6.31 Waiting for the Train. A commuter train arrives punctually at a station every half hour. Each morning, a commuter named John leaves his house and casually strolls to the train station. The time, in minutes, that John waits for the train is a variable with density curve $y=1/30$for $0<x<30$, and $y=0$ otherwise.
a. Graph the density curve of this variable.
b. Show that the area under this density curve to the left of any number $x$between 0 and 30 equals $x/30$.

What percentage of the time does John wait for the train
c. less than 5 minutes?
d. between 10 and 15 minutes?
e. at least 20 minutes?

To graph the density curve of this variable.

The random variable $x$represents the amount of time John waits for the train in minutes.$y=\left\{\begin{array}{l}\frac{1}{30}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{for}0<x<30\\ 0\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\hspace{0.25em}\text{otherwise}\end{array}\right.$
The density function graph can be drawn as:

To show that the area under this density curve to the left of any number $x$between 0 and 30 equals $x/30$.

The base of the rectangle is $x$, and the height of the rectangle is $1/30$, as seen above.
As a result, the rectangle's area is
$\text{Area}=\text{Base}\times \text{Height}$
$=\left(x\right)\left(\frac{1}{30}\right)$
$=\frac{x}{30}$
Since, it is proved

To find the percentage of the time does John wait for the train less than 5 minutes.

Determine the percentage of time spent waiting for the train by John is less than 5 minutes, as follows:
Area less than 5 minutes as:

$=\frac{5}{30}$

$=0.1667$

$=16.67\%$

As a result, the percentage of the time does John wait for the train less than 5 minutes is $16.67\%$.

To find the percentage of the time does John wait for the train between 10 and 15 minutes.

Determine the percentage of the time for which the John waits for the train is between 10 and 15 minutes as follows:
Area between 10 and 15= Area to the left of 15 - Area to the left of 10
$=\frac{15}{30}-\frac{10}{30}$
$=\frac{5}{30}$
$=16.67\%$
As a result, the percentage of the time for which the John waits for the train is between 10 and 15 minutes is $16.67\%$.

To determine the percentage of the time for which the John waits for the train is at least 20 minutes.

Determine the percentage of the time for which the John waits for the train is at least 20 minutes as follows:
Area to the right of 20 = 1 - Area to the left of 20
$=1-\frac{20}{30}$
$=\frac{10}{30}$
$=33.33\%$

Asa result, the percentage of the time for which the John waits for the train is at least 20 minutes is $33.33\%$.