what is the boats speed in still water when the current is 3 miles per hour and the boat goes 36 miles upstream and back down the 36 miles in 5 hours.

Answer 1

t is the time upstream

R is the speed of the boat

distance = rate * time

36 = (R+3)*t <-- upstream

36 = (R-3)*(5-t) <-- downstream

36 = Rt + 3t <--- distributive property for 1st equation

36 = 5R - Rt - 15 + 3T <--- FOIL method for 2nd equation

Rt + 3t = 5R - Rt - 15 + 3t <--- sets them equal, as they both equal 36

Rt = 5R - Rt - 15 <--- subtracts 3t from both sides, so 3t cancels out

2Rt = 5R - 15 <--- adds Rt to both sides, so it goes from right to left, changing sign

2Rt = 5(R-3) <--- factors out 5 on right side

(2/5)t = (R-3)/R <-- divides both sides by 5

t = (5/2)(R-3)/R <--- multiplies both sides by 5/2

36 = (R+3)(5/2)(R-3)/R <--- substitutes into the first equation

36 = (5/2)(R^2 - 9) / R <--- FOIL

36R = (5/2)(R^2 - 9) <--- multiplies both sides by R

72R = 5(R^2 - 9) <--- multiplies both sides by 2

72R = 5R^2 - 45 <--- distributive

0 = 5R^2 - 45 - 72R <--- subtracts 72R from both sides

0 = 5R^2 - 72R - 45

Factors,

0 = ( 5R + 3)( R - 15 )

5r+3=0 results in negative speeds

R-15=0 ---> R=15

The speed of the boat is 15 mph

downstream, it goes 36 miles in 15+3=18 mph, or 2 hours.

upstream, it goes 36 miles in 15-3 = 12 mph , or 3 hours.

That is a total of 5 hours!

The speed of the boat is 15 mph