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:

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<reasoning>

Express orbital period in terms of distance

Using the Keplers Third Law and Rational Exponents knowledge points
\[

$$\begin{aligned} &T^2 = A^3 \\ &T = A^{3/2} \end{aligned}$$

\]

Set up the ratio for planet Y and planet X

Using the Rational Exponents knowledge point
\[

$$\begin{aligned} &A_Y = k \cdot A_X \\ &T_X = A_X^{3/2} \\ &T_Y = A_Y^{3/2} = (k \cdot A_X)^{3/2} \end{aligned}$$

\]

Determine the scaling factor

Using the Exponent Rules knowledge point
\[

$$\begin{aligned} &T_Y = k^{3/2} \cdot A_X^{3/2} \\ &T_Y = k^{3/2} \cdot T_X \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(k^{\frac{1}{3}}\)</mcq-option>
<mcq-option>(B) \(k^{\frac{1}{2}}\)</mcq-option>
<mcq-option>(C) \(k^{\frac{2}{3}}\)</mcq-option>
<mcq-correct>(D) \(k^{\frac{3}{2}}\)</mcq-correct>
</answer>

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Answer:

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</pre_analysis>

<reasoning>

Express orbital period in terms of distance

Using the Keplers Third Law and Rational Exponents knowledge points
\[

$$\begin{aligned} &T^2 = A^3 \\ &T = A^{3/2} \end{aligned}$$

\]

Set up the ratio for planet Y and planet X

Using the Rational Exponents knowledge point
\[

$$\begin{aligned} &A_Y = k \cdot A_X \\ &T_X = A_X^{3/2} \\ &T_Y = A_Y^{3/2} = (k \cdot A_X)^{3/2} \end{aligned}$$

\]

Determine the scaling factor

Using the Exponent Rules knowledge point
\[

$$\begin{aligned} &T_Y = k^{3/2} \cdot A_X^{3/2} \\ &T_Y = k^{3/2} \cdot T_X \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>(A) \(k^{\frac{1}{3}}\)</mcq-option>
<mcq-option>(B) \(k^{\frac{1}{2}}\)</mcq-option>
<mcq-option>(C) \(k^{\frac{2}{3}}\)</mcq-option>
<mcq-correct>(D) \(k^{\frac{3}{2}}\)</mcq-correct>
</answer>

<post_analysis>
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"question_type": "Multiple Choice",
"knowledge_point": [
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