The long “tail” on a detergent molecule is made up mostly of carbon atoms bonded to other carbon atoms. Why would you expect the tail to be nonpolar?

Detergents are amphilic in nature having hydrophilic head (Polar) and hydrophobic long tail (non-polar).  The nonpolar hydrophobic tail is mostly made up of of carbon atoms bonded to other carbon atoms.

Tail plays a very important function of detergent by break down the surface tension that exists between grease and water. The tail is required to be non-polar because it attaches to fat (grease) or dirt and repels water molecules and form micelle, which get rinse of with detergent. The non-polar end then move towards the surface and weakens the hydrogen bonds in water, which breaks the break in the surface tension of the water.

Hence, tail of detergent is required to be nonpolar to function by break down the surface tension that exists between grease and water and do the proper cleansing of clothes.