A piece of spaghetti has length \(2 c\), where \(c\) is a positive constant. It is cut into two pieces at a random point. The continuous random variable \(X\) represents the length of the longer piece and is uniformly distributed over the interval \([ c , 2 c ]\).
Sketch the graph of the probability density function of \(X\)
Use integration to prove that \(\operatorname { Var } ( X ) = \frac { c ^ { 2 } } { 12 }\)
Find the probability that the longer piece is more than twice the length of the shorter piece.