Solve Each Problem Algebraically. Show All Work. Draw a Diagram If Applicable

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ALGEBRA II H – WS Applications of Conics

Solve each problem algebraically. Show all work. Draw a diagram if applicable.

1. A commemorative parabolic steel arch is planned, with its axis vertical and its feet 300 m apart. If the focus of the parabola is to be 80 m above ground, how high must the arch be?

2. A parabolic reflector is in the shape made by revolving an arc of a parabola, starting at the vertex, about the axis of the parabola. If the focus is 9 inches from the vertex, and the parabolic arc is 16 inches deep, how wide is the opening of the reflector?

3. How high is a parabolic arch, of span 24 ft and height 18 ft, at a distance 8 ft from the center of the span?

4. An arrow is shot over a lake from a cliff that is 20 feet above a lake. The arrow returns to the same height as the cliff at a distance of 160 feet from the cliff. Its path traces a parabola and reaches a height of 60 feet above the lake.

a. Find the equation which relates the height of the arrow above the lake to its horizontal distance from the cliff.

b. What is the arrow’s height above the water after it has traveled a horizontal distance of 40 feet?

c. At what distance from the cliff will the arrow strike the lake?

5. When the load is uniformly distributed, the cable of a suspension bridge hangs in the shape of an arc of a parabola. The supporting towers are 60 ft high and 500 ft apart, and the lowest point on the cable is 10 ft above the roadway. Using the road as the x-axis and the axis of symmetry of the parabola as the y-axis, find the equation of the parabola which the cable assumes. Find the length of a supporting rod 80 ft from the center of the bridge.

6. The Statuary Hall in the U.S. Capitol is elliptical. It measure 46 feet wide and 96 feet long. If a person is standing at one focus, her whispers can be heard by a person standing at the other focus.

a. How far apart are the two people?

b. How far will they be from the closest wall?

7. The base of an auditorium is in the form of an ellipse 200 feet long and 160 feet wide. A pin dropped at one focus can be heard at the other focus. How far apart are the foci?

8. An arch in the form of a semi-ellipse has a span of 150 feet and its greatest height is 45 feet. There are two vertical supports equidistant from each other and the ends of the arch. Find their height.

9. The face of a one-lane tunnel is a square with a semi-circle above it. The semi-circle has a diameter or 18 feet. A truck that is 15 feet wide and 22 feet tall tries to drive through the tunnel.

a. Will it make it?

b. How much over or under will the truck be?

10. The face of a one-lane tunnel is a square with a semi-circle above it. The semi-circle has a diameter of 5 m. A truck that is 4.5 meters wide and 6.5 meters tall tries to drive through the tunnel.

a. Will it make it?

b. How much over or under will the truck be?

11. A semi-circular arch over a street has a diameter of 12 meters. A sign is placed on the ceiling 2 meters from the edge. How far above the street will the sign be?

12. A miniature golf hole 3 meters wide and 8 meters long consists of a wall in the shape of an ellipse with both the tee and the cup at foci. A golf ball hit in any direction will bounce off the wall and roll into the cup.

a. Find the equation of the ellipse which describes the hole.

b. How far apart are the tee and the cup?

13. A 12-foot wide track surrounds an elliptical field that is 70 feet long and 30 feet wide. The area of an ellipse is A = ab, where 2a is the length of the major axis and 2b is the length of the minor axis. Find the area of the track.

14. A baseball is thrown from a height of 2 meters and is caught at the same height 40 meters away. During its parabolic path, it reaches a maximum height of 10 meters.

a. Find the equation which relates the height of the ball to the horizontal distance that it has traveled.

b. What is the height of the ball after it has travelled a horizontal distance of 30 meters?

c. If the ball is not caught, at which distance will it strike the ground?

ALGEBRA II H – WS Applications of Conics

Solve each problem algebraically. Show all work. Draw a diagram if applicable.

1. A commemorative parabolic steel arch is planned, with its axis vertical and its feet 300 m apart. If the focus of the parabola is to be 80 m above ground, how high must the arch be?

a. Find the equation which relates the height of the arrow above the lake to its horizontal distance from the cliff.

b. What is the arrow’s height above the water after it has traveled a horizontal distance of 40 feet?

c. At what distance from the cliff will the arrow strike the lake?

a. Will it make it?

b. How much over or under will the truck be?

11. A semi-circular arch over a street has a diameter of 12 meters. A sign is placed on the ceiling 2 meters from the edge. How far above the street will the sign be?