SOLUTION: An arch is in the shape of a parabola. It has a span of 208 meters and a maximum height of 26 meters. Find the equation of the parabola (assuming the origin is halfway between t

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: An arch is in the shape of a parabola. It has a span of 208 meters and a maximum height of 26 meters. Find the equation of the parabola (assuming the origin is halfway between t      Log On


   

Question 1175726: An arch is in the shape of a parabola. It has a span of 208 meters and a maximum height of 26 meters.
Find the equation of the parabola (assuming the origin is halfway between the arch's feet).
Determine the height of the arch 55 meters from the center.
Found 2 solutions by ewatrrr, ikleyn:
Answer by ewatrrr(24785) About Me  (Show Source):
You can put this solution on YOUR website!

Hi
An arch is in the shape of a parabola. 
It has a span of 208 meters and a maximum height of 26 meters.
 y = ax^2 + 26   (Sry, no clue where the 42 came from)
  a = -26/(104)^2 = -25/10816 = -1/416
 y = %28%28-1%2F416%29%29%28x%29%5E2+%2B+26m+
 y(55) =  18.73m rounded to nearest hundredth

Wish You the Best in your Studies.


Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
.
An arch is in the shape of a parabola. It has a span of 208 meters and a maximum height of 26 meters.
Find the equation of the parabola (assuming the origin is halfway between the arch's feet).
Determine the height of the arch 55 meters from the center.
~~~~~~~~~~~~


            The solution by @ewatrrr is  INCORRECT.

            I came to bring a correct one. 


You have this instruction to place the origin at halfway between the arch's feet).


It means that the parabola is open downward and its vertex is at the point  (0,26),

while the x-intercepts are 104 = 208%2F2 meters away from the origin.



THEREFORE, you write the equation of the parabola in the form

    y(x) = -ax^2 + 26


and you find then the value of the coefficient "a" from the condition  y(104) = 0, which is


    -a*104^2 + 26 = 0,   or


     a = 26%2F104%5E2 = 1%2F%284%2A104%29 = 1%2F416.


So, the parabola equation is  

    y(x) = - y%5E2%2F416 + 26      ANSWER


CHECK.   y(0) = 26;    y(104) = - 104%5E2%2F416 + 26 = - 104%2F4 + 26 = -26 + 26 = 0.    ! Correct !



    


                                   y(x) = - y%5E2%2F416 + 26


55 meters from the center, the height of the arch is

          y(55) = - 55%5E2%2F416 + 26 = 18.73 meters (rounded).

Solved (correctly).