91Ó°ÊÓ

Use a calculator in degree mode and assume that air resistance is negligible. A ball is thrown from a height of 5 feet above the ground with an initial velocity of 60 feet per second at an angle of \(50^{\circ}\) with the horizontal. (a) Graph the ball's path. (b) When and where does the ball hit ground?

Use a calculator in degree mode and assume that air resistance is negligible. A medieval bowman shoots an arrow which leaves the bow 4 feet above the ground with an initial velocity of 88 feet per second at an angle of \(48^{\circ}\) with the horizontal. (a) Graph the arrow's path. (b) Will the arrow go over the 40 -foot-high castle wall that is 200 feet from the archer?

Halley's Comet has an elliptical orbit with the sun as one focus and a major axis that is 1,636,484,848 miles long. The closest the comet comes to the sun is 54,004,000 miles. What is the maximum distance from the comet to the sun?

A satellite is to be placed in an elliptical orbit, with the center of the earth as one focus. The satellite's maximum distance from the surface of the earth is to be \(22,380 \mathrm{km},\) and its minimum distance is to be \(6540 \mathrm{km} .\) Assume that the radius of the earth is \(6400 \mathrm{km},\) and find the eccentricity of the satellite's orbit.

Show that the length of the latus rectum of the parabola with equation \(\left.y^{2}=4 p x \text { or } x^{2}=4 p y \text { is } 4|p| . \text { [ Hint: Exercise } 73 .\right]\)

Prove that the coordinate conversion formulas are valid when \(r < 0 .[\text {Hint: If } P \text { has coordinates }(x, y) \text { and }(r, \theta), \text { with } r < 0\) verify that the point \(Q\) with rectangular coordinates \((-x,-y)\) has polar coordinates \((-r, \theta) .\) since \(r < 0,-r\) is positive and the conversion formulas proved in the text apply to \(Q .\) For instance, \(-x=-r \cos \theta, \text { which implies that } x=r \cos \theta .]\)