A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 50 cm/s. Find the rate at which the area within the circle is increasing after each of the following. (a) after 1 s
(b) after 3 s
(c) after 7 s

1.) 15,708cm^2/s

2) 47,124cm^2/s

3.) 109,956cm^2/s

Step-by-step explanation:

Given the following:

50cm/s is the radius of the ripple per second , that is, radius(r) of ripple after t seconds = speed * time(t)

Speed = 50cm/s

r = 50t

.area of a circle(A) = πr^2

Rate of change of area with radius :

dA/dt = π2r . dr/dt

Speed of ripple created = 50cm/s; this is the rate at which the radius changes with time (dr/dt)

dr/dt = 50cm/s

Rate at which area is increasing with time:

dA/dt = π2r . dr/dt

dA/dt = π2(50t).50

dA/dt = 5000πt

After 1 second:

dA/dt = 5000π(1)

= 15,707.963cm^2/s

After 3 second:

dA/dt = 5000π(3)

= 47,123.889cm^2/s

After 7 second:

dA/dt = 5000π(7)

= 109,955.74cm^2/s