Optimizing Job Choices: Mathematical Calculation Report

Introduction:

This mathematical report aims to investigate three job opportunities commencing on January 1, 2020, with the objective of accumulating $2000 in a savings account by September 1, 2020. The report will employ mathematical formulas, tables, and graphs to analyze the three job prospects. Data will be presented both numerically (in formulas) and graphically (in tables and graphs). The primary goal of this investigation is to identify the most efficient job option for achieving the $2000 savings target within the shortest period.

Method:

The optimal job will be determined by analyzing graphs and tables to identify which job can reach the $2000 goal in the quickest manner.

To accomplish this, we will utilize the Desmos graphing calculator to create visual representations of the data. The formula y = mx + c will be employed, where 'm' represents the hourly pay rate, and 'c' represents the initial savings amount ($135). All calculations will be double-checked with a calculator to ensure accuracy.

Before proceeding with the investigation, it is essential to define certain parameters and assumptions:

Parameters:

• Wednesday cannot be worked due to Scouts.
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• Thursday is reserved for homework and STEM Club.
• $2000 should be in the savings account by September 1, 2020.

Assumptions:

• No workdays will be missed.
• Each week will consist of exactly the specified number of work hours.
• No additional pay will be received for public holidays, overtime, or any other reasons.
• Job commencement date is January 1 and concludes on September 1, 2020 (34 weeks and 6 days).
• No expenses will be incurred for other purposes.
• The pay rate will remain constant without changes due to promotions, demotions, or other pay-affecting scenarios.
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Results:

Before proceeding, let's define the following variables:

m = gradient (hourly pay rate)

c = y-intercept (initial savings amount of $135)

The following calculations determine the weekly salary for each job:

To determine the slope of the visual line in the following graph (Figure 2), we use the formula 'y = mx + c':

Game of Scones Bakery: y = 11.40x + 135

A Latte Fun Café: y = 13.50x + 135

Crust Pizza Cairns: y = 17.50x + 135

Verification:

Verification is employed to ensure the integrity of the calculations presented above. The formula for the gradient (m) is provided as evidence, along with visual graphs to support the validity of the calculations:

Crust Pizza Cairns: m = rise/run = 17.50/1

A Latte Fun Café: m = rise/run = 13.50/1

Game of Scones Bakery: m = rise/run = 11.40/1

Discussion/Evaluation:

A solution has been reached to address the initial question. It has been determined that Crust Pizza Cairns offers the best job opportunity and will reach the $2000 goal in the shortest time. This conclusion is well-supported by verification methods (verification calculations and Figure 3). All calculations have been presented, further validating the soundness of this conclusion.

The strengths of the method are evident in the thorough description of the process, including the tools, formulas, and variables used to find the solution. However, there are areas in which the method could be improved, such as specifying additional assumptions and parameters. The assumptions impacted the investigation by restricting workdays or hours, assuming no pay raises, missed workdays, or overtime. While simplifying calculations, this limited scope may not fully represent real-world scenarios. To enhance future investigations, considering a wider range of job opportunities and employing additional verification methods would be beneficial.

Conclusion:

This mathematical investigation aimed to assess three distinct job opportunities to save $2000 for a family holiday in September 2020. The analysis involved calculations to determine the weekly salary, and the Desmos Graphing Calculator was utilized to visually represent the formula y = mx + c for each job. The conclusion is that Crust Pizza Cairns is the optimal job choice as it enables the quickest attainment of the $2000 goal. For future investigations, it is recommended to consider higher salary options to provide a more competitive comparison and avoid biased initial assumptions.