A scarce resource is to be leased for an engineering project. There is a graduated cost associated with using the resource at a certain percentage level, \(U\). A step function has been defined for the cost rates for different levels of utilization. The step cost function is presented as follows:

If \(U<50 \%\), cost \(=\$ 0\)

If \(50 \% \leq U<60 \%, \$ 5,000\)

If \(60 \% \leq U<67 \%, \$ 20,000\)

If \(67 \% \leq U<85 \%, \$ 45,000\)

If \(U \geq 85 \%, \$ 55,000\)

Suppose the utilization level is a random variable following a triangular distribution with a lower limit of \(40 \%\), a mode of \(70 \%\), and an upper limit of \(95 \%\).

a. Plot the step function for the cost of leasing the resource.

b. Plot the triangular probability density function for the utilization level.

c. Find the expected cost of using this resource.