Science:Math Exam Resources/Courses/MATH100 B/December 2023/Question 11/Solution 1

Consider the picture below. Note that it is drawn at the point in time when $d$ is 3, but must remember to treat $d$ as a function of time. The quantity that we are looking for is the rate of change of the distance ${\textstyle \ell }$ between the car and the police officer, when the car is ${\textstyle 3}$m away from the point on the road closest to the officer (labelled ${\textstyle A}$).

We need to come up with an equation for how ${\textstyle \ell }$ changes with respect to the distance ${\textstyle d}$, from the car to point ${\textstyle A}$. This is given through the relation

$$\ell ^{2}=d^{2}+4^{2},$$

where ${\textstyle d}$ and ${\textstyle \ell }$ are both functions of time. Using implicit differentiation, we can find ${\textstyle \ell '}$

$${\begin{aligned}2\ell \ell '&=2dd'\\\ell '&={\frac {dd'}{\ell }}\\&={\frac {dd'}{\sqrt {d^{2}+16}}}.\end{aligned}}$$

Now, substituting in ${\textstyle d=3}$, ${\textstyle d'=-v=-20}$, we have ${\textstyle \ell '=-12}$m/s, so the speed at which the distance $\ell$ is changing at this point in time is 12m/s.

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