Chapter 4.1, Problem 30E

a.

To determine

To write: A linear model that represents the total cost of recording an album as a function of studio time.

Answer to Problem 30E

The linear model is $y=75x+50$ .

Explanation of Solution

Given information:Initial fees is $\$50$ , thereafter $\$75$ per hour is charged.

Calculation:

Slope intercept form: $y=mx+b$

Let $x$ represent time and $y$ represent the total cost.

As $\$50$ is the initial price, it will be considered as cost at $x=0$ .

At $x=1$ , total cost will become $\$125$ .

  $\therefore$ The points are $(0,50)$ and $(1,125)$ .

Solving for $m$ ,

  $m=\frac{y_2-y_1}{x_2-x_1}$

  $\begin{array}{l}m=\frac{125-50}{1-0}\\ \\ m=75\end{array}$

  $y-\mathrm{intercept}=50$   $($ when $x=0$ $)$

Substituting derived values in slope-intercept form,

Thus, the linear model is $y=75x+50$ .

b.

To determine

To find:Will it be less expensive to buy $12\mathrm{hrs}.$ of studio time or purchase a music software program?

Answer to Problem 30E

It will be less expensive $\left(by\$200\right)$ to buy a music software program $\left(worth\$750\right)$ than to spend $\$200$ extra and record at studio.

Explanation of Solution

Given information:Recording time to be purchased is $12\mathrm{hours}$ , its cost is to be compared with $\$750$ .

Calculation:

Using linear model derived in previous part,

  $y=75x+50$

Putting $x=12$ in above equation,

  $\begin{array}{l}y=75(12)+50\\ \\ y=900+50\\ \\ y=950\end{array}$

Thus, the total cost will be $\$950$ .

  $\therefore$ It will be less expensive $\left(by\$200\right)$ to buy a music software program $\left(worth\$750\right)$ than to spend $\$200$ extra and record at studio.