Science:Math Exam Resources/Courses/MATH102/December 2011/Question 07 (iii)

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MATH102 December 2011

  •  Q1 (a) i  •  Q1 (a) ii  •  Q1 (b) i  •  Q1 (b) ii  •  Q1 (b) iii  •  Q1 (c)  •  Q2 (a) i  •  Q2 (a) ii  •  Q2 (a) iii  •  Q2 (b) i  •  Q2 (b) ii  •  Q2 (b) iii  •  Q2 (c)  •  Q3  •  Q4 (i)  •  Q4 (ii)  •  Q4 (iii)  •  Q4 (iv)  •  Q4 (v)  •  Q4 (vi)  •  Q5  •  Q6  •  Q7 (i)  •  Q7 (ii)  •  Q7 (iii)  •  Q8 (i)  •  Q8 (ii)  •  Q8 (iii)  •  Q9  •  

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Question 07 (iii)
A particle is moving in the (x,y)-plane along the curve $y=x^{2}$. As it passes through the point P = (2, 4), its y coordinate changes at a rate of 8 units/sec. iii) What is the rate of change of the angle $\theta$ between the positive x-axis and the line connecting the origin and the particle?

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!

Hint
We need to relate the angle with the x and y coordinates. Try using $\displaystyle \tan(\theta )={\frac {y(t)}{x(t)}}$

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.

Solution

Notice that $\displaystyle \tan(\theta )={\frac {y(t)}{x(t)}}$ Taking derivatives with respect to time gives $\displaystyle \sec ^{2}(\theta ){\frac {d\theta }{dt}}={\frac {{\frac {dy}{dt}}\cdot x(t)-{\frac {dx}{dt}}\cdot y(t)}{x(t)^{2}}}$ From the previous parts, we have that $\displaystyle {\frac {dx}{dt}}{\bigg |}_{x=2}=2$ and we are given that $\displaystyle {\frac {dy}{dt}}{\bigg |}_{x=2}=8$ and so plugging in the point $\displaystyle (x(t),y(t))=(2,4)$ into the triangle, we see that $\displaystyle \sec(\theta )={\frac {1}{\cos(\theta )}}={\frac {1}{\frac {2}{\sqrt {2^{2}+4^{2}}}}}={\frac {\sqrt {20}}{2}}={\sqrt {5}}$ Using all this information, we plug in the point $\displaystyle (2,4)$ and see that ${\begin{aligned}\sec ^{2}(\theta ){\frac {d\theta }{dt}}&={\frac {{\frac {dy}{dt}}\cdot x(t)-{\frac {dx}{dt}}\cdot y(t)}{x(t)^{2}}}\\({\sqrt {5}})^{2}{\frac {d\theta }{dt}}&={\frac {(8)(2)-(2)(4)}{(2)^{2}}}\\5{\frac {d\theta }{dt}}&=2\\{\frac {d\theta }{dt}}&={\frac {2}{5}}\\\end{aligned}}$ and this completes the question.

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