Question

How do I find the rate at which the angle of elevation is increasing for a hot air balloon tracked by a range finder 1000ft from the liftoff point?

Answer 1

To find the rate at which the angle of elevation is increasing for a hot air balloon tracked by a range finder 1000 feet away from the liftoff point, we can use trigonometry and related rates in calculus.

Let's denote the following:
- \( x = 1000 \) ft (the horizontal distance from the range finder to the liftoff point, which is constant),
- \( h \) = height of the hot air balloon in feet (a variable),
- \( \theta \) = angle of elevation from the range finder to the balloon.

Using right triangle relationships, we can write:

\[
\tan(\theta) = \frac{h}{x}
\]

Since \( x \) is constant at 1000 ft, we can rewrite this as:

\[
\tan(\theta) = \frac{h}{1000}
\]

To find the rate of change of the angle \( \theta \) with respect to time \( t \), we will differentiate both sides of the equation with respect to \( t \):

Using the chain rule for differentiation, the derivative of \( \tan(\theta) \) is \( \sec^2(\theta) \cdot \frac{d\theta}{dt} \), and \( h \) is a function of time as the balloon rises, so we also have \( \frac{dh}{dt} \):

\[
\sec^2(\theta) \cdot \frac{d\theta}{dt} = \frac{1}{1000} \cdot \frac{dh}{dt}
\]

Now, we can solve for \( \frac{d\theta}{dt} \):

\[
\frac{d\theta}{dt} = \frac{1}{1000 \sec^2(\theta)} \cdot \frac{dh}{dt}
\]

Here, \( \frac{d\theta}{dt} \) is the rate at which the angle of elevation is increasing and \( \frac{dh}{dt} \) is the rate at which the height of the balloon is increasing (which you will need to know to calculate \( \frac{d\theta}{dt} \)).

To find \( \sec^2(\theta) \), recall that:

\[
\sec(\theta) = \frac{1}{\cos(\theta)} = \sqrt{1 + \tan^2(\theta)}
\]

And since we have \( \tan(\theta) = \frac{h}{1000} \):

\[
\sec^2(\theta) = 1 + \left(\frac{h}{1000}\right)^2
\]

Thus, you can substitute back into your equation:

\[
\frac{d\theta}{dt} = \frac{1}{1000 \left(1 + \left(\frac{h}{1000}\right)^2\right)} \cdot \frac{dh}{dt}
\]

With this formula, you can compute \( \frac{d\theta}{dt} \) if you know the height \( h \) of the balloon at a particular time and the rate \( \frac{dh}{dt} \) at that same moment.