Problem 2 - Algebra 2

Question

For Exercises \(2-4,\) use the following information. A radioisotope is used as a power source for a satellite. The power output \(P\) (in watts) is given by \(P=50 e^{-\frac{t}{250}},\) where \(t\) is the time in days. Is the formula for power output an example of exponential growth or decay? Explain your reasoning.

Step-by-Step Solution

Answer

The formula is an example of exponential decay because the exponent is negative.

Step 1: Understand the Formula

The given formula is \( P = 50 e^{-\frac{t}{250}} \). In this context, \( P \) represents the power output in watts, \( e \) is the base of the natural logarithm (approximately equal to 2.718), and \( t \) is time in days.

Step 2: Identify the Exponential Component

Look at the exponential part of the formula: \( e^{-\frac{t}{250}} \). Since the exponent is negative \( -\frac{t}{250} \), this indicates that the function is an exponential decay model rather than exponential growth.

Step 3: Analyze the Exponent Significance

In an exponential function \( a e^{kt} \), if \( k < 0 \), the function describes exponential decay because the value of \( e^{kt} \) decreases over time.

Step 4: Conclusion on Growth vs. Decay

The exponential component \( e^{-\frac{t}{250}} \) decreases as \( t \) increases because of the negative exponent. Hence, the power output \( P \) decreases over time, confirming it is exponential decay.

Key Concepts

Radioisotope Power

Radioisotopes are incredibly useful in many scientific and practical applications due to their ability to emit energy. This is particularly true for radioisotope power systems used in satellites. These systems provide a reliable and steady source of power over extended periods, often enduring harsh space conditions. The power generated from radioisotopes comes from the decay of radioactive isotopes, transforming them into a stable form while releasing energy.

• Spacecraft and satellites often employ radioisotope thermoelectric generators (RTGs) for power.
• RTGs convert the heat released by the natural decay of radioisotopes into electricity.
• The steady decay rate allows for consistent energy output, suitable for missions far from the Sun where solar power is impractical.

In the context of the given problem, the radioisotope's power output is a function of time, decreasing exponentially due to the decay process. This characteristic decay over time is essential for understanding the longevity and power capabilities of such power sources.

Exponential Functions

Exponential functions are a fundamental concept in mathematics, used to describe processes that increase or decrease at a constant rate. In our example with the radioisotope power, the function given is:
\[ P = 50 e^{-\frac{t}{250}} \]
This formula is a clear representation of exponential decay. Here's why:

• The function involves the expression \( e^{kt} \), where \( e \) is Euler's number, approximately 2.718.
• If \( k < 0 \), as in this case with \(-\frac{1}{250}\), the function describes a decaying process.
• The power output \( P \) decreases over time because the exponent of \( e \) is negative, leading to the decay characteristic.

Exponential functions are powerful for modeling real-world scenarios that show rapid changes, like population growth, radioactive decay, and investment returns. In the satellite power example, it helps predict how power decreases over time, ensuring optimal resource management for extended missions.

Natural Logarithm

The natural logarithm is deeply connected to exponential functions and is denoted as \( \ln \). It is a logarithm with the base \( e \). This connection is crucial when dealing with exponential decay and growth problems, as they often involve the base \( e \) that Euler discovered.

• The natural logarithm \( \ln(x) \) is the inverse function of \( e^x \).
• It is useful in solving for time \( t \) when manipulating the form \( e^{kt} \).
• In exponential decay contexts, like radioisotope power, the natural logarithm helps understand the rate and clarity of decay over time.

The formula in the exercise links closely to the natural logarithmic concepts, as it is driven by the base \( e \). This relationship helps express the decay proportional to a continuous process, like radioactive decay. Understanding the natural logarithm and its relation to exponential functions can simplify finding the exact time it takes for the power to decrease to a certain level or how quickly that decay occurs.