Step-by-Step Guidance for Linear Functions and Equations (Air Conditioning Cost Example)

Q1. Complete the table below. Use it to estimate the number of years until the total cost of gas is equal to the total cost of electricity. (Numerical Approach)

Background

Topic: Linear Functions and Equations

This question is testing your ability to model real-world situations with linear functions, evaluate those functions at specific values, and use a table to estimate when two quantities are equal.

Key Terms and Formulas

• Linear Function: A function of the form $f(t) = mt + b$, where $m$ is the rate of change (slope) and $b$ is the initial value (y-intercept).

• Total Cost Function: For each system (Electric and Gas), you will write a linear equation that models the total cost after $t$ years.

• Table of Values: Substitute each year value into the cost functions to fill in the table.

Step-by-Step Guidance

1. Identify or recall the linear cost functions for both the Electric and Gas systems. These should be of the form $C_{electric}(t) = m_1 t + b_1$ and $C_{gas}(t) = m_2 t + b_2$, where $t$ is the number of years.

2. For each year listed in the table (0, 3, 6, 9, 12, 15, 18), substitute the value of $t$ into both cost functions to calculate the total cost for Electric and Gas at that year.

3. Fill in the table with the calculated values for each year under the appropriate columns (Electric and Gas).

4. Compare the values in the Electric and Gas columns for each year. Look for the year where the costs are closest to being equal, or where the values cross over (one becomes greater than the other).

5. Estimate the number of years when the total cost of gas is equal to the total cost of electricity by identifying the interval where the switch occurs, and interpolate if needed.

Try solving on your own before revealing the answer!

Final Answer:

The table is completed by substituting each year value into the given linear cost functions for Electric and Gas. The year when the total cost of gas is approximately equal to the total cost of electricity is the point where their values in the table are closest or cross over. You can interpolate between the two closest years for a more precise estimate.

For example, if $C_{electric}(12) = 5000$ and $C_{gas}(12) = 4800$, and at $t=15$, $C_{electric}(15) = 6000$ and $C_{gas}(15) = 6200$, then the costs are equal somewhere between 12 and 15 years. You can estimate the exact year by linear interpolation.