What determines whether a carbon atom's covalent bonds to other atoms are in a tetrahedral configurationor a planar configuration

Answer

B - the presence or absence of double bonds between the carbon atom and other atoms.

Explanation:

The presence of single, double or triple bond between carbon atoms can dramatically alter the geometry of molecules (hydro-carbons).

For example, Ethane is a hydrocarbon (C2H6) that contains single bond between all carbon atoms. Here, two carbon atoms around form two tetrahedral shapes around 3 hydrogen atoms each. These tetrahedrons can rotate freely around C-C bond.

However, Ethene is a hydrocarbon (C2H4) that has a planar configuration and the rotation around C-C bond is not possible for it. This is because the C-C bond here are double bond, and it is a common feature of double bond that they prevent rotation of bond (Please see figure).

So anytime you see a molecule with double bond, just keep it in mind that it will have a planar configuration.

Hope it helps!