Problem $1$

A contractor tends to bid on a job installing $50$ airport security systems. Because this will be a new line of work for the contractor, he believes there will be a learning effect for the job. After reviewing time records for similar types of activity, the contractor is convinced that an $85$ percent learning rate is appropriate. He estimates that the first job will take his crew $8$ days to install. Calculate the following:

a$.$ The time estimate $($in days$)$ for producing the ${10}^{th}$ installation.

b$.$ The cumulative time estimate $($in days$)$ for the first $10$ installations.

c$.$ The cumulative time estimate $($in days$)$ for the second $10$ installations.

d$.$ The cumulative time estimate $($in days$)$ for all $50$ installations.

e$.$ If each worker assigned to this project works $20$ days per month, how many workers should be assigned to this project to complete it in a month?