Question

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 arrives within \(t\) minutes of 12: 00 PM. $$ F(t)=1-e^{-0.1 t} $$ (a) Determine the probability that a car arrives within 10 minutes of 12: 00 PM (that is, before 12: 10 PM). (b) Determine the probability that a car arrives within 40 minutes of 12: 00 PM (before 12: 40 PM). (c) What 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 \(50 \%\).

Short answer

(a) 63.21%, (b) 98.17%, (c) 1, (d) Use a graphing utility, (e) 6.93 minutes

Step-by-step solution

Determine the probability within 10 minutes

Use the given formula to find the probability that a car arrives within 10 minutes. Substitute t = 10 into the formula: \[ F(t) = 1 - e^{-0.1t} \] \[ F(10) = 1 - e^{-0.1 \times 10} \] Calculate the value of the exponential function: \[ F(10) = 1 - e^{-1} \] \[ e^{-1} \approx 0.3679 \] \[ F(10) = 1 - 0.3679 \approx 0.6321 \] The probability that a car arrives within 10 minutes is approximately 63.21%.

Determine the probability within 40 minutes

Substitute t = 40 into the formula: \[ F(t) = 1 - e^{-0.1t} \] \[ F(40) = 1 - e^{-0.1 \times 40} \] Calculate the value of the exponential function: \[ F(40) = 1 - e^{-4} \] \[ e^{-4} \approx 0.0183 \] \[ F(40) = 1 - 0.0183 \approx 0.9817 \] The probability that a car arrives within 40 minutes is approximately 98.17%.

Determine the limit of F(t) as t approaches infinity

As t becomes very large, the exponential term \( e^{-0.1t} \) approaches 0, hence: \[ \lim_{t \to \infty} F(t) = \lim_{t \to \infty} (1 - e^{-0.1t}) = 1 - 0 = 1 \] So, as \( t \to \infty \), \( F(t) \) approaches 1.

Graph F(t) using a graphing utility

Use a graphing utility to plot the function \( F(t) = 1 - e^{-0.1t} \). The x-axis represents time in minutes, and the y-axis represents the probability.

Determine the time for the probability to reach 50%

Set the equation equal to 0.5 and solve for t: \[ 0.5 = 1 - e^{-0.1t} \] Subtract 1 from both sides: \[ -0.5 = - e^{-0.1t} \] Simplify: \[ e^{-0.1t} = 0.5 \] Take the natural logarithm of both sides: \[ -0.1t = \ln(0.5) \] Solve for t: \[ t = \frac{\ln(0.5)}{-0.1} \] Using a calculator, \[ \ln(0.5) \approx -0.6931 \] \[ t = \frac{-0.6931}{-0.1} \approx 6.931 \] Therefore, it takes approximately 6.93 minutes for the probability to reach 50%.

Key concepts

Exponential Function

The exponential function is a mathematical function of the form \( f(x) = e^x \), where \( e \) is the base of the natural logarithm. Exponential functions are crucial in modeling growth or decay processes, such as population growth, radioactive decay, or in our case, car arrivals.
In our problem, the number of cars arriving is described using an exponential function: \( F(t) = 1 - e^{-0.1t} \). This specific type of function shows how the probability of an event changes over time.
Key points about exponential functions:

• They have a constant relative growth rate—meaning the function's growth rate is proportional to its current value.
• They exhibit rapid increases (or decreases) depending on the value of the exponent.
• They approach infinity as \( t \) becomes very large (for positive growth rates) or zero (for negative growth rates).

Understanding how exponential functions work is fundamental for solving various real-world problems related to probability and growth/decay.

Probability

Probability measures how likely an event is to occur. It ranges from 0 (impossible event) to 1 (certain event). For example, in our problem, we're interested in the probability that a car arrives within a certain time frame.
The formula used is \( F(t) = 1 - e^{-0.1 t} \). Here,

• \( t \) represents time in minutes.
• \( e^{-0.1 t} \) reflects how the likelihood changes over time.
• \( F(t) \) represents the cumulative probability up to time \( t \).

By substituting different values of \( t \) into the function, we can calculate the probability of a car arriving within 10 minutes, 40 minutes, or any other specified time frame. This approach provides a clear, quantitative way to understand and predict events over time.

Limit

The concept of a limit helps us understand the behavior of functions as they approach a particular point. In the context of our problem, we are interested in what happens to \( F(t) \) as \( t \) approaches infinity.
Our formula \( F(t) = 1 - e^{-0.1 t} \) contains an exponential component \( e^{-0.1 t} \). As \( t \) becomes very large (approaches infinity), \( e^{-0.1 t} \) gets closer and closer to 0. Thus,
\( \ \ \ \lim_{t \to \infty} F(t) = 1 - 0 = 1 \)Therefore, as \( t \) approaches infinity, the probability \( F(t) \) approaches 1. This indicates that over a very long period, the certainty of a car arriving becomes almost absolute, which makes intuitive sense.

Natural Logarithm

The natural logarithm, denoted as \( \ln(x) \), is the inverse function of the exponential function \( e^x \). It's a valuable tool in solving equations involving exponential growth or decay. In our example, we need to determine how long it takes for the probability to reach 50%. This involves solving the equation:

• Start with \( 0.5 = 1 - e^{-0.1 t} \)
• Simplify to \( e^{-0.1 t} = 0.5 \)
• Apply the natural logarithm: \( \ln(e^{-0.1 t}) = \ln(0.5) \)
• Thus, \( -0.1 t = \ln(0.5) \)
• Solve for \( t \): \( t = \frac{\ln(0.5)}{-0.1} = \approx 6.931 \)

The calculation shows it takes approximately 6.93 minutes for the probability to reach 50%. Understanding how to manipulate natural logarithms is crucial in working with exponential functions and solving real-world problems.