A state trooper is hidden 30 feet from a highway. One second after a truck passes, the angle $\theta$between the highway and the line of observation from the patrol car to the truck is measured. See the illustration.

(a) If the angle measures 15°, how fast is the truck traveling? Express the answer in feet per second and in miles per hour.

(b) If the angle measures 20°, how fast is the truck traveling? Express the answer in feet per second and in miles per hour.

(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 angles should the trooper issue a ticket?

Given that a state trooper is hidden 30 feet from a highway. One second after a truck passes, the angle $\theta$between the highway and the line of observation from the patrol car to the truck is measured.

If the angle measures $15°.$

We can draw a figure of the given problem as: -

We have to find $x.$

In $△ABC,\angle B=90°,AB=30ft,\angle C=15°.$

Now,

$$\begin{gathered} \tan \left(C\right)=\frac{AB}{BC} \\ \tan (15°)=\frac{30}{x} \\ x=\frac{30}{\tan (15°)} \\ x=\frac{30}{0.267} \\ x=112.36feet/sec \end{gathered}$$

Now, we covert x in miles per hour.

$$\begin{gathered} 112.36feet/sec=112.36\times \frac{3600}{1hour}\times \frac{1miles}{5280} \\ 112.36feet/sec=112.36\times \frac{3600}{5280}\frac{miles}{hour} \\ 112.36feet/sec=76.61\frac{miles}{hour} \end{gathered}$$

If the angle measures 20°.

We can draw a figure of the given problem as: -

In $△ABC,\angle B=90°,\angle C=20°,AB=30feet.$

$$\begin{gathered} \tan \left(C\right)=\frac{AB}{BC} \\ \tan (20°)=\frac{30}{x} \\ x=\frac{30}{\tan (20°)} \\ x=\frac{30}{0.363} \\ x=82.645feet/sec \end{gathered}$$

Now, we covert x in miles per hour.

$$\begin{gathered} 82.645feet/sec=82.645\times \frac{3600}{1hour}\times \frac{1miles}{5280} \\ 82.645feet/sec=82.645\times \frac{3600}{5280}\times \frac{miles}{hour} \\ 82.645feet/sec=56.348\frac{miles}{hour} \end{gathered}$$

If the speed limit is 55 miles per hour and a speeding ticket is issued for speeds of 5 miles per hour.

So the speed is 60 miles per hour.

NOw, convert miles per hour in feet per second.

$$\begin{gathered} 60\frac{miles}{hour}=60\times \frac{5280}{3600}\times \frac{feet}{sec} \\ 60\frac{miles}{hour}=88\frac{feet}{sec} \end{gathered}$$

We can draw a figure of the given problem as: -

Now,

$$\begin{gathered} \tan \left(x\right)=\frac{AB}{BC} \\ \tan \left(x\right)=\frac{30}{88} \\ \tan \left(x\right)=0.341 \\ x={\tan }^{-1}(0.341) \\ x=18.83° \end{gathered}$$

So, the angle is$18.83°.$