 Suggested languages for you:

Europe

Answers without the blur. Sign up and see all textbooks for free! Q. 53

Expert-verified Found in: Page 772 ### Calculus

Book edition 1st
Author(s) Peter Kohn, Laura Taalman
Pages 1155 pages
ISBN 9781429241861 # Measurements indicate that Earth’s orbital eccentricity is $0.0167$ and its semimajor axis is $1.00000011$ astronomical units. (a) Write a Cartesian equation for Earth’s orbit. (b) Give a polar coordinate equation for Earth’s orbit, assuming that the sun is the focus of the elliptical orbit.

Part (a) The cartesian equation is $\frac{{x}^{2}}{\left(1.00000011{\right)}^{2}}+\frac{{y}^{2}}{\left(0.9998606552{\right)}^{2}}=1$.

Part (b) The polar equation is $r=\frac{1.00000011}{1+0.0167\mathrm{cos}\theta }$.

See the step by step solution

## Part (a) Step 1. Given information.

The eccentricity is $0.0167.$

The semi-major axis is role="math" localid="1649526544989" $1.00000011\text{astronomical units.}$

## Part (a) Step 2. Explanation.

Now,

$e=\frac{\sqrt{{A}^{2}-{B}^{2}}}{A}\phantom{\rule{0ex}{0ex}}A=1.00000011,e=0.0167\phantom{\rule{0ex}{0ex}}0.0167=\frac{\sqrt{\left(1.00000011{\right)}^{2}-{B}^{2}}}{1.00000011}\phantom{\rule{0ex}{0ex}}0.0167\left(1.00000011\right)=\sqrt{\left(1.00000011{\right)}^{2}-{B}^{2}}\phantom{\rule{0ex}{0ex}}0.0167000018=\sqrt{1.00000022-{B}^{2}}\phantom{\rule{0ex}{0ex}}\left(0.0167000018{\right)}^{2}={\left(\sqrt{\left(1.00000022\right)-{B}^{2}}\right)}^{2}\phantom{\rule{0ex}{0ex}}0.00027889006=\left(1.00000022\right)-{B}^{2}\phantom{\rule{0ex}{0ex}}{B}^{2}=0.99972133\phantom{\rule{0ex}{0ex}}{A}^{2}=\left(1.00000011{\right)}^{2}=1.00000022\phantom{\rule{0ex}{0ex}}Therefore,\phantom{\rule{0ex}{0ex}}\frac{{x}^{2}}{1.00000022}+\frac{{y}^{2}}{0.99972133}=1\text{or}\frac{{x}^{2}}{\left(1.00000011{\right)}^{2}}+\frac{{y}^{2}}{\left(0.9998606552{\right)}^{2}}=1$

## Part (b) Step 1. Explanation.

Now, the standard form,

$r=\frac{eu}{1+e\mathrm{cos}\theta }\phantom{\rule{0ex}{0ex}}Therefore,\phantom{\rule{0ex}{0ex}}r=\frac{1.00000011}{1+0.0167\mathrm{cos}\theta }$ ### Want to see more solutions like these? 