Single Variable Calculus, Chapter 3, 3.8, Section 3.8, Problem 43

At what rate is the distance between the friends changing when the distance between them is 200m?
Given: Radius of track = 100m
Speed of the runner = $\displaystyle 7\frac{\text{m}}{s}$
Distance of the runners friend to the center of the track = 200m
Required: rate of change of distance between the two when distance between them is 200m






We use the law of cosines to relate all the given parameters
$a^2 = b^2 + c^2 - 2 (b)(c) \cos \theta \qquad \Longleftarrow \text{ Equation 1}$

$
\begin{equation}
\begin{aligned}
\text{we let } a &= \text{ distance between the runner and his friend}\\
b &= \text{ distance of the runner's friend to the center of the circle}\\
c &= \text{ radius of the circle}
\end{aligned}
\end{equation}
$


By analyzing, you will found out that the distance between the runner to the runner's friend and the angle $\theta$ are the only changing parameters while others are constants. Taking the derivative of the Equation 1 with respect to time and with respect to the changing parameters, we obtain

$\displaystyle \cancel{2a}\frac{da}{dt} = -\cancel{2}(b)(c) (-\sin \theta) \left(\frac{d \theta}{dt} \right)$
$\displaystyle \frac{da}{dt} = \frac{bc}{a}\sin\theta \left(\frac{d \theta}{dt} \right) \qquad \Longleftarrow \text{ Equation 2}$; $b$ is constant because the runners friend is not moving. Also, $c$ is constant because the distance between the center and the number is always fixed because it is the radius of the circle.

To solve for $\theta$, we will use the law of cosine given that $a = 200$

$
\begin{equation}
\begin{aligned}
a^2 &= b^2 + c^2 - 2(b)(c) \cos \theta\\
200^2 &= 200^2+100^2 - 2(200)(100) \cos \theta\\
\theta &= 75.52^\circ
\end{aligned}
\end{equation}
$

To solve for $\displaystyle \frac{d\theta}{dt}$, we know that the runner is travelling around the circle at 7$\displaystyle \frac{\text{m}}{s}$. That is, in each second, he covers an arc of 7m on the race track

Given the formula for arc,


$
\begin{equation}
\begin{aligned}
s = r \theta \text{ ;where } s &= \text{ arc}\\
r &= \text{ radius}\\
\theta &= \text{ angle subtended by the radius}
\end{aligned}
\end{equation}
$

so, $\displaystyle \theta = \frac{s}{100}$

Taking the derivative with respect to time

$
\begin{equation}
\begin{aligned}
\frac{d\theta}{dt} &= \frac{d\theta}{ds} \cdot \frac{ds}{dt} = \frac{1}{100}(7)\\
\\
\frac{d\theta}{dt} &= \frac{7}{100} \frac{\text{rad}}{s}
\end{aligned}
\end{equation}
$


To solve for the required, we will use Equation 2, plugging in the given and computed parameters to obtain $\displaystyle \frac{da}{dt}$


$
\begin{equation}
\begin{aligned}
\frac{da}{dt} & = \frac{bc}{a} \sin \theta \left( \frac{d\theta}{dt} \right)\\
\\
\frac{da}{dt} & = \frac{(200)(100)}{100} \sin (75.52) \left( \frac{7}{100} \right)
\end{aligned}
\end{equation}
$

$\boxed{\displaystyle \frac{da}{dt} = 6.78 \frac{\text{m}}{s}}$