Problem 61 - 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. Mercury (.24 years)

Step-by-Step Solution

Answer

Answer: The approximate distance from the sun to planet Mercury is 35.986 million miles.

Step 1: Insert the given orbit time into the equation

We are given the equation \(y=92.8935 \cdot x^{-6669}\) and the information that Mercury takes 0.24 Earth years to complete one orbit of the sun. We will substitute \(x=0.24\) into the equation as follows: $$y = 92.8935 \cdot (0.24)^{-6669}$$

Step 2: Simplify the equation and solve for \(y\)

We'll now simplify and solve for \(y\) using the given value of \(x\) by calculating the exponent and then multiplying by the constant 92.8935: $$y = 92.8935 \cdot (0.24)^{-6669} \approx 35.986$$ The distance from the sun to Mercury, denoted as \(y\), is approximately 35.986 million miles.

Key Concepts

Orbit Time

In astronomy, **orbit time** is a crucial concept that denotes how long a planet takes to complete one full revolution around the sun. For our solar system, these periods vary significantly from one planet to another.
The concept is closely related to the measurement of a year on Earth but differs for each planet. For example, Earth takes exactly one year to orbit the sun, while Mercury, being closer to the sun, completes its orbit in just 0.24 Earth years.
This variation in orbit time is pivotal for understanding planetary movements and their relationship with the sun. Calculating orbit time helps scientists predict planetary positions and study the mechanics of our solar system. It also aids in calculating planetary distances, which is essential for astronomical observations and research.

Distance Calculation

Calculating the distance from the sun to a planet involves understanding how **exponential functions** can describe natural phenomena. In the given exercise, the distance is determined using the formula: \[ y = 92.8935 \times x^{-6669} \]Here, **x** represents the orbit time or the number of Earth years a planet takes to orbit the sun, and **y** is the resulting distance in millions of miles.
The exponent \(-6669\) reflects a relationship specific to this calculation for estimating the planet's distance based on the orbit time. Substituting x with the given orbit time of Mercury, which is 0.24 years, allows us to compute the distance. Such calculations help us quantify and compare planetary distances and understand their spatial positioning in our solar system.

Planetary Orbits

**Planetary orbits** are the pathways that planets trace around the sun. Shaped like ellipses, these orbits vary from nearly circular to highly elongated. This variance influences factors like orbit time and how close or far a planet is from the sun during its orbit.
Understanding these orbits is essential for observing how planets move and predicting their positions over time. Key principles such as Kepler's laws of planetary motion, which detail the speed and position of planets along their orbits, provide foundational knowledge that enhances our grasp of planetary dynamics.
In our exercise, the focus on calculating distance using orbit time ties back to the nature of these orbits. Knowing how much time a planet takes to orbit the sun helps in mapping these elliptical paths, leading to insights about their interactions within the solar system.