SOLUTION: A semi-ellipse and a parabola rests on the same base 60 meters wide and 20 meters high. Using the common base as x-axis, compute the difference of ordinates at points 25 meters fro

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Question 1159790: A semi-ellipse and a parabola rests on the same base 60 meters wide and 20 meters high. Using the common base as x-axis, compute the difference of ordinates at points 25 meters from the center of the base.
Found 2 solutions by greenestamps, Edwin McCravy:
Answer by greenestamps(13200)   (Show Source): You can put this solution on YOUR website!


Draw a sketch of the graphs with the origin at the center of the base. Then....

The ellipse has its major axis with endpoints (-30,0) and (30,0); the semi-minor axis has endpoints (0,0) and (0,20).

The parabola has its vertex at (0,20) and passes through the points (-30,0) and (30,0).

The standard form of the equation for the ellipse with center at the origin is



We have all the numbers we need to write that equation:



For the semi-ellipse, the equation is then



The vertex form of the equation for the parabola is



where the vertex is (h,k) and the coefficient a determines the steepness of the parabola.

We have the vertex (0,20); to calculate the coefficient a we use one of the other known points on the parabola.






The equation for the parabola is



Here is a graph of the two curves (parabola red, semi-ellipse green):



Calculate the ordinates at x=25 for both equations and find the difference.

Since the numbers don't work out "nicely", the easiest way to do that is with a graphing calculator. My TI-83 calculator gives

parabola: (25,6.111)
ellipse: (25,11.055)

difference between ordinates at x=25: 4.944
Answer by Edwin McCravy(20060)   (Show Source): You can put this solution on YOUR website!
We want to know the differences in heights at the green line:



The ellipse's equation is of the form:



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:





The parabola's equation is of the form:



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





It goes through (30,0)











 <-- equation of parabola
 
We find the ordinate of the ellipse at x=25



Substitute x=25










We find the ordinate of the parabola at x=25










We subtract the two y-coordinates:





 <--exact difference

 <--approximate difference

Edwin
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