Radioactive Activity of a Gamma source

[MWoot]

Homework Statement

The issue I have is that I am asked to estimate the activity of a small source of gamma radiation; the question is written below:

'A small source of gamma radiation is placed at a distance of 160mm from a detector of area 18mm^2. The count recorded on the detector after 30 minutes was 15804. Estimate the activity of the source.

2. The attempt at a solution

As I understand it, activity in Becquerels is just the rate of decay of a given nucleus, i.e. 1Bq is one nucleus decay per second. I have calculated the 'counts-per-minute' of the source, simply as (15804/30)=526.8CPM, or 8.78 Counts per second. I have read in various places that the CPM can be converted into Becquerels 'if one applies a number of significant conversions that take into account the radiation background, the detector efficiency, the counting geometry, the sample size, and the self-absorption of the radiation by the sample.' The only relevant information to take into account in this example if the sample size/counting geometry, since the area of the detector and the distance of the source from the detector is given. The total volume of the area between the detector and the γ source is (160*18)=2880mm^3, or 2.88*10^-6m^3 as I understand it.

The answer given is 1.57*10^5 Bq.

Help would be massively appreciated, as I am completely stumped

 

[Astronuc]

One has to consider the geometric effect of the detector with respect to the source. Assuming that the gammas are emitted isotropically, the gamma flux at some distance would be uniform. The detector captures only a fraction of the total number of gammas at that distance.

One has to determine the solid angle of the spherical surface (defined by the distance between source and detector) that the detector area subtends.

Is the counting instantenous, e.g., a few seconds compared to 30 minutes, or is the counting occurring over 30 minutes.

One would not simply divide the number of counts per counting time. That would only give an average count rate, not a count rate as a function of time. One would have to integrate the decay curve between the time counting started and the time it stopped in order to determined the decay function. However, in this case, that approach does not apply.

The question appears to ask "what is the decay rate after 30 minutes?".