Answer
(a) $V = \frac{4}{3}πh[27 + 9h + h^2]$
(b) Domain $0 < h < ∞$
Work Step by Step
(a) Suppose the radius of the uncoated ball is $r$ and that of the coated ball is $r + h$. Then the plastic has volume
equal to the difference of the volumes, i.e. $V = \frac{4}{3}π(r +h)^3 − \frac{4}{3} πr^3 = \frac{4}{3}
πh[3r^2 + 3rh+h^2] in^3$
But $r = 3$ and hence $V = \frac{4}{3}πh[27 + 9h + h^2]$
(b) $0 < h < ∞$
