Problem $2.$ company plans to launch a new consuner electronic device on Jannary $1$ next year. It estimates that the market size for this device is $170,000-1$ its, with coeflicients of imovation and imitation $0\mathrm{.}07$ and $0\mathrm{.}31,$ respectively, when time is measured in years.

$($a$)$ When will the demand rate peak? What will be the demand rate at that time, and how many units will have been sold by that time?

$($b$)$ Plot the demand rate and cumulative demand as a function of time in years.

$($c$)$ Roughly when will the cumulative sales reach $90\%$ of the market size?