Step 1: Let $V$ be the volume and $r$ be the radius of the spherical balloon. Volume of sphere is:

$$V = \frac{4}{3}\pi r^3$$

Step 2: Since volume changes at a constant rate, $\frac{dV}{dt} = k$ (constant). Differentiating $V = \frac{4}{3}\pi r^3$ with respect to $t$:

$$\frac{dV}{dt} = 4\pi r^2 \times \frac{dr}{dt} = k$$

Step 3: Separating variables:

$$r^2 \, dr = \frac{k}{4\pi} \, dt$$

Integrating both sides:

$$\frac{r^3}{3} = \frac{k}{4\pi}t + C_1$$

or $r^3 = at + b$ (where $a$ and $b$ are constants)

Step 4: Applying initial conditions:

At $t = 0$, $r = 3$:

$$3^3 = a(0) + b, \quad \text{so } b = 27$$

At $t = 3$, $r = 6$:

$$6^3 = a(3) + 27$$ $$216 = 3a + 27, \quad \text{so } a = 63$$

Step 5: Therefore,

$$r^3 = 63t + 27$$ $$r = (63t + 27)^{1/3} = 3(7t + 3)^{1/3} \text{ units}$$