Question

always use significant figure rules. remember that these rules apply to all numbers that are measurements.
the earth takes 1 year to revolve around the sun at 1 a.u. distance (an astronomical unit = 93,000,000 miles). if a planet were 4 a.u. from the sun, how many years would it take to make 1 orbit?
years
8
64

Explanation:

Step1: Aplicar la tercera ley de Kepler

La tercera ley de Kepler establece que \( T^{2}\propto a^{3} \), donde \( T \) es el período orbital y \( a \) es la distancia media al Sol. Para la Tierra, \( T_{1} = 1\) año y \( a_{1}=1\) AU. Para el planeta, \( a_{2} = 4\) AU.

Step2: Establecer la proporción

$$ \frac{T_{2}^{2}}{T_{1}^{2}}=\frac{a_{2}^{3}}{a_{1}^{3}} $$

Sustituyendo los valores: \( T_{1} = 1\), \( a_{1}=1\), \( a_{2} = 4\)

$$ \frac{T_{2}^{2}}{1^{2}}=\frac{4^{3}}{1^{3}} $$

$$ T_{2}^{2}=64 $$

Step3: Calcular \( T_{2}\)

Tomando la raíz cuadrada de ambos lados:

$$ T_{2}=\sqrt{64}=8 $$

Answer:

8