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Algebra II: Strand 7. Conic Sections; Topic 4. Applications of Conic Sections; Task 7.4.1
TASK 7.4.1: RADIO ASTRONOMY
Solutions
Your cousin, Jake, has been studying radio astronomy. He wants to build a radio antenna. He
understands the concepts necessary to build the radio equipment, but asks for your help in
building the parabolic antenna. He has salvaged a recycled C-Band satellite television-receiving
antenna that is shown in Figure 1. He wants you to help him build a structure like the one
shown in Figure 2. The cross section of the antenna is a parabola. The receiver must be placed
at the focus of the parabola (marked with a star on the diagram).
Figure 1:
Figure 2:
You are able to take dimensions of the cross-section that is a parabola.
5 ft
1.8 ft
5 ft
How many feet from the vertex must he place the receiver? The radio receiver should be placed
3.472 feet from the vertex of the parabola.
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
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Algebra II: Strand 7. Conic Sections; Topic 4. Applications of Conic Sections; Task 7.4.1
How many feet of material will he need for the support struts? 15.861 ft. of material
The parabola may be positioned on a coordinate axis.
(-1.8,5)
(0,0)
The equation of the parabola is of the form x =ay2.
(-1.8,5) is a point on the parabola.
a =
-1.8
25
= - 0. 072
-1.8 = a(5)2
x = - 0.072y2
y2 = 13.889x
In a parabola with equation y2 = 4ax, the focus is located a units from the vertex.
In this situation
4a = 13.888
a = 3.472 ft.
The radio receiver should be placed 3.472 feet from the vertex of the parabola.
The depth of the paraboloid to the vertex is 1.8 feet.
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
3
Algebra II: Strand 7. Conic Sections; Topic 4. Applications of Conic Sections; Task 7.4.1
3.472 – 1.8 = 1.672 feet beyond cross section ( the bold segment in the diagram)
5 ft
1.8 ft
5 ft
The cross section perpendicular to the axis of the parabloid is a circle. Each of the struts (the
dashed line) should be connected to the circular edge of the paraboloid. The length of the
strut can be determined using the Pythagorean Theorem.
s2 = 1.6722 + 52
s = 5.272 feet
If there are three struts, then 3*5.272=15.861 ft. is the amount of material need to build them.
Teaching notes
Hold up an object that is shaped like a paraboloid (plastic eggs, headlight)
Briefly discuss the reflective properties of a paraboloid.
Describe the problem as it is presented on the activity sheet.
Ask the following questions to guide the discussion.
Q What might be the first step in analyzing the problem? Q
Position the cross section on an axis.
Q What do you know about the location of the focus? Q
It would be the focus of any cross section of the paraboloid.
Q Will it be inside or outside of the antenna? Q
Until the parabola is analyzed it cannot be determined.
Work through the solution together as a class. Be sure to point out that the answers must
be framed in the context of the problem.
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.
4
Algebra II: Strand 7. Conic Sections; Topic 4. Applications of Conic Sections; Task 7.4.1
TASK 7.4.1: RADIO ASTRONOMY
Your cousin, Jake, has been studying radio astronomy. He wants to build a radio antenna. He
understands the concepts necessary to build the radio equipment, but asks for your help in
building the parabolic antenna. He has salvaged a recycled C-Band satellite television-receiving
antenna that is shown in Figure 1. He wants you to help him build a structure like the one
shown in Figure 2. The cross section of the antenna is a parabola. The receiver must be placed
at the focus of the parabola (marked with a star on the diagram).
Figure 1:
Figure 2:
You are able to take dimensions of the cross-section that is a parabola.
5 ft
1.8 ft
5 ft
How many feet from the vertex must he place the receiver?
How many feet of material will he need for the support struts?
December 20, 2004. Ensuring Teacher Quality: Algebra II, produced by the Charles A. Dana Center at The University of Texas
at Austin for the Texas Higher Education Coordinating Board.