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 6 hours. Suppose the round trip is a total of 34 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?

34/2 = 17

Upstream: 17 = b + c
Downstream: 17 = 6(b - c)

Rearrange the first equation....
c = 17 - b
Substitute that into equation 2.
17 = 6(b - (17 - b))

Solve for b, after you do that, then you plug in the value for b and solve for c using this.
c = 17 - b

b = boat speed
c = current speed.

To determine the boat's speed and the current's speed, we can use the concept of relative velocity. Let's break down the problem step by step.

Step 1: Define the variables:
Let's assume the boat's speed is represented by 'B' (in miles per hour) and the speed of the current is represented by 'C' (in miles per hour).

Step 2: Determine the formulas:
We know that the boat's speed relative to the current is the sum of the boat's speed and the speed of the current (B + C). Similarly, the boat's speed relative to the current in the opposite direction is the difference between the boat's speed and the speed of the current (B - C).

Step 3: Set up equations using the given information:
From the problem, we have the following information:
- Trip up (with the current) takes 1 hour.
- Trip down (against the current) takes 6 hours.
- The total round trip distance is 34 miles.

Using the formula d = rt (distance = rate × time), we can set up two equations:

Equation 1: (B + C) × 1 = 34
Equation 2: (B - C) × 6 = 34

Step 4: Solve the equations:
Let's solve Equation 1 for B:
B + C = 34 / 1
B + C = 34 --- Equation 3

Now, solve Equation 2 for B:
6B - 6C = 34
B - C = 34 / 6
B - C = 17 / 3 --- Equation 4

Step 5: Solve the system of equations:
To solve the system of equations (Equations 3 and 4), we can use the method of substitution.
From Equation 3, we have B = 34 - C.
Substituting this value of B into Equation 4, we get:
(34 - C) - C = 17 / 3

Now, solve for C:
34 - 2C = 17 / 3
34 - 17 / 3 = 2C
(102 - 17) / 3 = 2C
85 / 3 = 2C
C = 85 / (3 * 2)
C = 85 / 6
C = 14.16 (approximately)

Step 6: Find the value of B:
Substitute the value of C (14.16) into Equation 3:
B + 14.16 = 34
B = 34 - 14.16
B = 19.84 (approximately)

Therefore, the boat's speed is approximately 19.84 miles per hour, and the current's speed is approximately 14.16 miles per hour.