Problem 16 - Algebra 2 and Trigonometry

Question

The orbital speed in kilometers per second and the distance from the sun in millions of kilo- meters of each of six planets is given in the table. $$ \begin{array}{|c|c|c|c|c|c|c|}\hline \text { Planet } & {\text { Venus }} & {\text { Earth }} & {\text { Mars }} & {\text { Jupiter }} & {\text { Saturn }} & {\text { Uranus }} \\ \hline \text { Orbital Speed } & {34.8} & {29.6} & {23.9} & {12.9} & {9.6} & {6.6} \\ \hline \text { Distance from the Sun } & {108.2} & {149.6} & {227.9} & {778.0} & {1,427} & {2,871} \\\ \hline\end{array} $$ a. Find the regression equation that appears to be the best fit for the data with the coefficient rounded to three decimal places. b. Neptune has an orbital speed of 5.45 \(\mathrm{km} / \mathrm{sec}\) and is \(4,504\) million kilometers from the sun. Does the equation found for the six planets given in the table fit the data for Neptune?

Step-by-Step Solution

Answer

The regression equation is \( y = -0.006x + 35.159 \); it does not fit Neptune's data well.

Step 1: Understand the Problem

We need to find a relationship between the orbital speed (in km/s) and the distance from the Sun (in millions of km) for six planets using a regression equation. The goal is to see if this equation can predict the orbital speed of Neptune based on its distance from the Sun.

Step 2: Determine the Type of Regression

Given that both variables (orbital speed and distance from the Sun) are continuous, we will use linear regression to find a relationship between them. We will use the orbital speed as the dependent variable (y) and the distance from the Sun as the independent variable (x).

Step 3: Calculate the Regression Equation

To find the linear regression equation of the form \( y = mx + b \), where \( y \) is the orbital speed and \( x \) is the distance from the Sun, we calculate the slope \( m \) and intercept \( b \) using the given data. Use the formula:\[ m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2} \]\[ b = \frac{(\sum y) - m(\sum x)}{N} \]where \( N = 6 \) is the number of planets.

Step 4: Compute Necessary Sums

Calculate the following based on given data:1. \( \sum x \) (sum of distances): 5561.7 2. \( \sum y \) (sum of orbital speeds): 117.4 3. \( \sum xy \) (sum of the product of distances and speeds): 95022.8 4. \( \sum x^2 \) (sum of squares of distances): 6446341.5

Step 5: Calculate Slope (m)

Substitute the values into the slope formula:\[ m = \frac{6 \times 95022.8 - 5561.7 \times 117.4}{6 \times 6446341.5 - (5561.7)^2} \approx -0.006 \] (rounded to three decimal places)

Step 6: Calculate Intercept (b)

Substitute \( m = -0.006 \), \( \sum x = 5561.7 \), and \( \sum y = 117.4 \) into the intercept formula:\[ b = \frac{117.4 - (-0.006 \times 5561.7)}{6} \approx 35.159 \] (rounded to three decimal places)

Step 7: Linear Regression Equation

Combine the calculated slope \( m \) and intercept \( b \) to form the equation:\[ y = -0.006x + 35.159 \]

Step 8: Evaluate Neptune's Data

Given Neptune's distance from the Sun is 4504 million km, calculate the predicted orbital speed using the regression equation:\[ y = -0.006(4504) + 35.159 \approx 8.135 \]Compare this with Neptune's actual speed of 5.45 km/s.

Step 9: Conclusion on Fit

The predicted orbital speed for Neptune using the regression equation is 8.135 km/s, which is significantly different from its actual speed of 5.45 km/s. Therefore, the regression equation does not fit Neptune's data well.

Key Concepts

Orbital Speed

Orbital speed is a fascinating concept that denotes the speed at which a planet travels around the sun. It is usually measured in kilometers per second (km/s). Understanding orbital speed helps us make sense of how different planets move through space. For example, the closer a planet is to the sun, such as Venus, the faster its orbital speed. This speed is vital for maintaining the planet's orbit and preventing it from being pulled into the sun by gravity.

Orbital speed can be influenced by several factors:

• Mass of the sun: The gravitational force exerted by the sun affects how fast a planet must travel to stay in orbit.
• Distance from the sun: Generally, the further a planet is from the sun, the slower its orbital speed. This is because the gravitational pull weakens with distance.
• Planet's mass: While less significant than the sun's mass, a planet's mass can influence its speed.

In the exercise, we see varying orbital speeds for planets like Earth, Mars, and Jupiter, illustrating how distance and other factors impact their motion.

Distance from the Sun

The distance from the sun is a crucial factor in determining a planet's orbital characteristics. It is expressed in millions of kilometers. The planets in our solar system orbit at various distances, leading to differences in orbital speed and year length. For example, Earth's average distance from the sun is about 149.6 million kilometers, allowing a perfect balance for life as we know it.

The distance affects planetary motion in the following ways:

• Gravitational pull: A planet closer to the sun, like Mercury or Venus, experiences a stronger gravitational pull.
• Orbital period: The further a planet is from the sun, the longer it takes to complete an orbit. Saturn and Uranus, for instance, have much longer years compared to Earth.
• Temperature and environment: Distance from the sun also influences the planet's climate and possibility of hosting life.

In our exercise, we observe that Venus, being closer to the sun, has a higher orbital speed than Uranus, which is farther away.

Regression Equation

A regression equation is a mathematical formulation used to describe the relationship between two variables. In this case, the exercise seeks to link the orbital speed of planets to their distance from the sun. We use linear regression because both variables are continuous, which involves finding a line of best fit through a set of data points.

With linear regression, we determine:

• Slope (\( m \)): Represents the rate of change of orbital speed with respect to distance.
• Intercept (\( b \)): Indicates the value of orbital speed when the distance is zero. While this scenario might not physically occur in our context, it mathematically anchors the line of best fit.

The regression equation we derived from the given dataset is \( y = -0.006x + 35.159 \) , where \( y \) represents the orbital speed and \( x \) is the distance from the sun.

This equation allows us to predict the orbital speed for other distances, although, as we saw with Neptune, predictions can differ if a planet's characteristics deviate from those of the sample data.