Question

Question #acc14

Answer 1

Find a relationship between `theta` and the horizontal distance.

Explanation:

Suppose the car is north and west of the observer, moving toward and past him/her, with angle `theta` between 0 and `pi`.

Draw a triangle connecting the following three points:
location of observer (Bob).
location of car
the point 10 meters north of Bob (where the car will pass).
Label the length of the side of the triangle that measures 10 meters.
Label as x the length of the other leg of the triangle.
You do not need to label the hypotenuse.

Observe that there is a relationship between x and `theta`, given by `10/x = tantheta`.
We may also write...
`x/10 = cottheta`.
That is,
`x = 10cottheta`.
This gives us a relationship between their derivatives:
`dx/dt = -10csc^2(theta)*(d(theta))/dt`
Since the car is approaching Bob right now, `dx/dt = -20`, so...

`-20 = -10csc^2(theta)*(d(theta))/dt`
Solve for `(d(theta))/dt`.
`2sin^2(theta) = (d(theta))/dt`

Now, for what value of `theta` is this largest? Doesn't it occur where `sin(theta) = 1`? Hmm... draw a conclusion.