Final answer:
The rate of change of the radius when the area of the circle is 47 square meters.
Step-by-step explanation:
Solution:
Let's denote the radius of the circle by r and the area of the circle by A.
We know that the area of a circle is given by the formula A = πr².
If the area of the circle is decreasing at a constant rate of 61 square meters per minute, then the rate of change of the area is -61 m²/min (negative sign indicates decrease).
We are given that the area of the circle is 47 square meters. Substituting this value into the area formula, we have:
47 = πr²
Solving for r, we get:
r = sqrt(47/π)
To find the rate of change of the radius, we need to differentiate the equation A = πr² with respect to time:
dA/dt = 2πr dr/dt
Substituting the given values: dA/dt = -61 m²/min and r = sqrt(47/π), we get:
-61 = 2π(sqrt(47/π)) dr/dt
Simplifying further, we have:
dr/dt = -61/(2π(sqrt(47/π)))
Calculating this expression will give us the rate of change of the radius.
Learn more about Rate of change, Circle geometry