Final answer:
The radius of the metal sphere is decreasing at a rate of approximately 0.715 cm/year.
Step-by-step explanation:
We can determine how fast the radius of the metal sphere is decreasing by relating the change in surface area with the change in radius. The formula for the surface area of a sphere is A = 4πr², where A is the surface area and r is the radius. We are given that the surface area is decreasing at a rate of 450 cm²/year, so we can differentiate the formula with respect to time to find the rate of change of the surface area with respect to time:
dA/dt = 8πr(dr/dt)
Since we are interested in finding how fast the radius is decreasing (dr/dt), we can rearrange the formula to solve for it:
dr/dt = (dA/dt) / (8πr)
Substituting the given rate of change of the surface area (dA/dt = -450 cm²/year) and the radius (r = 25 cm), we can calculate the rate of change of the radius:
dr/dt = (-450 cm²/year) / (8π(25 cm))
dr/dt ≈ -0.715 cm/year