11. The sum S of the first n consecutive positive integers, 1, 2, 3, ... , n, is given by the formula S=0.5n(n+1). Find n when S=120 and check your answer by addition.
We need two numbers whose sum is 1 and multiply to -240, so two numbers whose difference is 1 and multiply to 240, 15 and 16. We factor:
We reject the negative root.
n=15
Check:
1+2+3+4+5+6+7+8+9+10+11+12+13+14+15
=1+15 + 2+14 + 3+13 + 4+12 + 5+11 +6+10 + 7+9 + 8
= 7(16)+8
= 120 good
13. A boat travels downstream (with the current) for 36 miles and then makes the return trip upstream (against the current). The trip downstream took 3/4 of an hour less than the upstream trip. If the rate of the current is 4 mph, find the rate of the boat in still water and the time for each part of the trip.
Let v be the speed of the boat in still water.
The speed downstream is v+4 so the time to travel 36 miles downstream is 36/(v+4)
The speed upstream is v-4 so the time to travel 36 miles upstream is 36/(v-4)
Writing our equation,
3/4 = 36/(v-4) - 36/(v+4)
3 = 144 (1/(v-4) - 1/(v+4))
3(v-4)(v+4) = 144( (v+4) - (v-4) )
v^2 - 16 = 48(8)
v^2 = 48(8) + 16 = 400
v = 20 mph, the rate of the boat in still water
Time downstream is 36/(20+4)=36/24 = 3/2 = 1.5 hours
Time upstream is 36/(20-4)=36/16=9/4 = 2.25 hours
3/4 hour difference, good