Question 63860
the racers must complete 210 mi. One cyclist, traveling 10 mph faster than a second cyclist, covers the distance in 2.4 hrs less time than the second cyclist. Find the rate of the first cyclist.

:
Let s = the speed of the 1st cyclist
Let (s-10) = speed of the 2nd cyclist
:
Write a time equation: t = Dist/speed:
:
1st cyclist time + 2.4 hr = 2nd cyclist time
   210/s + 2.4 = 210/{s-10)
:
Mult the equation by s(s-10) to get rid of the denominators:
   210(s-10) + 2.4(s(s-10) = 210s
210s - 2100 + 2.4s^2 - 24s = 210s
2.4s^2 - 24s + 210s - 210s - 2100 = 0
2.4s^2 - 24s - 2100 = 0
:
Solve for s using the quadratic equation: a=2.4, b = -24; c = -2100
:
{{{x = (-b +- sqrt( b^2-4*a*c ))/(2*a) }}} 
{{{s = (-(-24) +- sqrt( (-24)^2- 4*2.4*-2100 ))/(2*2.4) }}} 
{{{s = (+24 +- sqrt(576 - (-20160)))/(4.8) }}} 
{{{s = (24 +- sqrt(576 + 20160))/(4.8) }}} 
Do the math and you should get a positive solution of:
s = + 35 mph is the speed of the 1st cyclist
:
Check solution using the time difference:
210/35 - 210/25 = 2.4 hrs