University:
University of Michigan
Course:
MATH 316 | Differential Equations
Academic year:

2022
Views:

220

Pages:

1
Author:

Tatum B

Evaporation of a raindrop. Hint: evaporation refers to the change of volume over time, so you need to relate volume to surface area to solve this problem. Solution: Call the volume of the spherical raindrop V and the surface area S. We are told is that the evaporation rate dV /dt is proportional to S. So, there is some constant k > 0 so that dV = −kS dt where we have the minus sign because evaporation amounts to a decrease of the total volume. Next express S in terms of V . For a sphere of radius r, we have V = 4 3 πr 3 and S = 4πr2 . Use the first to express r as a function of V , r = 3V 1/3 4π , so, The differential equation becomes S = 4π 3V 2/3 4π = 62/3 π 1/3 V 2/3 . dV = −k̃V 2/3 dt where k̃ = 62/3π1/3k is a new positive constant. Here is a puzzle to make sure you’ve got this. If length is measured in centimeters and time in seconds, what are the units of k? 1

Evaporation of a Raindrop

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