Pre-Calculus...please help.?

Question

I have 2 questions that I cannot understand. Help is greatly appreciated.
1)Physics:Uniform Motion-
A motorboat maintained a constant speed of 15mph relative to the water in going 10miles upstream and then returning.The total time for the trip was 1.5 hours.Use this information to find the speed of the current.
2)Computing Average speed
In going from Chciago to Atlanta, a car averages 45mph and in going from Atlanta to Miami, it averages 55mph.If Atlanta is halfway between Chicago and Miami,what is the average speed from Chicago to Miami? Discuss an intuitive solution. Write a paragraph defending your intuitive solution.Then solve the problem algebraically.Is the intuitive solution the same as the algebraic one?If not find the flaw.
Thanks in advance for any help.

Answer 1

First Problem: Rate x Time = Distance

Trip out: 10 miles = ( 15 + v ) ( T1 )
Trip back: 10 miles = ( 15 - v ) ( T2 )
Total time: T1 + T2 = 1.5 hours

You have three equations and three unknowns. Use the first two equations to develop substitutions for T1 and T2, drop them into the third, and solve for v, the speed of the current.

Second Problem: work it out using the same technique as above. The false "intuitive solution" is to average speeds over distances, when speeds can only be meaningfully averaged over time. That is, 45 mph for one hour and 55 mph for one hour is indeed an average of 50 mph for two hours. But 45 mph for 100 miles, then 55 mph for 100 miles, is NOT 50 mph for 200 miles. Instead work out the times per segment, then divide distance by time to get the actual average speed.

Answer 2

dist/vel = time

10/(15+x) + 10/(15-x) = 1.5 x = velocity of current
calc it

intuitively - (45+55)/ 2 = 50
algebraically - 2L / (L/45 + L/55) = 49.5
the flaw is that time taken to cover the distances in both cases is different