Problem 69 - Precalculus : Building Concepts and Connections

Question

Fill in the table according to the given rule and find an expression for the function represented by the rule. Salary The annual salary of an employee at a certain company starts at \(\$ 10,000\) and is increased by \(5 \%\) at the end of every year. $$\begin{aligned} &\begin{array}{cc} \text { Years at } & \text { Annual } \\ \text { Work } & \text { Salary } \end{array}\\\ &\begin{array}{l} 0 \\ 1 \\ 2 \\ 3 \\ 4 \end{array} \end{aligned}$$

Step-by-Step Solution

Answer

The annual increase in salary forms a geometric progression with an initial value of \$10000 and a ratio (rate) of 1.05. The salary for different years at work can be calculated using the formula \(Salary(n) = \$10000 \times 1.05^n\). The table filled according to this rule will have the following values for 0, 1, 2, 3, and 4 years at work: \$10000, \$10500, \$11025, \$11576.25, and \$12155.06 respectively.

Step 1: Identify the initial value and the rate

The initial value of the salary is \$10000 and the rate is \(1 +0.05 = 1.05\) or 105%.

Step 2: Predicate the salary function

It is seen that the annual salary is a function of the number of years at work, starting with an initial value and increased by a factor (rate) by the power of the years at work. Therefore it can be expressed as \(Salary(n) = Initial \ Value * Rate^n\), where \(n\) is the number of years the employee has worked.

Step 3: Apply the rule to fill in the table

For \(Years = 0, Salary(0) = Initial Value * Rate^0 = \$10000 * 1.05^0 = \$10000\). For \(Years = 1, Salary(1) = Initial Value * Rate^1 = \$10000 * 1.05 = \$10500\). For \(Years = 2, Salary(2) = Initial Value * Rate^2 = \$10000 * 1.05^2 = \$11025\). For \(Years = 3, Salary(3) = Initial Value * Rate^3 = \$10000 * 1.05^3 = \$11576.25\). For \(Years = 4, Salary(4) = Initial Value * Rate^4 = \$10000 * 1.05^4 = \$12155.06\). Thus, the salary for the 0th, 1st, 2nd, 3rd, and 4th year is \$10000, \$10500, \$11025, \$11576.25, and \$12155.06 respectively.

Key Concepts

Annual Salary Calculation

When we talk about calculating an annual salary, especially one that increases each year by a certain percentage, we're examining a practical application of exponential functions in finance. Consider this: an employee begins with a salary of \(\$10,000\) and receives a 5% raise at the end of every year. How do we calculate their salary after, say, three years?While the initial impulse might be to simply add 5% of the initial \(\$10,000\) each year, the reality is more complex due to the nature of exponential growth - the raise applies to the new salary, not just the initial amount. This cumulative process, where each year's salary becomes the base for the next year's 5% increase, is precisely what's at work in our problem.To articulate this process mathematically, we use the formula \(Salary(n) = Initial\ Value \times (1 + Rate)^n\), where \(n\) is the number of years worked. By plugging in the numbers, we can see how the salary is recalculated at each year, taking into account the exponential increase.

Geometric Sequences in Finance

The situation of the employee's annually increasing salary can be framed within the concept of geometric sequences—a foundational topic in finance. Geometric sequences are a series of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. In finance, they appear in the calculations of compound interest, investment growth, and, as in our example, salary increases.

• In our scenario, the initial salary is the first term of this sequence.
• The common ratio is the percentage increase converted into a multiplication factor, in this case, 1.05.
• Each subsequent term represents the salary at the end of each year.

Understanding these sequences helps in predicting long-term financial growth and planning for the future, as their exponential nature can produce significantly different outcomes compared to linear growth over time.

Exponential Growth Rate

Exponential growth, as seen in our salary problem, describes a situation where the increase in a quantity over time is proportional to its current value. With a 5% annual salary increase, each year's growth is relative to the salary of the previous year, not just the initial amount. This sort of growth is everywhere in finance, manifesting in areas like inflation, population growth, and, notably, in investments such as retirement funds.The remarkable thing about exponential growth in finance is that even seemingly small percentages, over long periods, can lead to substantial increases. The exponential growth rate here is encapsulated by the 1.05 multiplier—the salary doesn't just grow; it grows increasingly faster each year. Recognizing this principle is critical for long-term financial planning and evaluates the real value of investments or saving strategies across the span of several years or decades.