Chapter 9.4, Problem 6CYU

To determine

To find: The equation relating Earth’s orbit around the Sun in millions of miles.

Answer to Problem 6CYU

The equation relating Earth’s orbit around the Sun in millions of miles is $\frac{x^2}{{\left(93.385\right)}^2}+\frac{y^2}{{\left(93.37\right)}^2}=1$

Explanation of Solution

Given information:

Earth’s orbit is about $91.4\text{millon}$ miles at perihelion and about $94.5\text{millon}$ miles at aphelion.

Concept used:

Write the standard form of the equation of the ellipse whose orientation is along the horizontal axis is,

  $\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1$

The coordinate of the foci is $\left(h\pm c,k\right)$

The value of $c$ is calculated as,

  $c=\sqrt{a^2-b^2}$

The vertices are at $\left(h\pm a,k\right)$ and the co-vertices are at $\left(h,k\pm b\right)$ .

Write the standard form of the equation of the ellipse whose orientation is along the vertical axis is,

  $\frac{{\left(y-k\right)}^2}{a^2}+\frac{{\left(x-h\right)}^2}{b^2}=1$

The coordinate of the foci is $\left(h,k\pm c\right)$

The vertices are at $\left(h,k\pm a\right)$ and the co-vertices are at $\left(h\pm b,k\right)$ .

The value of $c$ is calculated as,

  $c=\sqrt{a^2-b^2}$

The diameter of the Sun is $0.87\text{million}$ miles.

The value of $a$ is calculated as,

  $\begin{array}{l}2a=0.87+91.4+94.5\\ 2a=186.77\\ a=93.385\text{million miles}\end{array}$

The value of $c$ is calculated as,

  $\begin{array}{c}c=93.385-\left(91.4+\frac{0.87}{2}\right)\\ =1.55\text{million miles}\end{array}$

The value of $b$ is calculated as,

  $\begin{array}{l}c=\sqrt{a^2-b^2}\\ 1.55=\sqrt{{\left(93.385\right)}^2-b^2}\\ b=93.37\text{million miles}\end{array}$

The standard form of the equation of the given ellipse is,

  $\begin{array}{l}\frac{{\left(x-h\right)}^2}{a^2}+\frac{{\left(y-k\right)}^2}{b^2}=1\\ \frac{{\left(x-0\right)}^2}{{\left(93.385\right)}^2}+\frac{{\left(y-0\right)}^2}{{\left(93.37\right)}^2}=1\\ \frac{x^2}{{\left(93.385\right)}^2}+\frac{y^2}{{\left(93.37\right)}^2}=1\end{array}$

Thus, the equation for the given ellipse is $\frac{x^2}{{\left(93.385\right)}^2}+\frac{y^2}{{\left(93.37\right)}^2}=1$ .