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A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s.
(a) Express the radius $ r $ of this circle as a function of the time $ t $ (in seconds).(b) If $ A $ is the area of this circle as a function of the radius, find $ A \circ r $ and interpret it.
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03:26
Jeffrey Payo
Calculus 1 / AB
Calculus 2 / BC
Calculus 3
Chapter 1
Functions and Models
Section 3
New Functions from Old Functions
Functions
Integration Techniques
Partial Derivatives
Functions of Several Variables
Ajla K.
October 17, 2021
How do we get r = 60t? I can't figure that part out.
Missouri State University
Campbell University
Oregon State University
Harvey Mudd College
Lectures
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In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.
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all right after the stone has dropped the circular ripples traveling outward at a speed of 60 centimeters per second, and we want to find the radius as a function of time. So to get the radius, you would take the rate of travel in centimeters per second and multiply it by the time in seconds. And that would give you centimeters. And we know the rate of travel is 60 and we'll let t stand for time so the radius will be 60 t Now we'll find the area so we know the area of a circle is pi r squared. But let's replace the are with 60 t and we get pi times the quantity 60 t squared. And if we simplify that, we get 3600 pie times t squared. So what is that? That gives us the area as a function of time rather than the area as a function of radius
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