Science:Math Exam Resources/Courses/MATH102/December 2015/Question 16/Solution 1

Let $x=x(t)$ be the distance from the coyote to the tree and $\theta =\theta (t)$ be the angle between the tree and the line connecting the squirrel and the coyote.

Then, since the height of the tree is 6, these variables are related to each other by $\tan \theta ={\frac {x}{6}}$.

From the question, the rate of change of $\theta$ at $x=8$ is given by $\theta '|_{x=8}={\frac {1}{12}}{\text{rad/sec}}$.

On the other hand, taking the derivative on both sides of the equation gives$\tan \theta ={\frac {x}{6}}$, we have

${\begin{aligned}&(\sec ^{2}\theta )\theta '=(\tan \theta )'=({\frac {x}{6}})'={\frac {1}{6}}x'\implies \\&x'=6(\sec ^{2}\theta |_{x=8})\theta '|_{x=8}=6\cdot (1+\tan ^{2}\theta |_{x=8})\cdot {\frac {1}{12}}=6\cdot \left(1+\left({\frac {4}{3}}\right)^{2}\right)\cdot {\frac {1}{12}}={\frac {25}{18}}\end{aligned}}$

Therefore, the coyote is walking at the speed $\color {blue}{\frac {25}{18}}m/sec$.