Answer:
The time will be 25 minutes in which snowball be completely melted.
Step-by-step explanation:
Given : The rate of change of the volume of a snowball that is melting is proportional to the surface area of the snowball. Suppose the snowball is perfectly spherical.
Then the volume (in centimeters cubed) of a ball of radius r centimeters is ![V=\frac{4}{3}\pi r^3](https://tex.z-dn.net/?f=V%3D%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3)
The surface area is ![S=4\pi r^2](https://tex.z-dn.net/?f=S%3D4%5Cpi%20r%5E2)
Set up the differential equation for how r is changing. Then, suppose that at time t = 0 minutes, the radius is 10 centimeters. After 5 minutes, the radius is 8 centimeters.
To find : At what time t will the snowball be completely melted?
Solution :
Using given condition,
![\frac{dV}{dt}\propto S](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdt%7D%5Cpropto%20S)
....(1)
![V=\frac{4}{3}\pi r^3](https://tex.z-dn.net/?f=V%3D%5Cfrac%7B4%7D%7B3%7D%5Cpi%20r%5E3)
![\frac{dV}{dt}=\frac{4}{3}\pi 3r^2\frac{dr}{dt}](https://tex.z-dn.net/?f=%5Cfrac%7BdV%7D%7Bdt%7D%3D%5Cfrac%7B4%7D%7B3%7D%5Cpi%203r%5E2%5Cfrac%7Bdr%7D%7Bdt%7D)
Substitute in (1),
Now, t=0 , r=10
So,
i.e.
After 5 minutes, t=5 , r=8
The equation form is
The snowball be completely melted means radius became zero.
The time will be 25 minutes in which snowball be completely melted.