Water drips into a circular puddle such that the radius of the puddle, in centimeters, at time t, in seconds, is given by the equation r(t) = root t. a. write an equation for the area of the puddle as a function of t. b. what is the aroc of the area of the puddle with respect to time between t = 0 and t = 16? c. what is the aroc of the area of the puddle with respect to the radius between t = 0 and t = 16? d. what is the aroc of the area of the puddle with respect to the circumference of the puddle between t = 0 and t = 16?

Part A

Given that the puddle is circular in shape and that the radius of the puddle, in centimeters, at time t, in seconds, is given by the equation .

Then the area of the puddle is given by the area of a circle =
But, given that , then

Therefore, the equation for the area of the puddle as a function of t is given by

Part B

The average rate of change of a function f(x) between x = a and x = b is given by .

Thus, the average rate of change of the area of the puddle with respect to time between t = 0 and t = 16 is given by

Therefore, the average rate of change of the area of the puddle with respect to time between t = 0 and t = 16 is π.

Part C

The area of the puddle with respect to the radius is given by 

Given that , thus when t = 0, and when t = 16,

Thus, the average rate of change of the area of the puddle with respect to the radius between r = 0 and r = 4 is given by



Therefore, the average rate of change of the area of the puddle with respect to the radius between t = 0 and t = 16 is 4π.

Part D

The circumference of a circle is given by

Thus, the radius of the puddle in terms of circumference is given by

Thus, the area of the puddle with respect to the circumference, C, of the puddle is given by

Since, and , thus when t = 0, r = 0 and C = 0; when t = 16, r = 4 and C = 8π.

Thus the area of the puddle with respect to the circumference, C, of the puddle between C = 0 and C = 8π is given by

Therefore, the average rate of change of the area of the puddle with respect to the circumference of the puddle between t = 0 and t = 16 is 2.

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