Question

A spherical raindrop evaporates at a rate proportional to its surface area. Write a differential equation for the volume of the raindrop as a function of time.

Short answer

Question: Write a differential equation describing the volume of a spherical raindrop as a function of time, knowing that the rate of evaporation is proportional to its surface area. Answer: The differential equation is dV/dt + k * (3^(2/3) / (4^(2/3) * pi^(1/3))) * V^(2/3) = 0, where V(t) represents the volume of the raindrop at time t, and k is a positive constant of proportionality.

Step-by-step solution

Define variables and write down given information

Let V(t) represent the volume of the spherical raindrop at time t, and A(t) represent its surface area at time t. We are given that the rate of evaporation (which is the rate of decrease of volume) is proportional to the surface area, i.e., dV/dt = -k * A(t), where k is a positive constant of proportionality.

Express surface area in terms of volume

The volume and surface area of a sphere are given by the following formulas: Volume: V = (4/3) * pi * r^3 Surface Area: A = 4 * pi * r^2 First, we need to find the relationship between the surface area and the volume. We will do this by expressing the radius r as a function of V: r = (3 * V / (4 * pi))^(1/3) Now, substitute this expression for r in the surface area formula: A = 4 * pi * ((3 * V / (4 * pi))^(1/3))^2

Write the differential equation

Now, we need to write a differential equation for the volume of the raindrop with respect to time, knowing that dV/dt = -k * A(t). Substitute the expression for surface area we found in Step 2: dV/dt = -k * 4 * pi * ((3 * V / (4 * pi))^(1/3))^2

Simplify the equation

Simplify the differential equation by canceling out common terms: dV/dt = -k * 4 * pi * (3^(2/3) / (4^(2/3) * pi^(2/3))) * V^(2/3) Cancel the "4 * pi" on the right-hand side: dV/dt = -k * (3^(2/3) / (4^(2/3) * pi^(1/3))) * V^(2/3) This is the required differential equation for the volume of the raindrop as a function of time: dV/dt + k * (3^(2/3) / (4^(2/3) * pi^(1/3))) * V^(2/3) = 0

Key concepts

spherical raindrop

Imagine a perfectly round raindrop, like a small sphere falling from the sky. When we talk about a "spherical raindrop," we're referring to a droplet that has a shape where every point on the surface is equidistant from the center. This symmetry makes it a sphere.
The properties of a sphere, such as volume and surface area, depend on its radius. The larger the radius, the bigger the raindrop will be in terms of volume and surface area. Some important formulas relating to a sphere include:

• Volume (\( V \)) of a sphere: \( V = \frac{4}{3} \pi r^3 \), where \( r \) is the radius.
• Surface Area (\( A \)) of a sphere: \( A = 4 \pi r^2 \).

These formulas are crucial in calculus, especially when studying how raindrops change over time due to processes like evaporation.

evaporation rate

Evaporation rate is a term used to describe how quickly a raindrop loses its volume over time, turning into vapor and disappearing. In the context of our exercise, this rate is crucial because it tells us how the raindrop's size changes with time.
For a spherical raindrop, the evaporation rate is said to be proportional to its surface area. This means:

• The larger the surface area, the faster the raindrop evaporates.
• A smaller raindrop, with less surface area, will evaporate more slowly.

Mathematically, this relationship is expressed as a differential equation where the rate of change of volume (\( \frac{dV}{dt} \)) depends on the surface area (\( A(t) \)):\[\frac{dV}{dt} = -k \cdot A(t)\]where \( k \) is the proportionality constant. This equation shows that evaporation speeds up with larger surfaces but also depends on \( k \), which could vary with conditions like temperature or humidity.

surface area

The surface area of a spherical raindrop is an important factor in determining how quickly it will evaporate. The surface area represents the total area that the droplet exposes to its environment, typically leading to faster evaporation due to more exposure.The formula for calculating the surface area of a sphere is:\[A = 4 \pi r^2\]Where \( r \) is the radius of the sphere. In the case of the raindrop, this means that as the raindrop gets smaller (as \( r \) decreases), its surface area also decreases.
Why is this significant?

• As the evaporation depends on the surface, a smaller surface means slower evaporation.
• The relationship between surface area and volume helps us understand this process better.

Understanding and expressing this surface area in terms of volume helps define the evaporation rate using the spherical formulas provided. It's a key step in forming the differential equation needed for modeling how raindrops diminish over time.

proportionality constant

The proportionality constant, often denoted as \( k \), acts as a scaling factor in our equation that determines the strength of the relationship between surface area and the evaporation rate of the raindrop. It's a vital element that modifies how quickly or slowly the raindrop loses its volume.The use of \( k \) in this context forms the differential equation:\[\frac{dV}{dt} = -k \cdot A(t)\] This shows us that:

• The larger the value of \( k \), the faster the raindrop evaporates.
• A smaller \( k \) results in slower evaporation.
• The constant \( k \) can depend on various external factors such as ambient temperature and wind speed.

Understanding the role of \( k \) helps us model real-world scenarios with accuracy and provides insight into how various conditions can alter the rate at which raindrops evaporate.