Question 1029075: Could someone please check my work on this? I apologize for the length, but would really appreciate some help! "The shape of the Gateway Arch can be approx modeled with the equations of 2 parabolas: one for the outer/upper surface, and one for the inner/lower surface. The height of the arch is 630 ft, and at ground level the outsides of the bases are 630 ft 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 ft. suppose you are standing at the origin which is at ground level directly underneath the center of the arch.
a) find the vertex and x-intercepts for both the outer and inner parabolas.
**My answer*** Outer parabola vertex: (0,630), x-intercepts (-315,0)(315,0) and inner parabola vertex: (0,615), x-intercepts: (-270,0) (270,0). Is this correct???
b) find the height above the ground of the focus and directrix for each parabola. Round your answer to the nearest tenth of a foot.
***my answer** outer parabola focus height: 590.625 and directrix height: 1220.625 feet.
Inner parabola focus height: 585.37 ft and directrix height: 1200.37 ft.
c) Find the equation for both outer and inner parabolas in standard form
**my answer** Inner: y=-(41/4860)x^2 + 615 and outer: y=-(2/315)x^2+630. I'm not really sure if this is even standard form???
I appreciate any help and again, I'm sorry it's such a long and involved question!
Answer by Fombitz(32388)   (Show Source):
You can put this solution on YOUR website! Vertex and intercepts are correct.
The graphs looks correct based on your equations.
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For focus, use the focus form,
Outer,
When  ,
The focus is then,
( , )
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Directrix,
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Focus is correct, directrix is incorrect for outer.
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Inner,
When  ,
The focus is then,
(,)
(, )
(, )
(, )
Directrix,
(,)
(, )
(, )
(, )
Focus is correct, directrix is incorrect for inner.
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You've got most of it down.
Just need to work on the directrix.
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