I just wanted to check my work on this one to make sure Im doing it right.
Suppose we have an overpass described by the points (0,0), ( 3,0) for the start and end of the parbola, and (1.5, 2.5) for the position of the arch.
now suppose we want to see can we fit a golf buggy of dimensions 1.8 m height by 1.4 m width.
first of all I found the function :
$f(x)=ax^2+bx+c$ in general
so
$f(0)=c=0$
$f(3)=9a+3b=0$
or $3a +b=0$
$f(1.25)=2.25a+1.25b=2.5$
or $9a+6b=10$
doing a simulateous I got $b=10/3$ and $a=-10/9$
the next thing I did was find the roots at the point where the height of the tunnel is 1.8 and I wanted to investigate is the distance between them greater than 1.4
When I used the -b formula on
$f(x)=\tfrac{-10x^2}{9}+\tfrac{10x}{3}-1.8$
I got $x=2.29$ and $x=0.706$
this gives a width of 1.53 which is greater than the neccesary width of 1.4.
Does that look correct to you guys ?