A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm/s.

Question

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 o r and interpret it.

Answer 1

(a) Since the ripple is traveling outward in all directions at a speed of 60 cm/s, the radius is increasing at that same linear speed of 60 cm/s. The radius' initial value is 0 and it is increasing by 60 centimeters every second, so we can model this with a linear function: r = 60t, where 60 is the slope, or the rate of change, of our linear function.

(b) The area of a circle is pi * r². A composed with r, where r is the radius of the circle as a function of time and A is the area of the circle as a function of radius, is a function that gives us the area within the boundary of the circular ripple at some time t. To find this function, substitute our linear function, 60t, for r in the equation A = pi * r². We get:

A = pi * (60t)², or A = pi * 3600 * t²