Problem 21 - Algebra 1: Concepts and Skills

Question

Write an exponential decay model for the investment. A bond is purchased for $70. Then the value decreases by 1% per year.

Step-by-Step Solution

Answer

The exponential decay model which represents the value of the bond decreasing by 1% each year is \( P(t) = $70 \times (0.99)^t \)

Step 1: Identify the Initial Value

The initial value is the starting amount, in this case, the cost of the bond. Here, the bond is initially worth $70. So, \( P_0 = $70 \)

Step 2: Convert the rate of decay to a decimal

The value of the bond decreases by 1% every year. To express this as a decimal, divide 1 by 100 which gives \( r = 0.01 \)

Step 3: Substitute the Initial Value and the Rate of Decay Into the Exponential Decay Formula

The exponential decay formula is \( P(t) = P_0 \times (1 - r)^t \). Substitute the initial value and the rate of decay into this formula to get the exponential decay model: \( P(t) = $70 \times (1 - 0.01)^t \) which simplifies to: \( P(t) = $70 \times (0.99)^t \)

Key Concepts

Initial Value in Exponential Decay

In the context of an exponential decay model, the 'initial value' is a pivotal concept, representing the beginning quantity before any decay has occurred. In our example, we look at a bond purchased for \(70, which serves as our initial value denoted as \( P_0 \). This initial value is fundamental as it sets the baseline for the prediction of future values. When considering any exponential decay scenario, the first step is always to ascertain the initial value. It's important to note that the entire model's accuracy heavily depends on the correct identification of this initial amount.

Think of the initial value as the starting line in a race; everything that follows is in relation to this point. As time passes, the value of the bond decreases by a given percentage, not from an arbitrary number, but from this very specific initial value of \)70.

Rate of Decay to Decimal

Converting the 'rate of decay' to a decimal is a critical step in the construction of an exponential decay model. In our exercise, the bond decreases in value by 1% each year. To use this percentage in mathematical formulas, it needs to be expressed as a decimal. This is a straightforward conversion––simply divide the percentage by 100. Thus, the decay rate of 1% becomes \( r = 0.01 \). It’s crucial to make this conversion because percentages are not just numbers, but a ratio based on 100, which needs standardization to be used in any exponential formula.

To illustrate, think of this conversion as translating a language; we must move from the common language of percentages into the more precise 'mathematical language' of decimals. This allows us to accurately apply the rate of decay in calculations without altering its meaning or value.

Exponential Decay Formula

The exponential decay formula is the foundation for modeling situations where a quantity decreases at a rate proportional to its current value. The formula is generally expressed as \( P(t) = P_0 \times (1 - r)^t \), where \( P(t) \) is the amount after time \( t \), \( P_0 \) is the initial amount, \( r \) is the decay rate, and \( t \) is time. In our bond example, we substitute the initial value of \(70 and the decay rate of 0.01 to get \( P(t) = \)70 \times (0.99)^t \). This mathematical representation enables one to predict the future value of the bond at any given time, capturing the essence of exponential decay––the idea that the quantity decreases by a consistent percentage over each time period. This formula is powerful in its ability to model natural phenomena and financial calculations alike.

Understanding the exponential decay formula allows students to grasp the conceptual nature of decay over time, visualizing how the bond's value dwindles not linearly, but at a rate that slows down, mirroring many real-world processes.