Question $2$: Principal$-$Agent Games: Adverse Selection

In October $2021,$ Tesla announced an upgrade to its "full self$-$driving" feature to automate driving on highways and city streets. Because it was in beta mode, Tesla released the upgrade only to its safety$-$conscious owners. To qualify for the upgrade, Tesla introduced a screening mechanism called "Safety Score" that ranks one's driving on a scale from $0$ to $100$ based on factors such as hard braking, aggressive turning, and unsafe following. The goal was to release the upgrade to defensive drivers and not to aggressive drivers.

Suppose that the value of driving a Tesla with the full self$-$driving upgrade is $200($measured as generic units$)$ while the value of driving a Tesla without the upgrade is $100.$ Maintaining a high Safety Score is based on natural driving abilities and effort. Suppose that the average Safety Score among all drivers is $90.$ However, a defensive driver is able to achieve a higher score with less effort: it costs $8$ units of value to raise the Safetyl Score by $1$ point. Aggressive drivers, on the other hand, endure a greater cost of $15$ units to raise the Safety Score by $1$ point $($because aggressive drivers find defensive driving to be boring$).$

a$)$ What is the minimum Safety Score $($provide a whole number that satisfies the condition$)$ that would achieve Tesla's goal of releasing the upgrade only to defensive drivers?