Reflecting Property of a Parabola Problem: Figure 12-7f shows the parabolic cross

Chapter 12, Problem C.1

(choose chapter or problem)

Reflecting Property of a Parabola Problem: Figure 12-7f shows the parabolic cross section of the television dish antenna shown in Figure 12-7e. The equation of the parabola is x = 0.01y2. a. Confirm that the focus of the parabola is at point (0, 25), as shown in Figure 12-7f. b. A television signal ray comes in parallel to the x-axis at y = 20. Find the coordinates of the point at which the ray strikes the parabola. c. Calculate the angle A between the incoming ray and a line to the focus from the point where the incoming ray strikes the parabola. d. An incoming ray and its reflected ray make equal angles with a line tangent to the curved surface. Use your answer to part c to calculate angles B and C if these angles have equal measure. With a protractor, measure A, B, and C in Figure 12-7f, thus confirming your calculations. Figure 12-7f e. Find the particular equation of the tangent line. Use angle C to find its slope. Plot the parabola and the tangent line on your grapher. Zoom in on the point of tangency. What do you notice about the tangent line and the graph as you zoom in? f. Pick another incoming ray. Calculate the measure of angle A. Let C be half of the supplement of A, as it was in part d. Is the line at angle C with the incoming ray tangent to the graph at the point where the incoming ray strikes the graph? How do you know? g. Write a conjecture about the direction the reflected ray takes whenever the incoming ray is parallel to the axis of the parabola. Your conjecture should give you insight into why the name focus is used and why television satellite antennas and other listening devices are made in the shape of a paraboloid.