Parabola problem involving depth.

Question

The equation for the parabola, I found, is

$$y^2=3.2x$$

What is the depth of the reflector?

I'm getting $\dfrac {25}{3.2}$.

Answer 1

Biggest width is $10$, which means $y = 5$

$$y^2 = 3.2x$$

$$\implies x = \dfrac {25}{3.2}$$

Looks good to me.

Answer 2

The vertex is at $(0,0)$, the focus is at $(0.8,0)$ and by the position the distance from the vertex to the focus is $p=0.8$

Since the axis of simmetry is horizontal the equation is: $$(y-0)^2=4\cdot0.8(x-0)$$ $$y^2=3.2x$$

Then if the widest part of the reflector is $10$, then by symmetry and the fact the directrix is in the $x$ axis is equal to the distance between the coordinate $y_0$ and its negative:
$$10=y_0-(-y_0)$$ $$y_0=5$$ Substituting this into the equation for x would give you $x_0$ which is the distance from the vertex to the widest part which gives you the depth of the reflector: $$5^2=3.2x_0$$ $$x_0=\frac{25}{3.2}=7.8125$$