Suppose a boat has to make a round trip up and down a river. The trip up takes 1 hour and is with the current. The trip back takes 9 hours. Suppose the round trip is a total of 32 miles, and that the boat's speed and the speed of the current are constant. What is the boat's speed and what is the current's speed?
Speed of boat with no current --- x mph
speed of current ----- y mph
Assuming 16 miles each way,
time with the current = 1 hr
16/(x+y) = 1
x+y = 16 ----> #1
time against the current = 9 hrs
16/(x-y) = 9
9x - 9y = 16----> 2
you now have two simple equations in 2 unknowns
Use your usual method solving these equations.
Hint: the speeds are not whole numbers
To solve this problem, let's assume that the boat's speed is represented by 'b' and the speed of the river's current is represented by 'c'.
Let's break down the information given:
1. The trip up the river takes 1 hour, and the trip back takes 9 hours.
2. The total round trip distance is 32 miles.
Using this information, we can formulate two equations based on the concept of relative speed:
1. When going up the river: Distance = (Speed of Boat + Speed of Current) x Time
2. When going down the river: Distance = (Speed of Boat - Speed of Current) x Time
Given that the distance for the round trip is 32 miles, we can set up the following equations:
Equation 1: 32 = (b + c) x 1
Equation 2: 32 = (b - c) x 9
Let's solve these equations simultaneously to find the boat's speed (b) and the current's speed (c).
First, we can simplify Equation 1:
32 = b + c
Solving for b, we have:
b = 32 - c
Substitute b = 32 - c into Equation 2:
32 = (32 - c - c) x 9
32 = (32 - 2c) x 9
32 = 288 - 18c
18c = 288 - 32
18c = 256
c = 256 / 18
c ≈ 14.22
Now that we have found the speed of the current (c ≈ 14.22), we can substitute it back into Equation 1 to find the boat's speed (b):
b = 32 - c
b = 32 - 14.22
b ≈ 17.78
Therefore, the boat's speed is approximately 17.78 mph, and the current's speed is approximately 14.22 mph.