Question

A state trooper is hidden 30 feet from a highway. One second after a truck passes; the angle θ between the highway and the line of observation from the patrol car to the truck is measured.

a. If the angle measures 15˚, how fast is the truck traveling?

b. If the angle measures 20˚, how fast is the truck traveling?

c. If the speed limit is 55 miles per hour and a speeding ticket is issued for speeds of 5 miles per hour or more over the limit, for what angle should the trooper issue a ticket?

Answer 1

Givens

• The state trooper is 30 feet from a highway.
• The trooper observes the truck 1 second after it passes.

This problem models a right triangle, which is attached.

a.

So, if
`\theta = 18\°`, we apply trigonometric reasons to find the speed of the truck

`tan 38\°= (30ft)/(x)`

Where
`x` is the horizontal distance of the truck

`x=(30ft)/(tan 38\°)\\ x=(30ft)/(0.78)\\ x=38.46ft`

So, if time of the truck is one second, assuming that is a constant movement, the speed would be

`s=(x)/(t)\\ s=(38.46ft)/(1sec)=38.46ft/s`

b.

If
`\theta =20 \°`, doing the same process, we have

`x=(30ft)/(tan 20\°)\\ x=(30ft)/(0.36)\\ x=83.33ft`

So, the speed would be

`s=83.33 ft/s`

c.

First, we need to tranform the speed into feet per second. We know that 1 mile equals 5280 feet, and 1 hour equals 3600 seconds,

`55(mi)/(hr)(1hr)/(3600sec)(5280ft)/(1mi)= 80.67 ft/sec`

So, the horizontal distance of the truck in this case is 80.67 feet, because, the trooper saw it after 1 second.

Now, we use the trigonometric relation to find the angle

`tan \theta= (30ft)/(80.67ft)\\\theta= tan^(-1)( 0.37)\\\theta \approx 20.4\°`

Therefore, the angle limit is 20.4° approximately, that means all angles more than this issue a ticket for excess of speed.

Answer 2

The following are the answers to the questions presented:

a. 5.5 mph

b. 7.45 mph

c. 71.17528 degrees

I am hoping that these answers have satisfied your queries and it will be able to help you in your endeavors, and if you would like, feel free to ask another question.