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

MATH100 B December 2023

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[hide] Question 11
A car is driving at ${\textstyle 20}$m/s along a straight road. A police officer is standing ${\textstyle 4}$m to the side of the road with a radar gun measuring speeds. The measured speed is the rate of change of the distance between car and officer. What is the measured speed when the car has not yet passed the officer and is 3m away from the closest point on the road to the officer?

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

[show] Hint
Draw a picture! This is a related rates question.

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[show] Solution

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. Q11 MATH100B solution related rates