Question 870540
{{{drawing(600,300,-50,50,-25,25,
grid(0),
red(arc(0,0,100,33.33,-180,0)),
arrow(40,0,40,10),arrow(40,10,40,0),
locate(0.5,9,b),locate(40.5,5.5,10)
)}}} Here's the half ellipse passing through (-50,0) and (50,0), so its span is 100.
Point (40,10) is 10 feet above point (40,0), which is on the ground, 40 feet from the center of the span.
The height at the center, at point (0,b), is b.
all measurements are in feet
The equation of that ellipse is
{{{x^2/50^2+y^2/b^2=1}}} <---> {{{(x/50)^2+(y/b)^2=1}}}
Substituting the coordinates of point (40,10) we get
{{{(40/50)^2+(10/b)^2=1}}}
{{{(0.8)^2+(10/b)^2=1}}}
{{{0.64+(10/b)^2=1}}}
{{{(10/b)^2=1-0.64}}}
{{{(10/b)^2=0.36}}}
{{{(10/b)^2=(0.6)^2}}}
{{{10/b=0.6}}}
{{{10/0.6=b}}}
{{{b=approximately16.7)}}} or {{{b=100/6=50/3=16&2/3}}} .
The height at the center is {{{16&2/3}}}{{{feet=highlight(16feet8inches)}}}