$1.$ Advanced Robotics Corporation $($ARC$)$ has developed a series of high$-$precision robotic arms used in advanced manufacturing processes. ARC has produced a total of $25$ units funded by both internal R&D and a government contract.

Testing has demonstrated that the robotic arms significantly increase manufacturing efficiency. The government has asked ARC to submit a bid for an additional $150$ units, with the requirement that they be modified for increased durability and easier integration into existing manufacturing lines.

The $25$ units produced so far have the following costs $($in millions$)$ listed in the order of production:

Unit Number Cost $($$ Million$)$ Unit Number Cost $($$ Million$)$

$115144\mathrm{.}5$

$213154\mathrm{.}2$

$310164\mathrm{.}0$

$49\mathrm{.}5173\mathrm{.}8$

$59\mathrm{.}0183\mathrm{.}7$

$68\mathrm{.}5193\mathrm{.}6$

$78\mathrm{.}0203\mathrm{.}5$

$87\mathrm{.}5213\mathrm{.}4$

$97\mathrm{.}0223\mathrm{.}3$

$106\mathrm{.}8233\mathrm{.}2$

$116\mathrm{.}5243\mathrm{.}1$

$126\mathrm{.}2253\mathrm{.}0$

$135\mathrm{.}0$

Questions: $($Please use the appropriate chart in the appendix.$)$

a$.$ Based on past production data, what is the learning rate for ARC's robotic arm production? $(10$ marks$)$

b$.$ What bid should ARC submit for the total order of $150$ units, assuming the learning rate continues? $(5$ marks$)$

c$.$ What is the cost expected to be for the last unit $($the $150$th unit$)$ under the estimated learning rate? $(5$ marks$)$