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 $\sqrt{x^3}$.