Chapter 1.8, Problem 58E

(a)

To determine

To find: The reason for $T$ being a function of $t$ .

Answer to Problem 58E

The $T$ is a function of $t$ because the temperature of the house varies with the time.

Explanation of Solution

Given:

The temperature in the house is $T$ .

The given graph is shown in Figure 1

  

Figure 1

Calculation:

The given graph shows that the temperature of the house varies with the passing time, hence the $T$ is the function of the time.

(b)

To determine

To find: The value of $T\left(4\right)$ and $T\left(15\right)$ .

Answer to Problem 58E

The value of $T\left(4\right)=60°\text{F}$ and $T\left(15\right)=72°$ .

Explanation of Solution

Given:

Thegiven temperatures are $T\left(4\right)$ and $T\left(15\right)$ .

Calculation:

The graph shows that at $T\left(4\right)$ the value of temperature is,

  $T\left(4\right)=60°\text{F}$

Consider the value of $T\left(15\right)$ from the graph is,

  $T\left(15\right)=72°$

(c)

To determine

To find: The change in temperature from the function $H\left(t\right)=T\left(t-1\right)$ .

Answer to Problem 58E

The function $H\left(t\right)=T\left(t-1\right)$ shifts the graph between temperature and time by 1 unit.

Explanation of Solution

Given:

The given function is 0 $H\left(t\right)=T\left(t-1\right)$ .

Calculation:

Consider the given function is,

  $H\left(t\right)=T\left(t-1\right)$

The above functions shows that the there is a shift in the graph to the right by one unit.

(d)

To determine

To find: The change in temperature from the function $H\left(t\right)=T\left(t\right)-1$ .

Answer to Problem 58E

The function $H=T\left(t\right)-1$ shifts the graph in downward direction by 1 unit.

Explanation of Solution

Given:

The given function is $H=T\left(t\right)-1$ .

Calculation:

Consider the given function is,

  $H=T\left(t\right)-1$

The above functions shows that the there is a shift in the graph in downward direction by 1 unit.

(e)

To determine

To find: The piecewise defined function that represents the graph.

Answer to Problem 58E

The require list of the piece wise function is,

  $T\left(t\right)=\{\begin{array}{l}60t\le 6\\ 60+12t6<t<7\\ 72c7\le t\le 20\\ 72-12t20<t<21\\ 6021<t\end{array}$

Explanation of Solution

The piece wise function that represents the graph is,

` $T\left(t\right)=\{\begin{array}{l}60t\le 6\\ 60+12t6<t<7\\ 72c7\le t\le 20\\ 72-12t20<t<21\\ 6021<t\end{array}$