The dollar valuation $V(t)$ of Batcoin as a function of time $t,$ measured in days, is governed

by the differential equation:

$\frac{dV}{dt}=-0\mathrm{.}04V^2+80V$

$($a$)$ For what initial valuations would Batcoin remain priced at the same level?

$($b$)$ Sketch a direction field describing this situation. You may assume that the Batcoin

valuation never dips below zero.

$($c$)$ Would Batcoin ever become essentially worthless? Justify your answer.

$($d$)$ If you purchased a single Batcoin at its market price of $$2500$ today, approximately how

much would your investment be worth six months from now? Use graphical analysis and

explain your answer.