Finding the Relationship Between the Variables

Oct 13, 2025 by talynt.org 61 views

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Introduction

In this article, we will delve into a classic problem involving a boat traveling in a river with a current. The problem requires us to use the concept of relative speed and the relationship between the speed of the boat, the speed of the current, and the time taken to travel a certain distance. We will use mathematical equations to solve the problem and understand the relationship between the variables involved.

Understanding the Problem

A boat can travel at an average speed of 10 miles per hour in still water. This means that if there is no current in the river, the boat will travel at a speed of 10 miles per hour. However, when the boat is traveling with the current, it can travel 6 miles in the same amount of time it takes to travel 4 miles upstream. This implies that the speed of the boat with the current is greater than the speed of the boat against the current.

Defining the Variables

Let's define the variables involved in the problem:

B: The speed of the boat in still water (in miles per hour)
C: The speed of the current (in miles per hour)
D: The distance traveled by the boat with the current (in miles)
D': The distance traveled by the boat against the current (in miles)
T: The time taken to travel the distance with the current (in hours)
T': The time taken to travel the distance against the current (in hours)

Formulating the Equations

We can formulate the following equations based on the given information:

B + C = 10 (The speed of the boat with the current is the sum of the speed of the boat and the speed of the current)
D = (B + C)T (The distance traveled by the boat with the current is the product of the speed of the boat with the current and the time taken)
D' = BT' (The distance traveled by the boat against the current is the product of the speed of the boat and the time taken)
D = 6 (The distance traveled by the boat with the current is 6 miles)
D' = 4 (The distance traveled by the boat against the current is 4 miles)

Solving the Equations

We can solve the equations to find the values of the variables involved. Let's start by substituting the values of D and D' into the equations:

D = (B + C)T => 6 = (B + C)T
D' = BT' => 4 = BT'

We can also substitute the value of B + C from the first equation into the second equation:

B + C = 10 => B = 10 - C

Substituting this value of B into the second equation, we get:

4 = (10 - C)T'

Finding the Relationship Between the Variables

We can now use the equations to find the relationship between the variables involved. Let's start by finding the relationship between the speed of the boat and the speed of the current.

B + C = 10 => C = 10 - B

This equation shows that the speed of the current is equal to the difference between the speed of the boat and 10.

Finding the Value of the Speed of the Current

We can now use the equation to find the value of the speed of the current. Let's substitute the value of D into the equation:

D = (B + C)T => 6 = (B + C)T

Substituting the value of B + C from the first equation, we get:

6 = (10 - C)T

Solving for the Speed of the Current

We can now solve for the speed of the current. Let's start by isolating C on one side of the equation:

6 = (10 - C)T => C = 10 - 6/T

This equation shows that the speed of the current is equal to the difference between 10 and 6/T.

Finding the Value of the Speed of the Boat

We can now use the equation to find the value of the speed of the boat. Let's substitute the value of C into the equation:

B + C = 10 => B = 10 - C

Substituting the value of C from the previous equation, we get:

B = 10 - (10 - 6/T)

Solving for the Speed of the Boat

We can now solve for the speed of the boat. Let's start by simplifying the equation:

B = 10 - (10 - 6/T) => B = 6/T

This equation shows that the speed of the boat is equal to 6/T.

Finding the Value of the Time Taken

We can now use the equation to find the value of the time taken. Let's substitute the value of B into the equation:

D = BT => 6 = (6/T)T

Solving for the Time Taken

We can now solve for the time taken. Let's start by simplifying the equation:

6 = (6/T)T => T = 1

This equation shows that the time taken is equal to 1 hour.

Conclusion

In this article, we have used mathematical equations to solve the river current problem. We have found the relationship between the speed of the boat, the speed of the current, and the time taken to travel a certain distance. We have also found the value of the speed of the current and the speed of the boat. The time taken to travel the distance with the current is equal to 1 hour.

Final Answer

The final answer is:

C = 2 (The speed of the current is 2 miles per hour)
B = 8 (The speed of the boat is 8 miles per hour)
T = 1 (The time taken to travel the distance with the current is 1 hour)

Note: The final answer is based on the assumption that the time taken to travel the distance with the current is equal to 1 hour.

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Introduction

In our previous article, we solved the river current problem using mathematical equations. In this article, we will provide a Q&A guide to help you understand the problem and its solution.

Q: What is the river current problem?

A: The river current problem is a classic problem in mathematics that involves a boat traveling in a river with a current. The problem requires us to use the concept of relative speed and the relationship between the speed of the boat, the speed of the current, and the time taken to travel a certain distance.

Q: What are the variables involved in the problem?

A: The variables involved in the problem are:

B: The speed of the boat in still water (in miles per hour)
C: The speed of the current (in miles per hour)
D: The distance traveled by the boat with the current (in miles)
D': The distance traveled by the boat against the current (in miles)
T: The time taken to travel the distance with the current (in hours)
T': The time taken to travel the distance against the current (in hours)

Q: What is the relationship between the speed of the boat and the speed of the current?

A: The relationship between the speed of the boat and the speed of the current is given by the equation:

B + C = 10 (The speed of the boat with the current is the sum of the speed of the boat and the speed of the current)

Q: How do we find the value of the speed of the current?

A: We can find the value of the speed of the current by using the equation:

C = 10 - B

Q: How do we find the value of the speed of the boat?

A: We can find the value of the speed of the boat by using the equation:

B = 10 - C

Q: What is the relationship between the distance traveled by the boat with the current and the time taken?

A: The relationship between the distance traveled by the boat with the current and the time taken is given by the equation:

D = (B + C)T (The distance traveled by the boat with the current is the product of the speed of the boat with the current and the time taken)

Q: How do we find the value of the time taken?

A: We can find the value of the time taken by using the equation:

T = 1 (The time taken to travel the distance with the current is 1 hour)

Q: What is the final answer to the problem?

A: The final answer to the problem is:

C = 2 (The speed of the current is 2 miles per hour)
B = 8 (The speed of the boat is 8 miles per hour)
T = 1 (The time taken to travel the distance with the current is 1 hour)