Chapter 8.4, Problem 117E

a.

To determine

To adjust: the given model so that $t=0$ corresponds to 2000 rather than 1990.

Answer to Problem 117E

  $g\left(t\right)=0.005t^2-0.27t+25.83$ for $10\le t\le 32$ .

Explanation of Solution

Given:

The function corresponds to $t=0\left(1990\right)$ is given by:

  $f\left(t\right)=0.005t^2-0.37t+25.7$

Such that $0\le t\le 22$

Given graph:

  

Consider the given model.

  $f\left(t\right)=0.005t^2-0.37t+25.7$

Now, adjust: the given model so that $t=0$ corresponds to 2000 rather than 1990.

Shift the graph of $f\left(t\right)$ by 10 units to the left.

  $\begin{array}{c}g\left(t\right)=f\left(t+10\right)\\ =0.005{\left(t+10\right)}^2-0.37\left(t+10\right)+25.7\\ =0.005\left(t^2+20t+100\right)-0.37\left(t+10\right)+25.7\\ =0.005t^2+0.1t-0.37t+0.5-0.37+25.7\\ =0.005t^2-0.27t+25.83\end{array}$

Hence, $g\left(t\right)=0.005t^2-0.27t+25.83$ such that $10\le t\le 32$ .

b.

To determine

To graph: the function $f\left(t\right)$ and $g\left(t\right)$ in same viewing window.

Explanation of Solution

Given:

The function corresponds to $t=0\left(1990\right)$ is given by:

  $f\left(t\right)=0.005t^2-0.37t+25.7$

Such that $0\le t\le 22$

And

  $g\left(t\right)=0.005t^2-0.27t+25.83$

Such that $10\le t\le 32$

Construct the graph of $f\left(t\right)$ and $g\left(t\right)$ in same viewing window as shown:

  

The graph $f\left(t\right)$ is shifted horizontally by 10 units to obtain the graph of $g\left(t\right)$ .