Math Problem: A reflecting telescope has a parabolic mirror...(please read)?

Question

Hi. I have a really important math assignment due tomorrow. My teacher was absent when we were given it and he hasn't done any problems like this in class (nor are there any similar ones in the book). So, I'm clueless as to where to start. I'm not looking for someone to just tell me the answer, I really want an explanation. Help?

"A reflecting telescope has a parabolic mirror for which the distance from the vertex to the focus is 26 feet. If the distance across the top of the mirror is 72 inches, how deep is the mirror in the center?"

My immediate thought is that the 72 inches should be converted to 6 feet before starting the problem, but after that I've got nothing.

Answer 1

You are correct about changing 72 inches to 6 feet.
Let the vertex be at the origin, and let Let the focus be
F(0, 26), and let D be the directrix with equation y = -26
Let P(x, y) be any point on the parabola, then the distance from P to D is y + 26 and the distance from P to F is:
√(x^2 + (y - 26)^2)
Equating the two distances gives you the equation of the parabola:
y + 26 = √(x^2 + (y - 26)^2)
Square both sides:
y^2 + 52y + 676 = x^2 + y^2 - 52y + 676
104y = x^2
y = (1/104)x^2
The diameter of the mirror is 6 feet. That gives us the x coordinate of a point on the parabola is 3, and we need the y coordinate:
y = (1/104)3^2
y = 9/104 = 0.086538 feet = 1.03846 inches.