Between 700 AM and 800 AM trains arrive at a subway station

Between 7:00 AM and 8:00 AM, trains arrive at a subway station at a rate of 15 trains per hour (0.25 trains per
minute). The following formula from statistics can be used to determine the probability that a train will arrive
within t minutes of 7:00 AM.
F(t) = 1 - e-0.25t
Determine how many minutes are needed for the probability to reach 90%.

a. 4.53 min
b. 0.38 min
c. 2.97 min
d. 9.21 min

Solution

90% means 90/100=0.9 is the probability
F(t)=0.9
so,
0.9=1-e^(-0.25t)
adding e^(-0.25t) both sides
0.9+e^(-0.25t)=1-e^(-0.25t)+e^(-0.25t)
0.9+e^(-0.25t)=1
subtracting 0.9 from both sides
0.9+e^(-0.25t)-0.9=1-0.9
e^(-0.25t)=0.1
taking ln both sides
-0.25t lne=ln0.1
-0.25t=-2.3026
dividing both sides by -0.25
-0.25t/-0.25=-2.3026/-0.25
t=9.21 approx
so option D is our answer