Calculating the Minimum Speed Required

Oct 13, 2025 by leolavanderia.com.br 97 views

Introduction

When planning a boat tour, several factors need to be considered to ensure a successful and enjoyable experience for the group. One of the most critical factors is the speed at which the boat travels. In this article, we will explore the minimum speed required for a boat tour to complete a round trip within a certain time frame, taking into account the effects of the current on the boat's speed.

Understanding the Problem

A boat tour guide expects his tour to travel at a rate of $x$ mph on the first leg of the trip. On the return route, the boat travels against the current, decreasing the boat's rate by 10 mph. The group needs to travel an average of at least $y$ mph to complete the round trip within the desired time frame.

Calculating the Minimum Speed Required

To calculate the minimum speed required for the boat tour, we need to consider the effects of the current on the boat's speed. When traveling with the current, the boat's speed is $x$ mph. However, when traveling against the current, the boat's speed is reduced by 10 mph, resulting in a speed of $(x - 10)$ mph.

The Average Speed Formula

The average speed of the boat tour can be calculated using the formula:

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}

In this case, the total distance is the sum of the distances traveled on the first leg and the return leg. Let's assume the distance traveled on the first leg is $d_1$ and the distance traveled on the return leg is $d_2$ . The total distance is then $d_1 + d_2$ .

The Time Formula

The time taken to travel a certain distance can be calculated using the formula:

\text{Time} = \frac{\text{Distance}}{\text{Speed}}

In this case, the time taken to travel the first leg is $\frac{d_1}{x}$ and the time taken to travel the return leg is $\frac{d_2}{x - 10}$ .

The Total Time Formula

The total time taken to complete the round trip is the sum of the times taken to travel the first leg and the return leg. Therefore, the total time is:

\text{Total Time} = \frac{d_1}{x} + \frac{d_2}{x - 10}

The Average Speed Formula with Total Time

Substituting the total time formula into the average speed formula, we get:

\text{Average Speed} = \frac{d_1 + d_2}{\frac{d_1}{x} + \frac{d_2}{x - 10}}

Simplifying the Average Speed Formula

To simplify the average speed formula, we can multiply both the numerator and the denominator by $x(x - 10)$ :

\text{Average Speed} = \frac{(x - 10)d_1 + xd_2}{d_1(x - 10) + d_2x}

The Minimum Speed Required

The minimum speed required for the boat tour is the average speed that is at least $y$ mph. Therefore, we can set up the inequality:

\frac{(x - 10)d_1 + xd_2}{d_1(x - 10) + d_2x} \geq y

Solving the Inequality

To solve the inequality, we can cross-multiply and simplify:

(x - 10)d_1 + xd_2 \geq y(d_1(x - 10) + d_2x)

Expanding and simplifying the inequality, we get:

(x - 10)d_1 + xd_2 \geq yd_1(x - 10) + yd_2x

Rearranging the Inequality

Rearranging the inequality, we get:

(x - 10)d_1 + xd_2 - yd_1(x - 10) - yd_2x \geq 0

Simplifying the Inequality

Simplifying the inequality, we get:

(x - 10)d_1 - yd_1(x - 10) + xd_2 - yd_2x \geq 0

Factoring the Inequality

Factoring the inequality, we get:

(x - 10)(d_1 - yd_1) + (x - yd_2)(d_2) \geq 0

The Final Inequality

The final inequality is:

(x - 10)(d_1 - yd_1) + (x - yd_2)(d_2) \geq 0

Conclusion

In conclusion, the minimum speed required for a boat tour to complete a round trip within a certain time frame can be calculated using the inequality:

(x - 10)(d_1 - yd_1) + (x - yd_2)(d_2) \geq 0

This inequality takes into account the effects of the current on the boat's speed and ensures that the average speed of the boat tour is at least $y$ mph.

Example

Let's consider an example where the distance traveled on the first leg is 100 miles and the distance traveled on the return leg is 150 miles. The boat tour guide wants to ensure that the average speed of the boat tour is at least 20 mph.

Substituting the Values

Substituting the values into the inequality, we get:

(x - 10)(100 - 20 \times 100) + (x - 20 \times 150)(150) \geq 0

Simplifying the Inequality

Simplifying the inequality, we get:

(x - 10)(-1800) + (x - 3000)(150) \geq 0

Expanding and Simplifying

Expanding and simplifying the inequality, we get:

-1800x + 18000 + 150x - 450000 \geq 0

Combining Like Terms

Combining like terms, we get:

-1650x - 432000 \geq 0

Dividing by -1650

Dividing both sides of the inequality by -1650, we get:

x + 261.81818181818182 \leq 0

Solving for x

Solving for x, we get:

x \leq -261.81818181818182

Conclusion

In conclusion, the minimum speed required for the boat tour is $x \leq -261.81818181818182$ mph.

Final Answer

The final answer is $\boxed{-261.81818181818182}$ .

Introduction

In our previous article, we explored the minimum speed required for a boat tour to complete a round trip within a certain time frame, taking into account the effects of the current on the boat's speed. In this article, we will answer some of the most frequently asked questions related to this topic.

Q: What is the average speed formula used to calculate the minimum speed required for a boat tour?

A: The average speed formula used to calculate the minimum speed required for a boat tour is:

\text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}}

Q: How do you calculate the total distance traveled by the boat tour?

A: The total distance traveled by the boat tour is the sum of the distances traveled on the first leg and the return leg. Let's assume the distance traveled on the first leg is $d_1$ and the distance traveled on the return leg is $d_2$ . The total distance is then $d_1 + d_2$ .

Q: How do you calculate the total time taken to complete the round trip?

A: The total time taken to complete the round trip is the sum of the times taken to travel the first leg and the return leg. Therefore, the total time is:

\text{Total Time} = \frac{d_1}{x} + \frac{d_2}{x - 10}

Q: What is the minimum speed required for a boat tour to complete a round trip within a certain time frame?

A: The minimum speed required for a boat tour to complete a round trip within a certain time frame can be calculated using the inequality:

(x - 10)(d_1 - yd_1) + (x - yd_2)(d_2) \geq 0

Q: How do you simplify the inequality to find the minimum speed required?

A: To simplify the inequality, we can multiply both the numerator and the denominator by $x(x - 10)$ :

\text{Average Speed} = \frac{(x - 10)d_1 + xd_2}{d_1(x - 10) + d_2x}

Q: What is the final inequality used to calculate the minimum speed required?

A: The final inequality used to calculate the minimum speed required is:

(x - 10)(d_1 - yd_1) + (x - yd_2)(d_2) \geq 0

Q: How do you solve the inequality to find the minimum speed required?

A: To solve the inequality, we can cross-multiply and simplify:

(x - 10)d_1 + xd_2 \geq y(d_1(x - 10) + d_2x)

Q: What is the minimum speed required for the boat tour in the example given?

A: In the example given, the distance traveled on the first leg is 100 miles and the distance traveled on the return leg is 150 miles. The boat tour guide wants to ensure that the average speed of the boat tour is at least 20 mph. The minimum speed required for the boat tour is $x \leq -261.81818181818182$ mph.

Q: What is the final answer to the minimum speed required for the boat tour?

A: The final answer to the minimum speed required for the boat tour is $\boxed{-261.81818181818182}$ .

Conclusion

In conclusion, the minimum speed required for a boat tour to complete a round trip within a certain time frame can be calculated using the inequality:

(x - 10)(d_1 - yd_1) + (x - yd_2)(d_2) \geq 0

This inequality takes into account the effects of the current on the boat's speed and ensures that the average speed of the boat tour is at least $y$ mph.

Final Answer

The final answer is $\boxed{-261.81818181818182}$ .