Problem 85 - Algebra and Trigonometry

Question

If you have been hired at an annual salary of \(\$ 42,000\) and expect to receive annual increases of \(3 \%,\) what will your salary be when you begin your fifth year?

Step-by-Step Solution

Answer

Approximately \textdollar 47,271

Step 1: Determine the annual increase factor

First, identify the percentage increase each year. The salary increases by 3%, which means the increase factor is 1 + 0.03 = 1.03.

Step 2: Write the formula to calculate the salary after n years

To find the salary after a certain number of years with a constant annual increase, use the formula: \( \text{Salary\textunderscore{n}} = \text{Initial Salary} \times (\text{Increase Factor})^{n-1} \).

Step 3: Substitute the values into the formula

Substitute the initial salary (\text{\textdollar 42,000}), the increase factor (1.03), and the number of years minus one (5-1=4) into the formula: \( \text{Salary\textunderscore{5}} = 42,000 \times (1.03)^4 \).

Step 4: Calculate (1.03)^4

Calculate the value of 1.03 raised to the power of 4: \( 1.03^4 \approx 1.1255 \).

Step 5: Calculate the final salary

Multiply the initial salary by the computed factor: \( 42,000 \times 1.1255 \approx 47,271 \). Thus, at the beginning of the fifth year, the salary will be approximately \textdollar 47,271.

Key Concepts

Annual Salary

Understanding annual salary is crucial in many job-related financial calculations. Annual salary refers to the total amount of money you earn before deductions over a full year of employment. If you have been hired at an annual salary of \(\$42,000\), this means that without taking any increases into account, you would earn \(\$42,000\) each year.
Annual salaries are often used as a basis for calculating benefits, bonuses, and taxation. It's essential to understand your annual salary to manage your finances accurately and plan for any pay raises effectively.

Percentage Increase

A percentage increase means you are adding a certain percentage of your original value to itself. In terms of a salary increase, it’s often expressed as an annual percentage increase.
If you earn \(\$42,000\) annually and receive a 3% increase each year, this increase factor is \frac{3}{100} = 0.03\. So, the total factor each year is \1 + 0.03 = 1.03\.
This factor is crucial for calculating future salaries as it helps determine how much your salary will grow over time.

Exponential Growth

Exponential growth represents a situation in which an amount increases by a consistent percentage over equal time increments. This model fits well when dealing with compound interest and salary increases. For salary calculations, exponential growth means your pay increases by a certain percentage each year.
In the given exercise, your \(\$42,000\) annual salary sees a 3% increase every year. Over multiple years, the salary grows exponentially rather than linearly. This growth is calculated using the formula \(\text{Salary_{n}} = \text{Initial Salary} \times (\text{Increase Factor})^{n-1}\).
The term \^{n-1}\ represents the power to which you raise the increase factor. This shows how the salary multiplies and grows significantly over time.

Formula Application

Applying the salary growth formula correctly is the key to accurate calculations. Here are the steps from the given exercise:

• Identify the initial salary: \$42,000\
• Determine the increase factor: \1.03\
• Calculate for the 5th year: Apply \(n-1\), so \ = 5\ gives \(5-1 = 4\)
• Write the formula: \(\text{Salary_{5}} = 42,000 \times (1.03)^4\)
• Calculate \(1.03^4 \approx 1.1255\)
• Calculate the final salary: \42,000 \times 1.1255 \approx 47,271\

Understanding and applying this formula allows you to forecast future earnings precisely, which is useful for budgeting and financial planning.