Question 147112
Restated, the problem is, when will the sum of their
distances from the starting point exceed 250 mi?
Assume I have a stopwatch. I will start the stopwatch 
when Alan has already driven foor 1 hr.
By then, Alan has driven {{{d[a] = 55*1}}}
{{{d[a] = 55}}}mi
I want to find {{{d[a] + d[d] = 250}}}
But now, {{{d[a]}}}, Allan's distance from the starting
point is {{{r[a]*t + 55}}} where {{{t}}} is the time
showing on my stopwatch.
Now I can write {{{r[a]*t + 55 + r[d]*t = 250}}}
{{{55t + 55 + 65t = 250}}}
{{{120t = 190}}}
{{{t = 12/19}}} hr
{{{t = .6316}}}hr
This is the time after Alan's 1 hr headstart, so they can
talk on their cellphones for
{{{t = 1.6316}}} hr
t = 1 hr 37.895 min
t = 1 hr 37 min 54 sec answer
That's it, unless I goofed.
check:
{{{55t + 55 + 65t = 250}}}
{{{55*1.632 + 55 + 65*1.632 = 250}}}
{{{89.76 + 55 + 106.08 = 250}}}
{{{250.84 = 250}}}
pretty close