Science:Math Exam Resources/Courses/MATH110/December 2012/Question 10

MATH110 December 2012

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

   Other MATH110 Exams

  •  December 2011  •  December 2010  •  April 2012  •  April 2011  •  December 2012  •  April 2013  •  December 2013  •  April 2014  •  December 2014  •  December 2015  •  April 2017  •  April 2016  •  December 2016  •  December 2017  •  April 2018  •  

Question 10
Tsunamis are large ocean waves produced by underwater seismic activity. In deep water, they can travel at extremely high speeds. A tsunami wave spreads outward in a widening circle. Suppose the radius of the circle is increasing at a rate of $900$ km/h. Calculate how fast the area of the circle is increasing when the radius is equal to $100$ km.

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
The formula for the area of a circle is $A=\pi r^{2}$. Try implicitly differentiating this formula with respect to time.

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

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The question is asking about change in area, so we will start with the formula for area of a circle: $A=\pi r^{2}$. Because the question involves rates, we will be implicitly differentiating with respect to time. This gives: ${\frac {dA}{dt}}=\pi (2r){\frac {dr}{dt}}$ Where dA/dt is the rate at which the area of the circle is changing, and dr/dt the rate at which the radius is changing. Plugging in the information from the statement of the question, we have: ${\frac {dA}{dt}}=\pi (2*100)(900)=180,000\pi$ Thus, the rate at which the area of the circle is increasing is $180,000\pi$ km/h.

Click here for similar questions

MER QGH flag, MER QGQ flag, MER QGS flag, MER RT flag, MER Tag Implicit differentiation, MER Tag Related rates, Pages using DynamicPageList3 parser function, Pages using DynamicPageList3 parser tag