Question #d2356

1 Answer
Oct 9, 2016

Answer:

Equation of coaster's path: #y=1/64(x-56)^2-9#

Height of the peak: #40# feet

Explanation:

[Although this was asked under Geometry:Right-Angled Triangle, my tools were strictly Algebraic. I anyone has a Geometric solution I would like to see it posted here.]
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Measuring (in feet) the horizontal distance from the peak as the x-coordinate and the distance above ground as the y-coordinate:

The coaster will pass through #(x,y)# intercept points #(32,0)# and #(80,0)#.
The vertex of its path will occur at #((32+80)/2,-9)=(56,-9)#
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The general vertex form of a parabola is:
#color(white)("XXX")y=color(green)(m(x-color(red)a)^2+color(blue)b#
for a parabola with vertex at #(color(red)a,color(blue)b)#.

Therefore the coaster's path must be modeled by
#color(white)("XXX")y=color(green)(m)(x-color(red)(56))^2color(blue)(-9)#
and
#(x,y)=(color(magenta)(32),color(orange)(0))# must be a solution to this equation.

#color(orange)(0)=color(green)(m)(color(magenta)(32)-color(red)(56))^2color(blue)(-9)#

#color(green)(m)=9/(24^2) = 1/64#

So the coaster's parabolic path equation is #y=color(green)(1/64)(x-color(red)(56))^2color(blue)(-9)#

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If #y# represents the height above ground (in feet) when #x# is the vertical distance from the peak,
when #color(maroon)(x=0)#, #color(orangered)(y)# will be the height of the peak

#color(white)("XXX")color(orangered)(y)=color(green)(1/64)(color(maroon)0-color(red)(56))^2color(blue)(-9)#

#color(white)("XXX")color(orangered)(y)=color(indigo)40#