Chapter 5.3, Problem 107AYU

(a)

To determine

To find:The probability that the car will arrive within 10 minutes of $12:00\text{PM}$ .

Explanation of Solution

The expression for the probability that a car will arrive within $t$ minutes of $12:00$ PM is given by,

  $F\left(t\right)=1-e^{-0.1t}$

The probability that the car will arrive within 10 minutes of $12:00\text{PM}$ is calculated as,

  $\begin{array}{c}F\left(10\right)=1-e^{-0.1\left(10\right)}\\ =0.6321\\ \approx 0.63\end{array}$

Therefore, the probability that the car will arrive within 10 minutes of $12:00\text{PM}$ is $0.63$ .

(b)

To determine

To find:The probability that the car will arrive within 40 minutes of $12:00\text{PM}$ .

Explanation of Solution

The expression for the probability that a car will arrive within $t$ minutes of $12:00$ PM is given by,

  $F\left(t\right)=1-e^{-0.1t}$

The probability that the car will arrive within 40 minutes of $12:00\text{PM}$ is calculated as,

  $\begin{array}{c}F\left(40\right)=1-e^{-0.1\left(40\right)}\\ =0.9816\\ \approx 0.98\end{array}$

Therefore, the probability that the car will arrive within 40 minutes of $12:00\text{PM}$ is $0.98$ .

(c)

To determine

To find:The value $F$ approach as $t$ becomes unbounded in the positive direction.

Explanation of Solution

The expression for the probability that a car will arrive within $t$ minutes of $12:00$ PM is given by,

  $F\left(t\right)=1-e^{-0.1t}$

Let $t\to \infty$

  $\begin{array}{c}F\left(t\right)=\underset{t\to \infty }{\lim }\left(1-e^{-0.1t}\right)\\ =1\end{array}$

Therefore, the value $F$ approach as $t$ becomes unbounded in the positive direction is $1$ .

(d)

To determine

To graph :The plot of $F$ .

Explanation of Solution

The expression for the probability that a car will arrive within $t$ minutes of $12:00$ PM is given by,

  $F\left(t\right)=1-e^{-0.1t}$

The graph of $F\left(t\right)$ is shown in figure below.

  

Figure (1)

Therefore, the graph of $F$ is shown in figure (1).

(e)

To determine

To find:The minutes required for the probability to reach $50\%$ .

Explanation of Solution

The expression for the probability that a car will arrive within $t$ minutes of $12:00$ PM is given by,

  $F\left(t\right)=1-e^{-0.1t}$

And,

  $F\left(t\right)=0.5$

The graph is plotted below.

  

Figure (2)

By intersection, the minutes requires is $6.93$ minutes

Therefore, the minutes required is $6.93$ minutes.