Question

Rewrite the function in the form \(y=a(1+r)^t\) or \(y=a(1-r)^t\). Then state the growth or decay rate. \(y=a\left(\frac{2}{3}\right)^{t / 10}\)

Short answer

The function can be rewritten as \(y = a (1 - \frac{1}{30})^t\) and the decay rate is \(3.33\%\).

Step-by-step solution

Identify the function type

The given function \(y = a\left(\frac{2}{3}\right)^{t / 10}\) is a form of a geometric function where the ratio (common ratio) is \(2/3\), which is less than 1. So, it is a decay function.

Rewrite the function

It's necessary to rewrite the function in the form \(y = a(1 - r)^t\). The expression inside the bracket should be written as \(1 - r\), where \(r\) represents the decay rate. That is \(1 - r = \frac{2}{3}\). Solving for \(r\), we will get \(r = 1 - \frac{2}{3} = \frac{1}{3}\). The power \(t/10\) should be converted to \(t\) by adjusting the decay rate. Thus the function \(y = a \left(\frac{2}{3}\right)^{t / 10}\) can be rewritten as \(y = a (1 - \frac{1}{30})^t\).

Determine the decay rate

It's clear now that the decay rate \(r\) is \(1/30\) or \(0.0333\) or \(3.33\%\) when expressed as a percentage.

Key concepts

Decay Rate

Understanding the decay rate is pivotal when working with exponential decay functions. In this context, the decay rate is the factor by which a quantity decreases over a specific period. We often represent this rate as a decimal or percentage, making it easier to interpret. For any function in the form of \(y = a(1 - r)^t\), the value \(r\) indicates how quickly the original amount \(a\) is reducing.
For example, consider the function given in the exercise: \(y = a\left(\frac{2}{3}\right)^{t / 10}\). Here, we need to rewrite this function to identify the decay rate correctly. By setting it into the form \(1 - r\), we find:

• \(1 - r = \frac{2}{3}\), implying that \(r = \frac{1}{3}\).
• This calculation implies a decay rate of \(\frac{1}{30}\) (approximately 0.0333 or 3.33%).

Knowing the decay rate helps predict how the quantity diminishes over time, making it a valuable metric in various fields like physics, biology, and economics.

Geometric Function

A geometric function is a powerful mathematical tool used to express growth or decay patterns. It follows a specific structure that defines how values change over a uniform interval. The core idea is centered around multiplication by a constant ratio, which is either greater than one (growth) or less than one (decay).
In our example, the function \(y = a\left(\frac{2}{3}\right)^{t / 10}\), we note a common ratio of \(\frac{2}{3}\). This indicates the process is a decay because the ratio is below 1.

• Here, \(a\) represents the initial value before any multiplicative changes.
• The ratio, represented by \(\frac{2}{3}\), defines the decay's "step," and effectively, the amount diminishes over time.

The understanding of geometric functions is integral to solving real-life problems where patterns and predictable change are subject to analysis, such as population studies, finance, and natural processes.

Rewriting Exponential Equations

Rewriting exponential equations can initially seem complex, but with a systematic approach, it becomes manageable. The goal is to simplify the function into a more recognizable form, making calculations and interpretations straightforward.
Starting with our exercise function \(y = a\left(\frac{2}{3}\right)^{t / 10}\), the task is to conform it to the standard form \(y = a(1 - r)^t\).

• First, equate the expression inside the exponent to \(1 - r\) (\(\frac{2}{3} = 1 - r\)), to find \(r\).
• Converting the power from \(t/10\) to \(t\) demands manipulating \(r\), resulting in \(r = \frac{1}{30}\).

This conversion not only reveals the decay rate but also aids in simplifying subsequent analysis or applications of the function. Rewriting such equations makes it easier to communicate findings and apply them practically.