Problem 112 - College Algebra Essentials

Question

For each planet in our solar system, its year is the time it takes the planet to revolve once around the sun. The formula $$E=0.2 x^{\frac{3}{2}}$$ models the number of Earth days in a planet's year, \(E,\) where \(x\) is the average distance of the planet from the sun, in millions of kilometers. There are approximately 88 Earth days in the year of the planet Mercury. What is the average distance of Mercury from the sun? Use a calculator and round to the nearest million kilometers.

Step-by-Step Solution

Answer

After performing calculations, the average distance of Mercury from the sun is approximately 58 million kilometers.

Step 1: Identify Known Values

We know that there are approximately 88 Earth days in the year of planet Mercury. It's given in the problem description.

Step 2: Solve Inverse Equation

Given, the equation \(E=0.2 x^{\frac{3}{2}}\). We aim to find 'x', which means that we need to isolate 'x' on one side of the equation. First, divide both sides of the equation by 0.2 to get \(\frac{E}{0.2} = x^{\frac{3}{2}}\). Then, to remove the fractional exponents, we will raise both sides of the equation to the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\). Consequently, the equation becomes \(\left(\frac{E}{0.2}\right)^{\frac{2}{3}} = x\).

Step 3: Calculate Result

We substitute 'E' with '88' in our new equation: \(x = \left(\frac{88}{0.2}\right)^{\frac{2}{3}}\). Use a calculator to calculate this value and round to the nearest integer. That will be the distance of Mercury from the sun in millions of kilometers.

Key Concepts

Inverse Equation

In solving for the unknown variable in an equation, an inverse equation is a powerful tool. For the given exercise, the main task is to express the variable 'x' in terms of 'E' using inverse operations. This involves reversing the operations applied to 'x' to isolate it.

• Start with the given equation: \(E = 0.2 x^{\frac{3}{2}}\).
• To isolate 'x', divide both sides by 0.2: \(\frac{E}{0.2} = x^{\frac{3}{2}}\).
• The next step is important: remove the fractional exponent by raising each side to the reciprocal of \(\frac{3}{2}\), which is \(\frac{2}{3}\).
• The final form of the equation will be: \(x = \left(\frac{E}{0.2}\right)^{\frac{2}{3}}\).

Inverse equations are all about undoing arithmetic operations to solve for your unknowns.

Fractional Exponents

Fractional exponents are an elegant way of representing powers and roots simultaneously. Understanding them can help simplify and solve complex expressions, such as the one in this exercise.

• In \(x^{\frac{3}{2}}\), the numerator 3 indicates that 'x' is cubed.
• The denominator 2 means that after cubing 'x', you then take the square root.
• Thus, \(x^{\frac{3}{2}}\) is the same as \((x^3)^{\frac{1}{2}}\).

When dealing with fractional exponents, remember that the number over the denominator (the base) is raised to the power of the numerator, and then you find the root indicated by the denominator. This understanding is key to tackling algebraic manipulations involving powers.

Solar System

The solar system is home to numerous planets, each orbiting the Sun at varying distances. A planet's year is the duration it takes to complete one orbit around the Sun—this varies based on the planet's average distance from the Sun. The given exercise uses this concept to relate the number of Earth days in a year on Mercury to its distance from the Sun.

• Planets closer to the Sun travel faster, resulting in shorter years (like Mercury).
• Conversely, planets farther away, like Neptune, have longer years.
• This relationship underscores the concept of planetary motion in our solar system.

Exploring these differences gives insight into how gravity and distance impact planetary orbits and year lengths.

Earth Days Calculation

Calculating the number of Earth days in another planet's year requires understanding the dynamics of planetary motion and distance. Here, the task was to find Mercury’s distance from the Sun based on its year.

• First, identify the number of Earth days in the given planetary year, in this case, Mercury’s, which is 88 days.
• Using the formulated equation, it relates days to distance: \(E = 0.2 x^{\frac{3}{2}}\).
• By solving for 'x' when \(E = 88\), we compute the distance Mercury is from the Sun.
• This results in calculating \(x = \left(\frac{88}{0.2}\right)^{\frac{2}{3}}\) using a calculator to find the correct rounded distance.

This approach allows translating calendar time on another planet into a measurable distance, bridging our understanding of Earth's and other planets' orbits.