Chapter 8.2, Problem 55AYU

To determine

To find: The possible distances between Earth and Mercury.

Answer to Problem 55AYU

The possible distances between Earth and Mercuryare $101439972\text{km}\text{or }187565035\text{km}$ .

Explanation of Solution

Given information:

The distance from the Sun to Earth is approximately $149600000\text{km}$ . The distance from the Sun to Mercury is approximately $57910000\text{km}$ , and the elongation angle for Mercury is $15°$ .

The given figure is,

  

Concept used:

Law of Cosines:

If ABC is a triangle with sides $a$ , $b$ , and $c$ , then

  $\begin{array}{l}\cos A=\frac{b^2+c^2-a^2}{2bc}\\ \cos B=\frac{a^2+c^2-b^2}{2ac}\\ \cos C=\frac{a^2+b^2-c^2}{2ab}\end{array}$

The given figure is,

  

From the figure, calculate thedistance of Mercury from Earth,

  $\begin{array}{l}\cos \alpha =\frac{b^2+c^2-a^2}{2bc}\\ \cos 15°=\frac{b^2+{\left(149600000\right)}^2-{\left(57910000\right)}^2}{2\left(b\right)\left(149600000\right)}\\ b=101439972\text{km}\text{or }187565035\text{km}\end{array}$

Therefore, the possible distances between Earth and Mercuryare $101439972\text{km}\text{or }187565035\text{km}$ .