Problem 40 - College Algebra and Calculus: An Applied Approach

Question

Salary You accept a new job with a starting salary of \(\$ 48,000\). You receive a \(4 \%\) raise at the start of your second year, a \(5.5 \%\) raise at the start of your third year, and an \(11.4 \%\) raise at the start of your fourth year. (a) Find your salary for the second year. (b) Find your salary for the third year. (c) Find your salary for the fourth year.

Step-by-Step Solution

Answer

The salary in the second year is \$49,920, in the third year \$52,665.6, and in the fourth year roughly \$58,669.48.

Step 1: Calculating the Second Year's Salary

To find the salary for the second year, a \(4\%\) increase on the first year's salary of \$48,000 is needed. This can be calculated as \(48000 \times \frac{4}{100} = 1920\). This increase should be added to the initial salary, resulting in \(48000 + 1920 = 49920\).

Step 2: Calculating the Third Year's Salary

The salary for the third year is a \(5.5\%\) increase on the second year's salary. Apply the previous process: find the increase by calculating \(49920 \times \frac{5.5}{100} = 2745.6\), and add that to the second year's salary of \$49,920, yielding \(49920 + 2745.6 = 52665.6\).

Step 3: Calculating the Fourth Year's Salary

Finally, to find the salary for the fourth year, apply an \(11.4\%\) increase on the third year's salary. Again, calculate the increase as \(52665.6 \times \frac{11.4}{100} = 6003.8776\). Add this to the third year's salary, so the fourth year's salary is \(52665.6 + 6003.8776 = 58669.4776\).

Key Concepts

Understanding Percent Increase

A percent increase is a common way to express how much a quantity grows over time. When we talk about salary raises in terms of percentages, it's an example of a percent increase. To calculate a percent increase, you need to find out what percentage of the original amount you are adding to that amount. For instance, a 4\% raise on a \\(48,000 salary means you're adding 4\% of \\)48,000 to itself.Here's how you do it:

• Convert the percentage to a decimal, like 4\% becomes 0.04.
• Multiply the original amount by this decimal (\\(48,000 \times 0.04 = \\)1,920).
• Add this increase to the original amount (\\(48,000 + \\)1,920).

The final sum (\$49,920) is the increased amount. Knowing how to calculate percent increases helps you not just in salary computations, but in any situation where percentages describe growth.

Annual Raise Calculations

An annual raise is a specific example of a percent increase that happens each year. It represents the growth of your salary over time, reflecting performance, inflation, or company policies.Every raise builds on the last year's new salary, not the original one. Thus, when calculating an annual raise:

• Use the new salary figure from the previous year.
• Apply the new percentage increase to this updated salary.
• For example, after a 5.5\% increase on the second year's salary of \\(49,920, your salary gets raised by \\)2,745.6, resulting in \$52,665.6.

This process makes your salary grow progressively, ensuring your income keeps pace with whatever criteria set the raises.

Mathematical Modeling of Salary Growth

Mathematical modeling is the tool we use to predict and calculate changes like salary increases over time. By using equations and formulas, we can model many real-life situations, including how salaries evolve over several years.Consider the exercise at hand:

• Each year's salary is modeled based on the previous year's salary and a percentage increase.
• This can be written as a formula: New Salary = Last Year's Salary \( \times \) (1 + Increase Percentage as a Decimal).
• For the fourth year, you take the third year's salary of \\(52,665.6 and multiply it by (1 + 0.114) for an 11.4\% raise.
• This yields the fourth year’s salary of \\)58,669.48 (rounded from \$58,669.4776).

Using mathematical modeling, you can systemically tackle complex calculations, avoiding errors and understanding financial projections better.