91Ó°ÊÓ

Write an equation for the path of each of the following elliptical orbits. Then use a graphing utility to graph the two ellipses in the same viewing rectangle. Can you see why early astronomers had difficulty detecting that these orbits are ellipses rather than circles? \(\cdot\) Earth's orbit: Length of major axis: 186 million miles Length of minor axis: 185.8 million miles \(\cdot\) Mars's orbit: Length of major axis: 283.5 million miles Length of minor axis: 278.5 million miles

An Earth satellite has an elliptical orbit described by $$\frac{x^{2}}{(5000)^{2}}+\frac{y^{2}}{(4750)^{2}}=1$$ (All units are in miles.) The coordinates of the center of Earth are \((16,0)\) a. The perigee of the satellite's orbit is the point that is nearest Earth's center. If the radius of Earth is approximately 4000 miles, find the distance of the perigee above Earth's surface. b. The apogee of the satellite's orbit is the point that is the greatest distance from Earth's center. Find the distance of the apogee above Earth's surface.

Use the Law of sines to solve triangle \(A B C\) if \(A=35^{\circ}, a=11,\) and \(b=15 .\) Assume \(B\) is acute. Round lengths of sides to the nearest tenth and angle measures to the nearest.

Exercises 105–107 will help you prepare for the material covered in the next section. Simplify and write the equation in standard form in terms of \(x^{\prime}\) and \(y^{\prime}\) $$ \left[\frac{\sqrt{2}}{2}\left(x^{\prime}-y^{\prime}\right)\right]\left[\frac{\sqrt{2}}{2}\left(x^{\prime}+y^{\prime}\right)\right]=1 $$