$.$ The Nova Tech Solutions Company is considering launching a new app aimed at helping

students with STEM education. The company has gathered the following data:

According to educational statistics, one of the regions where it aims to introduce the app has a

student population of $500,000$ between the ages of twelve to seventeen years. Research indicates

that $35$ percent of students are interested in science and technology$-$related subjects, and surveys

suggest that $15$ percent of students engaged in STEM subjects are interested in educational apps.

Nova Tech Solutions has observed from past market analyses that it can establish a dominant

position in a niche market for up to three years, reaching $100$ percent market share. In the best$-$

case scenario, it plans to achieve a $5$ percent market penetration in year $1,10$ percent in year $2,$

and $15$ percent in year $3.$

A$)$ What will be the peak number of app users in the best$-$case scenario over the next three years?

B$)$ What will be the number of app users over the next three years under a worst$-$case scenario?

Assume that the worst$-$case scenario has a penetration rate of $2$ percent in year $1,4$ percent in

year $2,$ and $6$ percent in year $3.$

C$)$ If the most likely case scenario involves half of the best$-$case scenario penetration rates,

calculate the number of app users for years $1,2,$ and $3$