Question 121556
First thing is to realize what 'not to exceed' means.  More than 400 feet exceeds 400 feet, but exactly 400 feet or anything less does not.  So you want to determine v such that {{{d<=400}}}


{{{d=v+(v^2)/20}}} is the given model.  This is a reasonably good model presuming you have a better than average reaction time, dry pavement, new tires, etc.  Be that as it may, you need to determine v such that:


{{{v+(v^2)/20<=400}}}


Multiply by 20


{{{20v + v^2<=8000}}}  (Since you multiplied by a positive number, the sense of the inequality remains the same)


Add -8000 to both sides


{{{v^2+20v-8000<=0}}}


Factor


{{{(v-80)(v+100)<=0}}}


So {{{v<=80}}} or {{{v<=-100}}}  Since {{{-100<80}}} we can exclude this root as redundant.


Check:


{{{d=80 + (80^2)/20=80+320=400}}},
{{{90+(90^2)/20=495>400}}}, and
{{{70+(70^2)/20=315<400}}}.


Answer checks.