Alice arrives at a train station at a time uniformly distributed between 8 p. m. and 9 p. m., and boards the first train that arrives. There are two types of trains that arrive between 7 a. m. and midnight: those that head to town A arrive every 15 minutes starting at 7 a. m., and those that head to town B arrive every 15 minutes starting at 7:05 a. m..

a. How long does Alice wait, on the average?
b. What proportion of days does she go to A?

a) She waits in average 4 minutes and 10 seconds.

b) She goes to A 2/3 of the days

Step-by-step explanation:

The trains reach the station at 8 p.m, 8:05, 8:15, 8:20, 8:30, 8:35, 8:45, 8:50 and 9 p.m between 8 and 9 pm.

Lets analyze how much does Allice waits in average in each 5 minutes interval:

between 8 and 8:05: She waits in average 2.5 minutes

between 8:05 and 8:10: she waits in average 7.5 minutes (she will have to wait at least 5 minutes and any amount uniformly between 5 and 10).

between 8:10 and 8:15: in average she waits 2.5 minutes.

Other intervals where she waits 2.5 minutes in average: 8:15-8:20, 8:25 - 8:30, 8:30 - 8:35, 8:40-8:45, 8:45-8:50 and 8:55 and 9.

Other intervals where she waits in average 7.5 minutes: 8:20-8:25, 8:35-8:40, 8:50-8:55

There are a total of 4 intervals where she waits in average 7.5 minutes and 8 intervals where she waits in average 2.5 minutes. Taking the average over the intervals we have that she waits in average

(8*2.5 + 4*7.5)/12 = 4.16667 minutes = 4 minutes and 10 seconds

b) She goes to A whenever she arrives in the following 10-minute periods:

8:05-8:15, 8:20-8:30, 8:35-8:45, 8:50-9

So, out of the 60 minute period where she can arrive uniformly, in 40 of those 60 minutes she goes to A, thus she goes to A with a proportion of 40/60 = 2/3.

"image"

step-by-step explanation:

in most of these problems, the notation is that the point with a "prime" is the image of the point without. that is, b' is the image of b.

here, there is no dilation or reflection. there is simply translation (with no direction shown). which figure you call the "preimage" and which one you call the "image" is up to you. as a matter of convention, the naming is as described above.