Problem 75
Find the area of the surface generated by revolving \(x=t^{2}, y=2 t, 0 \leq t \leq 4\) about the \(x\) -axis.
Problem 76
The mirror in an automobile headlight has a parabolic cross section, with the lightbulb at the focus. On a schematic, the equation of the parabola is given as \(x^{2}=4 y .\) At what coordinates should you place the lightbulb?
Problem 76
Find the surface area generated by revolving \(x=t^{2}, y=2 t^{2}, 0 \leq t \leq 1\) about the \(y\) -axis.
Problem 77
A satellite dish is shaped like a paraboloid of revolution. The receiver is to be located at the focus. If the dish is 12 feet across at its opening and 4 feet deep at its center, where should the receiver be placed?
Problem 79
A searchlight is shaped like a paraboloid of revolution. A light source is located 1 foot from the base along the axis of symmetry. If the opening of the searchlight is 3 feet across, find the depth.
Problem 80
Whispering galleries are rooms designed with elliptical ceilings. A person standing at one focus can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet and the foci are located 30 feet from the center, find the height of the ceiling at the center.
Problem 82
For the following exercises, determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbits of the comets or planets. Distance is given in astronomical units \((A U)\). $$ \text { Halley's Comet: length of major axis }=35.88 \text { , eccentricity }=0.967 $$
Problem 83
For the following exercises, determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbits of the comets or planets. Distance is given in astronomical units \((A U)\). Hale-Bopp Comet: length of major axis \(=525.91\), eccentricity \(=0.995\)
Problem 84
For the following exercises, determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbits of the comets or planets. Distance is given in astronomical units \((A U)\). Mars: length of major axis \(=3.049\), eccentricity \(=0.0934\)
Problem 85
For the following exercises, determine the polar equation form of the orbit given the length of the major axis and eccentricity for the orbits of the comets or planets. Distance is given in astronomical units \((A U)\). Jupiter: length of major axis \(=10.408\), eccentricity \(=0.0484\)
