What is the expected rate of reactions in the detector?

[unscientific]

Homework Statement

A reactor fires out $10^{21}$ neutrinos per second. A detector containing $10m^3$ of liquid which contains 30 carbon atoms every 60 hydrogen atoms. The detector-reactor distance is $1000$. The cross section for the reaction is $\sigma = 10^{-46} m^2$, and the density of the liquid is $870 kg m^{-3}$. Find the expected rate of reactions.

Homework Equations

The Attempt at a Solution

I know that the rate of reactions is $W = \sigma J n \delta x$ where $J$ is incoming particles per second and $n$ is number density and $\delta x$ is thickness.
Number of carbon atoms per unit volume in the detector is $\frac{60}{90} \frac{(870)(10.3)}{(1.67 \times 10^{-27}) (10.3)} = 3.5 \times 10^{29} m^{-3}$. Can I take $J$ to be $10^{21}$? What is the thickness in this case?

 

[unscientific]

Or is it better to use $W = \sigma \frac{J}{A} (nA\delta x) = \sigma \frac{J}{A} N$ where I can find the flux from using $\frac{J}{4\pi r^2}$?