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Think of two real-life quantities that are related to each other and construct a function that models the relationship between the two quantities. The function should be a piecewise function where not all components are linear. Describe the scenario you are modeling, the definitions of the variables involved in the function, and a graph or equation of the function.

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Scenario

Let's consider the relationship between the time spent studying for an exam and the score achieved on the exam. This scenario is realistic in educational settings, where students often see diminishing returns in exam performance as they increasingly invest time in studying.

Definitions of Variables

  • Let \( T \) represent the amount of time spent studying (in hours).
  • Let \( S \) represent the score achieved on the exam (out of 100 points).

Function Model

The relationship can be modeled with a piecewise function that captures different behaviors based on the amount of time studied:

  1. For \( T \leq 2 \):

    • Here, studying for up to 2 hours provides a positive linear effect on scores.
    • \( S(T) = 10T + 50 \)
    • A student can generally increase their score a great deal by spending a little time studying.

  2. For \( 2 < T \leq 5 \):

    • In this range, the score continues to increase but at a diminishing rate, reflecting the idea that after a couple of hours, the value of each additional hour decreases.
    • \( S(T) = -2(T - 2)^2 + 80 \)
    • This is a downward parabola that peaks at \( T = 2 \) and gradually decreases its slope, meaning while you can still improve your score, the incremental benefits reduce.

  3. For \( T > 5 \):

    • Beyond 5 hours, too much studying might lead to fatigue or burnout, resulting in no significant improvement in scores.
    • \( S(T) = 90 \)
    • Here, even if a student studies more than this time, their score caps out at 90, reflecting the maximum potential score that can be achieved.

Piecewise Function Representation

Putting it all together, we get the following piecewise function:

\[ S(T) = \begin{cases} 10T + 50 & \text{if } T \leq 2 \ -2(T - 2)^2 + 80 & \text{if } 2 < T \leq 5 \ 90 & \text{if } T > 5 \end{cases} \]

Visual Representation

The graph of this function would show a linear increase in score for small amounts of studying (up to 2 hours), a concave downward curve for moderate study times (2 to 5 hours), and a flat line representing a maximum score for any study time over 5 hours. The key features to note would be:

  • The score starts at 50 when \( T = 0 \),
  • Increases rapidly at first, then starts to plateau,
  • And finally, reaches a maximum score of 90 for more than 5 hours of study.

This piecewise model effectively captures the practical relationship between study time and exam success while factoring in both the positive impacts of studying and the negative consequences of over-studying.