João And Fernanda Decided To Train For An Hour, Turning Around The Athletics Track. During This Workout, They Met A Few Times At The Starting Point Of The Track. Analyze How Many Times This Meeting Has Happened And Select The Correct Alternative.a)
Introduction
João and Fernanda embarked on a 1-hour workout, navigating the circumference of an athletics track. As they ran, they occasionally converged at the starting point of the track. In this article, we will delve into the mathematics behind their encounters, exploring the factors that influence the frequency of their meetings.
Understanding the Problem
To analyze the frequency of João and Fernanda's meetings, we need to consider the following:
- The duration of their workout (1 hour)
- The distance around the athletics track
- The speed at which they are running
- The time interval between their meetings
The Mathematics Behind the Meeting
Let's assume that João and Fernanda are running at a constant speed. We can represent their speed as v meters per second. The distance around the athletics track is d meters.
When João and Fernanda meet at the starting point, they have covered a distance equal to the circumference of the track. We can express this distance as d = 2πr, where r is the radius of the track.
The time it takes for João and Fernanda to cover the distance around the track is given by the formula t = d/v. Substituting the expression for d, we get t = 2πr/v.
Since João and Fernanda are running for 1 hour, the total time they spend on the track is t = 3600 seconds.
Calculating the Frequency of Encounters
To determine the frequency of their meetings, we need to find the time interval between their encounters. Let's denote this time interval as Δt.
When João and Fernanda meet at the starting point, they have covered a distance equal to the circumference of the track. The time it takes for them to cover this distance is t = d/v.
Since they are running for 1 hour, the total time they spend on the track is t = 3600 seconds. We can express the time interval between their meetings as Δt = t/n, where n is the number of meetings.
Solving for the Number of Meetings
To find the number of meetings, we need to solve for n. We can do this by substituting the expression for t into the equation for Δt:
Δt = (2πr/v) / n
Since João and Fernanda are running for 1 hour, the total time they spend on the track is t = 3600 seconds. We can express the time interval between their meetings as Δt = 3600/n.
The Correct Alternative
Now that we have analyzed the frequency of João and Fernanda's meetings, we can select the correct alternative:
a) The number of meetings is equal to the total time spent on the track divided by the time interval between meetings.
b) The number of meetings is equal to the distance around the track divided by the speed at which they are running.
c) The number of meetings is equal to the time interval between meetings divided by the total time spent on the track.
d) The number of meetings is equal to the circumference of the track divided by the time interval between meetings.
Conclusion
In conclusion, the frequency of João and Fernanda's meetings is influenced by the duration of their workout, the distance around the athletics track, and the speed at which they are running. By analyzing the mathematics behind their encounters, we can determine the correct alternative.
The Final Answer
The correct alternative is:
a) The number of meetings is equal to the total time spent on the track divided by the time interval between meetings.
Q: What is the formula to calculate the number of meetings between João and Fernanda?
A: The formula to calculate the number of meetings between João and Fernanda is:
n = 3600 / Δt
where n is the number of meetings and Δt is the time interval between meetings.
Q: How do I calculate the time interval between meetings?
A: To calculate the time interval between meetings, you need to know the distance around the athletics track (d) and the speed at which João and Fernanda are running (v). You can use the formula:
Δt = d / v
Q: What if João and Fernanda are running at different speeds?
A: If João and Fernanda are running at different speeds, you need to calculate the time interval between meetings for each person separately. Then, you can use the formula:
n = 3600 / (Δt1 + Δt2)
where n is the number of meetings and Δt1 and Δt2 are the time intervals between meetings for João and Fernanda, respectively.
Q: Can I use this formula for any type of track?
A: Yes, you can use this formula for any type of track, including oval tracks, rectangular tracks, and even tracks with obstacles.
Q: What if João and Fernanda are running in different directions?
A: If João and Fernanda are running in different directions, you need to calculate the time interval between meetings for each person separately. Then, you can use the formula:
n = 3600 / (Δt1 + Δt2)
where n is the number of meetings and Δt1 and Δt2 are the time intervals between meetings for João and Fernanda, respectively.
Q: Can I use this formula to calculate the number of meetings for multiple people?
A: Yes, you can use this formula to calculate the number of meetings for multiple people. Simply add the time intervals between meetings for each person and use the formula:
n = 3600 / (Δt1 + Δt2 + ... + Δtn)
where n is the number of meetings and Δt1, Δt2, ..., Δtn are the time intervals between meetings for each person.
Q: What if I want to calculate the number of meetings for a specific time interval?
A: If you want to calculate the number of meetings for a specific time interval, you can use the formula:
n = (time interval) / Δt
where n is the number of meetings and Δt is the time interval between meetings.
Conclusion
In conclusion, the formula to calculate the number of meetings between João and Fernanda is:
n = 3600 / Δt
where n is the number of meetings and Δt is the time interval between meetings. You can use this formula for any type of track, including oval tracks, rectangular tracks, and even tracks with obstacles.