We will assume that the rate at which the snowball is melting is proportional to its surface area. Show that the radius of the snowball is changing at a constant rate. {{CMD_68}} First, we {{CMD_69}}introduce the variables{{CMD_70}} {{MATH_71}}, {{MATH_72}}, and {{MATH_73}}, denoting the volume, the radius, and the surface area of the snowball, in that order . Then, we {{CMD_74}}identify{{CMD_75}} the given rate {{MATH_76}} and the unknown rate {{MATH_77}}, the rate to be determined. This problem is a bit unusual, because``the given rate" is not explicitly given. We will deal with this issue below. Next, we {{CMD_78}}draw a picture{{CMD_79}}. {{CMD_80}} {{ENV_81}} {{CMD_82}} In order to {{CMD_83}}find equations{{CMD_84}} that relate all relevant variables, we recall formulas for the volume and the surface area of the sphere, {{MATH_85}} Now the key words are ``the rate at which the snowball is melting (the given rate) is proportional to its surface area.'' From this we have the following equation: {{CMD_86}} {{ENV_87}} {{CMD_88}} We need to compute {{MATH_89}}. Since {{MATH_90}} is a function of {{MATH_91}}, we differentiate both sides of equation {{MATH_92}}. So {{MATH_93}} On the other hand, {{MATH_94}} Therefore, {{MATH_95}} Now, we {{CMD_96}}solve{{CMD_97}} for {{MATH_98}} and obtain that {{ENV_99}} Hence, the radius is changing at a constant rate. Notice, in this example we did not have to evaluate quantities at particular time, because the unknown rate did not depend on time. {{CMD_100}} {{CMD_101}}