A firm wants to determine the learning curve rate for its manufacturing process. The production time for the fifth unit is $1,192$ hours, and the production time for the tenth unit is $953\mathrm{.}6$ hours.

Calculate the following:

a$)$ The learning curve rate and the progress rate for the manufacturing process.

b$)$ The time required to produce the first unit.

c$)$ The cumulative average time per unit for the first five units. $($Ans: $1,495\mathrm{.}2)$

d$)$ The expected time to produce the $10$th unit.

e$)$ The cumulative average time per unit for the first $10$ units. $($Ans: $1,263)$