Ripples in a Pond

Calculus Level 1

If I throw a stone into a pond, then a ripple will emanate from the point of impact of the stone with the water's surface. As the ripple begins to spread outward, the radius of the ripple grows at 7 inches per second. How large will the circumference of the ripple be (in inches) after 5 seconds?

100 π 100\pi 70 π 70\pi 140 π 140\pi 35 π 35\pi

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Clara Blackstone by Clara Blackstone , 32, USA

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2 solutions

Clara Blackstone
Oct 5, 2015

The circumference of a circle is given by C = 2 π r C = 2\pi r . Differentiating this with respect to time gives us d C d t = 2 π d r d t \displaystyle\frac{dC}{dt} = 2\pi \displaystyle\frac{dr}{dt} . We're given that d r d t = 7 \displaystyle\frac{dr}{dt} = 7 , so this means d C d t = 14 π \displaystyle\frac{dC}{dt} = 14\pi inches per second.

After 5 seconds, the ripple's circumference will be 70 π 70\pi inches.

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Julian Yu
Mar 7, 2016

Since the ripple grows at 7 7 inches per second, after 5 5 seconds the radius of the ripple will be 35 35 inches.

Using the formula C = 2 π r C=2\pi r , the circumference of the ripple is 70 π 70\pi inches.

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