A spherical snowball melts at a rate proportional to its surface area. Show that the rate of change of the radius is constant. (Hint: Surface area=4πr².)

Verified step by step guidance

1. Start by expressing the surface area of the snowball in terms of its radius using the formula for the surface area of a sphere: A = 4πr².
2. Since the snowball melts at a rate proportional to its surface area, we can write the rate of change of the volume V of the snowball as dV/dt = -kA, where k is a positive constant of proportionality.
3. Substituting the expression for surface area into the volume change equation gives us dV/dt = -k(4πr²).
4. Next, recall the formula for the volume of a sphere, V = (4/3)πr³, and differentiate it with respect to time t to find dV/dt in terms of dr/dt: dV/dt = 4πr²(dr/dt).
5. Now, set the two expressions for dV/dt equal to each other: 4πr²(dr/dt) = -k(4πr²), and simplify to find the relationship between dr/dt and the constants involved.