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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 4/3πr3. The surface area is 4πr2. 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. At what time t will the snowball be completely melted?

User Mickyjtwin
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2 votes

Answer:

The time will be 25 minutes in which snowball be completely melted.

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=(4)/(3)\pi r^3

The surface area is
S=4\pi r^2

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,


(dV)/(dt)\propto S


(dV)/(dt)=\lambda S ....(1)


V=(4)/(3)\pi r^3


(dV)/(dt)=(4)/(3)\pi 3r^2(dr)/(dt)

Substitute in (1),


(4)/(3)\pi 3r^2(dr)/(dt)=\lambda 4\pi r^2


(dr)/(dt)=\lambda


r=\lambda t+c

Now, t=0 , r=10

So,
10=\lambda(0)+c


c=10

i.e.
r=\lambda t+10

After 5 minutes, t=5 , r=8


8=\lambda (5)+10


5\lambda=-2


\lambda=-(2)/(5)

The equation form is
r=-(2)/(5)t+10

The snowball be completely melted means radius became zero.


0=-(2)/(5)t+10


(2)/(5)t=10


t=(10* 5)/(2)


t=25

The time will be 25 minutes in which snowball be completely melted.

User Dmitry Galchinsky
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