SOLUTION: Owachomo Natural Bridge is found in Natural Bridges National Monument in Utah. If the origin is located at one end of the natural arch, the curve can be modelled by the equation.

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Question 134020: Owachomo Natural Bridge is found in Natural Bridges National Monument in Utah. If the origin is located at one end of the natural arch, the curve can be modelled by the equation.
h = -0.043d sqaured + 2.365d
where h metres is the height of the arch, and d metres is the horizontal distance from the origin.
a) What is the maximum height of the arch, to the nearest hundredth of a metre?
b) What is the width of the arch at the base?
Answer by solver91311(24713) About Me  (Show Source):
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h=.043d%5E2%2B2.365d

Maximum height is at the vertex. For a parabola y=f%28x%29=ax%5E2%2Bbx%2Bc, the vertex is located at (-b%2F2a,f%28-b%2F2a%29). For your equation, d is the independent variable, so a = .043 and b = 2.365.

-b%2F2a=-2.365%2F2%28.043%29=27.5 and f%28-b%2F2a%29=f%2827.5%29=.043%2827.5%29%5E2%2B2.365%2827.5%29=97.556

Therefore maximum height is 97.556 feet.

The base is the horizontal axis, so the parabola intersects the base at the roots of .043d%5E2%2B2.365d=0.

Factoring: d%28.043d%2B2.365%29=0

Hence d=0 or .043d%2B2.365=0 => d=55, so the coordinates of the intersection of the parabola with the base on the d-h plane are (0,0) and (55,0).

The distance between two points on a horizontal line is the difference between their x (rather d, in this example) coordinates. 55-0=55