Marika Is Training For A Track Race. She Starts By Sprinting 100 Yards And Gradually Increases Her Distance By Adding 4 Yards Each Day For 21 Days.The Explicit Formula That Models This Situation Is $a_n = 100 + (n-1) \cdot 4$.How Far Does She

Feb 27, 2025 by ADMIN 243 views

Marika's Track Training: A Mathematical Approach to Distance and Progress

Marika is a dedicated athlete training for a track race. Her training regimen involves sprinting a certain distance each day, gradually increasing the distance over time. In this article, we will explore the mathematical model that describes Marika's training, and use it to determine how far she will have sprinted after 21 days.

The explicit formula that models Marika's training is given by:

a_n = 100 + (n-1) \cdot 4

where $a_n$ is the distance sprinted on the $n$ th day, and $n$ is the day number (starting from 1).

Understanding the Formula

Let's break down the formula to understand what it represents. The formula consists of two parts:

The first part, $100$ , represents the initial distance sprinted on the first day (100 yards).
The second part, $(n-1) \cdot 4$ , represents the additional distance sprinted each day, starting from the second day (day 2).

The formula $(n-1) \cdot 4$ can be interpreted as follows:

The $(n-1)$ term represents the number of days that have passed since the first day (day 1).
The $\cdot 4$ term represents the additional distance sprinted each day (4 yards).

Calculating the Distance

To calculate the distance sprinted on a given day, we simply plug in the day number ( $n$ ) into the formula:

a_n = 100 + (n-1) \cdot 4

For example, to calculate the distance sprinted on the 5th day, we plug in $n=5$ :

a_5 = 100 + (5-1) \cdot 4 = 100 + 4 \cdot 4 = 100 + 16 = 116

Therefore, on the 5th day, Marika will have sprinted a distance of 116 yards.

Finding the Total Distance

To find the total distance sprinted over 21 days, we need to calculate the distance sprinted on each day and add them up. However, using the formula, we can simplify the calculation:

\text{Total Distance} = \sum_{n=1}^{21} a_n = \sum_{n=1}^{21} (100 + (n-1) \cdot 4)

Using the formula for the sum of an arithmetic series, we get:

\text{Total Distance} = 21 \cdot 100 + 4 \cdot \frac{21 \cdot 20}{2} = 2100 + 840 = 2940

Therefore, over 21 days, Marika will have sprinted a total distance of 2940 yards.

In this article, we explored the mathematical model that describes Marika's track training. We used the explicit formula to calculate the distance sprinted on each day and the total distance sprinted over 21 days. The formula provided a clear and concise way to model Marika's training, and we were able to use it to determine the total distance sprinted over the 21-day period.

Day	Distance Sprinted
1	100 yards
2	104 yards
3	108 yards
4	112 yards
5	116 yards
...	...
21	316 yards

| Total Distance Sprinted | 2940 yards |

[Insert chart or graph showing Marika's training progress]

Note: The chart or graph can be created using a spreadsheet or a graphing tool, and can be customized to show the distance sprinted on each day and the total distance sprinted over 21 days.
Marika's Track Training: A Mathematical Approach to Distance and Progress - Q&A

In our previous article, we explored the mathematical model that describes Marika's track training. We used the explicit formula to calculate the distance sprinted on each day and the total distance sprinted over 21 days. In this article, we will answer some frequently asked questions about Marika's training and provide additional insights into her progress.

Q: What is the initial distance sprinted on the first day? A: The initial distance sprinted on the first day is 100 yards.

Q: How much distance is added each day? A: 4 yards are added each day.

Q: What is the formula for calculating the distance sprinted on a given day? A: The formula is:

a_n = 100 + (n-1) \cdot 4

where $a_n$ is the distance sprinted on the $n$ th day, and $n$ is the day number (starting from 1).

Q: How can I calculate the total distance sprinted over 21 days? A: You can use the formula for the sum of an arithmetic series to calculate the total distance sprinted over 21 days:

\text{Total Distance} = 21 \cdot 100 + 4 \cdot \frac{21 \cdot 20}{2} = 2100 + 840 = 2940

Q: What is the total distance sprinted over 21 days? A: The total distance sprinted over 21 days is 2940 yards.

Q: Can I use this formula to calculate the distance sprinted on any day? A: Yes, you can use the formula to calculate the distance sprinted on any day by plugging in the day number ( $n$ ) into the formula.

Q: How can I visualize Marika's training progress? A: You can create a chart or graph to show the distance sprinted on each day and the total distance sprinted over 21 days.

Q: What is the significance of the 4-yard increase each day? A: The 4-yard increase each day represents the gradual increase in distance that Marika is training for. This increase is designed to help her build up her endurance and prepare her for the track race.

Q: Can I use this formula to calculate the distance sprinted on a specific day? A: Yes, you can use the formula to calculate the distance sprinted on a specific day by plugging in the day number ( $n$ ) into the formula.

Q: How can I modify the formula to calculate the distance sprinted on a different number of days? A: You can modify the formula by changing the number of days (21) to the desired number of days.

In this article, we answered some frequently asked questions about Marika's track training and provided additional insights into her progress. We hope that this Q&A article has been helpful in understanding the mathematical model that describes Marika's training and how it can be used to calculate the distance sprinted on each day and the total distance sprinted over 21 days.

[Insert chart or graph showing Marika's training progress]

Note: The chart or graph can be created using a spreadsheet or a graphing tool, and can be customized to show the distance sprinted on each day and the total distance sprinted over 21 days.

[Marika's Track Training: A Mathematical Approach to Distance and Progress](link to previous article)
[Marika's Training Progress Chart](link to chart or graph)
[Mathematical Model for Track Training](link to mathematical model)

We hope that this Q&A article has been helpful in understanding Marika's track training and how it can be used to calculate the distance sprinted on each day and the total distance sprinted over 21 days.