Question

the orbital period, \\(p\\), of a planet and the planets distance from the sun, \\(a\\), in astronomical units is related by the formula \\(p = a^{\frac{3}{2}}\\). if saturns orbital period is 29.5 years, what is its distance from the sun?

9.5 au
19.7 au
44.3 au
160.2 au

Explanation:

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<reasoning>

Set up the equation

Using the Keplers Third Law knowledge point
\[
P = a^{\frac{3}{2}}
\]
\[
29.5 = a^{\frac{3}{2}}
\]

Solve for the distance

Using the Rational Exponents knowledge point
\[
a = (29.5)^{\frac{2}{3}}
\]
\[
a \approx 9.547\text{ AU}
\]

Match with the options

The calculated value \(a \approx 9.5\text{ AU}\) matches the first option.
</reasoning>

<answer>
<mcq-correct>(A) 9.5 AU</mcq-correct>
<mcq-option>(B) 19.7 AU</mcq-option>
<mcq-option>(C) 44.3 AU</mcq-option>
<mcq-option>(D) 160.2 AU</mcq-option>
</answer>

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Answer:

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<reasoning>

Set up the equation

Using the Keplers Third Law knowledge point
\[
P = a^{\frac{3}{2}}
\]
\[
29.5 = a^{\frac{3}{2}}
\]

Solve for the distance

Using the Rational Exponents knowledge point
\[
a = (29.5)^{\frac{2}{3}}
\]
\[
a \approx 9.547\text{ AU}
\]

Match with the options

The calculated value \(a \approx 9.5\text{ AU}\) matches the first option.
</reasoning>

<answer>
<mcq-correct>(A) 9.5 AU</mcq-correct>
<mcq-option>(B) 19.7 AU</mcq-option>
<mcq-option>(C) 44.3 AU</mcq-option>
<mcq-option>(D) 160.2 AU</mcq-option>
</answer>

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