x) = 7x − 1 for 8 < x ≤ 15, and B(x) = 11x − 61 for x > 15.
    Plan A is the $12 base plus $6 per 1,000 gallons, so A(x) = 12 + 6x. On the first block, B(x) = 27 + 3.5x, so B(8) = 27 + 28 = 55. On the second block, the bill is the $55 already reached at 8,000 gallons plus $7 for each 1,000 gallons above 8: B(x) = 55 + 7(x − 8) = 55 + 7x − 56 = 7x − 1. At x = 8 this gives 7(8) − 1 = 55, which matches the first piece. At x = 15, B(15) = 7(15) − 1 = 104. On the third block, B(x) = 104 + 11(x − 15) = 104 + 11x − 165 = 11x − 61. At x = 15 this gives 11(15) − 61 = 165 − 61 = 104, which matches the second piece, so the graph of B has no jumps.

  2. Answer: At 4,000 gallons, A = $36 and B = $41 (Plan A cheaper by $5); at 10,000 gallons, A = $72 and B = $69 (Plan B cheaper by $3); at 18,000 gallons, A = $120 and B = $137 (Plan A cheaper by $17).
    At x = 4: A(4) = 12 + 6(4) = 12 + 24 = 36, and x = 4 is on the first piece, so B(4) = 27 + 3.5(4) = 27 + 14 = 41. The difference is 41 − 36 = 5. At x = 10: A(10) = 12 + 60 = 72, and x = 10 is on the second piece, so B(10) = 7(10) − 1 = 69. The difference is 72 − 69 = 3. At x = 18: A(18) = 12 + 108 = 120, and x = 18 is on the third piece, so B(18) = 11(18) − 61 = 198 − 61 = 137. The difference is 137 − 120 = 17. The cheaper plan changes twice across these three usages, which is a sign that the two graphs cross twice.

  3. Answer: Two valid break-evens: 6,000 gallons at $48 and 13,000 gallons at $90. The third-piece solution, x = 14.6, is rejected.
    First piece (0 ≤ x ≤ 8): 12 + 6x = 27 + 3.5x, so 2.5x = 15 and x = 6. Since 0 ≤ 6 ≤ 8, it is valid. Check: A(6) = 12 + 36 = 48 and B(6) = 27 + 21 = 48.
    Second piece (8 < x ≤ 15): 12 + 6x = 7x − 1, so x = 13. Since 8 < 13 ≤ 15, it is valid. Check: A(13) = 12 + 78 = 90 and B(13) = 91 − 1 = 90.
    Third piece (x > 15): 12 + 6x = 11x − 61, so 73 = 5x and x = 14.6. Since 14.6 is not greater than 15, this solution is rejected. The line 11x − 61 does meet Plan A's line at 14.6, but Plan B is not billed by that formula at 14,600 gallons. The real Plan B bill there is B(14.6) = 7(14.6) − 1 = 101.20, while A(14.6) = 12 + 87.6 = 99.60, so the plans do not cost the same at that usage.

  4. Answer: Plan B is cheaper on (6, 13); Plan A is cheaper on [0, 6) ∪ (13, ∞); the plans tie at x = 6 and x = 13.
    On 0 ≤ x ≤ 8: D(x) = (27 + 3.5x) − (12 + 6x) = 15 − 2.5x. On 8 < x ≤ 15: D(x) = (7x − 1) − (12 + 6x) = x − 13. On x > 15: D(x) = (11x − 61) − (12 + 6x) = 5x − 73.
    Plan B is cheaper where D(x) < 0. On the first piece, 15 − 2.5x < 0 gives x > 6, so the solutions there are 6 < x ≤ 8. On the second piece, x − 13 < 0 gives x < 13, so the solutions there are 8 < x < 13. On the third piece, 5x − 73 < 0 gives x < 14.6, which has no overlap with x > 15. Joining the pieces gives 6 < x < 13, or (6, 13).
    Plan A is cheaper where D(x) > 0: 0 ≤ x < 6 on the first piece, 13 < x ≤ 15 on the second piece, and every x > 15 on the third piece, since 5x − 73 > 0 whenever x > 14.6. Joining these gives [0, 6) ∪ (13, ∞).

  5. Answer: Plan B's corners are at (8, 55) and (15, 104); the graphs cross at (6, 48) and (13, 90).
    Plan A is a single straight line from (0, 12) to (20, 132), since A(20) = 12 + 120 = 132. Plan B starts at (0, 27), bends upward at (8, 55) and again at (15, 104), and ends at (20, 159), since B(20) = 11(20) − 61 = 220 − 61 = 159. A vertical axis from $0 to $160 in steps of $10 and a horizontal axis from 0 to 20 in steps of 1 fit every point. Plan B's graph starts above Plan A's, drops below it at (6, 48), stays below it until (13, 90), and stays above it for the rest of the window. The only stretch where Plan B's graph is lower is between the two crossings, which matches the interval (6, 13) from problem 4. The graphs never meet near x = 14.6, which confirms that the third-piece solution in problem 3 was not a real break-even.

  6. Answer: Plan A: $6.00 at all three usages. Plan B: $3.50 at 4,000 gallons, $7.00 at 10,000 gallons, and $11.00 at 18,000 gallons. Plan B sends the stronger signal at 10,000 and 18,000 gallons.
    Plan A's marginal price is its slope, 6, at every usage. Plan B's marginal price is the slope of the piece the household is on: 3.5 on the first piece (x = 4), 7 on the second piece (x = 10), and 11 on the third piece (x = 18). Plan B charges more than Plan A for the next 1,000 gallons only when the household is already using 8,000 gallons or more. Below 8,000 gallons, Plan B's signal to conserve is weaker than Plan A's, because each extra 1,000 gallons costs $3.50 instead of $6.00.

  7. Answer: Average use is 10,000 gallons per month; the shortcut gives $864 for Plan A and $828 for Plan B, so it recommends Plan B as $36 cheaper.
    Total use: 6 + 5 + 6 + 7 + 9 + 14 + 19 + 20 + 15 + 8 + 6 + 5 = 120 thousand gallons. Average: 120 ÷ 12 = 10 thousand gallons. A(10) = 72, and 12 × 72 = 864. B(10) = 69, and 12 × 69 = 828. The shortcut's difference is 864 − 828 = 36.

  8. Answer: Plan A costs $864.00 for the year and Plan B costs $909.50, so Plan A is cheaper by $45.50.
    Plan A: the twelve base fees are 12 × 12 = 144, and the usage charge is 6 × 120 = 720, so the year costs 144 + 720 = 864. Billing each month separately gives the same total.
    Plan B, month by month: January (6) 27 + 21 = 48.00; February (5) 27 + 17.5 = 44.50; March (6) 48.00; April (7) 27 + 24.5 = 51.50; May (9) 7(9) − 1 = 62.00; June (14) 7(14) − 1 = 97.00; July (19) 11(19) − 61 = 148.00; August (20) 11(20) − 61 = 159.00; September (15) 7(15) − 1 = 104.00; October (8) 27 + 28 = 55.00; November (6) 48.00; December (5) 44.50.
    Running total: 48 + 44.5 = 92.5; + 48 = 140.5; + 51.5 = 192; + 62 = 254; + 97 = 351; + 148 = 499; + 159 = 658; + 104 = 762; + 55 = 817; + 48 = 865; + 44.5 = 909.5. Plan B costs $909.50, and 909.50 − 864.00 = 45.50.
    Check by monthly differences B − A: January 0, February 2.50, March 0, April −2.50, May −4.00, June 1.00, July 22.00, August 27.00, September 2.00, October −5.00, November 0, December 2.50. These add to 45.50.

  9. Answer: Plan A's yearly cost depends only on total use, while Plan B's monthly bill rises faster at higher use, so its heavy summer months cost more than the average-month shortcut counts.
    Plan A is one straight line, so its yearly cost is 12 × 12 + 6 × (total use) = 144 + 6 × 120 = 864 however the 120 thousand gallons are spread across the year. That is why 12 × A(10) gives the exact $864. Plan B's slope increases from 3.5 to 7 to 11, so its high months add more than its low months save. August, 10 thousand gallons above average, costs B(20) − B(10) = 159 − 69 = $90 more than an average month under Plan B, while under Plan A the same extra use adds A(20) − A(10) = 132 − 72 = $60. February, 5 thousand gallons below average, saves only B(10) − B(5) = 69 − 44.50 = $24.50 under Plan B, compared with A(10) − A(5) = 72 − 42 = $30 under Plan A. The shortcut bills every month at $69 and never charges the $11.00 third-block price that July and August actually pay. July and August alone cost 22 + 27 = $49 more under Plan B than Plan A, which is more than the whole $45.50 yearly gap. As a result, the shortcut's $828 for Plan B is 909.50 − 828 = $81.50 below the actual yearly cost.

  10. Answer: $7.25 per 1,000 gallons for the second block; the third-piece equation then gives x = 14.25, which is rejected.
    The first block is unchanged, so B(8) is still 55 and the lower break-even stays at 6,000 gallons. With a second-block price of r, the second piece is B(x) = 55 + r(x − 8). The board wants B(12) = A(12), and A(12) = 12 + 72 = 84. So 55 + r(12 − 8) = 84, which gives 4r = 29 and r = 7.25. Check: B(12) = 55 + 7.25(4) = 55 + 29 = 84.
    The new value at the end of the second block is B(15) = 55 + 7.25(7) = 55 + 50.75 = 105.75, so the third piece becomes B(x) = 105.75 + 11(x − 15) = 11x − 59.25. Setting 11x − 59.25 = 12 + 6x gives 5x = 71.25 and x = 14.25. Since 14.25 is not greater than 15, it is rejected, and Plan B is now cheaper only on (6, 12).

Differentiation

Support

Give students a partly filled table of both plans' bills at x = 0, 4, 6, 8, 10, 13, 15, 18, and 20, with the Plan B piece used for each row named, so they can see the two crossings before solving algebraically. For problem 1, provide the frame "B(x) = (bill at the start of the block) + (block price)(x − start of the block)" and have students simplify it to mx + b. For problem 8, allow students to work from the monthly B − A differences instead of full bills, as long as they explain how the differences give the yearly gap.

Extension

Halbrook considers a third plan for large-lot customers: a $60 base fee plus $4.00 per 1,000 gallons, so C(x) = 60 + 4x. Ask students to find every break-even it creates with Plans A and B, reject any that fall outside the relevant Plan B piece, and state the interval on which each of the three plans is cheapest. For checking: A and C break even at x = 24; B and C give rejected solutions on the first two pieces and a valid one on the third piece at x = 121/7, about 17.3, where Plan A is already cheaper than both. Plan A is cheapest on [0, 6) and (13, 24), Plan B on (6, 13), and Plan C for x > 24. Students should explain why the valid B-and-C break-even does not change which plan is cheapest.

ELL supports

Pre-teach "block," "inclining block rate," "break-even," and "marginal price" with a labeled sketch of Plan B's graph showing three straight pieces that get steeper at 8,000 and 15,000 gallons. Provide sentence frames for problems 3, 4, and 9, such as "The solution x = ___ is rejected because it is not inside ___," "Plan ___ is cheaper for x in ___," and "The shortcut works for Plan ___ because ___." Allow students to label intervals and crossing points directly on their graph before writing them in interval notation.