Q. 108 - Precalculus Enhanced with Graphing Utilities

Question

Between $5:00$ pm and $6:00$ pm, cars arrive at Jiffy Lube at the rate of $9$ cars per hour ($0.15$ car per minute). The following formula from probability can be used to determine the probability that a car will arrive within t minutes of $5:00$ pm:

$F\left(t\right)=1-e^{-0.15t}$

Part (a): Determine the probability that a car will arrive within $15$minutes of $5:00$ pm (that is, before $5:15$ pm).

Part (b): Determine the probability that a car will arrive within $30$  minutes of $5:00$ pm (before $5:30$ pm).

Part (c): What value does F approach as t becomes unbounded in the positive direction?

Part (d): Graph F using a graphing utility.

Part (e): Using INTERSECT, determine how many minutes are needed for the probability to reach $60\%$.

Step-by-Step Solution

Answer

Part (a): The probability that a car will arrive within $15$ minutes of $5:00 pm$ is $0.894$.

Part (b): The probability that a car will arrive within $30$ minutes of $5:00pm$ is $0.988$.

Part (c): At $1$ F approach as t becomes unbounded in the positive direction.

Part (d): The required graph of the function is given below,

Part (e): Using intersection, we see $6.1minutes$ is needed for the probability to reach $60\%$.

Part (a) Step 1. Given information.

Consider the given function,

$$\begin{gathered} F\left(t\right)=1-e^{-0.15t}, \\ t=15 \end{gathered}$$

Substitute the value in the given function,

$$\begin{gathered} F\left(15\right)=1-e^{-0.15\times 15} \\ F\left(15\right)=0.894 \end{gathered}$$

Part (b) Step 1. Determine the probability within 30 minutes.

Consider the given function,

$$\begin{gathered} F\left(t\right)=1-e^{-0.15t}, \\ t=30 \end{gathered}$$

Substitute the value in the given function,

$$\begin{gathered} F\left(15\right)=1-e^{-0.15\times 30} \\ F\left(15\right)=0.988 \end{gathered}$$

Part (c) Step 1. When F approach as t becomes unbound in positive direction.

Consider the given function,

$F\left(t\right)=1-e^{-0.15t}$

Assume $t\to \infty$.

Substitute the value in the function,

$$\begin{gathered} F\left(\infty \right)=\underset{t\to \infty }{\lim }1-e^{-0.15t} \\ F\left(\infty \right)=1 \end{gathered}$$

Part (d) Step 1. Plot the graph.

Consider the given function,

$F\left(t\right)=1-e^{-0.15t}$

Plot the function,

Part (e) Step 1. Plot the graph.

Let us consider $F\left(t\right)=0.6$, then we get,

By the intersection we see the time is $6.1minutes$.