Final answer:
The fraction of the initial mass of a melting snowball that remains after half an hour is 1/8 or 0.125, calculated by comparing the initial and reduced volumes due to the snowball's shrinking radius.
Step-by-step explanation:
The question asks what fraction of the initial mass of a melting snowball remains after half an hour. The volume of a sphere is given by the formula V = (4/3)πr3. The radius of the snowball after t hours is represented as 1−t meters. To calculate the volume after half an hour, we plug r = 1 - 0.5 = 0.5 meters into the formula.
Initial volume (when t = 0): Vinitial = (4/3)π(13) = (4/3)π
Volume after half an hour: Vhalf-hour = (4/3)π(0.53) = (4/3)π(0.125)
The fraction of the volume remaining is Vhalf-hour / Vinitial = ((4/3)π*0.125) / ((4/3)π) = 0.125.
Since the mass is directly proportional to the volume for a given density, the fraction of the initial mass that remains after half an hour is also 1/8 or 0.125.