Question

A spherical raindrop evaporates at a rate proportional to its surface area with (positive) constant of proportionality k; i.e. the rate of the change of the volume exactly equals -k times the surface area. Write differential equations for each of the quantities below as a function of time. Find the volumne of the drop, dV/dt.

Answer 1

The rate of change of volume of a spherical raindrop can be represented by the differential equation dV/dt = -k * 4πr^2, where V is the volume, t is time, and r is the radius of the raindrop.

Step-by-step explanation:

To write the differential equation for the rate of change of volume of a spherical raindrop as a function of time, we need to consider the given information. The rate of change of volume is exactly equal to the negative constant of proportionality (k) times the surface area of the raindrop. The surface area of a sphere is given by the formula A = 4πr^2, where r is the radius of the sphere.

So, we can write the differential equation for the rate of change of volume (dV/dt) as:

dV/dt = -k * 4πr^2

where V is the volume of the raindrop, t is time, and r is the radius of the raindrop.