Marika Is Training For A Track Race. She Starts By Sprinting 100 Yards And Gradually Increases Her Distance By Adding 4 Yards A Day For 21 Days.Which Explicit Formula Models This Situation?A. A N = 100 + ( N − 1 ) × 21 A_n = 100 + (n-1) \times 21 A N ​ = 100 + ( N − 1 ) × 21 B. $a_n = 4 +

Introduction

Marika is a dedicated athlete training for a track race. To improve her performance, she starts by sprinting a short distance and gradually increases it over time. In this scenario, we are tasked with finding an explicit formula that models Marika's training situation. An explicit formula is a mathematical expression that describes the relationship between the input (in this case, the number of days) and the output (the distance Marika sprints).

Understanding the Situation

Marika begins by sprinting 100 yards on the first day. Each subsequent day, she adds 4 yards to her previous distance. This means that on the second day, she sprints 100 + 4 = 104 yards, on the third day, she sprints 104 + 4 = 108 yards, and so on. We need to find a formula that represents this pattern.

Analyzing the Data

Let's analyze the data for the first few days:

Day	Distance (yards)
1	100
2	104
3	108
4	112
5	116

We can see that the distance increases by 4 yards each day. To find the explicit formula, we need to identify the pattern and express it mathematically.

Finding the Explicit Formula

Let's examine the formula options provided:

A. $a_n = 100 + (n-1) \times 21$ B. $a_n = 4 + (n-1) \times 4$

We can test each formula by plugging in the values for the first few days:

Option A: $a_n = 100 + (n-1) \times 21$

Day	Distance (yards)	Formula (A)
1	100	100 + (1-1) × 21 = 100
2	104	100 + (2-1) × 21 = 121 ( incorrect)
3	108	100 + (3-1) × 21 = 142 (incorrect)

The formula A does not accurately represent the situation.

Option B: $a_n = 4 + (n-1) \times 4$

Day	Distance (yards)	Formula (B)
1	100	4 + (1-1) × 4 = 4 ( incorrect)
2	104	4 + (2-1) × 4 = 8 (incorrect)
3	108	4 + (3-1) × 4 = 12 (incorrect)

The formula B also does not accurately represent the situation.

Deriving the Correct Formula

Let's re-examine the situation and derive the correct formula. We know that Marika starts by sprinting 100 yards and increases her distance by 4 yards each day. This means that on the nth day, she will sprint 100 + (n-1) × 4 yards.

The correct formula is:

$a_n = 100 + (n-1) \times 4$

However, this formula is not among the options provided. Let's simplify the formula by factoring out the common term:

$a_n = 100 + 4(n-1)$

$a_n = 100 + 4n - 4$

$a_n = 96 + 4n$

This is the correct explicit formula that models Marika's training situation.

Conclusion

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Introduction

In our previous article, we explored Marika's track training situation and derived an explicit formula that models her distance. In this article, we will answer some frequently asked questions related to the topic.

Q: What is an explicit formula?

A: An explicit formula is a mathematical expression that describes the relationship between the input (in this case, the number of days) and the output (the distance Marika sprints).

Q: How does the explicit formula work?

A: The explicit formula $a_n = 96 + 4n$ works by taking the initial distance (96 yards) and adding 4 times the number of days (n) to it. This results in the total distance Marika sprints on the nth day.

Q: What is the significance of the initial distance (96 yards)?

A: The initial distance (96 yards) represents the starting point of Marika's training. It is the distance she sprints on the first day, and it serves as the foundation for the subsequent distances.

Q: How does the explicit formula account for the increase in distance?

A: The explicit formula accounts for the increase in distance by multiplying the number of days (n) by 4. This results in a linear increase in distance, where each day Marika sprints 4 more yards than the previous day.

Q: Can the explicit formula be used for other scenarios?

A: Yes, the explicit formula can be used for other scenarios where a linear increase in distance is involved. For example, if Marika were to increase her distance by 5 yards each day, the formula would be $a_n = 96 + 5n$.

Q: How can the explicit formula be applied in real-world situations?

A: The explicit formula can be applied in real-world situations where a linear increase in distance is involved. For example, in construction, the formula can be used to calculate the total length of a road or a building, given the initial length and the rate of increase.

Q: What are some common applications of explicit formulas?

A: Explicit formulas have numerous applications in various fields, including:

• Physics: to model the motion of objects under constant acceleration
• Engineering: to calculate the total length of a road or a building
• Economics: to model the growth of a population or an economy
• Biology: to model the growth of a population or a species

Conclusion

In this article, we answered some frequently asked questions related to Marika's track training situation and the explicit formula that models her distance. We hope that this article has provided a better understanding of the concept of explicit formulas and their applications in real-world situations.

Additional Resources

For more information on explicit formulas and their applications, please refer to the following resources:

• Mathematics textbooks: for a comprehensive introduction to explicit formulas and their applications
• Online resources: for interactive examples and exercises on explicit formulas
• Professional journals: for research articles on the applications of explicit formulas in various fields