Final answer:
The approximate increase in volume of the snowball after picking up an additional 0.25 inches of snow around its 4-inch diameter is about 3.14159 cubic inches.
Step-by-step explanation:
When your snowball, which originally has a 4 inch diameter, picks up an additional 0.25 inches of snow all the way around, its radius increases by 0.125 inches (as diameter increase is twice the radius increase). Using differentials to approximate the increase in volume of the snowball, we need to first write the formula for the volume of a sphere, which is V = (4/3)πr^3. The differential of volume dV is given by dV = 4πr^2dr, where dr is the increase in radius. Given that the original radius r is 2 inches and dr is 0.125 inches, the approximate increase in snowball volume is dV = 4π(2^2)(0.125) cubic inches.
To calculate this, multiply to get dV = 4π(4)(0.125) = π cubic inches. Numerically, with π approximately equal to 3.14159, the increase in volume is approximately 3.14159 cubic inches.