Respuesta :
The diameter of the ripple changes by the following equation = [tex]\frac{dA}{dt} = \frac{\pi }{2} D . \frac{d(D)}{dt}[/tex]
Explanation:
When a pebble or a rock is dropped in the water then the ripples or waves are formed which starts moving radially outward. As the time passes, the radius of the circle increases.
Let D be the diameter of the circle.
[tex]\frac{dD}{dt}[/tex] will be the rate of change of diameter of the circle with respect to time.
Let A be the area of the circle.
[tex]\frac{dA}{dt}[/tex] will be the rate of change of area of the circle with respect to time.
Area of the circle = [tex]\frac{\pi }{4} D^{2}[/tex]
So,
[tex]\frac{dA}{dt} = \frac{d [\frac{\pi }{4} D^{2}] }{dt}[/tex]
[tex]\frac{dA}{dt} = \frac{\pi }{4} \frac{d[ D(t)^{2}] }{dt}[/tex]
Using chain rule,
[tex]\frac{dA}{dt} = \frac{\pi }{4} X \frac{d[D(t)^{2} ]}{d[D(t)]} X \frac{d[D(t)]}{dt}[/tex]
[tex]\frac{dA}{dt} = \frac{\pi }{4} . 2D(t) . \frac{dD}{dt}[/tex]
[tex]\frac{dA}{dt} = \frac{\pi }{2} X D(t) X \frac{d(D)}{dt}[/tex]
[tex]\frac{dA}{dt} = \frac{\pi }{2} D . \frac{d(D)}{dt}[/tex]
Thus, the following equation determines the change in area in terms of diameter with respect to time : [tex]\frac{dA}{dt} = \frac{\pi }{2} D . \frac{d(D)}{dt}[/tex]
