Problem 52 - Calculus: Graphical, Numerical, Algebraic

Question

Salary Negotiation A welder's contract promises a 3.5\(\%\) salary increase each year for 4 years and Luisa has an initial salary of \(\$ 36,500\) . (a) Show that Luisa's salary is given by $$y=36,500(1.035)^{\text { int } t}$$ where \(t\) is the time, measured in years, since Luisa signed the contract. (b) Graph Luisa's salary function. At what values of \(t\) is it continuous?

Step-by-Step Solution

Answer

Luisa's salary is modeled by the function y = 36,500(1.035^t), where \(t\) is the time in years since the contract signing which is continuous for all \(t \geq 0\).

Step 1: Understanding the Mathematical Model

The salary of Luisa is modeled by an exponential function, where the initial salary of $36500 is multiplied by \(1.035^t\), and \(t\) is the number of years since the contract signing. The 1.035 in the equation represents a 3.5% raise each year (1+3.5/100=1.035).

Step 2: Validation of the Mathematical Model

Replace y with 36,500 and t with 0, as per the initial conditions. The result is 36,500 = 36,500 * \(1.035^0\) which simplifies to 36,500 = 36,500 validating the initial condition. Thus, the provided salary model is accurate.

Step 3: Sketching the Function y = 36500 * \(1.035^t\)

When graphing the function, we see it takes the form of an exponential growth curve starting at $36,500 when t = 0 (the point at which Luisa signed the contract) and increases every year due to the annual salary increase.

Step 4: Continuity of the Function

The function y = 36,500(1.035^t) is continuous for all values of \(t\), because \(t\) is the time measured in years since signing the contract, and the domain of this exponential function is all real numbers. Therefore, it is continuous for all \(t \geq 0\).

Step 5: Interpreting the Function

The function shows that Luisa's salary is continuously increasing each year by 3.5\%. This type of salary increase is common in many job contracts where a fixed percentage increase is applied each year.

Key Concepts

Salary Increase

When it comes to a salary increase, it's fascinating to understand how a simple percentage can significantly impact earnings over time. In Luisa's case, a 3.5% annual raise is guaranteed by her contract.
A salary increase like this can be calculated using an exponential formula. Initially, Luisa's salary is $36,500. Each year, her salary gets multiplied by 1.035, which represents a 3.5% rise. This means, her new salary each year is not just the previous salary with an extra 3.5%, but a whole 3.5% on top of that prior increase.
This multiplicative effect is why exponential growth is so powerful. Over several years, even a seemingly small percentage like 3.5% can lead to significant growth in income. Understanding this concept helps in planning finances and knowing the true value of raises over the long term.

Continuous Function

The concept of a continuous function is essential in understanding Luisa's salary increase over time. In mathematics, a continuous function means that the value can flow smoothly without any breaks or jumps. In this context, Luisa's salary over time is modeled by the equation \[ y = 36,500(1.035)^t \] where \(t\) represents the years elapsed.
Continuous functions are crucial, as they reassure us that there's no abrupt change in Luisa's salary at any point within the years. As each year passes, the salary neatly transitions to the next without any disruption.
For Luisa's scenario, this implies her salary will gradually increase, ensuring a steady financial progression without surprises each year.

Graphing Exponential Functions

Graphing exponential functions can seem daunting, but they offer a clear visual representation of growth. For Luisa's salary, the function \[ y = 36,500(1.035)^t \] represents exponential growth—this is when values rise rapidly over time, and it's what we see in her annual salary increases.
When you plot this function on a graph, starting at \(t = 0\), you'll observe an upward curve starting at $36,500. Over time (\(t > 0\)), the function smoothly ascends, showcasing the impact of each year's 3.5% salary rise.
For students, graphing such functions helps visualize how small annual increases compound over time. It is a useful tool for seeing the progression of any exponential growth situation, whether it's salaries, investments, or even populations. The steady climb reflects the compounded growth rate, offering an intuitive glance at how exponential functions operate.