Problem 58 - College Algebra with Corequisite Support

Question

Keisha devised a week-long study plan to prepare for finals. On the first day, she plans to study for 1 hour, and each successive day she will increase her study time by 30 minutes. How many hours will Keisha have studied after one week?

Step-by-Step Solution

Answer

Keisha will have studied for 17.5 hours after one week.

Step 1: Define the Study Plan

Keisha starts studying for 1 hour on the first day and increases the study time by 0.5 hours (30 minutes) each day for a total of 7 days.

Step 2: Determine the Daily Study Times

List out the study times for each day: - Day 1: 1 hour - Day 2: 1 + 0.5 = 1.5 hours - Day 3: 1.5 + 0.5 = 2 hours - Day 4: 2 + 0.5 = 2.5 hours - Day 5: 2.5 + 0.5 = 3 hours - Day 6: 3 + 0.5 = 3.5 hours - Day 7: 3.5 + 0.5 = 4 hours

Step 3: Calculate the Total Study Hours

Add all the daily study hours to find the total study time over the week:\[1 + 1.5 + 2 + 2.5 + 3 + 3.5 + 4 = 17.5 \text{ hours}\]

Key Concepts

Daily Study Time

Understanding Keisha's study plan begins by looking at her daily commitment. Each day she plans to study for a specific time. On the first day, she starts with just 1 hour. This is a manageable amount of time that can gradually build into a solid study routine. Every day, she adds 30 minutes, meaning by day two, she's studying 1.5 hours. By slowly increasing the time each day, Keisha can adapt to longer study sessions without feeling overwhelmed. This approach also helps in maintaining a steady pace towards her goal.

To summarize her daily study sessions:

• Day 1: 1 hour
• Day 2: 1.5 hours
• Day 3: 2 hours
• Day 4: 2.5 hours
• Day 5: 3 hours
• Day 6: 3.5 hours
• Day 7: 4 hours

This arithmetic sequence shows a clear pattern of growth each day, making it predictable and easy for Keisha to follow.

Total Study Hours

Once Keisha's daily study hours are established, the next step is to calculate her total study time over the entire week. By adding up all the hours studied each day, we find the amount of time Keisha dedicates to her preparation. This involves simple addition of each day's study period.

• Day 1: 1 hour
• Day 2: 1.5 hours
• Day 3: 2 hours
• Day 4: 2.5 hours
• Day 5: 3 hours
• Day 6: 3.5 hours
• Day 7: 4 hours

The arithmetic sum of these values is 17.5 hours. This total is crucial, as it reflects the cumulative study effort put forth by Keisha throughout the week. It not only lets her gauge the effectiveness of her study plan but also helps in adjusting her future strategies for improvement.

Incremental Growth

Incremental growth is a key concept in making Keisha's study strategy effective. Instead of cramming her schedule with many hours on the first day, she uses an arithmetic increase. By adding 30 minutes each subsequent day, her study time grows gradually. This method helps her to progressively adapt to longer study times without experiencing burnout.

Incremental increases have several benefits:

• They provide a realistic and sustainable way to increase productivity.
• Enable steady progress without demanding too much at once.
• Help in maintaining motivation, as the task seems achievable each day.
• Allow monitoring of performance and identifying areas of improvement.

Using an arithmetic sequence for her study plan, Keisha's study time growth is both measured and effective. It allows her to build confidence and efficiency in her study habits over time, making her preparation for finals more thorough and less daunting.