Problem 41 - College Algebra with Corequisite Support

Question

For the following exercises, determine whether the equation represents continuous growth, continuous decay, or neither. Explain. \(y=2.25(e)^{-2 t}\)

Step-by-Step Solution

Answer

The equation represents continuous decay due to the negative exponent.

Step 1: Identify the Structure

The given equation is \(y = 2.25e^{-2t}\). It follows the general form of an exponential function, \(y = ae^{rt}\), where \(a\) is the initial value, \(e\) is the base of the natural logarithm, and \(r\) is the rate of growth or decay.

Step 2: Analyze the Exponential Component

In the equation \(y = 2.25e^{-2t}\), the exponent is \(-2t\). The key part to determine growth or decay is the sign of the exponent. Here, the exponent is \(-2t\), which implies a negative rate.

Step 3: Determine Growth or Decay

If the rate \(r\) is negative, as it is here with \(-2\), it indicates continuous decay. This is because the presence of a negative sign in the exponent leads to the function decreasing over time.

Key Concepts

Continuous Growth and Decay

Continuous growth and decay refer to how quantities change over time in a smooth and unbroken manner. These concepts are often modeled using exponential functions, which can either increase or decrease in value at a constant percentage rate:

• Continuous Growth: This occurs when a quantity increases steadily over time. If you look at the equation in the form \(y = ae^{rt}\), a positive rate \(r\) signifies growth.
• Continuous Decay: This happens when a quantity steadily decreases. A negative rate \(r\) in the equation indicates decay, meaning the quantity decreases over time at a constant percentage rate.

The specific case of the equation \(y = 2.25e^{-2t}\) illustrates continuous decay due to the negative rate, \(-2\). This indicates that the value of \(y\) decreases as time increases. By understanding the sign of the rate, you can determine whether a function models either continuous growth or decay.

Exponential Decay

Exponential decay describes a process where a quantity diminishes at a rate proportional to its current value. This type of process is widely observed in nature and science, from radioactive decay to cooling of objects:

• The formula \(y = ae^{rt}\) serves as the standard for exponential functions, where \(a\) represents the starting quantity, \(e\) is the base of the natural logarithm, \(r\) is the decay rate, and \(t\) is time.
• A negative \(r\) results in a diminishing exponential function, characterized by a curve that starts at a value \(a\) and decreases, approaching zero but never quite reaching it.

For example, in the equation \(y = 2.25e^{-2t}\), \(-2\) signifies a decay rate, indicating that the value of \(y\) fades rapidly over time as \(t\) progresses. Exponential decay is all around us, from depreciation of assets in finance to the fading of colors in dyed materials.

Mathematical Expressions

Mathematical expressions are combinations of numbers, variables, and operations that represent a particular value or set of operations. In mathematics, expressions can take many forms:

• Constants and Variables: Constants like 2.25 in our example remain unchanged, while variables denote unknowns that can vary, such as \(t\) for time.
• Operations: Operations include addition, subtraction, multiplication, division, and exponentiation, which in our case is represented by \(e^{rt}\).
• Functions: Functions denote what you're solving for, like \(y\) in the equation \(y = 2.25e^{-2t}\), and they define the relationship between the variables and constants in the expression.

Expressions like \(y = 2.25e^{-2t}\) help students model real-world situations and perform calculations. Understanding how to manipulate and interpret these expressions is key to solving mathematical problems across various disciplines.