Question 518758
The shape of the gateway arch in st. Louis, Missouri, is a catenary curve, which closely resemble a parabola. The function{{{ y = -2x^2/315 + 4x }}} models the shape of the arch, where y is the height in feet and x is the horizontal distance from the base of the left side of the arch in feet.
:
Convert the coefficient of x^2 to a decimal {{{(-2)/315}}} = -.00635
:
a)Find the vertex
Find the axis of symmetry
x = {{{(-4)/(2*-.00635)}}}
x = 315 ft, from the left side of the base
Find y (the height)
{{{ y = -.00635(315)^2 + 4(315) }}}
y = -630 + 1260 
y = 630 ft high
vertex, 315, 630
:
b) Describe a reasonable domain and range for the function. Explain.
domain 0 to 630; range 0 to 630 (only positive values here)
:
c) According to the model, what is the maximum height of the arch
630 ft
d) What is the width of the arch at the base?
2(315) = 630 ft
:
A graph of this equation illustrates this well
{{{ graph( 300, 200, -100, 700, -100, 700, -.00635x^2+4x) }}}