Problem 35 - Algebra 1: Concepts and Skills

Question

Write an exponential decay model for the situation. Then graph the model and use the graph to estimate the value at the end of the given time period. A 22,000 dollar investment decreases in value by \(9 \%\) per year for 8 years.

Step-by-Step Solution

Answer

The exponential decay model for the given situation is \(A = \$22,000e^{ln(0.91)t}\). By examining the graphical representation, or by substituting \(t = 8\) into the equation, the value of the investment at the end of 8 years can be estimated.

Step 1: Preliminary

Identify the initial value and the decay rate. For this problem the initial value is $22,000 — often called the ‘principal’ in financial problems — and the decay rate is \(9%\) per year. A decay rate of \(9%\) means the value reduces at an exponential rate of \(1 - 0.09 = 0.91\) per year.

Step 1: Formula formation

Write down the formula for exponential decay model, which is \(A = A_0e^{kt}\), where \(A_0\) is the initial value, \(k\) is the decay constant and \(t\) is the time. Given the problem, \(A_0 = \$22,000\) and \(k = ln(0.91)\), so that the equation becomes \(A = \$22,000e^{ln(0.91)t}\).

Step 2: Graphical Representation

Plot the exponential decay model on an appropriate graph to visualize the investment over time.

Step 3: Estimation

Look at the graph at the end of the 8-year period and estimate the final value of the investment. Alternatively, substitute \(t=8\) into the formula to calculate the exact value.

Key Concepts

Investment

Investing is a popular way to grow your wealth or achieve financial goals over time. One concept that comes up often in investment is understanding how your money can change in value, particularly when it decreases. In this example, you start with a $22,000 investment. However, instead of growing, this investment decreases in value every year. This is known as an investment in decline, where you lose a certain percentage each year, like the 9% decrease described here. Understanding this decline helps investors make informed decisions, balancing risk and potential reward.

Exponential Function

An exponential function is a mathematical tool that describes how a quantity changes over time. It is particularly useful for modeling situations where change happens at a consistent percentage rate. The standard form of an exponential decay function is

• \(A = A_0e^{kt}\)

where:

• \(A\) is the amount after time \(t\).
• \(A_0\) represents the initial amount, in this case, $22,000.
• \(k\) is the decay constant, derived from the percent decrease.

Exponential decay occurs when the function involves a decay constant reflecting a loss, like the 9% annual decrease. Using natural logarithms can help calculate the decay constant, \(ln(0.91)\) in this problem because the value retains only 91% of its previous total annually.

Percent Decrease

Percent decrease indicates the rate at which a value declines annually. In the context of an investment, it is crucial to understand how much is lost each year. A 9% decrease means the investment retains 91% of its value annually, calculated as follows:

• The decay factor is derived from \(1 - ext{percent decrease}\).
• For a 9% decrease, the decay factor is \(1 - 0.09 = 0.91\).

This figure helps determine how quickly an investment will reduce over time. The smaller the percentage of decrease, the slower the decline, influencing the overall strategy of managing investments effectively.

Graphing Models

Graphing is a powerful method to visualize changes over time, especially for functions describing exponential decay. By plotting the value of the investment over the 8 years, one can better understand its diminishing value.

• Start by drawing the axes, with time (years) on the x-axis and investment value on the y-axis.
• Plot the initial value at year 0, which is $22,000.
• Use the exponential decay function \(A = 22000e^{ln(0.91)t}\) to calculate points for each subsequent year.

The result is a smooth curve that steadily decreases, encapsulating the decay process. Observing this graph makes it easier to estimate future values and plan accordingly, highlighting both the inevitability of decline and the potential value at any given point in the timeline.