For each planet in a solar​ system, its year is the time it takes the planet to revolve around the center star. The formula Upper E equals 0.2 x Superscript 3 divided by 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 365 Earth days in the year of Planet Upper C. What is the average distance of Planet Upper C from the center​ star?
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Part 1
The average distance is approximately 
  
enter your response here million kilometers.
​(Do not round until the final answer. Then round to the nearest whole number as​ needed.) 

We are given the formula for the number of Earth days in a planet's year, \( E \), which is:

\[
E = 0.2x^{\frac{3}{2}}
\]

where:
- \( E \) is the number of Earth days in the planet's year,
- \( x \) is the average distance of the planet from the center star in millions of kilometers.

We are also told that Planet \( C \) has a year of approximately 365 Earth days. Thus, we can substitute \( E = 365 \) into the equation and solve for \( x \).

### Step 1: Substitute the known value of \( E \) into the equation.

\[
365 = 0.2x^{\frac{3}{2}}
\]

### Step 2: Isolate \( x^{\frac{3}{2}} \).

Divide both sides by 0.2:

\[
x^{\frac{3}{2}} = \frac{365}{0.2} = 1825
\]

### Step 3: Solve for \( x \).

To eliminate the exponent \( \frac{3}{2} \), raise both sides of the equation to the power of \( \frac{2}{3} \):

\[
x = 1825^{\frac{2}{3}}
\]

Now we can calculate \( 1825^{\frac{2}{3}} \).

Let me calculate that for you.The average distance of Planet \( C \) from the center star is approximately \( 149 \) million kilometers, rounded to the nearest whole number.

Would you like more details, or do you have any questions?

Here are 5 related questions to explore:

1. How does the exponent \( \frac{3}{2} \) affect the relationship between distance and orbital period?
2. What is the significance of the constant 0.2 in the formula?
3. How would you calculate the orbital period for a planet that is twice as far from the star as Planet C?
4. How does this formula compare to Kepler’s third law of planetary motion?
5. If the distance of a planet from the center star were halved, how would its year change?

**Tip:** When dealing with fractional exponents, remember that they represent both roots and powers: \( x^{\frac{3}{2}} \) is the same as \( \sqrt{x^3} \).

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 = 0.2x^(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 365 Earth days in the year of Planet C. What is the average distance of Planet C from the center star?

Solve for the average distance of Planet C from its center star using the formula E = 0.2x^(3/2). This step-by-step guide explores concepts like fractional exponents and exponential equations, suitable for high school students in Grades 10-12.

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