System of Equations

Question

A motorboat maintained a constant speed of 11 miles per hour relative to the water in going 10 miles upstream and then returning. The total time for the trip was 5.5 hours. Use this information to find the speed of the current.

Answer 1

v_b = speed of the boat relative to the current = 11 mi/hr

d = distance each way = 10 mi

t = total time of round trip = 5.5 hr

v_c = speed of the current that we're trying to solve for.

boat speed

boat speed upstream = v_b – v_c = 11 – v_c

boat speed downstream = v_b + v_c = 11 + v_c

d = 10 = vt so t = d/v = 10/v

time upstream = 10/(11 – v_c)

time downstream = 10/(11 + v_c)

total time

t = time upstream + time downstream = 5.5 hr, so

10/(11 – v_c) + 10/(11 + v_c) = 5.5

multiply both sides by (11 + v_c)(11 – v_c) to get

10(11 + v_c) + 10(11 – v_c) = 5.5(11 + v_c)(11 – v_c) = 5.5(121 – v_c²)

= 110 + 10v_c + 110 – 10v_c = 220 = 5.5(121 – v_c²)

divide both sides by 5.5 to get

220/5.5 = 40 = 121 – v_c²

Add v_c² – 40 to both sides to get

v_c² = 121– 40 = 81

Take the square root of both sides to get

v_c = √81 = 9

Answer

The speed of the current 9 mi/hr

Check

time upstream = 10/(11 – 9) = 10/2 = 5 hr

time downstream = 10/(11 + 9) = 10/20 = 0.5 hr

total time = 5 + 0.5 = 5.5 hr

Answer 2

11 mph = b = boat speed

9 mph = c = current speed

10 miles = (b+c)t = 20t,t= 1/2 hour downstream time

10 mile = (b-c)(5.5-t) = 2(5.5- .5) = 2(5) = 10, 5 hours up stream time

5+1/2 = 5.5 hours for 20 miles

current speed = 9 mph

Answer 3

Let x = speed of current.

Then, speed upstream = 11 - x and speed downstream = 11 + x.

Distance = (Rate)(Time), so Time = Distance / Rate

Time upstream + Time downstream = 5.5

10 / (11 - x) + 10 / (11 + x) = 11 / 2

20(11 + x) + 20(11 - x) = 11(11 - x)(11 + x)

440 = 11(121 - x²)

40 = 121 - x²

x² = 81

x = 9 mph