Question

between 12:00 pm and 1:00 pm, cars arrive at a local banks drive - thru at the rate of 4 cars per hour (0.07 car per minute). the formula below from probability can be used to determine the probability that a car will arrive within t minutes of 12:00 pm. use this information to answer the questions in parts (a)-(e)
f(t)=1 - e^{-0.07t}
(a) using intersect, determine how many minutes are needed for the probability to reach 50%?
5.96 (rounded to the nearest whole number as needed)

Explanation:

Step1: Set up the equation

We want to find $t$ when $F(t)=0.5$. So we set $0.5 = 1 - e^{- 0.07t}$.

Step2: Rearrange the equation

First, we rewrite the equation as $e^{-0.07t}=1 - 0.5=0.5$.

Step3: Take the natural - logarithm of both sides

Taking the natural logarithm of both sides, we get $\ln(e^{-0.07t})=\ln(0.5)$.
Since $\ln(e^{x}) = x$, the left - hand side simplifies to $-0.07t$. So, $-0.07t=\ln(0.5)$.

Step4: Solve for $t$

Dividing both sides by $- 0.07$, we have $t=\frac{\ln(0.5)}{-0.07}$.
We know that $\ln(0.5)\approx - 0.6931$. Then $t=\frac{-0.6931}{-0.07}\approx9.9$.
Rounding to the nearest whole number, $t = 10$.

Answer:

$10$