Some people call these kinds of problems "related rates" problems. If you do a web search for "related rates calculus" you may get some additional information.
I recommend following these steps for this kind of problem:
- Draw a diagram and label the different values that are given and unknown.
- Write an equation that shows the relationship between the changing values. In this case, the equation should include a variable for the horizontal distance between you and the bird, another variable for the angle of your head, and a constant for the 10 m height (constant because it doesn't change).
- Take the derivative of the equation with respect to time. This should give a new equation that has four variables. One variable will be the rate of change of the head angle, another will be the horizontal distance, the third will be the head angle, and the fourth will be the rate of change of the horizontal distance.
- Plug in the given horizontal distance, the calculated head angle, and rate of change of horizontal distance and solve for the rate of change of head angle.
Let's go through this problem. I won't upload a diagram in this case, but the diagram we would draw would be a right triangle. The top point of the triangle is the bird. The other point of the triangle that isn't the right angle is your head. We want to label the angle at the point where your head is - in this case I will call it x. The leg of the triangle opposite x is the height of the bird, so we will label that 10, and the leg of the triangle along the ground is the horizontal distance. We could put 170 there, but it is changing over time so instead we should use another variable - let's use d.
Next, we write our equation. We have to know trigonometry for this problem since our diagram is a triangle. We can relate the three values that we are working with using a tangent function. If we recall SOHCAHTOA, we remember that the tangent of an angle is equal to the length of the opposite leg divided by the length of the adjacent leg. In this case, that means:
tan(x) = 10/d.
Now we take the derivative of both sides with respect to time, because the question is asking for a rate. Notice we have the speed of the bird in meters per second. That is a clue that we will take a time derivative. Because neither of our variables is time, we will use the chain rule for both sides of the equation. If we remember (or look up) our derivatives, when we take the time derivative of both sides, we get:
sec2(x)(dx/dt) = -10/d2(dd/dt)
In this equation, (dx/dt) is the rate of head tilt that we want to find, (dd/dt) is the horizontal speed of the bird, d is the horizontal distance of the bird, and x is the angle of our head when the bird is 170 m away. We are given (dd/dt) as -3 (because as the bird flies it's getting closer which means d is getting smaller) and d as 170. We have to use trigonometry to find x, but that is our original equation, tan(x) = 10/d. Now want to plug in 170 for d and solve for x. Using my scientific calculator I get x = 3.366 radians (approximately). sec2(x) is so close to 1 I think it's ok to consider it 1 (I got 1.003460207).
Plugging in -3 for (dd/dt) and 170 for d, I get only .00103 radians/second. That seems very small. This is a case where I would want to show my work and I'm confident about the algebra and the calculus but probably made a mistake with the calculator. I know what I did wrong. I forgot that we have to be careful with the tan-1 function on the calculator. I should have subtracted pi from the answer I got, because we should have gotten a value close to zero, not close to pi.
Correcting, that makes x = 0.22486 (approximately). However, the secant of a very small angle is the same as the secant of an angle very close to pi, so this doesn't change our answer very much at all. After rounding, it's still 0.00103 radians/second.
But. this does make some sense because the bird is still pretty far away.
