Problem 84 - College Algebra and Calculus: An Applied Approach

Question

You accept a new job with a starting salary of \(\$ 28,800\). You are told that you will receive an annual raise of at least \(\$ 1500\). What is the maximum number of years you must work before your annual salary will be \(\$ 40,000\) ?

Step-by-Step Solution

Answer

To the exact solution, the equation will give two values for n, a positive and a negative one. The number of years needed to work to reach a total of \( $ 40,000 \) should be the nearest whole value (considering only the realistic, positive value).

Step 1: Understanding the problem

The yearly salary accumulation forms an arithmetic progression where the first term (a1) is \( $ 28,800 \), the difference (d) is \( $ 1500 \) and the total amount to reach (S) is at least \( $ 40,000 \). The goal is to find the number of terms n. The formula for the sum (S) of an arithmetic progression is \( S = n/2 * (2*a1 + (n-1)*d) \). The goal is to calculate n, the number of years to work until the salary will be \( $ 40,000 \).

Step 2: Formulating the Equation

The known values are a1 = 28800, d = 1500 and S = 40000. Substituting these values into the sum formula for an arithmetic progression, we have: 40000 = n/2 * (2*28800 + (n-1)*1500). This is the equation to solve for n.

Step 3: Solving the equation

To solve the equation, first simplify it by multiplying through by 2 to remove the fraction: 80000 = n * (57600 + (n-1)*3000),Then expand it and simplify further: 80000 = 57600*n + 3000*n^2 - 3000*n. Rearrange this to a quadratic equation: 3000*n^2 + 54600*n - 80000 = 0.Divide through by 1000 to make the calculations easier: 3*n^2 + 54.6*n - 80 = 0.To solve this, the quadratic formula is used: n = [-b ± sqrt(b^2 - 4ac)] / (2a),Substituting, we find n = [-54.6 ± sqrt((54.6^2) - 4*3*(-80))] / (2*3).

Step 4: Solving for n

Calculate the values inside the square root and get the values for n. Check both values to find the minimum positive integer (since it is not possible to work for a fraction of a year).

Step 5: Verifying the result

Verify the result by substituting n back into the original equation of salary accumulation. It should result in a sum of at least \( $ 40,000 \).

Key Concepts

Understanding Arithmetic Sequences

An arithmetic sequence is a series of numbers in which each term after the first is found by adding a constant, known as the common difference, to the previous term. For example, if we start with a number 3 and have a common difference of 2, our arithmetic sequence looks like 3, 5, 7, 9, and so on.

In the context of our exercise, the salary of \(28,800 with an annual raise of \)1,500 forms an arithmetic sequence since every year's salary is \(1,500 more than the year before. This common difference is a crucial component, as it allows us to predict the salary for any given future year.

The sum of the first 'n' terms of an arithmetic sequence is given by the formula: , where 'a1' is the first term, 'd' is the common difference, and 'n' is the number of terms. This is what allows us to solve the problem at hand – determining how long it will take for the salary to reach at least \)40,000.

Solving Quadratic Equations in Algebra

Quadratic equations are a foundational element in algebra with the general form of . These equations are solved by finding the values of 'x' that make the equation true. Two main methods for solving them include factoring and using the quadratic formula, . In our exercise, after we set up the equation based on the sum of an arithmetic sequence, we end up with a quadratic equation. To solve this, we simplify the terms and apply the quadratic formula. It's important to remember that the discriminant of the quadratic formula (the part under the square root) must be non-negative for real solutions to exist. In a scenario like our job salary problem, we're looking for a positive integer as a solution since the number of years worked can't be negative or fractional.

The Role of Financial Mathematics in Algebra

Financial mathematics applies algebraic methods to solve problems that relate to money and finances. In our exercise, understanding the arithmetic sequence and solving a quadratic equation are part of the financial mathematics toolkit. This further extends to other areas such as calculating loan repayments, investment returns, and more complex financial instruments. The principles of financial mathematics are grounded in the time value of money, which dictates that a sum of money today is worth more than the same sum in the future due to its earning capacity.

Besides arithmetic progressions, financial mathematics often deals with geometric progressions when considering compounded interest, annuities, and perpetuities. These concepts are vital for making informed decisions regarding investments, budgeting, and planning for future financial goals. Having a strong understanding of how algebra represents these financial scenarios can greatly benefit one's ability to manage real-life financial challenges.