We know,

Surface area of balloon (s) = 4$$\pi$$r²

$$\therefore$$ $${{ds} \over {dt}} = {d \over {dt}}(4\pi {r^2})$$

$$ \Rightarrow {{ds} \over {dt}} = 4\pi (2r) \times {{dr} \over {dt}}$$

$$ \Rightarrow {{ds} \over {dt}} = 8\pi r \times {{dr} \over {dt}}$$

Given that, surface area of balloon is increasing in constant rate.

$$\therefore$$ $${{ds} \over {dt}}$$ = constant = k (Assume)

$$ \Rightarrow k = 8\pi r\,.\,{{dr} \over {dt}}$$

$$ \Rightarrow \int {k\,dt = \int {8\pi r\,dr} } $$

$$ \Rightarrow kt = 8\pi \times {{{r^2}} \over 2} + c$$

$$ \Rightarrow kt = 4\pi {r^2} + c$$ ..... (1)

Given at t = 0, radius r = 3

So, $$0 = 4\pi ({3^2}) + c$$

$$ \Rightarrow c = - 36\pi $$

$$\therefore$$ Equation (1) becomes

$$kt = 4\pi {r^2} - 36\pi $$

Also given, at t = 5, radius r = 7

$$\therefore$$ $$k(5) = 4\pi {(7)^2} - 36$$

$$ \Rightarrow k = 32\pi $$

$$\therefore$$ Equation (1) is

$$32\pi t = 4\pi {r^2} - 36\pi $$

Now at $$t = 9$$

$$ \Rightarrow 32\pi (9) = 4\pi {r^2} - 36\pi $$

$$ \Rightarrow 8 \times 9 = {r^2} - 9$$

$$ \Rightarrow {r^2} = 81 \Rightarrow r = 9$$