Question

can find the orbital period p (in earth years) using the formula p = √d³, where units, abbreviated au) of the planet from the sun.
jupiter
sun
d = 5.2 au
a. simplify the formula.
p = □.
b. what is jupiters orbital period? round your answer to the nearest hundredth.
about □ earth years.

Explanation:

Step1: Simplify the formula

Given \(P = \sqrt{d^{3}}\), using the property \(\sqrt{a}=a^{\frac{1}{2}}\), we can rewrite it as \(P=(d^{3})^{\frac{1}{2}}\).
By the power - of - a - power rule \((a^{m})^{n}=a^{mn}\), so \(P = d^{\frac{3}{2}}\).

Step2: Find Jupiter's orbital period

We know \(d = 5.2\) AU. Substitute \(d = 5.2\) into the formula \(P=d^{\frac{3}{2}}\).
First, \(d^{\frac{3}{2}}=\sqrt{d^{3}}\), so \(d^{3}=5.2^{3}=5.2\times5.2\times5.2 = 140.608\).
Then \(\sqrt{140.608}\approx11.86\).

Answer:

a. \(P = d^{\frac{3}{2}}\)
b. About \(11.86\) Earth years.