Differentiation of a sphere -- raindrop evaporating as it falls

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SUMMARY

The discussion centers on the mathematical problem of a spherical raindrop evaporating as it falls through dry air, specifically addressing how the radius of the drop decreases at a constant rate. The evaporation rate is proportional to the surface area, represented by the formula S = 4πr². Participants are tasked with deriving the relationship between the rate of change of the radius (dr/dt) and the surface area, utilizing the chain rule to express this proportionality mathematically.

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  • Understanding of calculus, specifically differentiation and the chain rule.
  • Familiarity with the concept of surface area and volume of a sphere.
  • Knowledge of proportional relationships in mathematical modeling.
  • Basic grasp of rates of change in physics and mathematics.
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  • Explore the application of the chain rule in calculus with specific examples.
  • Study the relationship between surface area and volume in three-dimensional shapes.
  • Investigate differential equations related to rates of change in physical systems.
  • Learn about the principles of evaporation and its mathematical modeling in fluid dynamics.
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Students and educators in mathematics, particularly those studying calculus and its applications in physical phenomena, as well as professionals in fields related to fluid dynamics and environmental science.

moonwzrd
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< Moderator Note -- Thread moved from the technical PF Calculus forum >[/color]

I can't seem to grasp the idea of this problem, any help is much needed. The problem reads, "As a spherical raindrop falls, it reaches a layer of dry air and begins to evaporate at a rate that is proportional to its surface area (S=4πr^2). Show that the radius decreases at a constant rate.
 
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The problem says that the drop "begins to evaporate at a rate that is proportional to its surface area". This means that the volume is decreasing with time proportionally to its surface area S.

Can you write out this proportionality in terms of S?

They are asking you to find the rate at which the radius is changing. This means that they want you to find dr/dt. How can you use the chain rule and the proportionality you wrote out above to get dr/dt?
 

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