Chapter 1.1, Problem 55PS

Solution

To find: the value of $x$ at which the store’s revenue is maximized.

The store can maximize the revenue by charging $\$8.5$ for each song.

Given information:

The store sells 4000 songs each day when the price is $1, and for each $0.5 increase in the price, the sale decreases by 80 songs each day.

Concept used:

The vertex of the quadratic function represents the maximum/minimum point of the function.

The x-coordinate of the vertex of a parabola of the form $y=a{x}^2+bx+c$ is given $x=-\frac{b}{2a}$ .

Calculation:

Let the price be increased $x$ times.

Then, the equation modeling the situation can be given as $R(x)=(1+0.05x)\cdot (4000-80x)$ .

Rewrite the equation as follows:

  $\begin{array}{c}R(x)=(1+0.05x)\cdot (4000-80x)\\ R(x)=-4x^2+120x+4000\end{array}$

This represents a downward parabola. So, it has a maximum value at its vertex.

The x-coordinate of the vertex of a parabola of the form $y=a{x}^2+bx+c$ is given $x=-\frac{b}{2a}$ .

For $R(x)=-4x^2+120x+4000$ , $a=-4$ and $b=120$ . Use this information to find the x-coordinate of the vertex.

  $\begin{array}{c}x=-\frac{b}{2a}\\ =-\frac{120}{2\left(-4\right)}\\ =-\frac{120}{-8}\\ =15\end{array}$

This shows that the maximum revenue of the store occurs when $x=15$ .

Now, $15\times 0.5=7.5$ . That means, the store can maximize its revenue by increasing the price $7.5.

That is, the store’s revenue is maximum when it charges $\$\left(1+7.5\right)=\$8.5$ .

Conclusion:

The store can maximize the revenue by charging $\$8.5$ for each song.