Science:Math Exam Resources/Courses/MATH104/December 2010/Question 01 (j)

MATH104 December 2010

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Question 01 (j)
A racing car travels along the ellipse $\displaystyle {}x^{2}+2y^{2}=9$ where x and y are measured in kilometers. When the car is at the point (1,2), the x-coordinate of the point is changing at the rate of 3 kilometers per minute. How fast is the y-coordinate changing at that moment?

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 hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.

Hint 1
Here x and y are functions of time, so we have the implicit relation $\displaystyle {}x(t)^{2}+2y(t)^{2}=9.$

Hint 2
Use implicit differentiation!

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. We have that $x(t)^{2}+2y(t)^{2}=9$ and we want to find dy/dt when x=1, y=2 and dx/dt=3. Therefore, we implicitly differentiate and isolate for dy/dt, ${\begin{aligned}2x{\frac {{\textrm {d}}x}{{\textrm {d}}t}}+4y{\frac {{\textrm {d}}y}{{\textrm {d}}t}}&=0\\{\frac {{\textrm {d}}y}{{\textrm {d}}t}}&=-{\frac {x}{2y}}{\frac {{\textrm {d}}x}{{\textrm {d}}t}}.\end{aligned}}$ We then sub in x=1, y=2, and dx/dt=3 to get that $\left.{\frac {{\textrm {d}}y}{{\textrm {d}}t}}\right|_{x=1,y=2}=-{\frac {1}{4}}(3)=-{\frac {3}{4}}$ and so we conclude that the y-coordinate is changing at -3/4 kilometers per minute at that moment.

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