Question 536983: Car traveled 504 mi averaging a certain speed. If the car had gone 7 mph faster, the trip would have taken 1 hour less. Find the average speed.
Answer by lmeeks54(111)   (Show Source):
You can put this solution on YOUR website! This looks hinky, but is actually pretty straightforward. We need to recall the basic formula linking distance (d), time (t), and rate (r):
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d = r * t
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It is useful to think of this as two trips, one that is and one that could be. In both trips, d = 504 miles. We don't know r or t in either trip, but we are given the relationship between the two trips, so we can calculate r and t:
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in the case of multiple unknowns, it is best solved with a system of equations with as many equations as unknowns. In this case two unknowns: r and t, so we need to derive two equations. The easiest method is often substitution, in which in one of the equations, the equation needs to be written (or re-written) in terms of one of the unknowns.
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We are given:
d = 504. We know this is for both equal distance trips.
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Lets rewrite d = r * t in terms of r:
r = d / t
r = 504/t
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The 2nd equation relates the 2nd trip to the 1st: 7 mph faster shaves 1 hr off the time:
r + 7 and t - 1
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2nd equation:
d = (r + 7) (t - 1)
504 = (r + 7)(t - 1)
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Substitute r = 504/t into this 2nd equation:
504 = (504/t + 7)(t - 1)
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Simplify the terms:
504 = 504t/t - 504/t + 7t - 7
505 = 504 - 504/t + 7t - 7
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Subtract 504 from both sides:
0 = 7t - 504/t - 7
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Multiply both sides by t (so we can get rid of the 504/t term):
0 = 7t^2 - 7t - 504
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If we divide everything by 7 to simplify further, we end up with a standard form quadratic equation:
0 = t^2 - t - 72
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Factor the quadratic equation:
0 = (t + 8)(t - 9)
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t = -8 or 9 are the possible solutions to the quadratic equation. t = -8 is a nonsensical answer, so we can reject it. t = 9 hrs is how long the original trip takes. To figure out r, we just go back to the original equation with d = 504 and t = 9:
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d = r * t
504 = 9r
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divide both sides by 9:
504/9 = 9r/9
56 = r
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The original trip of 504 miles takes 9 hours at 56 mph.
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The notional trip means going 7 mph faster and shaving 1 hr off the time:
2nd trip: d = 504, r = 56 + 7, t = 9 - 1
r = 63 mph, t = 8 hrs
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Always check your work:
56 * 9 = 504 checks
63 * 8 = 504 checks
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The average speed for the notional trip is r = 63 mph
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Cheers,
Lee
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