Kim drove 84 miles to see her grandmother. Nine miles of the trip were inside the city limits. Her average speed outside the city was 12 mph faster than inside the city. The whole trip took 3 hours. How fast did Kim go while inside the city?
We don’t know r, and we don’t know the individual t‘s, but we know that the total t is 3. So we fill in the boxes like this:
d | r | t | |
in | 9 | r | 9/r |
out | 84 – 9 = 75 | r + 12 | 75/(r + 12) |
And we get this: 9/r + 75/(r + 12) = 3
9(r + 12) + 75r = 3r(r + 12)
9r + 108 + 75r = 3r^{2} + 36r
3r^{2} + 36r – 9r – 75r – 108 = 0
3r^{2} – 48r – 108 = 0
3(r^{2} – 16r – 36) = 0
(r – 18)(r + 2) = 0
r = 18 or -2 (extraneous)
r = 18 in the city and r + 12 = 30 outside the city.
9r + 108 + 75r = 3r^{2} + 36r
3r^{2} + 36r – 9r – 75r – 108 = 0
3r^{2} – 48r – 108 = 0
3(r^{2} – 16r – 36) = 0
(r – 18)(r + 2) = 0
r = 18 or -2 (extraneous)
r = 18 in the city and r + 12 = 30 outside the city.