91Ó°ÊÓ

Convert from polar coordinates to rectangular coordinates. A diagram may help. $$\left(5, \frac{5 \pi}{6}\right)$$

Find the equation of an ellipse (in standard form) that satisfies the following conditions: vertices at (-6,0) and (6,0)\(;\) foci at (-4,0) and (4,0)

Find the vertex, focus, and directrix for the parabolas defined by the equations given, then use this information to sketch a complete graph (illustrate and name these features). For Exercises 43 to 60 , also include the focal chord. $$y^{2}-2 y+8 x+9=0$$

Polar form of an ellipse with center at the pole: $$r^{2}=\frac{a^{2} b^{2}}{a^{2} \sin ^{2} \theta+b^{2} \cos ^{2} \theta}$$ If an ellipse in the \(r \theta\) -plane has its center at the pole (with major axis parallel to the \(x\) -axis), its equation is given by the formula here, where \(2 a\) and \(2 b\) are the lengths of the major and minor axes, respectively. (a) Given an ellipse with center at the pole has a major axis of length 8 and a minor axis of length \(4,\) find the equation of the ellipse in polar form and (b) graph the result on a calculator and verify that \(2 a=8\) and \(2 b=4\).

Planetary motion: The perihelion, aphelion, and orbital period of the planets Jupiter, Saturn, Uranus, and Neptune are shown in the table. Use the information to answer or complete the following exercises. The formula $$L=2 \pi \sqrt{0.5\left(a^{2}+b^{2}\right)}$$ can be used to estimate the length of the orbital path. Recall for an cllipse, \(c^{2}=a^{2}-b^{2}\). $$\begin{array}{|l|c|c|c|} \hline \text { Planet } & \begin{array}{c} \text { Perihelion } \\ \left(10^{6} \mathrm{mi}\right) \end{array} & \begin{array}{c} \text { Aphelion } \\ \left(10^{6} \mathrm{mi}\right) \end{array} & \begin{array}{c} \text { Period } \\ (y r) \end{array} \\ \hline \text { Jupiter } & 460 & 507 & 11.9 \\ \hline \text { Saturn } & 840 & 941 & 29.5 \\ \hline \text { Uranus } & 1703 & 1866 & 84 \\ \hline \text { Neptune } & 2762 & 2824 & 164.8 \\ \hline \end{array}$$ Find the eccentricity of the planets Jupiter and Saturn.

Planetary motion: The perihelion, aphelion, and orbital period of the planets Jupiter, Saturn, Uranus, and Neptune are shown in the table. Use the information to answer or complete the following exercises. The formula $$L=2 \pi \sqrt{0.5\left(a^{2}+b^{2}\right)}$$ can be used to estimate the length of the orbital path. Recall for an cllipse, \(c^{2}=a^{2}-b^{2}\). $$\begin{array}{|l|c|c|c|} \hline \text { Planet } & \begin{array}{c} \text { Perihelion } \\ \left(10^{6} \mathrm{mi}\right) \end{array} & \begin{array}{c} \text { Aphelion } \\ \left(10^{6} \mathrm{mi}\right) \end{array} & \begin{array}{c} \text { Period } \\ (y r) \end{array} \\ \hline \text { Jupiter } & 460 & 507 & 11.9 \\ \hline \text { Saturn } & 840 & 941 & 29.5 \\ \hline \text { Uranus } & 1703 & 1866 & 84 \\ \hline \text { Neptune } & 2762 & 2824 & 164.8 \\ \hline \end{array}$$ Find the eccentricity of the planets Uranus and Neptune.

Solve by setting up and solving a system of nonlinear equations. The surface area of a closed cylindrical tank is \(192 \pi \mathrm{m}^{2}\). Find the dimensions of the tank if the volume is \(320 \pi \mathrm{m}^{3}\) and the radius is as small as possible.

Find the equation of the parabola in standard form that satisfies the conditions given: vertex: (-3,-4) focus: (-3,-1)

Find the equation of the hyperbola (in standard form) that satisfies the following conditions: foci at \((-2,-3 \sqrt{2})\) and \((-2,3 \sqrt{2}) ;\) length of conjugate axis: 6 units

The midpoint formula in polar coordinates: \(M=\left(\frac{r \cos \alpha+R \cos \beta}{2}, \frac{r \sin \alpha+R \sin \beta}{2}\right)\) The midpoint of a line segment connecting the points \((r, \alpha)\) and \((R, \beta)\) in polar coordinates can be found using the formula shown. Find the midpoint of the line segment between \((r, \alpha)=\left(6,45^{\circ}\right)\) and \((R, \beta)=\left(8,30^{\circ}\right),\) then convert these points to rectangular coordinates and find the midpoint using the "standard" formula. Do the results match?