You are stationary on the ground,
and are watching a bird fly horizontally away,
at a rate of 10 meters per second.
The bird is located 12 meters above your head.
How fast does the angle of elevation change,
when the horizontal distance between you, and the bird,
is 35 meters?
Let's take a look at the diagram shown on the right.
First, the of elevation is the angle
formed from your line of sight, up to the bird,
this angle here, which we will call angle theta.
And because the bird is flying horizontally away
at a rate of 10 meters per second,
we know the bird is flying in this direction here.
So if we label this horizontal distance, let's say x,
x is increasing at a rate of 10 meters per second,
which we indicate as dx dt equals 10 meters per second.
We also know the bird is located 12 meters above your head,
which we would think of as this distance here.
But again, because we're looking for the angle of elevation,
if this distance up to the bird is 12 meters,
then this leg of the right triangle would also be 12 meters.
And notice how this isn't going to change,
because the bird is flying horizontally.
And let's go ahead and label the distance from your eyes
to the bird, z,
which would be the hypotenuse of the right triangle.
Because we're trying to find how fast
the angle of elation changes,
when the horizontal distance
between you and the bird is 35 meters,
we are looking for d theta dt,
when x, the horizontal distance, is exactly 35 meters.
And now we need to write an equation
that involves the angle theta.
We also want to include 12,
the vertical distance which doesn't change,
and also include x, because we know dx dt
is equal to 10 meters per second.
Notice, in relation to the angle theta,
12 is the opposite side, and x is the adjacent side,
and therefore we'll use the tangent function.
Our equation is going to be
tangent theta is equal to 12 divided by x.
And now from here, because we're looking for d theta dt,
we will differentiate both sides of the equation,
with respect to t.
On the left, we have the derivative of tangent theta,
with respect to t.
On the right,
we have the derivative of 12 divided by x with respect to t,
and let's write 12 divided by x,
as 12x to the power of negative one.
And now going back to the left,
when differentiating tangent of theta,
with respect to t, we need to apply the chain rule.
We first differentiate tangent theta with respect to theta
and then multiply by d theta dt.
The derivative of tangent theta, with respect to theta,
is secant squared theta.
And then we have, again, times d theta dt,
because we need to apply the chain rule,
equals, on the right, derivative of
12x to the power of negative one with respect to t,
is negative 12 times x the power of negative two,
times dx dt.
Again, we also have to apply the chain rule here,
because we have an x term,
and we're differentiating with respect to t.
Next, we need to solve the equation for d theta dt,
and therefore let's multiply both sides of the equation,
by one over secant squared theta.
Let's also write negative 12x the power of negative two,
as negative 12 divided by x squared.
On the left,
one divided by secant squared theta
times secant squared theta, simplifies the one,
leaving us with d theta dt equals, on the right,
one divided by secant theta is equal to cosine theta.
And therefore, let's write one
divided by secant squared theta,
as cosine squared data.
And we still have times negative 12,
divided by x squared times dx dt.
So now we need to determine cosine theta,
and then determine cosine squared theta.
So if we go back to our right triangle here,
remember cosine theta is equal to the ratio
of the adjacent side of the hypotenuse,
which is x divided by z.
And therefore cosine square theta
is equal to x squared divided by z squared.
This gives us d theta dt equals x squared
divided by z squared, times negative 12,
divided by x squared, times dx dt.
Notice here we can simplify.
X squared divided by x squared simplifies to one.
And now notice, in order to determine d theta dt,
we need to determine the value of z,
which is the hypotenuse of the right triangle.
And we can do this because we know one leg is 12,
and we're trying to find d theta dt,
when X is exactly 35 meters.
So if we know x is equal to 35 meters,
and the other leg is 12 meters,
using the Pythagorean theorem,
we know that z squared must equal
35 squared plus 12 squared.
And therefore z is equal to the square root
of the sum of 35 squared and 12 squared,
which gives us z equals 37.
And now we have all the information we need
to determine d theta dt,
when dx dt is equal to 10 meters per second,
and x equals 35 meters.
We have d theta dt equals,
we have negative 12 divided by the square of z,
which is the square of 37 times dx dt, which is 10.
And now let's go the calculator.
The exact value is negative 120 divided by 1,369.
And the decimal approximation, to four decimal places,
would be negative 0.0877.
Because this is measuring the change
in the angle of elevation, with respect to time,
the units here would be radians per second.
I hope you found this helpful.