Question

question
currently, the number of vehicles passing through a traffic light is 82. it is estimated that the number of vehicles passing through the traffic light will decrease by 5% each hour. find the total number of vehicles passing through the traffic light at the end of 7 hours.
select the correct answer below.
434
470
488
495

Explanation:

Step1: Identify the formula for exponential decay

The formula for exponential decay is $N = N_0(1 - r)^t$, where $N_0$ is the initial amount, $r$ is the rate of decay, and $t$ is the time. Here, $N_0 = 82$, $r=0.05$, and $t = 7$.

Step2: Calculate the number of vehicles each hour

For the first - hour, the number of vehicles $N_1=N_0(1 - r)=82\times(1 - 0.05)=82\times0.95$.
For the second - hour, $N_2=N_1\times(1 - r)=82\times(0.95)^2$.
In general, after $t = 7$ hours, $N = 82\times(0.95)^7$.

Step3: Calculate the value of $(0.95)^7$

$(0.95)^7=0.95\times0.95\times0.95\times0.95\times0.95\times0.95\times0.95\approx0.698337$.

Step4: Calculate the number of vehicles after 7 hours

$N = 82\times(0.95)^7\approx82\times0.698337\approx57.26$.
This is the number of vehicles at the end of 7 hours. To find the total number of vehicles that have passed through the traffic - light in 7 hours, we need to use the sum of a geometric series formula $S_n=\sum_{k = 0}^{n - 1}a\times r^k=\frac{a(1 - r^n)}{1 - r}$ (where $a = 82$, $r = 0.95$, and $n = 8$ since we start from the initial state at $t = 0$).
$S_8=\frac{82\times(1-(0.95)^8)}{1 - 0.95}=\frac{82\times(1 - 0.66342)}{0.05}=\frac{82\times0.33658}{0.05}=\frac{27.60956}{0.05}=434.1912\approx434$.

Answer:

A. 434