The excursion boat on the river takes 2½ hours to make the trip to a point 12 miles upstream and to return. If the rate at which the boat travels in still water is 5 times the rate of the river current, what is the rate of the current?
Let's denote the rate of the boat in still water as B and the rate of the river current as R.
When the boat is traveling upstream, the effective speed is reduced by the rate of the current, so the boat's speed is B - R. Similarly, when the boat is traveling downstream, the effective speed is increased by the rate of the current, so the boat's speed is B + R.
We are given that it takes 2½ hours to make the round trip, which consists of the trip upstream and the trip downstream. Since the distance traveled upstream and downstream is the same, we can set up the following equation:
(12 miles) / (B - R) + (12 miles) / (B + R) = 2½ hours.
Now we need to represent the relationship between the boat's speed in still water and the rate of the current. The problem tells us that the boat's speed in still water is 5 times the rate of the river current, i.e., B = 5R.
Substituting this expression for B into the earlier equation:
(12 miles) / (5R - R) + (12 miles) / (5R + R) = 2½ hours.
Simplifying the denominators:
(12 miles) / (4R) + (12 miles) / (6R) = 2½ hours.
Combining the fractions:
(36R + 24R) / (24R) = 2½ hours.
Simplifying the numerator:
60R / (24R) = 2½ hours.
Simplifying the fraction:
60R / 24R = 2½ hours.
Cancelling out the common factor of 12:
5 / 2 = 2½ hours.
Now we can solve for the unknown, which is the rate of the river current, R:
5 / 2 = 5/2 hours.
Dividing both sides by 2:
5 / 2 = 2½ hours.
Simplifying the fraction:
5 = 5 hours.
Therefore, the rate of the river current is 5 miles per hour.
To solve this problem, we'll need to set up a system of equations and solve for the rate of the current.
Let's denote the rate of the boat in still water as "B" and the rate of the river current as "C".
We're given that the boat takes 2½ hours to make the trip to a point 12 miles upstream and return. This means that the boat takes 2.5 hours (or 5/2 hours) to travel 12 miles upstream and the same time to travel 12 miles downstream.
Step 1: Finding the equation for the upstream trip
The speed of the boat relative to the ground is the difference between the boat's speed in still water and the speed of the current:
(B - C)
Since the boat takes 2.5 hours to travel 12 miles upstream, we can set up the equation:
(B - C) * 2.5 = 12
Step 2: Finding the equation for the downstream trip
The speed of the boat relative to the ground is the sum of the boat's speed in still water and the speed of the current:
(B + C)
Since the boat takes 2.5 hours to travel 12 miles downstream, we can set up the equation:
(B + C) * 2.5 = 12
Step 3: Solve the system of equations
Let's solve the system of equations to find the values of B and C.
Multiplying the equations by 2 to eliminate the fractions, we have:
2(B - C) = 24 --> Equation 1
2(B + C) = 24 --> Equation 2
Expanding the equations, we get:
2B - 2C = 24 --> Equation 1
2B + 2C = 24 --> Equation 2
Adding Equation 1 and Equation 2 together, the C terms cancel out:
4B = 48
Dividing both sides by 4, we get:
B = 12
Substituting B = 12 into Equation 1 or Equation 2, we can solve for C:
2(12) - 2C = 24
24 - 2C = 24
-2C = 0
C = 0
Therefore, the rate of the current is 0.
To solve this problem, let's break it down into smaller steps:
Step 1: Define the variables
Let's assume:
- The rate at which the boat travels in still water = B (in miles per hour)
- The rate of the river current = C (in miles per hour)
Step 2: Calculate the time taken for the boat to travel upstream
The boat travels upstream against the current. The effective speed of the boat can be calculated by subtracting the rate of the river current from the rate at which the boat travels in still water:
Effective speed upstream = B - C
To calculate the time taken to travel upstream, we can use the formula: time = distance / speed
The distance traveled upstream = 12 miles
The time taken to travel upstream = 12 miles / (B - C)
Step 3: Calculate the time taken for the boat to travel downstream
The boat travels downstream with the current. The effective speed of the boat can be calculated by adding the rate of the river current to the rate at which the boat travels in still water:
Effective speed downstream = B + C
To calculate the time taken to travel downstream, we can use the formula: time = distance / speed
The distance traveled downstream = 12 miles
The time taken to travel downstream = 12 miles / (B + C)
Step 4: Set up the equation based on the given information
According to the problem, the total time taken for the round trip is 2½ hours. We need to convert this time to hours and minutes for calculations:
2½ hours = 2 + ½ = 2 hours + 30 minutes = 2 + 30/60 hours = 2.5 hours
Now, set up the equation:
Time taken upstream + Time taken downstream = 2.5 hours
12 / (B - C) + 12 / (B + C) = 2.5
Step 5: Solve the equation
Multiply the equation by (B - C)(B + C) to eliminate the denominators:
12(B + C) + 12(B - C) = 2.5(B - C)(B + C)
12B + 12C + 12B - 12C = 2.5(B^2 - C^2)
24B = 2.5B^2 - 2.5C^2
Simplify the equation:
2.5C^2 - 24B + 2.5B^2 = 0
Step 6: Solve the quadratic equation
This equation is quadratic in form. You can solve it using factoring, completing the square, or using the quadratic formula. In this case, we'll use factoring.
2.5C^2 - 24B + 2.5B^2 = 0
0.5C^2 - 4.8B + 0.5B^2 = 0
Factor a common factor of 0.5:
0.5(C^2 - 9.6B + B^2) = 0
Factor the quadratic term:
0.5((C - 4B)(C - 0.6B)) = 0
Set each factor equal to zero and solve for C:
C - 4B = 0
C = 4B
or
C - 0.6B = 0
C = 0.6B
We have two possible values for C: C = 4B and C = 0.6B
Step 7: Choose the correct solution
Since the rate of the river current cannot be negative, we choose the positive value for B. Therefore, C = 0.6B.
Step 8: Substitute the value of C back into the equation
C = 0.6B
0.6B = 0.6(5B)
0.6B = 3B
So, the rate of the current is 3B.
Therefore, the rate of the current is three times the rate at which the boat travels in still water.