run the 16 mile course

Remember r*t=d so if the slower runner runs at a rate of r, the winne is r + 2.78. since they both ran 16 miles, we can represent each times as 16/(x+2.78) and 16/x for the winner and other runner. Since the winner ran in a shorter time by an hour, we can write the equation 16/(x+2.78) + 1 = 16/x . making lcd and canceling we get 16x + x^2 + 2.78x = 16x + 44.48 . This becomes

x^2 + 2.78x - 44.48 = 0. Plug into quadratic formula and you get two answers, one of which is negative and rejected. The other answer comes to be approximately 5.42 mph. plug in other expressions to get 8.2 mph and the two times result in 1.95 hr and 2.95 hr. Therefore winning time of 1.95 hr.