A stone is dropped into a lake, creating a circular

oliviayychengwh 2021-12-19 Answered

A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s. Find the rate at which the area within the circle is increasing after 1 s.

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Expert Answer

Deufemiak7
Answered 2021-12-20 Author has 34 answers
Step 1
The ripple travels outward at 60 cm/s, this is the rate of change of the radius with respect to time
drdt=60cms
From the problem description we can also get an equation that gives the radius at time t.
r=60t
Step 2
The area of a circle is A=πr2 , and we need to find the rate that this is increasing, which is dAdt. So differentiate the A equation with respect to time t.
A=πr2
dAdt=π2rdrdt
We know the value of drdt so we can substitute that in. Furthermore we can substitute r=60 t so that we have an equation where all we need to do is plug in the time t to get the rate of change of the area.
dAdt=π2(60t)(60)=7200π
Step 3
So for t=1
dAdt=7200π(1)22619cm2s

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