Question 1159790
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Draw a sketch of the graphs with the origin at the center of the base.  Then....<br>
The ellipse has its major axis with endpoints (-30,0) and (30,0); the semi-minor axis has endpoints (0,0) and (0,20).<br>
The parabola has its vertex at (0,20) and passes through the points (-30,0) and (30,0).<br>
The standard form of the equation for the ellipse with center at the origin is<br>
{{{x^2/a^2+y^2/b^2=1}}}<br>
We have all the numbers we need to write that equation:<br>
{{{x^2/900+y^2/400 = 1}}}<br>
For the semi-ellipse, the equation is then<br>
{{{y = sqrt(400(1-x^2/900))}}}<br>
The vertex form of the equation for the parabola is<br>
{{{y-k = a(x-h)^2}}}<br>
where the vertex is (h,k) and the coefficient a determines the steepness of the parabola.<br>
We have the vertex (0,20); to calculate the coefficient a we use one of the other known points on the parabola.<br>
{{{y-20 = a(x-0)^2}}}
{{{0-20 = a(30-0)^2}}}
{{{-20 = 900a}}}
{{{a = -20/900 = -1/45}}}<br>
The equation for the parabola is<br>
{{{y = (-1/45)x^2+20}}}<br>
Here is a graph of the two curves (parabola red, semi-ellipse green):<br>
{{{graph(800,400,-40,40,-1,30,(-1/45)x^2+20,sqrt(400(1-x^2/900)))}}}<br>
Calculate the ordinates at x=25 for both equations and find the difference.<br>
Since the numbers don't work out "nicely", the easiest way to do that is with a graphing calculator.  My TI-83 calculator gives<br>
parabola: (25,6.111)
ellipse: (25,11.055)<br>
difference between ordinates at x=25: 4.944<br>