Q 29 - Precalculus Enhanced with Graphing Utilities

Question

A small manufacturing firm collected the following data on advertising expenditures A (in thousands of dollars) and total revenue R (in thousands of dollars).

(a) Draw a scatter diagram of the data. Comment on the type of relation that may exist between the two variables.

(b) The quadratic function of best fit to these data is

$R\left(A\right) = -7.76A^2 + 411.88A + 942.72$

Use this function to determine the optimal level of advertising.

(c) Use the function to predict the total revenue when the optimal level of advertising is spent.

(d) Use a graphing utility to verify that the function given in part (b) is the

 quadratic function of best fit.

(e) Use a graphing utility to draw a scatter diagram of the data and then graph the quadratic function of best fit on the scatter diagram.

Step-by-Step Solution

Answer

(a) The scatter diagram of the data is 

(b) The optimal level of advertising is $A\approx 26.5$

(c) The total revenue when the optimal level of advertising is spent is $R\left(26.5\right)=6408.08$

(d) The graph of the quadratic function of best fit is 

(e) The graph of the quadratic function of best fit on the scatter diagram is 

Step 1. Given information

A small manufacturing firm collected the following data on advertising expenditures A (in thousands of dollars) and total revenue R (in thousands of dollars).

Part (a) of Step 1. Scatter plot

The scatter plot of the given data is 

Part (b) of Step 1. Optimal level of advertising.

From the given function it is clear that $a<0$, this indicates that the parabola opens down. The function has a maximum at the vertex of the parabola. The optimal level of advertising is A for which the function reaches its maximum value,

$$\begin{gathered} A=-\frac{b}{2a} \\ A=-\frac{411.88}{2·\left(-7.76\right)} \\ A\approx 26.5 \end{gathered}$$

Part (c) of Step 1. The total revenue

The optimal level of advertising is spent when ,$A=26.5$,

Substitute $R\left(26.5\right)$ in the given function $R\left(A\right) = -7.76A^2 + 411.88A + 942.72$

So the total revenue is 

$R\left(26.5\right)=6408.08$

Part (d) of Step 1. Verify the quadratic function of best fit.

Plot the given coordinates on a cartesian coordinate plane and in the same plane, and plug in the equation to see if the equation is the best fit.

The graph of the function together with the coordinate of the data is 

All the points seem to be located along with the graph of the $y=-7.76A^2+411.88A+942.72$

Therefore the given function is the best fit function.

Part (e) of Step 1 . Scatter diagram and the graph of the quadratic function

Using the information from the table and the given function   $R\left(A\right) = -7.76A^2 + 411.88A + 942.72$

The graph of the quadratic function of best fit on the scatter diagram is