Math Problem?
As part of my math homework I am assigned the following problem:
A spotlight has a parabolic cross section that is 6 ft wide at the opening and 2.5 ft deep at the vertex. How far from the vertex is the focus? Round answer to two decimal places.
I need help solving it. I'm kinda stuck. (Could you please show the procedure?)
Thank you!
Update:Can you also help me solve these 2 systems of matrices?
First one
1x+1y+1z=10
1x-1y+4z=23
2x+1y+1z=14
Second one:
3x+14y+13z=114
5x+7y+15z=95
9x+15y+6z=141
Appreciate it!
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Answers & Comments
When deriving the equation of a parabola, all points were created to be the same distance from directrix line y = - p as they were from the focus at (0, p). That includes the vertex, (set to be on the y axis), so we know the vertex is at the origin (0, 0). The resulting parabola has the equation
y = [1/(4p)]x^2 (remember this for next time)
Now we interpret " ...parabolic cross section 6 ft wide at the opening and 2.5 ft deep at the vertex"
as x = 3, y = 2.5 and substitute these to find p.
2.5 = [1/(4p)]*3^2
10p = 9, so p = 0.9 which means the focus is 0.9 feet, (10.8 inches), from the vertex
Extra: The equation of the parabola was y = (5/18)x^2
Regards - Ian H