Answer
A(r)=4$\pi$$r^{2}$+4$\pi$r+$\frac{4}{3}$$\pi$
Work Step by Step
Let the amount of air required as a function of the original radius of the balloon be A(r).
A(r) = V(r+1)-V(r)
= $\frac{4}{3}$$\pi$$(r+1)^{3}$-$\frac{4}{3}$$\pi$$r^{3}$
= $\frac{4}{3}$$\pi$[$(r+1)^{3}$-$r^{3}$]
= $\frac{4}{3}$$\pi$[$r^{3}$+3$r^{2}$+3r+1-$r^{3}$]
= $\frac{4}{3}$$\pi$[3$r^{2}$+3r+1]
= 4$\pi$$r^{2}$+4$\pi$r+$\frac{4}{3}$$\pi$
