Let $c$ Be a Positive Number $\frac { d y } { d t } = k y ^ { 1 + c }$

Multiple Choice

Let $c$ be a positive number. A differential equation of the form $\frac { d y } { d t } = k y ^ { 1 + c }$ where $k$ is a positive constant is called a doomsday equation because the exponent in the expression $k y ^ { 1 + c }$ is larger than the exponent 1 for natural growth. An especially prolific breed of rabbits has the growth term $k y ^ { 100 }$ . If 3 such rabbits breed initially and the warren has 27 rabbits after 3 months, then when is doomsday?

A) $47.03$ months
B) $97.03$ months
C) $67.03$ months
D) $137.03$ months
E) $117.03$ months