Answer:
10 square inches per second.
Explanation:
The radius of the circle is given by the equation:
r(t) = (1/π in/s)*t
Where time in seconds.
Remember that the area of a circle of radius R is written as:
A = π*R^2
Then the area of our circle will be:
A(t) = π*( (1/π in/s)*t)^2 = π*(1/π in/s)^2*(t)^2
Now we want to find the rate of change (the first derivation of the area) when the radius is equal to 5 inches.
Then the first thing we need to do is find the value of t such that the radius is equal to 5 inches.
r(t) = 5 in = (1/ in/s)*t
5in*(π s/in) = t
5*π s = t
So the radius will be equal to 5 inches after 5*π seconds, let's remember that.
Now let's find the first derivate of A(t)
dA(t)/dt = A'(t) = 2*(π*(1/π in/s)^2*t = (2*π*t)*(1/π in/s)^2
Now we need to evaluate this in the time such that the radius is equal to 5 inches, we will get:
A'(5*π s) = (2*π*5*π s)*((1/π in/s)^2
= (10*π^2 s)*(1/π^2 in^2/s^2) = 10 in^2/s
The rate of change is 10 square inches per second.