meg rowed her boat upstream a distance of 5 miles then rowed back to the starting point. the total time of the trip was 6 hours. if the rate of the current was 2mph, find the average speed of the boat relative to the water.

Meg, Mathematically the information given is:

Tu + Td = 6 hours

D = 5 miles

Current Speed is 2 mph ( assumed downstream of course!)

Boat Speed = S = ???

Avg Speed = Sa = ???

Su = S – 2 upstream speed diminished by current (0)

Sd = S + 2 downstream speed enhanced by current

Su*Tu = 5 miles Tu is the upstream travel time (1)

Sd*Td = 5 miles Td is the downstream travel time (2)

Isolate Tu from (0) & (1):

(S-2)*Tu =5

Tu = 5 / (S-2)

Use Tu + Td = 6 with the previous result:

5/(S-2) + Td = 6

Isolate Td:

Td = (6*S-17)/(S-2)

Substitute this Td expression back into (2), (recall that Sd=S+2):

Sd*Td = 5

(S + 2)*(6*S-17)/(S-2)=5

Simplify into a quadratic expression and factor:

6*S^2 -10*S -24 = 0

( 3*S + 4) * (2S – 6) =0 (Check: 6*S^2 + 8*S - 18*S – 24 = 0 OK!)

Solutions to the quadratic equation:

(3*S + 4) = 0 or S = -4/3 (an unusual speed!)

(2*S – 6) = 0 or S = 3

Check the answers:

S=3

Su = S-2 = 3-2 = 1 mph (3)

Sd = S+2= 3+2= 5 mph (4)

Tu = 5/1 = 5 hrs

Td = 5/5 = 1 hrs

Tot time 6 hrs OK!

S= -4/3

Su=S-2=(-4/3) -2=-10/3

Sd=S+2= (-4/3) +2 = -4/3+6/3 =2/3

Tu = 5/(-10/3)= -15/10

Td = 5/(2/3) = +75/10

Tot time (75-15)/10 = 60/10 + 6 hrs OK but STRANGE!

Caveat: The negative speeds are mathematically correct and lead to the correct total travel time. But note that the Tu=-10/15 is a negative time which is not a real answer. So the correct answers are (3) & (4) and the correct
**Avg Speed = Sa = 1 + 5 = 6 mph**.

*BruceS*