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# The average spped of the winner of a race was 2.78 miles per hour faster than the person who finished 1 hour behind him. find the time that the winner took to

run the 16 mile course

### 2 Answers by Expert Tutors

Ira S. | Bilingual math tutor and much moreBilingual math tutor and much more
5.0 5.0 (257 lesson ratings) (257)
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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.
Arthur D. | Effective Mathematics TutorEffective Mathematics Tutor
5.0 5.0 (7 lesson ratings) (7)
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d=r*t
16=(r+2.78)*t (the winner was 2.78 mph faster)
16=(r)*(t+1) ( the loser took 1 hour longer to finish)
they are both equal to 16 so set them equal to each other
(r+2.78)*t=(r)*(t+1)
rt+2.78t=rt+r
subtract rt from both sides
2.78t=r
substitute into either equation; I used the first
(2.78t+2.78)*t=16
2.78t2+2.78t-16=0
[(-2.78+(√(2.782+4*2.78*16)]/5.56
[(-2.78+√7.7284+177.92)]/5.56
[(-2.78+√185.648)]/5.56
(-2.78+13.6252)/5.56
10.84528/5.56
1.95 hr=t
t=1.95 hr for the winner
t=1.95+1=2.95 hr for the loser
16=(r+2.78)*1.95
16=1.95r+5.421
16-5.421=1.95r
10.579=1.95r
10.579/1.95=r
5.425=r
r=5.425 mph for the loser
r=5.425+2.78=8.205 mph for the winner
check:
16=8.205*1.95=15.9997 for the winner
16=5.425*2.95=16.0037 for the loser