Question

When Jeff starts a math assignment, he spends 5 minutes getting out his book and a sheet of paper, sharpening his pencil, looking up the assignment in his assignment notebook, and turning to the correct page in his book. The equation \(t=10 p+5\) models the time, \(t\) minutes, Jeff budgets for a math assignment with \(p\) problems. Which of the following statements is necessarily true according to Jeff's model? F. He budgets 15 minutes per problem. G. He budgets 10 minutes per problem. H. He budgets 5 minutes per problem. J. He budgets 10 minutes per problem for the hard problems and 5 minutes per problem for the easy problems. K. He budgets a 5 -minute break after each problem.

Short answer

Answer: G. He budgets 10 minutes per problem.

Step-by-step solution

Identify the given equation

The given equation is \(t = 10p + 5\), where \(t\) represents the total time in minutes, and \(p\) represents the number of problems.

Analyze the equation

The equation is a linear function with a slope of 10 and a y-intercept of 5. This means that each problem takes 10 minutes to solve, and Jeff spends an additional 5 minutes on the initial task setup (like getting a book, sharpening a pencil, etc.).

Compare the equation with the given statements

Now, let's analyze each statement to see if it is true according to the given equation: F. He budgets 15 minutes per problem - This statement is false since the slope of the equation is 10, meaning he budgets 10 minutes per problem. G. He budgets 10 minutes per problem - This statement is true, as it matches the slope of the equation. H. He budgets 5 minutes per problem - This statement is false since the initial setup time is 5 minutes, not the time per problem. J. The statement is false, as there is no differentiation between hard and easy problems in the given equation. K. The statement is false, as there is no mention of breaks in the equation, only the initial 5-minute setup.

Conclusion

Based on the analysis, the statement that is necessarily true according to Jeff's model is: G. He budgets 10 minutes per problem.

Key concepts

Linear Equations

Linear equations are foundational to algebra and represent a relationship between two variables in the form of a straight line when plotted on a graph. These equations generally look like \( ax + b = 0 \) where \( a \) and \( b \) are constants. The equation given in Jeff's time management scenario, \( t = 10p + 5 \) is a typical linear equation and represents how Jeff allocates his time for math assignments.

This equation can be expressed in the slope-intercept form as \( y = mx + c \) where \( m \) is the slope, representing the time budgeted per problem, and \( c \) is the y-intercept, representing the initial time spent preparing. Here, the slope is 10, indicating that Jeff budgets 10 minutes per problem. The intercept is 5, suggesting a one-time setup time of 5 minutes regardless of the number of problems.

In the context of time management, understanding that the slope defines the rate at which time is allocated per problem, and the intercept represents a starting value, is crucial. Such insights gleaned from linear equations can help students plan and allocate their time effectively.

Problem-Solving Strategy

When facing a problem, a strong problem-solving strategy is vital to find a solution efficiently. Jeff’s approach to managing his math homework can be seen as a simple problem-solving strategy. It involves clear steps, much like the 'Step by Step Solution' provided:

• Identify the problem or task at hand.
• Analyze the problem to understand the components.
• Find a formula or method that applies to the given scenario.
• Assess the available choices or compare outcomes.
• Conclude with the most logical option based on the analysis.

Whether you’re dealing with math assignments or everyday challenges, approaching problems methodically can minimize mistakes and enhance your ability to manage tasks efficiently. It’s about understanding the situation, breaking it down into manageable parts, and applying logical reasoning to arrive at a solution.

Time Budgeting

Time budgeting is an essential skill for effective study habits. It helps ensure that you allocate time to all necessary activities, allowing for thorough understanding and completion of assignments. Jeff's \( t = 10p + 5 \) equation demonstrates a time budgeting method for tasks with multiple, similar components.

Understanding your own pace, as Jeff does with the 10 minutes per problem, allows for realistic and effective time budgets. Starting each homework session by estimating the amount of time needed can reduce anxiety and improve focus. However, it is also important to remember to factor in time for setup and transitions, as represented by the 5-minute intercept in Jeff’s model.

In practice, you can apply time budgeting by:

• Evaluating how long each task will take.
• Setting priorities based on deadlines and difficulty.
• Creating a schedule that reflects your pace and time allocated per task.
• Building in extra time for unexpected interruptions or difficult problems.
• Reviewing and adjusting your time budget as necessary.

Good time management can make the difference between feeling rushed and staying in control of your studies, ultimately leading to better performance and less stress.