Problem 4 - 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. Ten watts of power are required to operate the equipment in the satellite. How long can the satellite continue to operate?

Step-by-Step Solution

Answer

The satellite can operate for approximately 402 days.

Step 1: Understand the Problem

You need to determine how long the radioisotope can provide at least 10 watts of power for the satellite equipment. The power output is given by the function \(P = 50e^{-\frac{t}{250}}\), where \(t\) is the time in days.

Step 2: Set Up the Equation

Set the power output \(P\) to 10 watts to find out when the power output becomes exactly 10 watts: \[ 10 = 50e^{-\frac{t}{250}} \]

Step 3: Isolate the Exponential Term

Divide both sides of the equation by 50 to isolate the exponential term:\[ e^{-\frac{t}{250}} = \frac{10}{50} = 0.2 \]

Step 4: Solve for Time \(t\)

To solve for \(t\), take the natural logarithm of both sides:\[ \ln(e^{-\frac{t}{250}}) = \ln(0.2) \]This simplifies to:\[ -\frac{t}{250} = \ln(0.2) \]

Step 5: Calculate the Value of \(t\)

Multiply both sides by -250 to solve for \(t\):\[ t = -250 \times \ln(0.2) \]Using a calculator, compute \(\ln(0.2)\) and then multiply by -250. You should find that:\[ t \approx -250 \times (-1.6094) \approx 402.35 \]

Step 6: Interpret the Result

The calculated value of \(t\) is approximately 402.35 days. This means the satellite can operate for about 402 days before the power drops below 10 watts, which is insufficient to run the equipment.

Key Concepts

Radioisotope

A radioisotope is a type of isotope that is radioactive. This means it emits radiation as it decays over time. In various scientific applications, particularly in space exploration, radioisotopes are critical for providing a reliable power source when solar energy or other methods are not feasible.

In this context, radioisotopes are used in radioisotope thermoelectric generators (RTGs), which convert the heat released from radioactive decay into electricity.

• Radioisotopes provide a consistent and long-lasting power source.
• They are crucial for missions in shadowy or distant parts of space, where sunlight is minimal.

This decay is predictable, which allows engineers to calculate the duration of power output, ensuring that crucial systems remain operational for extended periods.

Power Output

Power output is a measure of the amount of energy produced over time, typically expressed in watts. In the case of the exercise, power output is described by an exponential function, represented as \(P = 50 e^{-\frac{t}{250}}\).

In this formula:

• \(P\) is the power output in watts.
• The constant \(50\) represents the initial power output at \(t = 0\).
• The exponential factor \(e^{-\frac{t}{250}}\) describes how quickly the power diminishes over time, reflecting the process of radioactive decay.

As \(t\) (time) increases, the exponential term decreases, showing a drop in power output as the radioisotope decays, highlighting the importance of understanding exponential decay for power management in long-duration space missions.

Natural Logarithm

The natural logarithm, denoted usually as \(\ln\), is the logarithm to the base \(e\), where \(e\) is an irrational and transcendental number approximately equal to 2.71828. In the exercise, the natural logarithm is used to solve for \(t\), which is time in days.

Natural logarithms are particularly useful in solving equations that involve exponential decay. For instance:

• To find \(t\), you need to solve an equation in the form of \(e^x\), which can be neatly handled using logarithms.
• The equation simplifies the problem by translating exponential terms into linear forms, making it easier to isolate the variable of interest.

In problems of decay, growth, or scale, employing \(\ln\) helps to reveal the time or rate aspects more transparently, making calculations more straightforward in practical applications.

Satellite Operations

Satellites rely on highly stable and efficient power sources to maintain successful operations over time. Many satellites are designed to operate in environments where consistent power supply is critical. In addition to tasks such as data collection and transmission, satellites may also need to control onboard systems that ensure their proper function and orientation in space.

For such operations:

• Satellites require a minimum power level to ensure their systems are continuously operational.
• Power levels impact critical functions like data transmission, propulsion, and maintaining a stable orbit.

The capability to calculate a satellite's operational time based on power output helps engineers plan for the satellite's lifecycle, allowing them to prepare backup solutions or adjust mission durations accordingly. By understanding the expected longevity of the power source using mathematical calculations, missions can be optimized for duration and efficiency, ensuring high reliability and success in satellite operations.