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 km/h. Calculate how fast the area of the circle is increasing when the radius is equal to 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!
The formula for the area of a circle is . 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.
The question is asking about change in area, so we will start with the formula for area of a circle: . Because the question involves rates, we will be implicitly differentiating with respect to time. This gives:
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:
Thus, the rate at which the area of the circle is increasing is km/h.