# Question d2356

Oct 9, 2016

Equation of coaster's path: $y = \frac{1}{64} {\left(x - 56\right)}^{2} - 9$

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.]
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

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 $\left(x , y\right)$ intercept points $\left(32 , 0\right)$ and $\left(80 , 0\right)$.
The vertex of its path will occur at $\left(\frac{32 + 80}{2} , - 9\right) = \left(56 , - 9\right)$

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 $\left(\textcolor{red}{a} , \textcolor{b l u e}{b}\right)$.

Therefore the coaster's path must be modeled by
$\textcolor{w h i t e}{\text{XXX}} y = \textcolor{g r e e n}{m} {\left(x - \textcolor{red}{56}\right)}^{2} \textcolor{b l u e}{- 9}$
and
$\left(x , y\right) = \left(\textcolor{m a \ge n t a}{32} , \textcolor{\mathmr{and} a n \ge}{0}\right)$ must be a solution to this equation.

$\textcolor{\mathmr{and} a n \ge}{0} = \textcolor{g r e e n}{m} {\left(\textcolor{m a \ge n t a}{32} - \textcolor{red}{56}\right)}^{2} \textcolor{b l u e}{- 9}$

$\textcolor{g r e e n}{m} = \frac{9}{{24}^{2}} = \frac{1}{64}$

So the coaster's parabolic path equation is $y = \textcolor{g r e e n}{\frac{1}{64}} {\left(x - \textcolor{red}{56}\right)}^{2} \textcolor{b l u e}{- 9}$

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

If $y$ represents the height above ground (in feet) when $x$ is the vertical distance from the peak,
when $\textcolor{m a r \infty n}{x = 0}$, $\textcolor{\mathmr{and} a n \ge red}{y}$ will be the height of the peak

$\textcolor{w h i t e}{\text{XXX}} \textcolor{\mathmr{and} a n \ge red}{y} = \textcolor{g r e e n}{\frac{1}{64}} {\left(\textcolor{m a r \infty n}{0} - \textcolor{red}{56}\right)}^{2} \textcolor{b l u e}{- 9}$

$\textcolor{w h i t e}{\text{XXX}} \textcolor{\mathmr{and} a n \ge red}{y} = \textcolor{\in \mathrm{di} g o}{40}$