The Gateway Arch in St. Louis, MO, is 630 feet high and 630 feet wide at ground level and can be modeled with the equation of a parabola.

Question

a) Find the equation that provides a model for the Gateway Arch.

b) If a banner is to be hung across the arch at a height of 100 feet, how long will the

banner have to be?

c) How high is the arch 75 feet from either base?

d) If a coin was dropped straight down from the arch at a height of 400 feet, how far

from the center of the arch would it hit the ground (assuming its path not affected

by any outside force other than gravity)?

Answer 1

(a) If 1 unit on the x axis is 1 ft then the parabola goes through the points (0,0), (630,0) and (315, 630)

[ the vertex has y = 630 and is halfway between the zeroes ]

So the vertex is (h,k) h = 315 k = 630

Equation is y = a (x-h)² + k = a (x-315)² + 630

x = 0 y = 0 so a (315)² + 630 = 0 so a = - 630 / 315²

y = (-630/315²) (x-315)² + 630

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(b) Draw the function on Desmos

Draw the line y=100

Find the x values at the 2 points of intercept

Find the difference of these x values

(c) Draw x = 75

Find the y value where this intercepts the parabola

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(d) Draw y = 400

Find the x coordinate of one of the points of intersection with the parabola

Calculate (x-315) and take the absolute value of this