Question

the equation \\(t^2 = a^3\\) shows the relationship between a planets orbital period, \\(t\\), and the planets mean distance from the sun, \\(a\\), in astronomical units, au. if planet y is \\(k\\) times the mean distance from the sun as planet x, by what factor is the orbital period increased?

• \\(k^{\frac{1}{3}}\\)
• \\(k^{\frac{1}{2}}\\)
• \\(k^{\frac{2}{3}}\\)
• \\(k^{\frac{3}{2}}\\)

Explanation:

Express orbital period in terms of distance

Using the Keplers Third Law and Rational Exponents knowledge points

$$
LATEXBLOCK0
$$

Set up the ratio for planet Y and planet X

Using the Rational Exponents knowledge point

$$
LATEXBLOCK1
$$

Determine the scaling factor

Using the Exponent Rules knowledge point

$$
LATEXBLOCK2
$$

Answer:

• (A) \(k^{\frac{1}{3}}\)
• (B) \(k^{\frac{1}{2}}\)
• (C) \(k^{\frac{2}{3}}\)
• (D) \(k^{\frac{3}{2}}\) (Correct answer)