Final answer:
The rate of change of the radius of a circle whose area is increasing at a constant rate of √118 per minute and has an area of √25π is approximately 0.346 per minute, rounded to three decimal places.
Step-by-step explanation:
The area of a circle is given by the formula A = πr². To find the rate of change of the radius as the area increases at a constant rate, we use the derivative of the area with respect to time (dA/dt). Given that dA/dt = √118 per minute and at the instant the area of the circle is √25π, we can solve for dr/dt (the rate of change of the radius).
First, let's find the radius when the area is √25π. Since A = πr², we have πr² = 25π. Simplifying, we get r² = 25, which gives us r = 5. Now we can differentiate both sides of the area formula with respect to time. This gives us d(πr²)/dt = 2πr (dr/dt). We plug in the known values to get 2π(5) (dr/dt) = √118, which simplifies to 10π (dr/dt) = √118. Solving for dr/dt, we get dr/dt = √118 / (10π). Using a calculator, we find that dr/dt ≈ 0.346 (rounded to three decimal places).