Question

Question #d2356

Answer 1

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)`

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`