# Thread: Determine an expression for the instantaneous rate if change.

1. ## Determine an expression for the instantaneous rate if change.

A stone is tossed into a pond, creating a circular ripple on the surface. The radius of the ripple increases at the rate of 0.2m/s.

a) Determine the length of the radius at the following times.

i) 1s *calculated to 0.2m
ii) 3s *calculated to 0.6m
iii) 5s *calculated to 1m

b) Determine an expression for the instantaneous rate of change in the area outlined by the circular ripple with respect to the radius.

This is the part of the question i'm having trouble with. Can someone tell me how to show my work to create the following expression (the answer to b, which I don't know how to get to):

dA/dr = 2pir or (2)(pi)(r) m^2/m

What does dA mean.. delta something? And dr is delta radius I assume.

c) Determine the instantaneous rate of change of the area corresponding to each radius in part a).

Would I just use the expression from b) and sub in my values I got from a) into r to get the answers?

2. $\displaystyle \frac{dA}{dr}$ means the rate of change in the area with respect to the radius. $\displaystyle dA$ can be thought of as the difference in area.

To answer part (c), note that $\displaystyle \frac{dA}{dt} = \frac{dA}{dr} \cdot \frac{dr}{dt}$.

3. Would anyone happen to know how to get to the expression dA/dr = 2pir or (2)(pi)(r) m^2/m?