Chapter 1.3, Problem 91E

a.

To determine

To identify:

Appropriate model for the given condition.

Answer to Problem 91E

The second model is appropriate model.

Explanation of Solution

Give information:

Given:

  $\begin{array}{l}{C}_1\left(t\right)=1+0.50〚t-1〛\\ C_2\left(t\right)=10.50〚-\left(t-1\right)〛\end{array}$ .

Calculation:

The cost of parking is metered $\$1.00$ for the first hour and $\$0.50$ for each additional hour or the portion of the hour.

There are two models given for the parking cost in metered lot for $t-$ hours.

  $\begin{array}{l}{C}_1\left(t\right)=1+0.50〚t-1〛\\ C_2\left(t\right)=1-0.50〚-\left(t-1\right)〛\end{array}$

Both the models fit well if the additional cost of $\$050$ is calculated for a complete hour but the first model fails when he cost is calculated for an additional portion of hour.

For the second hour, when $t=1.25$ ,

Using first model,

  $\begin{array}{l}{C}_1\left(t\right)=1+0.50\Vert t-1\Vert \\ =1+0.50\Vert 0.25\Vert \\ =1+0.50\left(0\right)\\ =1+0\\ =1\end{array}$

Using second model,

  $\begin{array}{l}{C}_2\left(t\right)=1-0.50\Vert -\left(1.2-1\right)\Vert \\ =1-0.50\Vert -0.25\Vert \\ =1-0.50\Vert -1\Vert \\ =1+0.5\\ =1.5\end{array}$

Similarly for $t=2.5,C_1\left(2.5\right)=1.50,C_2\left(2.5\right)=2$ .

Therefore, the second model is appropriate model.

b.

To determine

To calculate:

cost of parking and graph the appropriate model.

Answer to Problem 91E

The cost of parking for $7$ hours and $10$ minutes is $\$4.50$

Graph for Appropriate model is $C_2\left(t\right)=1-0.50\Vert -\left(t-1\right)\Vert$ is,

  

Explanation of Solution

Give information:

Given:

  $\begin{array}{l}{C}_1\left(t\right)=1+0.50〚t-1〛\\ C_2\left(t\right)=10.50〚-\left(t-1\right)〛\end{array}$ .

Calculation:

When parking is metered for $7$ hours and $10$ minutes, then using model $C_2\left(t\right)=1-0.50\Vert -\left(t-1\right)\Vert$ ,

  $\begin{array}{l}{C}_2\left(7+\frac{10}{60}\right)=C_2\left(7.167\right)\\ =1-0.50\Vert -\left(7.167-1\right)\Vert \\ =1-0.50\Vert -6.617\Vert \\ =1-0.50\left(-7\right)\\ =1+3.5\\ =4.5\end{array}$

Therefore, the cost of parking for $7$ hours and $10$ minutes is $\$4.50$

Graph for Appropriate model is $C_2\left(t\right)=1-0.50\Vert -\left(t-1\right)\Vert$ is,