Problem 62 - Contemporary Precalculus

Question

Use the equation \(y=92.8935 \cdot x^{-6669}\) which gives the approximate distance \(y\) (in millions of miles) from the sun to a planet that takes \(x\) earth years to complete one orbit of the sun. Find the distance from the sun to the planet whose orbit time is given. Mars ( 1.88 years)

Step-by-Step Solution

Answer

Answer: The approximate distance from the Sun to Mars is 141.553 million miles.

Step 1: Write down the given equation

We have the equation for distance in millions of miles (y) from the Sun to a planet that takes x Earth years to complete one orbit: \[y = 92.8935 \cdot x^{-6669}\]

Step 2: Substitute x with given orbit time

The orbit time, x, for Mars is 1.88 years. Substitute this value into the equation: \[y = 92.8935 \cdot (1.88)^{-6669}\]

Step 3: Compute y (distance)

Now, calculate the value of y using the substituted value of x: \[y = 92.8935 \cdot (1.88)^{-6669} \approx 141.553\]

Step 4: Interpret the result

So, the approximate distance from the Sun to Mars is 141.553 million miles.

Key Concepts

Distance Calculation

When calculating distances in space, especially the vast expanses between planets and the Sun, exponential functions play a critical role. The given equation allows one to compute the distance between the Sun and a specific planet by substituting the planet's orbital period into the equation.

For Mars, whose orbital period is 1.88 Earth years, we substitute this value for the variable \(x\). This substitution allows for a specific distance calculation, giving an intuitive way to understand how different orbital periods affect distance in planetary systems.

Steps to calculate distances using exponential functions can be summarized as follows:

• Identify the exponential equation relating distance and orbital period.
• Substitute the orbital period into the equation.
• Solve the equation to find the distance.

These simple steps can help unravel the complexity of astronomical distances.

Planetary Orbits

Understanding planetary orbits is crucial for calculating distances and navigating space. Planetary orbits are the paths planets follow as they rotate around the Sun. The shape and duration of these orbits can inform us about the planet's distance from the Sun.

Most planetary orbits in our solar system are elliptical. However, they are often close to circular due to the Sun's gravitational force keeping them in a stable path. Knowing the time it takes for a planet to complete one orbit (its orbital period) is key to determining its average distance from the Sun.
Key aspects regarding planetary orbits include:

• Elliptical shape: Though close to circular, true planetary orbits are not perfect circles.
• Orbital period: This is the time a planet takes to complete one full orbit around the Sun.
• Uniform motion: Due to gravitational forces, planet speeds are relatively constant within their orbit.

These characteristics of planetary orbits influence how scientists and astronomers model distances within our solar system.

Mathematical Modeling

Mathematical modeling in astronomy allows us to represent complex systems with understandable equations and formulas. By using mathematical equations, such as the one given in the Mars distance problem, we can predict and explain planetary behaviors and characteristics.

Models serve functions such as predicting future positions of planets and understanding relationships between variables like orbital period and distance. Exponential models, like the one given, showcase the power of math to simulate vast and complex systems on Earth and beyond.
Key elements of mathematical modeling include:

• Equations: Representations of relationships among different variables.
• Substitution: Inserting known values to solve for unknowns.
• Prediction: Using models to foresee planetary positions and distances over time.

By applying mathematical models, we simplify the immensity of space into manageable calculations, which grow our understanding of the universe.