Chapter 4, Problem 11CT

a.

To determine

To determine: A new model S for the total number of new vehicle sales.

Answer to Problem 11CT

The required models are $S\left(t\right)=37t^4-660t^3+5900t^2-13400t+17900$ and $S\left(-t\right)=37t^4+660t^3+5900t^2+13400t+17900$ .

Explanation of Solution

Given information:

The number of new truck sales = T

The number of new car sales = C

The model for T : $T=23t^4-330t^3+3500t^2-7500t+9000$

The model for C : $C=14t^4-330t^3+2400t^2-5900t+8900$

The given equations are:

  $T=23t^4-330t^3+3500t^2-7500t$

  $C=14t^4-330t^3+2400t^2-5900t+8900$

Add the above two equations to expression the model S for the total number of new vehicle sales.

  $\begin{array}{c}S\left(t\right)=23t^4-330t^3+3500t^2-7500t+9000+14t^4-330t^3+2400t^2-5900t+8900\\ S\left(t\right)=37t^4-660t^3+5900t^2-13400t+17900\end{array}$

Also, $\begin{array}{c}S\left(-t\right)=37{\left(-t\right)}^4-660{\left(-t\right)}^3+5900{\left(-t\right)}^2-13400\left(-t\right)+17900\\ S\left(-t\right)=37t^4+660t^3+5900t^2+13400t+17900\end{array}$

Hence, the required models are $S\left(t\right)=37t^4-660t^3+5900t^2-13400t+17900$ and $S\left(-t\right)=37t^4+660t^3+5900t^2+13400t+17900$ .

b.

To determine

To determine: Whether the function S is even, odd or neither.

Answer to Problem 11CT

The function S is neither odd not even.

Explanation of Solution

Given information:

The number of new truck sales = T

The number of new car sales = C

The model for T :

  $T=23t^4-330t^3+3500t^2-7500t+9000$

The model for C : $C=14t^4-330t^3+2400t^2-5900t+8900$

The given equations are:

  $T=23t^4-330t^3+3500t^2-7500t$

  $C=14t^4-330t^3+2400t^2-5900t+8900$

Add the above two equations to expression the model S for the total number of new vehicle sales.

  $\begin{array}{c}S\left(t\right)=23t^4-330t^3+3500t^2-7500t+9000+14t^4-330t^3+2400t^2-5900t+8900\\ S\left(t\right)=37t^4-660t^3+5900t^2-13400t+17900\end{array}$

Also, $\begin{array}{c}S\left(-t\right)=37{\left(-t\right)}^4-660{\left(-t\right)}^3+5900{\left(-t\right)}^2-13400\left(-t\right)+17900\\ S\left(-t\right)=37t^4+660t^3+5900t^2+13400t+17900\end{array}$

Here, it is seen that $S\left(-t\right)\ne S\left(t\right)$ and $S\left(-t\right)\ne -S\left(t\right)$ .

Hence, the function S is neither odd not even.