Chapter 2.2, Problem 69E

To determine

To find: The amount charged by store per son to maximize daily revenue using verbal model and quadratic function.

Answer to Problem 69E

The store should charge $1.75 per song to maximize daily revenue of $4900.

Explanation of Solution

Given Information:

Total sales of music in a day is 4000 songs when it charges $1 per song. For each $0.05 increase in price, about 80 fewer songs were sold per day.

  $\begin{array}{c}\text{Revenue}=\text{Price}\times \text{Sales}\\ \text{R}\left(x\right)=\left(1+0.05x\right)\times \left(4000-8x\right)\end{array}$

Calculation:

The maximum sale of the model can be obtained by the y -coordinates. Write the model of revenue in standard from and find the vertex,

  $\begin{array}{c}\text{R}\left(x\right)=\left(1+0.05x\right)\times \left(4000-8x\right)\\ =4000+200x-80x-4x^2\\ =-4x^2+120x+4000\\ =-4\left(x^2-30x\right)+4000\\ =-4\left(x^2-30x+225-225\right)+4000\\ =-4\left(x^2-30x+225-225\right)+4\times 225+4000\\ =-4{\left(x-15\right)}^2+900+4000\\ =-4{\left(x-15\right)}^2+4900\end{array}$

Which is of the form $y=a{\left(x-h\right)}^2+k$

This gives the value of the vertex $\left(h,k\right)$ as $h=15,k=4900$ .

Therefore, the store should charge,

  $\begin{array}{c}1+0.05x=1+0.05\times 15\\ =\$1.75\end{array}$

Maximum daily revenue earned is $4900.