A parabolic headlight reflector is designed according to the specifications in this illustration. The light bulb needs to be at the focus. How far from the vertex is the focus?

Question

anonymous · Answer

attachment

anonymous · Answer

A) 3 inches
 B) 1 inch
 C) ¾ inch
 D) ½ inch

mathmale · Answer

I see that this is a VERTICAL parabola (it opens upward, not to the right or to the left).

When the vertex of a vertical parabola is at (0,0), the general equation of that parabola which involves p, the distance between the vertex and the focus, is

4py=x^2.

If you'd please go back to the illustration and choose a point on the curve, such as (3,3), and substitute these x- and y-coordinates into the above equation, you should be able to find p.  Please give this a try.

p = ?

Thus, the equation for the parabola becomes   ???

Solve this result for y and you'll be done.

anonymous · Answer

so  4py=x^2
substitute (3,3)    so     4p(3)=(3)^2   so  p=3/4  or 0.75 
am I correct

mathmale · Answer

You can check that yourself.  Substitute your p = 3/4 into the equation

4py=x^2.  What do you get?

Next, substitute the coordinates of the point (3,3) into this latest equation.  

Is the equation now true or false?  

Doing this will tell you whether or not your value of p is correct.

anonymous · Answer

ok so when I substitute p=3/4  in the equation, I got 9 which is (x)^2  when x=3 
so this is correct 
next when I substiute the (3,3) in the equation of the parabola 
4py=x^2    so that 4(3/4)(3)=(3)^2  this is correct 
so I'm doing this right

anonymous · Answer

now using this equation of the parabola  4py=x^2
I should solve for y ?

mathmale · Answer

Yes, but first please substitute the value you've found for p.

anonymous · Answer

ok so I found that p=3/4

so that     4py=x^2
               4(3/4)y=(3)^2
                3y=9
    then  y=9/3      so y=3

anonymous · Answer

which is in the given choices A) 3inches

mathmale · Answer

Getting there!  But WHAT is 3 inches?  Always a good idea to label your results / answers.

anonymous · Answer

that would be the focus which is equals 3 inches 
choice A

anonymous · Answer

the focus will be 3 inches

mathmale · Answer

"The distance between the vertex and the focus is 3 inches."

thanks for your involvement and perseverance.

anonymous · Answer

no thank you so much for your help. this was very helpful. thank you @mathmale

mathmale · Answer

My great pleasure, @jody999!  Hope to work with you again.

anonymous · Answer

I hope so, thanks a lot :)