Answer:
(d) 8.60 inches
Step-by-step explanation:
The shape of a reflecting telescope mirror is a parabola. The parent function of a parabola is f(x) = x². This function will have the form g(x) = 1/(4p)x^2 when its focus is at y=p.
The parent function goes through the point (x, f(x)) = (1, 1). The curve describing the mirror will go through the point (38, 42). We can make that happen by ...
- horizontal scaling by a factor of 38
- vertical scaling by a factor of 42
The resulting function will be ...
g(x) = 42·f(x/38)
g(x) = (42/38²)x² = (21/722)x²
For this problem, we are interested in finding the value of p such that
g(x) = 1/(4p)x² = (21/722)x²
Multiplying by 722p/(21x²), we find ...
p = 722/(4·21) = 8 25/42 ≈ 8.60
The focus of the mirror is 8.60 inches above its vertex.
__
Additional comment
The points that line on a parabola are the same distance from the focus as from the directrix. So, the horizontal line through the focus (latus rectum) will intersect the parabola the same distance from the focus as from the directrix. Since the vertex is halfway between the focus and directrix, the vertical distance from the vertex to the latus rectum will be half the horizontal distance from the focus to the latus rectum intersection with the parabola.
In other words, a line with slope 1/2 will intersect the parabola at the height of the latus rectum, hence the height of the focus. This is true for a parabola of any shape, so provides an easy way to find (or verify) the focus graphically.
