Chapter 3, Problem 47RE

(a)

To determine

To draw: a scatter diagram of the given data.

Answer to Problem 47RE

  

Explanation of Solution

Given:

Advertising	Total revenue
20	$6101
22	$6222
25	$6350
25	$6378
27	$6453
28	$6423
29	$6360
31	$6231

Calculation:

Scatter diagram of above data is shown below:

  

From above graph it appears that the data follows a quadratic equation with $a<0$ .

Conclusion:

Therefore, a scatter diagram is drawn.

(b)

To determine

the optimal level of advertising.

Answer to Problem 47RE

The optimum level of advertising is $27$

Explanation of Solution

Given:

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

Calculation:

Compare $R\left(A\right)=-7.76A^2+411.88A+942.72tof\left(x\right)=a{x}^2+bx+c$ and find the values. $a=-7.76,b=411.88and\text{ c}=942.72Since\text{ a}=-7.76\text{ <}0$ , the parabola opens down. so the vertex is a maximum point.

The maximum value occurs at

  $\begin{array}{l}x=-\frac{b}{2a}\\ =-\frac{411.88}{2\left(-7.76\right)}\\ =\frac{411.88}{15.52}\\ \approx 27\end{array}$

Therefore the optimum level of advertising is $27$

Conclusion:The optimum level of advertising is $27$

(c)

To determine

To predict: the total revenue when the optimal level of advertising is spent.

Answer to Problem 47RE

Total revenue is $\$6406.44$

Explanation of Solution

Calculation:

The peak income earned is

  $\begin{array}{l}f\left(-\frac{b}{2a}\right)=f\left(27\right)\\ =-7.76{\left(27\right)}^2+411.88\left(27\right)+942.72\\ =-5657.04+11120.76+942.72\\ =6406.44\end{array}$

Therefore total revenue is $\$6406.44$

Conclusion:

Total revenue is $\$6406.44$

(d)

To determine

To verify: the function given in part (b) is the quadratic function of best fit or not

Answer to Problem 47RE

The quadratic equation of best fit is $Y\left(x\right)=-7.75957x^2+411.875x+942.721$

Explanation of Solution

Calculation:

Upon executing the QUADraticREGression program.

The obtained results are shown below:

  

The output that the utility provides shows us the equation $y=a{x}^2+bx+c$ . The quadratic equation of best fit is

  $\boxed{Y\left(x\right)=-7.75957x^2+411.875x+942.721}$

Conclusion:

The quadratic equation of best fit is $Y\left(x\right)=-7.75957x^2+411.875x+942.721$

(e)

To determine

To draw: a graph the quadratic function of best fit

Answer to Problem 47RE

  

Explanation of Solution

Calculation:

Graph of the quadratic function found in part d) drawn on the scatter diagram is shown below:

  

Conclusion:

Thus, the graph the quadratic function is drawn.