Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Solve each problem. The formula \(f(x)=x^{1.5}\) calculates the number of years it would take for a planet to orbit the sun if its average distance from the sun is \(x\) times farther than that of Earth. If there were a planet located 15 times farther from the sun than Earth, how many years would it take for the planet to orbit the sun?

Step-by-Step Solution

Answer

The planet would take approximately 58 years to orbit the sun.

Step 1: Formula Substitution

Start with the given formula: \( f(x) = x^{1.5} \). Substitute \( x = 15 \) into the formula to find the time it takes for the planet to orbit the sun. This gives us: \( f(15) = 15^{1.5} \).

Step 2: Calculate Exponent

Calculate \( 15^{1.5} \). This is equivalent to \( 15^{3/2} \), which can be rewritten as \( \sqrt{15^3} \). First, calculate \( 15^3 = 3375 \).

Step 3: Square Root Extraction

Find the square root of \( 3375 \), which is \( \sqrt{3375} \). Use a calculator if necessary to find that \( \sqrt{3375} \approx 58.0948 \).

Step 4: Conclude the Solution

Now that we have calculated \( 15^{1.5} \approx 58.0948 \), this value represents the number of years it would take for the planet to orbit the sun.

Step 5: Rounding for Practical Use

Round \( 58.0948 \) to the nearest whole number. The number of years it would take for the planet to orbit is approximately \( 58 \) years.

Key Concepts

Exponential Functions

Exponential functions are a fundamental part of precalculus and calculus. These functions involve variables raised to a constant power, which can lead to rapid increases or decreases. In mathematical terms, an exponential function is typically expressed as \( f(x) = x^a \), where \( a \) is a constant. This essentially means that the output grows or shrinks at a rate proportional to its current value.

In the context of our exercise, the formula \( f(x) = x^{1.5} \) is an example of an exponential function. Here, the value of \( x \) is raised to the power of 1.5. This specific power implies a relationship where the variable not only scales directly but also accelerates due to the power of 1.5, often making calculations seem more complex.

To better understand this concept:

• The value of \( x \) represents the multiple of distance compared to Earth's orbit.
• The exponent reflects the gravitational effects and characteristics defined by the specific scenario or model.

Understanding how to manipulate such functions is crucial for solving real-world problems and modeling natural phenomena.

Orbital Mechanics

Orbital mechanics, also known as celestial mechanics, deals with the motion of celestial bodies under the influence of gravitational forces. It's a significant part of astrophysics and is essential for understanding how planets, moons, and other astronomical objects move through space.

The concept helps explain phenomena like planetary orbits, including why certain planets have longer or shorter years compared to Earth. According to Kepler's Third Law of Planetary Motion, the time a planet takes to orbit the sun is related to its distance from the sun. This is encapsulated in the exercise formula \( f(x) = x^{1.5} \), linking the orbital period of a planet to its solar distance.

Important points about orbital mechanics:

• The gravitational pull from the sun decreases with distance, affecting the speed and duration of a planet's orbit.
• When a planet is further away from the sun, the increase in distance results in a longer path, leading to a longer orbital period.

Understanding these principles allows us to calculate and predict the orbits of different celestial bodies accurately.

Mathematical Modeling

Mathematical modeling involves creating mathematical representations of real-world systems to predict and analyze their behavior. This approach uses equations and data to simulate physical, biological, or economic processes, providing insights into complex systems.

In our exercise, the equation \( f(x) = x^{1.5} \) serves as a model for determining the orbital period of a planet relative to Earth. By inputting the variable \( x \), which represents how much further the planet is from the sun compared to Earth, we can model the time it takes for the planet to complete its orbit.

Key aspects of mathematical modeling:

• It helps in understanding and predicting the behavior of systems that are too complex for practical experimentation.
• The accuracy of a model depends on how well the chosen mathematical relationships reflect real-world dynamics.

Through modeling, astronomers and scientists can explore new hypotheses, plan space missions, and better understand the universe.