Two soap bubbles in vacuum having radii 3 cm and 4cm respectively coalesce under isothermal conditions to form a single bubble.  The radius of the new bubble is _____cm.

Here, $r_1=30cm$ and $r_2=4cm$.

Since a saop bubble has two free surfaces, total surface area of the coalesced bubble=$2\times 4\pi r^2=8\pi r^2$

Potential energy of the coalesced bubble, i.e. $W=8\pi r^{2}T$

Similarly if $W_1$ and $W_2$ are the potential energies of the two given bubbles,

$W_1=8\pi r_1^{2}T$ and $W_2=\pi r_2^{2}T$

Since the total energy of the system is conserved.

$8\pi r^{2}T=8\pi r_1^{2}T+8\pi r_2^{2}T$

 or $r^2=r_1^2+r_2^2={\left(3\right)}^2+{\left(4\right)}^2=25$

 or $r=5cm$.