Answer:
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
\(tan 38\°= \frac{30ft}{x}\)
Where \(x\) is the horizontal distance of the truck
\(x=\frac{30ft}{tan 38\°}\\ x=\frac{30ft}{0.78}\\ x=38.46ft\)
So, if time of the truck is one second, assuming that is a constant movement, the speed would be
\(s=\frac{x}{t}\\ s=\frac{38.46ft}{1sec}=38.46ft/s\)
b.
If \(\theta =20 \°\), doing the same process, we have
\(x=\frac{30ft}{tan 20\°}\\ x=\frac{30ft}{0.36}\\ x=83.33ft\)
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,
\(55\frac{mi}{hr}\frac{1hr}{3600sec}\frac{5280ft}{1mi}= 80.67 ft/sec\)
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
\(tan \theta= \frac{30ft}{80.67ft}\\\theta= tan^{-1}( 0.37)\\\theta \approx 20.4\°\)
Therefore, the angle limit is 20.4° approximately, that means all angles more than this issue a ticket for excess of speed.
