Final answer:
The length of the radius of the circle at the given time is 10.
Step-by-step explanation:
To solve this problem, we need to set up an equation using the rates of change of the circumference and the area. Let's denote the radius of the circle as 'r'.
The rate at which the circumference is growing is equal to the derivative of the circumference with respect to time, which is 2πr '(prime) (since the circumference of a circle is given by C = 2πr).
The rate at which the area is growing is equal to the derivative of the area with respect to time, which is πr^2 '(prime) (since the area of a circle is given by A = πr^2).
According to the problem, the rate of change of the circumference is 5 times the rate of change of the area. So, we can set up the equation: 5 * 2πr ' = πr^2 '. Now, let's solve for the value of r.
5 * 2πr ' = πr^2 '
10πr ' = πr^2 '
10r ' = r^2
10r = r^2 ' (dividing both sides by r)
10 = r '(dividing both sides by r)
r = 10 '
The length of the radius of the circle at that time is given by r = 10. Therefore, the correct answer is Option A) r.