got a tough trig problem!!

[jordan83]

A state trooper sits 35 feet from a highway. One second after a vehicle passes, the angle (theta) between the highway and the line of observation from the patrol car to the vehicle is measured.

a.) If the angle measure 15 degrees, how fast is the vehicle traveling? Express in miles per hour.
b.) If the speed limit is 45 mph, and a speeding ticket is issued for speeds of 5 mph or more over the limit, for what angles should the trooper issue a ticket?

 

[stapel]

Draw a right triangle:

So we're looking at the same picture, have the trooper "above" the car on your paper. The angle "theta" (I'll use "@" for this) will go up here. Draw a vertical line down to indicate the distance from the trooper to the path of the car. From the end of this line, draw an horizontal line to the right indicating the path of the car. Connect "trooper" to "car" with a line for the hypotenuse.

Label the vertical line as "y = 35". Label the horizontal line as "x". Label the angle, up by the trooper, as "@ = 15°".

a) Use trig ratios to find the value of x. This gives you the distance in feet that were covered in one second. So how many feet are covered in sixty seconds? In sixty minutes? Then convert that number of feet to miles.

b) Working backwards, figure out the feet-per-minute that would be covered at 50 mph. Then use trig ratios to find the angle. (In other words, do part (a) in reverse.)

If you get stuck, please reply showing (or describing) your steps. Thank you.

Eliz.

 

[soroban]

Hello, jordan83!

My diagram is different . . .

A state trooper sits 35 feet from a highway.
One second after a vehicle passes, the angle (\(\displaystyle \theta\))
between the highway and the line of observation from the patrol car to the vehicle
is measured.

a) If the angle measure 15 degrees, how fast is the vehicle traveling? Express in miles per hour.
b) If the speed limit is 45 mph, and a speeding ticket is issued for speeds of 5 mph or more over the limit,
for what angles should the trooper issue a ticket?

      T
      *
      |   *                        The trooper is at T.
      |       *
   35 |           *                The car is at C. 
      |               *
      |               θ   *        Angle TCA = θ
      + - - - - - - - - - - - * 
      A          x            C

\(\displaystyle \tan\theta\:=\:\frac{35}{x}\;\;\Rightarrow\;\;x\:=\:\frac{35}{\tan\theta}\)

If \(\displaystyle \theta = 15^o:\;\;x\,=\,\frac{35}{\tan15^o}\,\approx\,130.6\) ft

The car had a speed of \(\displaystyle 130.6\text{ ft/s }\approx\:89\text{ mph.}\) . . . . Bus-ted!

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~

A speed of 45 mph is equivalent to 66 ft/sec.

The right triangle has angle \(\displaystyle \theta\), opp = 35, adj = 66.

Hence: .\(\displaystyle \tan\theta\,=\,\frac{35}{66}\;\;\Rightarrow\;\;\theta\,=\,\arctan\left(\frac{35}{66}\right)\:\approx\:28^o\)