please help

Question

please help

anonymous · Answer

The diameter of a parabolic mirror is 22 centimeters, and the mirror has a depth of 0.5 centimeters at the center. What is the equation of the cross–sectional parabola of the mirror?

directrix · Answer

Comments on this problem by @Phone-a-Friend:

I thought of 2 other possibilities ) but here is a thought. Since the problem asked for a parabolic cross-section (and the one I suggested earlier would be circular cross-sections), we might envision the "thang" as being upright like the y-axis and a very simple "parabolic" cross-section could be defined as y = (0.5-x_i)^2 where -11<x_i<11. 

 My other idea involved y=ax^2 and using (along the x-axis this time) (0,0) and (0.5,11) to solve for a. This idea, however, yields a circular cross-section with each section being pi (ax^2)^2.

directrix · Answer

I believe the problem has a few inherent problems.

 For example, without knowing the thickness of the mirror, how can one measure the (I assume) perpendicular distance to the center. So if the mirror has a depth of 0.5 cm at the center, it could be 0.25 thick and a very strange mirror. Now, if the 0.5 cm is a perpendicular measure to the table on which the mirror sits, we can go from there.

The problem (as I see it) is that there would be infinitely many such mirrors. We also have to assume that "the diameter" is the max. diameter or else the problem is too ambiguous. Again, I think we need to know something about the thickness of the mirror. If the problem is purely theoretical (no thickness), the wouldn't a cross-section just be a circle whose radius would vary from 0 to 11 cm.

 Now to get THE APPROPRIATE circular cross-section, we could enter (0,0) and (0.5,11) as ordered pair into the TI-84 and ask for a best-fit parabola... I just attempted to draw this thang by hand as a space capsule-looking thing along the x-axis with one point at (0,0) and one at (0.5, 11) with circular cross-sections determined in part by an 0<=x_i<=0.5 giving pi (x_i)^2 as the desired cross-section. I think we need one more point to make this a specific mirror.