Cost of Natural Gas In April 2009, Nicor Gas had the...

cost of Natural Gas In April $2009,$ Nicor Gas had the following rate schedule for natural gas usage in single family residences: Monthly customer charge 8.40 dollars Distribution charge 1st 20 therms 0.1473 therm Next 30 therms 0.0579 dollarstherm Over 50 therms $0.0519 /$ therm Gas supply charge 0.43 dollars therm (a) What is the charge for using 40 therms in a month? (b) What is the charge for using 150 therms in a month? (c) Develop a model that gives the monthly charge $C$ for $x$ therms of gas. (d) Graph the function found in part (c).

cost of Natural Gas In April $2009,$ Nicor Gas had the following rate schedule for natural gas usage in single family residences:
Monthly customer charge 8.40 dollars
Distribution charge 1st 20 therms 0.1473 therm Next 30 therms 0.0579 dollarstherm Over 50 therms $0.0519 /$ therm
Gas supply charge
0.43 dollars therm
(a) What is the charge for using 40 therms in a month?
(b) What is the charge for using 150 therms in a month?
(c) Develop a model that gives the monthly charge $C$ for $x$ therms of gas.
(d) Graph the function found in part (c).

Key Concepts

Graphing Piecewise Functions

Graphing piecewise functions requires plotting each segment of the function on its corresponding interval, understanding where the function’s rule changes, and indicating whether endpoints are included or excluded. This visual representation helps in understanding how the overall cost behaves in different consumption ranges and is useful for analysis and decision-making.

Cost Modeling

Cost modeling involves developing mathematical models to estimate total costs based on various components, such as fixed charges and variable charges that depend on consumption. This approach can include both constant and variable terms, and is essential in budgeting, economic analysis, and decision-making.

Piecewise Functions

Piecewise functions are mathematical functions defined by different expressions over distinct intervals of the independent variable. They are particularly useful in modeling scenarios where the relationship between variables changes at certain thresholds, such as in tiered pricing systems, allowing for accurate representation of the variable cost structure.

Tiered Pricing

Tiered pricing is a structure where the unit price of a good or service changes based on the amount purchased or consumed. In such systems, different consumption ranges have different rates, with lower rates often applying after a certain threshold is exceeded, which encourages efficient use and can lead to cost savings for lower usage levels.

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