Chapter 8.3, Problem 38E

(a)

To determine

To find: The unit price value.

Answer to Problem 38E

The unit price should be:

  $p=90\left(\$90\right)$

Explanation of Solution

Given information:

The given function as following below,

  $\begin{array}{l}R\left(n\right)\text{=}\left(\text{unit}\text{price}\right)\left(\text{units}\text{sold}\right)\\ \text{or}\\ R\left(n\right)=\left(120-6n\right)\left(80+8n\right)\end{array}$

Formula used:

  $n=-\frac{b}{2a}$

Calculation:

From the given, $\left(120-6n\right)$ represents the unit price so we need to find the value of n where R is maximum which is located at the vertex. Simplifying the revenue function, we have:

  $R\left(n\right)=-48n^2+480n+9600$

The x-coordinate is at $n=-\frac{b}{2a}$

Where: $a=-48\&b=480$

  $\begin{array}{l}n=-\frac{480}{2\left(-48\right)}\\ n=5\end{array}$

Substitute to the equation to find the y-coordinate of the vertex:

  $R=\left(120-6\left(5\right)\right)\left(80+8\left(5\right)\right)=10800$

The vertex is at (5, 10800)

Therefore, the unit price should be:

  $\begin{array}{l}p=120-6\left(5\right)\\ p=90\left(\$90\right)\end{array}$

Conclusion:

The unit price should be:

  $p=90\left(\$90\right)$

(b)

To determine

To analyse: The revenue model that results in greater maximum monthly revenue.

Answer to Problem 38E

The original revenue results in greater maximum monthly revenue.

Explanation of Solution

Given information:

The given function as following below,

  $\begin{array}{l}R\left(n\right)=\left(\text{unit}\text{price}\right)\left(\text{units}\text{sold}\right)\\ \text{or}\\ R\left(n\right)=\left(120-6n\right)\left(80+8n\right)\end{array}$

Formula used:

The x-coordinate is at $n=-\frac{b}{2a}$

Calculation:

Using the given revenue function as a pattern, the different revenue function will be:

  $\begin{array}{l}R\left(n\right)=\left(120-4n\right)\left(80+5n\right)\\ \text{or}\\ R\left(n\right)=-20n^2+280n+9600\end{array}$

The x-coordinate is at $n=-\frac{b}{2a}$

Where: $a=-20\&b=280$

  $\begin{array}{l}n=-\frac{280}{2\left(-20\right)}\\ n=7\end{array}$

Substitute to the equation to find the y-coordinate of the vertex:

  $R=\left(120-4\left(7\right)\right)\left(80+5\left(7\right)\right)=10580$

The vertex is at (7, 10580)

This means that the maximum revenue is $10580 which is less than the original revenue at $10800.

Therefore, the original revenue results in greater maximum monthly revenue.

Conclusion:

The original revenue results in greater maximum monthly revenue.