Let surface area of the spherical balloon $A=4\pi r^2$

$\frac{dA}{dt}=8\pi r\frac{dr}{dt}=k$ (let) $...\left(1\right)$

On integrating on both sides w.r.t $t$ , we get

$4\pi r^2=kt+C$ . 

Given that, at $t=0, r=3$.

$\Rightarrow 36\pi =C$

Also given that, at $t=5, r=7$

$\Rightarrow 4\pi \times 49=5k+36\pi$

$\Rightarrow 5k=4\pi \left(49-9\right)$   $\Rightarrow 5k=4\pi \times 40$ 

$\Rightarrow k=32\pi$

On substituting $k$ value in equation $\left(1\right)$we get, $4\pi r^2=32\pi t+36\pi$

$\Rightarrow r^2=8t+9$

Given $t=9$.

$\Rightarrow r^2=81\Rightarrow r=9$.