# Charlene's question at Yahoo! Answers regarding related rates

#### MarkFL

Staff member
Here is the question:

Rate of change of theta?

An airplane is flying at a constant altitude of 8 miles over a radar at a rate of 450mph. At what rate is the angle theta changing when s=25?
I have posted a link there to this topic so the OP can see my work.

#### MarkFL

Staff member
Hello Charlene,

I would first draw a diagram to represent the problem. I will assume the quantity $s$ represents the distance from the radar station directly to the plane. $R$ represents the location of the radar station, and $P$ the location of the plane. I have let $h$ represent the constant altitude of the plane. All linear measures are in miles. Since we are given:

(1) $$\displaystyle \left|\frac{dx}{dt} \right|=v\,\therefore\,\frac{dx}{dt}=\pm v$$

where $v$ is the speed of the plane, and we know $h$, we want to relate $\theta$, $x$ and $h$. We may do so by using the definition of the tangent function as follows:

$$\displaystyle \tan(\theta)=\frac{h}{x}$$

Now, implicitly differentiating with respect to time $t$, we obtain:

$$\displaystyle \sec^2(\theta)\frac{d\theta}{dt}=-\frac{h}{x^2}\cdot\frac{dx}{dt}$$

Multiplying through by $$\displaystyle \cos^2(\theta)$$ and using (1) we have:

$$\displaystyle \frac{d\theta}{dt}=\pm\frac{hv\cos^2(\theta)}{x^2}$$

Now, from our diagram, we see that:

$$\displaystyle \cos(\theta)=\frac{x}{s}$$

and so we find:

$$\displaystyle \frac{d\theta}{dt}=\pm\frac{hv\left(\frac{x}{s} \right)^2}{x^2}=\pm\frac{hv}{s^2}$$

As we are only asked for the magnitude of the rate of change in $\theta$ with respect to time, we may write:

$$\displaystyle \left|\frac{d\theta}{dt} \right|=\frac{hv}{s^2}$$

Now, using the given data:

$$\displaystyle h=8,\,v=450,\,s=25$$

we find, in radians per hour:

$$\displaystyle \left|\frac{d\theta}{dt} \right|=\frac{8\cdot450}{25^2}=\frac{144}{25}$$