Problem 41 - College Algebra

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 Base of the Exponential Function

The given equation is \( y = 2.25e^{-2t} \). Here, \( e \) is the base of the exponential function, often associated with continuous growth or decay, depending on the exponent.

Step 2: Analyze the Exponent

In the equation \( y = 2.25e^{-2t} \), the exponent is \(-2t\). The exponent is the key to determining growth or decay. Since it is preceded by a negative sign, the function incorporates a decreasing exponential factor.

Step 3: Determine Continuous Growth or Decay

For continuous decay, the exponent in the form of \( e^{kt} \) should be negative, which means \( k < 0 \). Here, \(-2 < 0\), which indicates that the function represents continuous decay.

Step 4: Conclusion

Since the exponent \(-2t\) is negative, \( y = 2.25e^{-2t} \) is a model of continuous decay.

Key Concepts

Exponential functions

Exponential functions play a vital role in mathematics, especially when modeling real-world scenarios involving rapid change. An exponential function is generally expressed in the form \( y = ab^{x} \), where \( a \) is a constant and \( b \) is the base of the exponential function. If \( b = e \), which is a special irrational number approximately equal to 2.718, the function often describes situations involving continuous growth or decay.

Here’s why exponential functions are so powerful:

• They can model phenomena that change at constant percentages or rates.
• They are used across disciplines, from finance to biology, to describe anything from population growth to radioactive decay.
• They have unique properties, such as the fact that the rate of change is proportional to the current value, making them ideal for continuous change.

In the exercise, the equation \( y = 2.25e^{-2t} \) is an exponential function with \( b = e \), indicating a focus on continuous change.

Continuous growth vs. continuous decay

Continuous growth and continuous decay describe how systems persistently increase or decrease over time. In mathematical models, the distinction between growth and decay often comes down to the sign of the exponent in the exponential function.

In the equation \( y = 2.25e^{-2t} \):

• Continuous growth: If the exponent is positive (\( kt > 0 \)), the function describes continuous growth, meaning the value of \( y \) increases as \( t \) increases.
• Continuous decay: Conversely, if the exponent is negative (\( kt < 0 \)), like with \( -2t \), the function models continuous decay, indicating that \( y \) decreases as \( t \) increases.

This framework helps predict behavior in various fields, allowing us to model how populations decrease over time in a controlled environment, or how investments shrink if affected by consistent negative interest rates.

Exponent analysis

The exponent in an exponential function determines its characteristics, signifying whether a function grows or decays over time. Analyzing the exponent is a critical step in understanding the nature of the function.

In \( y = 2.25e^{-2t} \), the exponent is \(-2t\), and you can infer:

• Negative exponent: As seen here, when the exponent \( -2t \) is negative, it confirms the function is a model of continuous decay.
• Exponent and time relationship: The exponent usually involves a constant and a variable like time (\( t \)), indicating how quickly the function changes.

Conducting an exponent analysis allows you to determine effective strategies for problems in economics, environmental sciences, or engineering, where decay rates need careful management. Understanding exponent details helps avoid assumptions and provides clarity on function behavior.