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Math problem set Grade 8 Math

Coldharbor Cove Desalination Costs: Building a Linear Function

A grade 8 problem set where students build a linear cost function for a fictional desalination plant, identify its rate of change and initial value, and evaluate it.

Objectives

  1. Students will construct a linear function from a description of a fixed cost plus a per-unit cost.
  2. Students will identify and interpret the rate of change and initial value of a linear function.

Vocabulary

Key words: linear function, rate of change, initial value

Teacher overview

A desalination plant's daily cost is a fixed amount plus an energy cost for each unit of water made, which is a linear function of production. This set has students build that function, read its rate of change and initial value, and use it to predict costs. A common mistake to watch for is treating the relationship as proportional; because the fixed cost is not zero, the graph does not pass through the origin, and the initial value carries real meaning.

Coldharbor Cove Desalination Costs: Building a Linear Function — Grade 8 Math
Name: ____________________________ Date: ______________

Problems

Coldharbor Cove is a fictional coastal town that runs a small reverse-osmosis plant that treats brackish groundwater. Making fresh water costs a fixed amount each day, plus an energy cost for every 1,000 gallons produced. The plant uses 4 kWh of electricity for every 1,000 gallons, electricity costs $0.15 per kWh, and the fixed cost is $900 per day. Let x be the number of thousands of gallons produced in one day.

  1. Find the energy cost to produce 1,000 gallons: multiply 4 kWh by $0.15 per kWh.
  2. The daily cost is $900 fixed plus the energy cost for each 1,000 gallons. Write the daily cost as a linear function C, in dollars, using x for thousands of gallons.
  3. Name the rate of change and the initial value of your function, and explain what each one means for the plant.
  4. Find the cost to produce 500,000 gallons in a day, where x = 500.
  5. Find the cost to produce 1,200,000 gallons in a day, where x = 1,200.
  6. One day the total cost was $1,500. How many thousands of gallons were produced? Solve 0.60x + 900 = 1,500.
  7. Explain why this cost is not a proportional relationship, even though the energy cost by itself is proportional.
  8. How many kWh does the plant use to produce 1,000,000 gallons, where x = 1,000?

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: $0.60.
    4 kWh × $0.15 per kWh = $0.60 for each 1,000 gallons.
  2. Answer: C = 0.60x + 900.
    The variable cost is $0.60 per thousand gallons and the fixed cost is $900, so C = 0.60x + 900.
  3. Answer: rate of change 0.60, initial value 900.
    The rate of change is $0.60 per thousand gallons, the energy cost of each unit. The initial value is $900, the fixed cost the plant pays even when x = 0.
  4. Answer: $1,200.
    C = 0.60(500) + 900 = 300 + 900 = $1,200.
  5. Answer: $1,620.
    C = 0.60(1,200) + 900 = 720 + 900 = $1,620.
  6. Answer: 1,000 thousand gallons.
    0.60x + 900 = 1,500, so 0.60x = 600 and x = 1,000. That is 1,000,000 gallons.
  7. Answer: The graph does not pass through the origin.
    A proportional relationship has an initial value of 0. Here the initial value is $900, so at x = 0 the cost is $900, not $0, and the relationship is linear but not proportional.
  8. Answer: 4,000 kWh.
    4 kWh per 1,000 gallons × 1,000 = 4,000 kWh to produce 1,000,000 gallons.

Differentiation

Support

Give students the completed energy-cost step ($0.60 per thousand gallons) so they can start at problem 2, and provide a two-row table showing the cost at x = 0 and x = 100 to make the fixed cost and the rate visible before they evaluate other values.

Extension

Suppose a second, fictional plant has a lower fixed cost of $600 but a higher energy cost of $0.90 per thousand gallons. Ask students to write its linear cost function, then find the production level where the two plants cost the same amount in a day.

ELL supports

Pre-teach "fixed cost," "per unit," and "initial value" with a labeled picture of the function showing the starting point on the y-axis and the climb per unit. Provide the sentence frame "The initial value is ___ because the plant pays it even when x is ___."