Find the sum of the volumes

Question

Let $d$ be the distance between the centers of two spheres which are in contact with each other.

Let $A$ be the sum of the surface areas of the two spheres.

Find the sum of the volumes of the two spheres(in terms of $d$ and $A$).

Answer 1

$$R+r=d$$ $$4π(R^2+r^2)=A$$ $$R^2+r^2=\frac{A}{4π}$$ $$(R+r)^2=\frac{A}{4π}+2Rr$$ $$\frac{d^2-\frac{A}{4π}}{2}=Rr$$ Sum of volumes, $$\frac{4π}{3}(R^3+r^3)=\frac{4π}{3}(R^2+r^2-Rr)(R+r)$$ $$ =\frac{4π}{3}(A-\frac{d^2-\frac{A}{4π}}{2})(d)$$ $$ =\frac{8πAd-4πd^3+Ad}{6}$$

Answer 2

Let their radii be $r$ and $R$. Then $A=4\pi(r^2+R^2)$ and $d=r+R$. The sum of volumes is $\dfrac{4}{3}\pi(r^3+R^3)$.

$$r^3+R^3=(r+R)(r^2-rR+R^2)=d\left(\dfrac{A}{4\pi}-\dfrac{1}{2}\left(d^2-\dfrac{A}{4\pi}\right)\right)$$ where we use that $d^2-\dfrac{A}{4\pi}=2rR$.