Chapter 4.7, Problem 40E

To determine

What is its average distance from the sun, if it takes Mars 687 days to orbit the sun?

Answer to Problem 40E

  $141,433,434\text{ miles}$

Explanation of Solution

Given information: Astronomy: Johann Kepler (1571-1630) determined the relationship of the time of revolution of two planets and their average distance from the sun. This relationship can be expressed as $\frac{T_a}{T_b}=\sqrt{{\left(\frac{r_a}{r_b}\right)}^3}$ . In this equation $T_a$ represents the time it takes planet a to orbit the sun and $r_a$ represents the average distance between planet a and the sun. Likewise, $T_b$ represents the time it takes planet b to orbit the sun and $r_b$ represents the average distance between planet b and the sun. The average distance between Venus and the sun is 67,200,000 miles and it takes Venus about 225 days to orbit the sun.

  

Calculation:

  $\frac{T_a}{T_b}=\sqrt{{\left(\frac{r_a}{r_b}\right)}^3}$

Replace values given

  $\frac{225}{687}=\sqrt{{\left(\frac{67,200,000}{r_b}\right)}^3}$

Reduce the left fraction

  $\frac{75}{229}=\sqrt{{\left(\frac{67,200,000}{r_b}\right)}^3}$

Square both sides

  ${\left(\frac{75}{229}\right)}^2=\frac{3.03\times {10}^{23}}{{\left(r_b\right)}^3}$

Solve for $r_b$

  $\begin{array}{l}{\left(r_b\right)}^3=2.83\times {10}^{24}\\ r\simeq 141,433,434\text{ miles}\end{array}$