Example 8 from Algebra and Trigonometry, 4.1 Linear Functions

Using a Linear Function to Determine the Number of Songs in a Music Collection

Marcus currently has 200 songs in his music collection. Every month, he adds 15 new songs. Write a formula for the number of songs, $N,$ in his collection as a function of time, $t,$ the number of months. How many songs will he own at the end of one year?

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
The initial value for this function is 200 because he currently owns 200 songs, so $N (0) = 200,$ which means that $b = 200.$
The number of songs increases by 15 songs per month, so the rate of change is 15 songs per month. Therefore we know that $m = 15.$ We can substitute the initial value and the rate of change into the slope-intercept form of a line.

Figure 10

We can write the formula $N (t) = 15 t + 200.$

With this formula, we can then predict how many songs Marcus will have at the end of one year (12 months). In other words, we can evaluate the function at $t = 12.$

$\begin{matrix} N (12) & = & 15 (12) + 200 \\ = & 180 + 200 \\ = & 380 \end{matrix}$
Marcus will have 380 songs in 12 months.

How did it go?

Solution

Solution