A rock dropped into a pond causes a circular wave of ripples whose radius increases at 6 inches per second.

Question

A rock dropped into a pond causes a circular wave of ripples whose radius increases at 6 inches per second. How fast is the area of the circle of ripples expanding at the instant that the radius of the circle is 12 inches? 24 inches? 100 inches?

 

find dA/dt|||r=12                      =             in2sec

dA/dt|||r=24                            =             in2sec

dA/dt|||r=100                          =             in2sec

A rock dropped into a pond causes a circular wave of ripples whose radius increases at 6 inches per second. How fast is the area of the circle of ripples expanding at the instant that the radius of the circle is 12 inches? 24 inches? 100 inches?

Answer 1

Liz, all you have to do is write the equation of the area of a circle and take the derivative of it with respect to time.

 

A=pi*r^2

 

dA/dt=2*pi*r*dr/dt

 

dr/dt=6 in/s 

 

Hence the equation for the rate of change of area with respect to time becomes:

 

dA/dt=2*pi*r*6=12*pi*r

 

To find these rates of areas at these given radius', you simply substitute the values of r into the area rate equation.

 

dA/dt| | |r=12          =12*pi*12=144*pi in^2/s

 

Kate, I think you got the idea, and I'll let you do the rest of the problem.