The shape of the Gateway Arch in St. Louis, Missouri, can be approximately model

Question

The shape of the Gateway Arch in St. Louis, Missouri, can be approximately modeled with the equations of two parabolas: one parabola for the outer/upper surface, and one for the inner/lower surface. The height of the arch is 630 feet, and at ground level the outsides of the bases are 630 feet apart. The arch narrows as it rises. Therefore, the insides of the bases are only 540 feet apart at ground level, but the inside of the arch has a height of 615 feet. Suppose you are standing at the origin which is at ground level directly underneath the center of the arch. Find the vertex and x-intercepts for both the outer and inner parabolas. Find the height above the ground of the focus and directrix for each parabola. Round your answers to the nearest tenth of a foot. Find the equation for both the outer and inner parabolas in standard form.

Explanation / Answer

Outer parabola :

vertax (h,k) = (0,630) , x-intecept is (-315, 0) and (315,0)

for inner parabola

vertax (h,k) =(0,615) , x-intercepts are (-270,0) and (270,0)

equation of inner parabola x2 = -(2702/615)(y-615) ----------------------------solution in the attached image

for outer parabola equation is

(x-0)2 = 4p(y-630) -----------------------------------------(2)

it passes through (315, 0) which gives eqaution of outer parabola

x2 = -(3152/630)(y-630) ----------------------------------------------------------------(2)

For outer parabola :

focus is at (h, k+p) = (0, 630+ (-3152/(4*630) ) = (0, 630 -39.375) = (0, 590.625)

hence focus height is = 590.625 feet

directrix is at y = k-p => y = 630-(-39.375) = 669.375feet

so directrix height is y = 669.375 feet

for inner parabola :

focus is at (0, 615 +(-2702/4*615)) = (0, 615+(-29.624146)) = (0,644.624146)

focus height is 585.375854 feet

directrix is at y = k- p = 615-(-29.624146) = 644.624146 feet

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