We might also guess that the volume V of a melting snowball...

We might also guess that the volume $V$ of a melting snowball decreases at a rate proportional to its surface area. Argue as in Exercise 55 to find a differential equation satisfied by $V$ . Suppose the snowball has volume 1000 $\mathrm{cm}^{3}$ and that it loses half of its volume after 5 min. According to this model, when will the snowball disappear?

Key Concepts

Differential Equation Modeling

This concept involves expressing a physical process—here, the melting of a snowball—as a differential equation. The idea is to relate the rate of change of a quantity (volume, in this case) to another property (surface area) that affects it, leading to a mathematical model that can be analyzed and solved.

Geometric Relationships for a Sphere

Understanding the relation between a sphere's volume and surface area is crucial. For a sphere, the volume is proportional to the cube of its radius and the surface area to the square of its radius. This implies that the surface area can be written as a constant times the volume raised to the 2/3 power. This connection provides the bridge from a rate proportional to the surface area to a differential equation involving the volume.

Separable Differential Equations

Once a differential equation is set up, the method of separation of variables is a common technique to solve such equations. By isolating terms involving the dependent variable on one side and the independent variable on the other, one can integrate both sides. This approach is key in solving the model derived from the melting snowball scenario.

Initial Value Problems and Finite-Time Extinction

After obtaining the general solution of the differential equation, specific initial conditions—like the initial volume and a volume at a given time—are used to determine the integration constant. This process is known as setting up an initial value problem. In scenarios like the melting snowball, the model predicts that the volume becomes zero in a finite time, a phenomenon referred to as finite-time extinction.

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