Final answer:
The rate at which the radius of the oil slick is increasing when it is 8 meters is approximately -0.063 meters per hour.
Step-by-step explanation:
In this problem, we are given that a circular oil slick of uniform thickness is caused by a spill of 1 cubic meter of oil. The thickness of the oil decreases at the rate of 0.1 cm/hr as the slick spreads.
To find the rate at which the radius of the slick is increasing when it is 8 meters, we can use the formula for the volume of a cylinder: V = πr²h.
Since the thickness of the oil slick is decreasing at a constant rate, we can use the derivative to find the rate at which the volume is changing concerning the radius: dV/dt = 2πrh * dr/dt.
Substituting the given values, we have dV/dt = 2π(8)(0.001) * dr/dt.
We know that DV/dt = -1 (since the volume is decreasing at a rate of 1 cubic meter per hour), so we can solve for dr/dt:
-1 = 2π(8)(0.001) * dr/dt.
Simplifying the equation, we have -1 = 0.016π * dr/dt.
Solving for dr/dt, we get dr/dt = -1 / (0.016π).
Converting the value to meters per hour, we have dr/dt = -0.063 meters per hour.