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

MATH110 December 2012

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[hide] 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.

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

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[show] Solution
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.