Which are the similarities between the planetary model of the atom and the solar system?
Among others: i) the planets/electrons travel around nucleus, which is the sun/atom nucleus ii) the planets/electrons travel in defined paths called planetary orbits / atomic orbits iii) most of the mass is located in the nucleus since in relative terms the planets/electrons are weightless.
How can I know if a function I made does really describe the real problem I am trying to solve?
A function always has a dependent and an independent variable, so the first thing is identify such variables in the problem. Then, I would suggest to determine the values of some pair of dependent and independent variables, which must coincide with the values of the dependent variable when you replace the independent variable in the determined function. If the values coincide this is a strong suggestion that the function actually describes the problem.
Why are the units of measurement so important and are mandatory to put them in front of the answer, when applicable?
The units of measurement describe the physical quantities that the problem involves and it is important to address them since they allow any person handling such quantities to compare the results with other previous experiences or problems regarding the same phenomenon or context and deduce whether a right or reasonable answer is or not.