Science:Math Exam Resources/Courses/MATH100/December 2018/Question 06

MATH100 December 2018

• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q2 (a) • Q2 (b) • Q2 (c) • Q3 • Q4 • Q5 (a) • Q5 (b) • Q5 (c) • Q5 (d) • Q6 • Q7 • Q8 • Q9 • Q10 • Q11 (a) • Q11 (b) •

Other MATH100 Exams

• December 2011 • December 2010 • December 2012 • December 2013 • December 2014 • December 2015 • December 2016 • December 2018 •

[hide] Question 06
Particle $A$ travels with a constant speed of 2 units per minute on the $x$ -axis starting at the point (4,0) and moving away from the origin, while particle $B$ travels with a constant speed of 1 unit per minute on the $y$ -axis starting at the point (0,8) and moving towards the origin. Find the rate of change of the distance between the two particles when the distance between the two particles is exactly 10 units.

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
Use Pythagoras' theorem to relate the positions of the two particles.

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.

If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

[show] Solution
Let $x$ be the distance from the origin to the particle $A$ and let $y$ be the distance from the origin to the particle $B$ . Also Let $a$ be the distance between $A$ and $B$ . Notice that $x,y$ and $a$ are functions of time and we shall thus write them as $x(t),y(t),a(t)$ respectively. By the given information, we have ${\frac {dx}{dt}}=2$ units/min and ${\frac {dy}{dt}}=-1$ units/min. Using the initial positions, then we see that $x(t)=4+2t$ and $y(t)=8-t$ . We can relate these two functions by Pythagoras' theorem: we have $x(t)^{2}+y(t)^{2}=a(t)^{2}$ , and therefore $a(t)^{2}=x(t)^{2}+y(t)^{2}=(4+2t)^{2}+(8-t)^{2}=16+16t+4t^{2}+64-16t+t^{2}=80+5t^{2}.$ Now, we will first determine the time $t$ at which the distance between the particles is $10$ units. We do this by setting $a(t)=10$ and solving for $t$ . This gives $100=80+5t^{2}$ and therefore $t=2$ (we can ignore the negative root since time cannot be negative). Next, we want to find $a'(t)$ when $a(t)=10$ . By differentiating both sides of the equation $a(t)^{2}=80+5t^{2}$ , we get $2a(t)a'(t)=10t$ . Now we can plug in $a(t)=10$ and $t=2$ , which gives $\color {blue}a'(2)=1{\text{ unit per min}}$ .