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Chapter 2: Problem 9

A residential customer in the Midwest purchases gas from a utility company that charges according to the formula \(C(g)=11+10.50(g)\), where \(C(g)\) is the cost, in dollars, for \(g\) thousand cubic feet of gas. a. Find \(C(0), C(5),\) and \(C(10)\). b. What is the cost if the customer uses no gas? c. What is the rate per thousand cubic feet charged for using the gas? d. How much would it cost if the customer uses 96 thousand cubic feet of gas (the amount an average Midwest household consumes during the winter months)?

Short Answer

Expert verified
a. \(C(0) = 11\), \(C(5) = 63.50\), \(C(10) = 116\). b. 11 dollars. c. 10.50 dollars per thousand cubic feet. d. 1019 dollars.

Step by step solution

01

Understand the Formula

The formula given is \(C(g)=11+10.50g\), where \(C(g)\) represents the cost in dollars and \(g\) represents the amount of gas in thousand cubic feet.
02

Calculate C(0)

To find \(C(0)\), substitute \(g = 0\) into the formula: \[ C(0) = 11 + 10.50(0) = 11 \]
03

Calculate C(5)

To find \(C(5)\), substitute \(g = 5\) into the formula: \[ C(5) = 11 + 10.50(5) = 11 + 52.50 = 63.50 \]
04

Calculate C(10)

To find \(C(10)\), substitute \(g = 10\) into the formula: \[ C(10) = 11 + 10.50(10) = 11 + 105 = 116 \]
05

Answer Part b

The cost when the customer uses no gas is simply \(C(0)\), already calculated as 11 dollars.
06

Answer Part c

The rate per thousand cubic feet of gas is the coefficient of \(g\) in the formula, which is 10.50 dollars per thousand cubic feet.
07

Calculate Cost for 96 Thousand Cubic Feet

To find the cost for 96 thousand cubic feet, substitute \(g = 96\) into the formula: \[ C(96) = 11 + 10.50(96) = 11 + 1008 = 1019 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Utility cost analysis
In utility cost analysis, we aim to understand how the cost of a utility, like gas or electricity, is calculated based on usage. Here, we examine a residential customer鈥檚 gas bill using a given formula. This formula helps to break down the cost into a fixed component and a variable component, depending on the amount of gas consumed.
The formula provided is: \(C(g) = 11 + 10.50g\), where:
  • \(C(g)\) represents the total cost in dollars.
  • \(g\) is the number of thousand cubic feet of gas consumed.

Fixed Costs: The fixed part of the cost is \(11\) dollars, which is the base charge.
Variable Costs: The variable part, \(10.50g\), depends on how much gas, measured in thousand cubic feet, is consumed. As the gas usage increases, the cost increases proportionately.
Linear functions in algebra
Linear functions are mathematical expressions where the variables are only raised to the first power, implying a constant rate of change. They are represented by the general form \(y = mx + b\), where:
  • \(y\) is the dependent variable (in this case, cost \(C(g)\)).
  • \(m\) is the slope or rate of change (here, 10.50).
  • \(x\) is the independent variable (gas usage \(g\)).
  • \(b\) is the y-intercept or fixed part of the function (here, 11).

In our gas cost formula \(C(g) = 11 + 10.50g\), we see a clear linear relationship between the gas consumed and the total cost. As the value of \(g\) (amount of gas used) increases, the cost \(C(g)\) also increases linearly.
Unit rate interpretation
The unit rate is a crucial concept that represents the cost per single unit of measurement. Here, it denotes the cost per thousand cubic feet of gas. According to the formula \(C(g) = 11 + 10.50g\):
  • The coefficient of \(g\), which is 10.50, represents the unit rate.

This means that for every additional thousand cubic feet of gas consumed, the customer pays an extra 10.50 dollars. Understanding this unit rate helps in predicting and controlling utility expenses efficiently.
For example, if a customer uses one thousand cubic feet of gas, they can expect to pay an additional 10.50 dollars on top of the fixed charge.
Substitution method in algebra
The substitution method in algebra involves replacing a variable in an equation with a given value to find the result. Using our gas cost formula as an example:\(C(g) = 11 + 10.50g\), we can substitute specific values of \(g\) to find corresponding values of the cost \(C(g)\).
  • To find \(C(0)\): \[ C(0) = 11 + 10.50(0) = 11 \]
  • To find \(C(5)\): \[ C(5) = 11 + 10.50(5) = 63.50 \]
  • To find \(C(10)\): \[ C(10) = 11 + 10.50(10) = 116 \]
  • To find the cost for 96 thousand cubic feet of gas: \[ C(96) = 11 + 10.50(96) = 1019 \]

This method is extremely useful for quickly determining the cost at various levels of usage, making it easier to budget and manage utility expenses.

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