Problem 86 - College Algebra and Calculus: An Applied Approach

Question

A principal \(P\), invested at \(4.85 \%\) interest and compounded continuously, increases to an amount that is \(K\) times the principal after \(t\) years, where \(t\) is given by \(t=\frac{\ln K}{0.0485}\) Use a graphing utility to graph this function.

Step-by-Step Solution

Answer

Upon graphing the function \(t=\frac{\ln K}{0.0485}\), an upward opening curve is obtained that begins at (1,0) and shoots up towards infinity as \(K\) increases. This reflects the nature of continuously compounded interest, where a longer time is required for an amount to grow by a larger factor.

Step 1: Identify the function

The function given in this exercise is \(t=\frac{\ln K}{0.0485}\). This function represents the time \(t\) it would take for an amount \(P\) to grow by a factor of \(K\) under continuous compounding at an annual interest rate of \(4.85\%\). The variable \(K\) represents the multiple of the original principal \(P\).

Step 2: Identify the domain and range

Because \(K\) represents a multiple of the principal, it can take on any positive real value. We would therefore consider the domain of the function to be \(K > 0\). Since time cannot be negative, the range of the function will be \(t \geq 0\).

Step 3: Graph the function

Using a graphing utility, input the function \(t=\frac{\ln K}{0.0485}\) and set the domain to be \(K > 0\). The function will be an upward opening curve with the y-intercept at \(t = 0\) (when \(K = 1\)), indicating that the principal amount doubles with no time elapsed. The graph reflects that the more the sum increases (higher \(K\)), the more time it takes for the investment to grow, reflecting the nature of compound interest.

Key Concepts

Exponential Growth

Exponential growth is a powerful concept where quantities increase rapidly over time. Benefitting from the principles of multiplication, exponential growth occurs when the growth rate of a value is directly proportional to its current size. This means bigger quantities grow faster compared to smaller ones. In the context of compound interest, this principle explains how investments grow over time. With continuous compounding, as seen in this exercise, the money grows at every instant, and this leads to an especially quick rise in value. Continuous compounding can be modeled by the formula \(A = Pe^{rt}\), where \(P\) is the initial principal, \(r\) is the rate of interest, and \(t\) is time.

• Exponential growth results in a steep rise in graph plots.
• It's essential to recognize that as time progresses, the growth accelerates due to the compounding effect.

This is why investments can significantly gain in value over long periods of time if left untouched, showing the power of exponential growth.

Compound Interest

Compound interest is a system where interest is calculated on the initial principal, which also includes all of the accumulated interest from previous periods. This is opposed to simple interest, where interest is computed on the principal alone. With compound interest, each period’s interest becomes part of the principal for the next period. Continuous compounding is its ultimate form, where this compounding happens at every moment in time, mathematically described using the natural exponential function. This can make a significant difference in long-term investments.

• It is used in many financial products like savings accounts and loans.
• Its formula is typically \(A = Pe^{rt}\), contributing to the exponential nature of financial growth.
• The continuous compounding formula \(t = \frac{\ln K}{0.0485}\) here determines the time to reach a certain growth factor \(K\).

The interest acts as a stimulant for quick growth, making it an attractive aspect of many investment strategies.

Graphing Functions

Graphing functions provides a visual representation of how variables in a function interact with each other. For this specific function \(t = \frac{\ln K}{0.0485}\), a graph depicts the relationship between \(K\) (the multiple of the principal) and \(t\) (the time required to reach \(K\) times the principal). By using a graphing utility, such as a graphing calculator or software, you can view how the timeline changes as \(K\) increases.

• The graph here will be an upward-facing curve.
• This visually demonstrates how more substantial growth factors \(K\) result in longer periods \(t\).
• The curve starts at \(t = 0\) when \(K = 1\), meaning no time is needed for the principal to be itself.

Graphing helps in understanding the behavior and dynamics of mathematical relationships and is crucial in analyzing continuous compound interest scenarios.

Domain and Range

In mathematical functions, the domain and range are crucial concepts that describe the set of possible input values (domain) and the corresponding output values (range). For a function like \(t = \frac{\ln K}{0.0485}\), the domain consists of all positive values for \(K\), meaning \(K > 0\). This is because \(K\) represents a factor by which the principal grows, and it cannot be a negative or zero value.

The range of this function is defined as \(t \geq 0\). This reflects that time cannot be negative—it always moves forward as an investment grows. For every positive value of \(K\), you'll have a corresponding positive or zero time \(t\) required to reach that growth.

• Domain: The set of possible \(K\) values determining principal growth (\(K > 0\)).
• Range: The resulting \(t\) values indicating time, which must be non-negative (\(t \geq 0\)).

Understanding domain and range is essential as it sets the limits and scope of how the function can be analyzed and interpreted.