The Speed Of A Stream Is 6 Mph. If A Boat Travels 94 Miles Downstream In The Same Time It Takes To Travel 47 Miles Upstream, What Is The Speed In Mph Of The Boat In Still Water?
Understanding the Problem
In this problem, we are given the speed of a stream, which is 6 mph. We are also given the information that a boat travels 94 miles downstream in the same time it takes to travel 47 miles upstream. Our goal is to find the speed of the boat in still water.
The Formula for Speed
The formula for speed is:
Speed = Distance / Time
We can use this formula to find the speed of the boat in still water.
Downstream Speed
When a boat travels downstream, the speed of the boat is the sum of the speed of the boat in still water and the speed of the stream. Let's call the speed of the boat in still water "b". Then, the downstream speed is:
Downstream Speed = b + 6
We know that the boat travels 94 miles downstream in the same time it takes to travel 47 miles upstream. This means that the time it takes to travel 94 miles downstream is equal to the time it takes to travel 47 miles upstream.
Upstream Speed
When a boat travels upstream, the speed of the boat is the difference between the speed of the boat in still water and the speed of the stream. So, the upstream speed is:
Upstream Speed = b - 6
Setting Up the Equation
We know that the time it takes to travel 94 miles downstream is equal to the time it takes to travel 47 miles upstream. We can use this information to set up an equation. Let's call the time it takes to travel 94 miles downstream "t". Then, the time it takes to travel 47 miles upstream is also "t".
We can use the formula for speed to write an equation for the downstream speed:
94 / t = b + 6
We can also write an equation for the upstream speed:
47 / t = b - 6
Solving the Equation
We can solve these two equations simultaneously to find the value of "b", which is the speed of the boat in still water.
First, let's multiply both sides of the first equation by "t" to get:
94 = bt + 6t
Next, let's multiply both sides of the second equation by "t" to get:
47 = bt - 6t
Now, let's add the two equations together to get:
141 = 2bt
Next, let's divide both sides of the equation by 2 to get:
70.5 = bt
Now, let's divide both sides of the equation by "t" to get:
b = 70.5 / t
Finding the Value of "t"
We know that the time it takes to travel 94 miles downstream is equal to the time it takes to travel 47 miles upstream. We can use this information to find the value of "t".
Let's call the time it takes to travel 94 miles downstream "t". Then, the time it takes to travel 47 miles upstream is also "t".
We can use the formula for speed to write an equation for the downstream speed:
94 / t = b + 6
We can also write an equation for the upstream speed:
47 / t = b - 6
We can solve these two equations simultaneously to find the value of "t".
First, let's multiply both sides of the first equation by "t" to get:
94 = bt + 6t
Next, let's multiply both sides of the second equation by "t" to get:
47 = bt - 6t
Now, let's add the two equations together to get:
141 = 2bt
Next, let's divide both sides of the equation by 2 to get:
70.5 = bt
Now, let's divide both sides of the equation by "b" to get:
t = 70.5 / b
Substituting the Value of "t"
We can substitute the value of "t" into the equation for "b" to get:
b = 70.5 / (70.5 / b)
Now, let's simplify the equation to get:
b = b^2 / 70.5
Next, let's multiply both sides of the equation by 70.5 to get:
70.5b = b^2
Now, let's divide both sides of the equation by "b" to get:
70.5 = b
The Final Answer
The final answer is 70.5. This is the speed of the boat in still water.
Conclusion
In this problem, we were given the speed of a stream, which is 6 mph. We were also given the information that a boat travels 94 miles downstream in the same time it takes to travel 47 miles upstream. Our goal was to find the speed of the boat in still water. We used the formula for speed and set up two equations to solve for the speed of the boat in still water. We found that the speed of the boat in still water is 70.5 mph.
References
- [1] "Speed and Time" by Math Open Reference
- [2] "Boat Speed and Stream Speed" by Khan Academy
Additional Information
- The speed of a stream is the speed of the water in the stream.
- The speed of a boat in still water is the speed of the boat when it is not moving with the stream.
- The downstream speed is the speed of the boat when it is moving with the stream.
- The upstream speed is the speed of the boat when it is moving against the stream.
Q&A: The Speed of a Stream and the Speed of a Boat in Still Water ==================================================================
Q: What is the speed of a stream?
A: The speed of a stream is the speed of the water in the stream. It is usually measured in miles per hour (mph) and can vary depending on the location and time of year.
Q: How does the speed of a stream affect the speed of a boat?
A: The speed of a stream can affect the speed of a boat in two ways. When a boat is moving with the stream, the speed of the boat is increased by the speed of the stream. When a boat is moving against the stream, the speed of the boat is decreased by the speed of the stream.
Q: What is the formula for the speed of a boat in still water?
A: The formula for the speed of a boat in still water is:
Speed = (Downstream Speed + Upstream Speed) / 2
Q: How do I calculate the speed of a boat in still water?
A: To calculate the speed of a boat in still water, you need to know the downstream speed and the upstream speed. You can use the following steps:
- Measure the distance traveled downstream and the time it takes to travel that distance.
- Measure the distance traveled upstream and the time it takes to travel that distance.
- Use the formula for speed to calculate the downstream speed and the upstream speed.
- Add the downstream speed and the upstream speed together and divide by 2 to get the speed of the boat in still water.
Q: What is the difference between downstream speed and upstream speed?
A: The downstream speed is the speed of the boat when it is moving with the stream. The upstream speed is the speed of the boat when it is moving against the stream.
Q: How do I calculate the downstream speed and the upstream speed?
A: To calculate the downstream speed and the upstream speed, you can use the following steps:
- Measure the distance traveled downstream and the time it takes to travel that distance.
- Use the formula for speed to calculate the downstream speed.
- Measure the distance traveled upstream and the time it takes to travel that distance.
- Use the formula for speed to calculate the upstream speed.
Q: What is the significance of the speed of a boat in still water?
A: The speed of a boat in still water is an important factor in determining the speed of a boat in different conditions. It is used in navigation, transportation, and other applications where the speed of a boat is critical.
Q: How do I apply the concept of speed of a boat in still water in real-life situations?
A: The concept of speed of a boat in still water can be applied in various real-life situations, such as:
- Navigation: Knowing the speed of a boat in still water can help navigators plan their route and estimate the time it will take to reach their destination.
- Transportation: The speed of a boat in still water is an important factor in determining the speed of a boat in different conditions, which can affect the transportation of goods and people.
- Recreation: The speed of a boat in still water can affect the speed of a boat in different conditions, which can impact the enjoyment of recreational activities such as boating and fishing.
Q: What are some common mistakes to avoid when calculating the speed of a boat in still water?
A: Some common mistakes to avoid when calculating the speed of a boat in still water include:
- Not accounting for the speed of the stream.
- Not using the correct formula for speed.
- Not measuring the distance and time accurately.
- Not considering the effects of wind, currents, and other external factors on the speed of the boat.
Q: How can I improve my understanding of the speed of a boat in still water?
A: To improve your understanding of the speed of a boat in still water, you can:
- Practice calculating the speed of a boat in still water using different scenarios and conditions.
- Study the formulas and concepts related to the speed of a boat in still water.
- Consult with experts and professionals in the field of navigation and transportation.
- Participate in hands-on activities and experiments to gain practical experience.