Rate of change (ice ball melting)

In summary, the radius of an iceball decreases as time progresses, but this change in radius is constant. The ball will eventually melt completely.
  • #1
jisbon
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Homework Statement
Assume a ice ball melts so that volume decreases proportionately to its surface area. It takes eight hours to melt down to 1/8 of its original volume. How much time does it take to melt completely?
Relevant Equations
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So to begin this question, I do know that volume =4/3 pi r cubed, while the surface area) 4 pi r squared.
I will like to clarify some things about the question:
1) does the first sentence means dv/dt is proportional to 4 pi r squared?
2) given the second sentence how am I able to construct an equation out of the values given to me?

Thanks
 
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  • #2
1) If you write v for volume then yes.
2) If you assume a certain proportionality factor (say, k) can you then write up a relationship between change in volume as function of change in time?
 
  • #3
My hint will be to consider the radius of the ball ##r## as a function of time ##r(t)##. Express volume as function of ##r(t)## then calculate ##\frac{dV}{dt}## with implicit differentiation, and setup the differential equation $$\frac{dV}{dt}=k4\pi r^2(t)$$
From what I can infer that differential equation which will have unknown the function ##r(t)## is pretty easy. Use then the information from the second sentence to find a relationship between ##r(8)## and ##r(0)## and then also using the solution of the diff. equation you should be able to determine the constant k. Then knowing k all you need to do is go back to the solution of differential equation and solve the algebraic equation##r(t_0)=0## that is to find the time ##t_0## for which the radius of the iceball becomes zero.
 
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  • #4
Delta2 said:
My hint will be to consider the radius of the ball ##r## as a function of time ##r(t)##. Express volume as function of ##r(t)## then calculate ##\frac{dV}{dt}## with implicit differentiation, and setup the differential equation $$\frac{dV}{dt}=k4\pi r^2(t)$$
From what I can infer that differential equation which will have unknown the function ##r(t)## is pretty easy. Use then the information from the second sentence to find a relationship between ##r(8)## and ##r(0)## and then also using the solution of the diff. equation you should be able to determine the constant k. Then knowing k all you need to do is go back to the solution of differential equation and solve the algebraic equation##r(t_0)=0## that is to find the time ##t_0## for which the radius of the iceball becomes zero.

Ok, from what I gathered.
##\frac{dv}{dt} = kA##
##\frac{dv}{dt} = 4\pi r^2\frac{dr}{dt}##
##4\pi r^2\frac{dr}{dt} = kA##
##4\pi r^2\frac{dr}{dt} =k (4\pi r^2)##
##\frac{dr}{dt} = k## (constant)

##V=\frac{4}{3}\pi R^3##
##V'=\frac{1}{8}V##
Let r be radius after 8 hours and R be original radius.
##\frac{4}{3}\pi r^3 =\frac{1}{8}*\frac{4}{3}\pi R^3##
##R= 2r##

So, since the radius decreased wrt time is constant, can I assume since it lost 0.5 of its original radius in 8 hours, is it true that it will take 16 hours for it to melt completely (aka lose all radius)?
 
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  • #5
I think that is correct.
 
  • #6
There may also be a shortcut to take, if one considers how radius will vary over time as the ball melts.

Edit: sorry, missed that the OP actually did derive that dr/dt is constant :oops:
 
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What is the rate of change of an ice ball melting?

The rate of change of an ice ball melting refers to how quickly the ice ball is melting over a specific period of time. It is usually measured in terms of how much volume or mass is lost per unit of time.

What factors affect the rate of change of an ice ball melting?

The rate of change of an ice ball melting can be affected by a variety of factors, such as the temperature of the surroundings, the initial size and shape of the ice ball, the composition of the ice, and the presence of any external forces or substances.

How can the rate of change of an ice ball melting be calculated?

The rate of change of an ice ball melting can be calculated by measuring the change in volume or mass of the ice ball over a specific period of time. This can be done by placing the ice ball in a controlled environment and recording its measurements at regular intervals.

Can the rate of change of an ice ball melting be predicted?

While the rate of change of an ice ball melting can be estimated based on known factors, it is difficult to accurately predict due to the complexity of the melting process and the potential for external factors to influence it.

How can the rate of change of an ice ball melting be slowed down or sped up?

The rate of change of an ice ball melting can be slowed down by lowering the temperature of the surroundings or by adding a layer of insulation around the ice ball. It can be sped up by increasing the temperature or by exposing the ice ball to external forces, such as heat or pressure.

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