Problem 492 - Precalculus

Question

Does the equation \(y=2.294 e^{-0.654 t}\) represent continuous growth, continuous decay, or neither? Explain.

Step-by-Step Solution

Answer

The equation represents continuous decay due to the negative exponent.

Step 1: Analyzing the Equation

First, look at the given equation: \(y = 2.294 e^{-0.654 t}\). This is an exponential equation of the form \(y = a e^{kt}\), where \(a = 2.294\) and \(k = -0.654\). Here, the coefficient \(a\) represents the initial value, and \(k\) is the growth/decay rate.

Step 2: Identifying the Growth Rate

Examine the value of \(k = -0.654\). In an exponential growth or decay equation, the sign of \(k\) determines if the equation represents growth or decay. If \(k > 0\), it represents continuous growth. If \(k < 0\), it represents continuous decay. If \(k = 0\), it is neither.

Step 3: Determining Continuous Growth or Decay

Since \(k = -0.654\) is negative, this indicates continuous decay. The negative value of \(k\) means the function is decreasing over time.

Step 4: Conclusion

The equation \(y = 2.294 e^{-0.654 t}\) represents continuous decay because the exponent of \(t\) is negative, showing that as \(t\) increases, \(y\) decreases.

Key Concepts

Exponential Functions

Exponential functions are a special type of mathematical equation that describe situations where a quantity changes at a rate proportional to its current value. The general form of an exponential function is \(y = a e^{kt}\), where \(y\) is the final amount, \(a\) is the initial value, \(e\) is the base of the natural logarithm approximately equal to 2.718, \(k\) is the rate of growth or decay, and \(t\) is time.
These functions can represent both growth and decay processes, such as population growth, radioactive decay, and interest accumulation.

• When \(k > 0\), the process is one of growth, meaning the quantity increases over time.
• When \(k < 0\), the process is one of decay, indicating the quantity decreases over time.
• The base \(e\) is special because its properties make calculations involving rates of change particularly straightforward.

Understanding exponential functions is crucial since they model many real-world phenomena in science, engineering, and finance.

Growth Rate Analysis

Growth rate analysis is essential in understanding how a quantity changes over time within exponential functions. The rate of growth or decay is represented by the parameter \(k\) in the equation \(y = a e^{kt}\).
Analyzing the sign of \(k\) allows us to determine the behavior of the function:
- **Positive \(k\):** Indicates exponential growth. As time \(t\) increases, the value of \(y\) grows exponentially. Examples include compound interest or unchecked population growth.- **Negative \(k\):** Indicates exponential decay. In this case, as time \(t\) progresses, \(y\) decreases. Instances of this include radioactive decay and depreciation of assets.The magnitude of \(k\) affects how fast the growth or decay occurs:
- A larger absolute value of \(k\) results in a steeper curve, meaning faster change.- A smaller absolute value means slower change, leading to a gradual curve.By understanding \(k\), you can predict how rapidly a quantity will change, which is vital for planning and forecasting in various fields.

Initial Value

The initial value, denoted by \(a\), in an exponential function \(y = a e^{kt}\), is the starting point of the quantity being studied. It represents the value of \(y\) at \(t = 0\).
This component is crucial because it sets the stage for how the exponential function behaves as \(t\) progresses:
- In growth scenarios, having a larger initial value means the function will naturally start from a greater amount.- For decay processes, the initial value represents what is present before the process begins reducing the quantity.
When analyzing exponential functions, it's essential to identify \(a\) to comprehend the initial conditions of the model.
Consider a practical example: In a savings account earning continuous interest, \(a\) includes your initial deposit. It plays a critical role and influences the total amount accumulated over time.
By understanding and identifying the initial value, you can connect real-world conditions to mathematical models, enhancing the practicality and relevance of the solutions.