Suppose that you irradiate a beam of monoenergetic alpha particles on a target. You can determine the saturation activity using

Saturation activity = $$\frac{R_a x \sigma \rho N_A}{A}$$

But how do you determine the activity after say half-an-hour of irradiation?

What is confusing me about this is that a radioactive isotope will start to decay as soon as it is produced so I don't know how to quantify this.

This is just a matter of writing down and solving a simple differential equation

Students are often taught all about solving differential equations, and very rarely how to write them down in the first pace, so here goes.

Suppose the number of unstable nuclei at time $$t$$ is $$n(t)$$, with $$n(0)=0$$. The activity is $$\lambda n(t)$$ where $$\lambda$$ is one over the mean lifetime of whatever isotope you're making.

In a short time interval $$\delta t$$, the $$\alpha$$ particles create $${R x \sigma \rho N_A \over A} \delta t$$ more of the unstable nuclei, but there are also $$\lambda n(t) \delta t$$ decays.

So we have $${dn \over dt}={R x \sigma \rho N_A \over A} - \lambda n$$

That's the equation. The easiest way to solve it is 'by inspection' - the answer is

$$n(t)={R x \sigma \rho N_A \over \lambda A}(1-e^{-\lambda t})$$

As you can see, this gives $$n(0)=0$$ as required, and at large $$t$$ it gives the saturation value.

please , see page 43 the general activation equation : https://www.nrc.gov/docs/ML1122/ML11229A714.pdf