Question 1159790
<pre>
We want to know the differences in heights at the green line:

{{{drawing(400,1200/7,-35,35,-5,25,

graph(400,1200/7,-35,35,-5,25,20-x^2/45),
green(line(25,11.05541597,25,0)),
red(arc(0,0,60,-40,0,180)) )}}}

The ellipse's equation is of the form:

{{{x^2/a^2+y^2/b^2=1}}}

where a=center to vertex = (0,0) to (30,0) = 30
and b = center to covertex = (0,0) to (0,20) = 20

Equation of ellipse:

{{{x^2/30^2+y^2/20^2=1}}}

{{{x^2/900+y^2/400=1}}}

The parabola's equation is of the form:

{{{y=a(x-h)^2+k}}}

where the vertex (h,k) = (0,20)

{{{y=a(x-0)^2+20}}}

{{{y=ax^2+20}}}

It goes through (30,0)

{{{0=a(30)^2+20}}}

{{{0=900a+20}}}

{{{-20=900a}}}

{{{(-20)/(900)=a}}}

{{{-1/45=a}}}

{{{y=(-1/45)x^2+20}}} <-- equation of parabola
 
We find the ordinate of the ellipse at x=25

{{{x^2/900+y^2/400=1}}}
{{{4x^2+9y^2=3600}}}
Substitute x=25
{{{4(25)^2+9y^2=3600}}}
{{{4(625)+9y^2=3600}}}
{{{2500+9y^2=3600}}}
{{{9y^2=1100}}}
{{{y^2=1100/9}}}
{{{y=sqrt(1100/9)}}}
{{{y=sqrt(1100)/sqrt(9)}}}
{{{y=sqrt(100*11)/3}}}
{{{y=10sqrt(11)/3}}}

We find the ordinate of the parabola at x=25

{{{y=(-1/45)x^2+20}}}

{{{y=(-1/45)(25)^2+20}}}
{{{y=(-1/45)(625)+20}}}
{{{y=-625/45+20}}}
{{{y=-125/9+20}}}
{{{y=-125/9+180/9}}}
{{{y=55/9}}}

We subtract the two y-coordinates:

{{{difference=10sqrt(11)/3-55/9}}}

{{{difference=30sqrt(11)/9-55/9}}}

{{{difference=(30sqrt(11)-55)/9}}} <--exact difference

{{{4.944304857}}} <--approximate difference

Edwin</pre>