Differentiation of a sphere -- raindrop evaporating as it falls

In summary, the problem states that a spherical raindrop falls and begins to evaporate at a rate proportional to its surface area. This means that the volume decreases at a rate proportional to the surface area. Using the chain rule and the given proportionality, we can find the rate of change of the radius, or dr/dt.
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moonwzrd
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< Moderator Note -- Thread moved from the technical PF Calculus forum >

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?
 

1. How does the size of a raindrop affect its rate of evaporation?

The size of a raindrop does not significantly affect its rate of evaporation. This is because the surface area to volume ratio of a raindrop remains relatively constant regardless of its size, meaning that the amount of water exposed to the surrounding air is the same. However, larger raindrops may take slightly longer to evaporate due to their increased mass and gravitational pull.

2. What factors influence the rate of evaporation for a falling raindrop?

The rate of evaporation for a falling raindrop is influenced by several factors, including temperature, humidity, wind speed, and the composition of the surrounding air. Higher temperatures, lower humidity, and stronger winds can all increase the rate of evaporation, while the presence of pollutants or other particles in the air can slow it down.

3. How does the shape of a raindrop affect its rate of evaporation?

The shape of a raindrop does not significantly affect its rate of evaporation. As long as the raindrop maintains a spherical shape, the surface area to volume ratio will remain constant. However, if the raindrop becomes distorted or flattened, it may increase the surface area and potentially increase the rate of evaporation.

4. Can a raindrop evaporate completely before it reaches the ground?

It is unlikely for a raindrop to completely evaporate before reaching the ground. This is because the time it takes for a raindrop to fall from the clouds to the ground is usually shorter than the time it takes for the raindrop to evaporate. However, in extreme weather conditions with high temperatures and low humidity, it is possible for a raindrop to evaporate completely before reaching the ground.

5. How does the rate of evaporation change as a raindrop falls through different layers of the atmosphere?

The rate of evaporation for a falling raindrop may vary as it passes through different layers of the atmosphere. In general, the lower layers of the atmosphere (closer to the ground) tend to have higher temperatures and lower humidity, which can increase the rate of evaporation. However, as the raindrop falls and reaches higher altitudes, the temperature and humidity may decrease, potentially slowing down the rate of evaporation. Additionally, wind speed and other environmental factors may also vary at different altitudes, further influencing the rate of evaporation.

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