Q:

In a factory, a parabolic mirror to be used in a searchlight was placed on the floor. it measured 50 centimeters tall and 90 centimeters wide. find the equation of the parabola.

Accepted Solution

A:
Answer:[tex]y = \frac{-2}{81}x^2 + 50[/tex]Step-by-step explanation:The parabola in Cartesian terms goes through the points:  (0, 50) [assuming center at 0](-45,0) and (45,0) (touches floor at both edges)Since it is roundSo, width = 90 cm = DiameterSo, Radius = [tex]\frac{Diameter}{2}=\frac{90}{2} = 45 cm[/tex] So, take points (0, 50) and (45,0) and (-45,0) Equation :[tex]y = ax^2 + bx + c[/tex]  --Asince x= 0 gives 50   So, [tex]50 = a(0)^2 + b(0) + c[/tex]So,c = 50  Since x= -45 and x = 45 gives the same result So, b has no effect i.e. b = 0So, plugging in b and c for the value 45 for the expression   [tex]a\times 45^2 +0 \times 45 + 50 = 0[/tex] [tex]2025a = -50[/tex][tex]a =\frac{ -50}{2025}  [/tex][tex]a =\frac{ -2}{81}  [/tex]Now substitute values of a,b and c in A[tex]y = \frac{-2}{81}x^2 + 50[/tex]Hence the equation of parabola is    [tex]y = \frac{-2}{81}x^2 + 50[/tex]