Final answer:
The rate at which the diameter decreases when the diameter is 10 cm is -0.119 cm/min.
Step-by-step explanation:
We can use the formula for the surface area of a sphere, which is A=4πr², where A is the surface area and r is the radius (or half the diameter) of the sphere.
Given that the surface area is decreasing at a rate of 3 cm²/min, we can find the rate at which the radius is decreasing.
Using the chain rule, we can relate the rates of change of the surface area and the radius as follows:
- dA/dt = (dA/dr)(dr/dt)
- -3 = (8πr)(dr/dt)
- (dr/dt) = -3/(8πr)
Substituting the given diameter of 10 cm, we can find the rate at which the diameter is decreasing:
- (dr/dt) = -3/(8π(10/2))
- (dr/dt) = -0.119 cm/min (rounded to three decimal places)