Chapter 2.2, Problem 56PS

Solution

a.

To state: The degree and the type of function

The function $M=-11.0t^3+267t^2-592t+21600$ is a polynomial function it is a $3^{rd}$ degree (cubic) polynomial function.

Given:

From $1987$ to $2003$ , the number of indoor movie screens $M$ in the United States can be modeled by $M=-11.0t^3+267t^2-592t+21,600$ where $t$ is the number of years since $1987$ .

Explanation:

A polynomial function is a function of the form $f\left(x\right)=a_n{x}^n+a_{n-1}{x}^{n-1}+\dots +a_{1}x+a_0$ where $a_n\ne 0$ , the exponents are all whole numbers and the coefficients are all real numbers.

Hence the function $M=-11.0t^3+267t^2-592t+21600$ is a polynomial function it is a $3^{rd}$ degree (cubic) polynomial function with leading coefficient $-11$

b.

To make: A table of values for the function $M=-11.0t^3+267t^2-592t+21,600$ .

Table is as follows.

  $\begin{array}{ccccccccc}t& 2& 4& 6& 8& 10& 12& 14& 16\\ M& 21396& 22800& 25284& 28320& 31380& 33936& 35460& 35424\end{array}$

Given:

From $1987$ to $2003$ , the number of indoor movie screens $M$ in the United States can be modeled by $M=-11.0t^3+267t^2-592t+21,600$ where $t$ is the number of years since $1987$ .

Calculation:

Consider the value of $t=2,4,6,8,10,12,14,16$ and substitute these values into the function, obtain the values and make the table as follows:

  $\begin{array}{ccccccccc}t& 2& 4& 6& 8& 10& 12& 14& 16\\ M& 21396& 22800& 25284& 28320& 31380& 33936& 35460& 35424\end{array}$

c.

To sketch: A graph of the function $M=-11.0t^3+267t^2-592t+21,600$

  

Given:

From $1987$ to $2003$ , the number of indoor movie screens $M$ in the United States can be modeled by $M=-11.0t^3+267t^2-592t+21,600$ where $t$ is the number of years since $1987$ .

Graph:

The following figure represents the graph of the given function.