Amanda is the manager of a venture capital firm in charge of raising funds for a new start$-$up$.$ She has $18$ days for this task and plans to divide them between a first phase of search of potential investors and a second phase of persuasion of the pool of investors to increase their funding. The number of investors she recruits is proportional to the duration of phase $1,$ and the amount that each of them invests is proportional to the duration of phase $2.$ Amanda is going to be compensate in proportion to the total funds she raises.a$)$ Show that the two phases will have the same duration.Amanda estimates that it takes $3$ days to recruit one investor and that each day spent on persuading the pool increases their individual funding by $1$ million $($that is$,$ if she spends k days on the persuasion phase, every single investor will invest k$).$b$)$ If everything goes as planned, how many investors will Amanda recruit? How much will they invest?For the next project, Amanda has the target of raising $75$ millions, no deadline, and will receive no additional compensation if she raises more than that. She still plans to divide the task into the two phases above, and munber of investors and amount of funds per investor are still proportional to the duration of the two phases, although the exact proportionality rates may differ.c$)$ Show that Amanda still plans to have phase $1$ lasting as much as phase $2.$d$)$ How long will each phase last if number of investors and funds are determined exactly as in the first project?e$)$ At the end of the planned duration of phase $1,$ Amanda realises that it took $5$ days instend of $3$ to recruit ench investor, and this becomes her new estimate of how long it takes to recruit one more. How will she readjust the duration of the two phases? Give a qualitative answer. Is the answer different from the anawer you would have given for the first project?