91Ó°ÊÓ

In Exercises 17–30, find the standard form of the equation of each parabola satisfying the given conditions. Focus: \(\quad(-10,0) ;\) Directrix: \(x=10\)

Describe a viewing rectangle, or window, such as [-30, 30, 3] by [-8, 4, 1], that shows a complete graph of each polar equation and minimizes unused portions of the screen. $$ r=\frac{16}{3+5 \cos \theta} $$

In Exercises \(25-36,\) find the standard form of the equation of each ellipse satisfying the given conditions. Foci: \((-2,0),(2,0) ;\) vertices: \((-6,0),(6,0)\)

In Exercises 49–56, identify each equation without completing the square. $$ 9 x^{2}+4 y^{2}-36 x+8 y+31=0 $$

Prove that the polar equation of a planet's elliptical orbit is $$ r=\frac{\left(1-e^{2}\right) a}{1-e \cos \theta} $$ where \(e\) is the eccentricity and \(2 a\) is the length of the major axis.

Two fire-lookout stations are 10 miles apart with station B directly east of station A. Both stations spot a fire on a mountain to the north. The bearing from station A to the fire is N39°E (39° east of north). The bearing from station B to the fire is N42°W (42° west of north). How far, to the nearest tenth of a mile, is the fire from station A?

A kite flies at a height of 35 feet when 60 feet of string is out. If the string is in a straight line, find the angle that it makes with the ground. Round to the nearest tenth of a degree.

The reflector of a flashlight is in the shape of a parabolic surface. The casting has a diameter of 8 inches and a depth of 1 inch. How far from the vertex should the light bulb be placed?

A satellite dish, like the one shown below, is in the shape of a parabolic surface. Signals coming from a satellite strike the surface of the dish and are reflected to the focus, where the receiver is located. The satellite dish shown has a diameter of 12 feet and a depth of 2 feet. How far from the base of the dish should the receiver be placed?

The towers of the Golden Gate Bridge connecting San Francisco to Marin County are 1280 meters apart and rise 160 meters above the road. The cable between the towers has the shape of a parabola and the cable just touches the sides of the road midway between the towers. What is the height of the cable 200 meters from a tower? Round to the nearest meter.