Q. 107 - Precalculus Enhanced with Graphing Utilities

Question

Exponential Probability Between 12:00 PM and 1:00 PM, cars arrive at Citibank's drive-thru at the rate of 6 cars per hour $(0.1$ car per minute$)$. The following formula from probability can be used to determine the probability that a car will arrive within t minutes of 12:00 PM:

$F\left(t\right)=1-e^{-0.1t}$

(a) Determine the probability that a car will arrive within 10 minutes of 12:00 PM (that is, before 12:10 PM).
(b) Determine the probability that a car will arrive within 40 minutes of 12:00 PM (before 12:40 PM).
(c) What value does F approach as t becomes unbounded in the positive direction?
(d) Graph F using a graphing utility.
(e) Using INTERSECT, determine how many minutes are needed for the probability to reach $50\%$. 

Step-by-Step Solution

Answer

(a)  $P\left(10\right)=0.632$

(b)  $P\left(40\right)=0.981$

(c)  $P(\infty )=1$

(d)  Graph of the function

(e)  Using INTERSECT, $6.93$ minutes are needed for the probability to reach 

Step 1.Given information

Let us consider the time t in minutes.
Also the probability of a car coming in that time be

$F\left(t\right)=1-e^{-0.1t}$

Step 2.(a) Determine the probability that a car will arrive within 10 minutes of 12:00 PM (that is, before 12:10 PM).

Let us consider t be 10 minutes
Then we get

$$\begin{gathered} F\left(t\right)=1-e^{0.1t} \\ F\left(10\right)=1-e^{0.1\left(10\right)} \\ =0.632 \end{gathered}$$

Step 3.(b) Determine the probability that a car will arrive within 40 minutes of 12:00 PM (before 12:40 PM).

Let us consider t be 40 minutes
Then we get

$$\begin{gathered} F\left(t\right)=1-e^{-0.1t} \\ F\left(40\right)=1-e^{-0.1\left(40\right)} \\ =0.981 \end{gathered}$$

Step 4.(c) What value does F approach as t becomes unbounded in the positive direction?

Let us consider $t\to \infty$
Then we get 

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

Step 5.(d) Graph F using a graphing utility.

Considering the function $F\left(t\right)=1-e^{-0.1t}$, the graph of the function is as follows

Step 6.Using INTERSECT, determine how many minutes are needed for the probability to reach 50 %

Let us consider $F\left(t\right)=0.5$

Using graphing utility we observe the following graph