Chapter 6.7, Problem 53PPS

(a)

To determine

To find: The number of positive real zeros, negative real zeros, and imaginary zeros.

Answer to Problem 53PPS

The $M\left(x\right)$ has either 3 real positive zeros or one real positive zeros and 2 imaginary zeros.

Explanation of Solution

Given:

The equation is $M\left(x\right)=-2.03x^3+50.1x^2-214x+4020$ .

Calculation:

Since, $M\left(x\right)$ is of degree 3, there are at most 3 zeros. There are 3 sign changes for the coefficients of $M\left(x\right)$ , hence the function has 3 or 1 positive real zeros.

If the equation is as follows.

  $M\left(x\right)=2.03x^3+50.1x^2+214x+4020$

There are 0 sign changes for the coefficients of $M\left(-x\right)$ , so $M\left(x\right)$ has 2 or negative real zeros.

Therefore, the $M\left(x\right)$ has either 3 real positive zeros or one real positive zeros and 2 imaginary zeros.

(b)

To determine

To plot: The graph of the function by using calculator.

Answer to Problem 53PPS

The graph of the function $M\left(x\right)=-2.03x^3+50.1x^2-214x+4020$ is shown in figure (3).

Explanation of Solution

Given:

The equation is $M\left(x\right)=-2.03x^3+50.1x^2-214x+4020$ .

Calculation:

Us the graphing calculator to plot the graph of the function by using following steps.

1. Press $\boxed{Y=}$ and enter $y_1=-2.03x^3+50.1x^2-214x+4020$ as shown below.

  

Figure (1)

2. Press the $\boxed{\text{WINDOW}}$ key and set the window size as shown below.

  

Figure (2)

3. Press the $\boxed{\text{GRAPH}}$ key to trace the graph as shown below.

  

Figure (3)

Therefore, the graph of the function $M\left(x\right)=-2.03x^3+50.1x^2-214x+4020$ is shown in figure (3).

(c)

To determine

To approximate: The zeros to the nearest tenth and explain the significance of each zeros.

Answer to Problem 53PPS

The approximation is $x\approx 23.7517$ and this indicated that the music hall will not earn any money after $2026$ .

Explanation of Solution

Given:

The equation is $M\left(x\right)=-2.03x^3+50.1x^2-214x+4020$ .

Calculation:

Equate the given equation to zero and simplify as follows.

  $\begin{array}{c}-2.03x^3+50.1x^2-214x+4020=0\\ -2.03\left(x-23.7517\right)\left(x^2-0.928095x+83.3749\right)=0\\ \left(x-23.7517\right)\left(x^2-0.928095x+83.3749\right)=0\end{array}$

From the above equation,

  $\begin{array}{c}x-23.7517=0\\ x\approx 23.7517\end{array}$

This indicates that the music hall will not earn any money after $2026$ .

Therefore, the approximation is $x\approx 23.7517$ and this indicated that the music hall will not earn any money after $2026$ .