Conics Application
Trigonometry
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PARABOLA
1) A cable TV receiving dish is in the shape of a paraboloid of revolution. Find the location of the
receiver, which is placed at the focus, if the dish is 6 feet across at its opening and 2 feet deep.
2) A bridge is to be built in the shape of a parabolic arch and is to have a span of 100 feet. The height of
the arch a distance of 40 feet from the center is to be 10 feet. Find the height of the arch at its center.
3) A searchlight is shaped like a paraboloid of revolution. If the light source is located 2 feet from the
base along the axis of symmetry and the depth of the searchlight is 4 feet, what should the width of the
opening be?
4) Suppose that a golf ball travels a distance of 600 feet as measured along the ground and reaches an
altitude of 200 feet. If the origin represents the tee and the ball travels along a parabolic path that
opens downward, find an equation for the path of the golf ball.
CIRCLE
5) A satellite in a stationary orbit rotates once each day about the Earth. Assume that the satellite is
22,300 miles above the surface of the Earth and the Earth’s radius is 3960 miles. Write an equation that
represents the orbit of this satellite on a coordinate plane with the origin representing the center of the
Earth.
6) Vibrations from a certain earthquake were noticeable up to 120 miles away from the earthquake’s
epicenter. The center of the closest city to the earthquake’s epicenter is 85 miles east and 90 miles
south of the epicenter. If Alexandra was 15 miles west of this city, could she have noticed the vibrations
at the time of the earthquake?
7) An earthquake transmits its energy in seismic waves that radiate from its underground focus in all
directions. A seismograph station determines that the earthquake’s epicenter (the point from the
Earth’s surface directly above the focus) is 100 miles from the station.
a) Write the equation for the possible locations of the epicenter, using (0, 0) as the location of
the seismograph station.
b) A second station is 120 miles east and 160 miles south of the first station and 100 miles from
the earthquake’s epicenter. Write a second equation for the possible locations of the epicenter.
ELLIPSE
8) An arch in the shape of the upper half of an ellipse is used to support a bridge that is to span a river
20 meters wide. The center of the arch is 6 meters above the center of the river. Write an equation for
the ellipse in which the x-axis coincides with the water level and the y-axis passes through the center of
the arch.
9) The arch of a bridge is a semi-ellipse with a horizontal major axis. The span is 30 feet, and the top of
the arch is 10 feet above the major axis. The roadway is horizontal and is 2 feet above the top of the
arch. Find the vertical distance from the roadway to the arch at 10 feet from the center along the axis of
symmetry.
10) An arch for a bridge over a highway is in the form of half an ellipse. The top of the arch is 20 feet
above the ground level (the major axis). The highway has four lanes, each 12 feet wide; a center safety
strip 8 feet wide; and two side strips, each 4 feet wide. What should the span of the bridge be (the
length of its major axis) if the height 28 feet from the center is to be 13 feet?
11) The moon orbits Earth in an elliptical path with the center of the Earth at one focus. The major axis
of the orbit is 774,000 kilometers, and the minor axis is 773,000 kilometers.
a) Using (0, 0) as the center of the ellipse, write the standard equation for the orbit of the Moon
around Earth.
b) How far from the center of Earth is the Moon at its closest point?
c) How far from the center of Earth is the Moon at its farthest point?
12) Earth orbits the Sun in an elliptical path with the Sun at one focus of the ellipse. The closest that
Earth gets to the Sun is 91.4 million miles and the farthest it gets is 94.6 million miles. Write an equation
for Earth’s orbit around the Sun with the center of the ellipse at the origin.
HYPERBOLA
13) Some nuclear power plants utilize “natural draft” cooling towers in the shape of a hyperboloid, a
solid obtained by rotating a hyperbola about its conjugate axis. Suppose that such a cooling tower has a
base diameter of 400 feet and the diameter at its narrowest point, 360 feet above the ground, is 200
feet. If the diameter at the top of the tower is 300 feet, how tall is the tower?