Problem 112 - College Algebra

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 Use the equation to solve. 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

The average distance of Mercury from the sun is approximately 57 million kilometers.

Step 1: Substitute the given value into the equation

First, replace \(E\) with 88 in the equation to get:\(88=0.2x^{\frac{3}{2}}\)

Step 2: Isolate the x term

To isolate the \(x\) term, we first divide both sides by 0.2 to get:\(440=x^{\frac{3}{2}}\)

Step 3: Solve for x

Next, raise both sides to the power of \(\frac{2}{3}\) to solve for \(x\). This gives us:\(x=(440)^{\frac{2}{3}}\)

Step 4: Calculate

Using a calculator, compute the value of \(x\) and round it to the nearest million. The calculated value is approximately 57 million kilometers.

Key Concepts

Solving Algebraic Equations

Solving algebraic equations is a fundamental skill in mathematics that helps us find the unknown variables in an equation. In our exercise, the formula \(88 = 0.2x^{\frac{3}{2}}\) is used to determine the average distance of Mercury from the sun. Here's how we approach it:

• Identify the equation and the variable you need to solve for, in this case, \(x\).
• Substitute any known values into the equation to simplify it. Here, \(E\) is 88 Earth days.
• Isolate the variable \(x\) by performing inverse operations. This includes dividing by 0.2 to clean up the equation.
• Finally, solve for \(x\) by raising both sides to the power that cancels out the exponent attached to \(x\).

Breaking down problems into these steps makes it manageable, allowing you to solve for the variable with confidence.

Planetary Motion

Planetary motion is a fascinating area of astronomy that deals with how planets move around the sun. Each planet has a unique orbit, and the distance from the sun affects the time it takes to complete one full orbit, or "year."

• Mercury, being closest to the sun, has a shorter orbital period of 88 Earth days, as reflected in our formula.
• The time a planet takes to orbit the sun increases with distance, which is why Earth, positioned further away, takes 365 days.

Understanding these patterns helps astronomers determine orbital paths and predict future positions of planets in our solar system.

Exponents

Exponents are a mathematical notation indicating the number of times a base is multiplied by itself. In the formula \(E = 0.2x^{\frac{3}{2}}\), the exponent \(\frac{3}{2}\) means that \(x\) is raised to the power of 1.5. Here's how you handle it:

• The fraction \(\frac{3}{2}\) can be understood as a square root followed by a cube operation.
• To solve \(440 = x^{\frac{3}{2}}\), raise both sides to the power of \(\frac{2}{3}\) to "undo" the exponent.

Calculators can help, especially with non-integer and complicated exponents, making these operations straightforward and manageable.

Step-by-Step Solutions

Step-by-step solutions are essential educational tools that simplify complex problems into digestible segments. Let's revisit the original solution:

• Step 1: Insert the known value, \(E = 88\), into the equation.
• Step 2: Divide using the constant 0.2 to adjust the equation to \(440 = x^{\frac{3}{2}}\).
• Step 3: Tackle the exponent by raising both sides to \(\frac{2}{3}\) power, simplifying your work to solve for \(x\).
• Step 4: Use a calculator to compute the result, rounding to 57 million kilometers.

Following these structured steps not only enhances understanding but also ensures accuracy in problem-solving. It's a methodical approach that builds confidence as students learn to solve increasingly complex equations.