Questions: For each planet in a solar system, its year is the time it takes the planet to revolve around the center star. The formula E=0.2 x^(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 center star, in millions of kilometers. There are approximately 224.8 Earth days in the year of Planet B. What is the average distance of Planet B from the center star?

For each planet in a solar system, its year is the time it takes the planet to revolve around the center star. The formula E=0.2 x^(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 center star, in millions of kilometers. There are approximately 224.8 Earth days in the year of Planet B. What is the average distance of Planet B from the center star?
Transcript text: For each planet in a solar system, its year is the time it takes the planet to revolve around the center star. The formula $E=0.2 x^{3 / 2}$ models the number of Earth days in a planet's year, $E$, where $x$ is the average distance of the planet from the center star, in millions of kilometers. There are approximately 224.8 Earth days in the year of Planet B. What is the average distance of Planet B from the center star?
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Solution

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Solution Steps

To find the average distance of Planet B from the center star, we need to solve for \( x \) in the given formula \( E = 0.2 x^{3/2} \). We know that \( E = 224.8 \) Earth days. We can rearrange the formula to solve for \( x \) and then use Python to compute the value.

Step 1: Given Information

We are given the formula \( E = 0.2 x^{3/2} \) which models the number of Earth days in a planet's year, \( E \), where \( x \) is the average distance of the planet from the center star in millions of kilometers. We know that for Planet B, \( E = 224.8 \) Earth days.

Step 2: Rearrange the Formula

To find the average distance \( x \), we need to solve the equation for \( x \): \[ 0.2 x^{3/2} = 224.8 \]

Step 3: Solve for \( x \)

Rearranging the equation to isolate \( x \): \[ x^{3/2} = \frac{224.8}{0.2} \] \[ x^{3/2} = 1124 \] Next, we solve for \( x \) by raising both sides to the power of \( \frac{2}{3} \): \[ x = 1124^{\frac{2}{3}} \]

Step 4: Calculate the Value of \( x \)

Using the calculation, we find: \[ x \approx 108.1046 \]

Final Answer

The average distance of Planet B from the center star is approximately: \[ \boxed{108.1 \text{ million kilometers}} \]

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