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) About Me  (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

x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=1

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

x%5E2%2F900%2By%5E2%2F400+=+1

For the semi-ellipse, the equation is then

y+=+sqrt%28400%281-x%5E2%2F900%29%29

The vertex form of the equation for the parabola is

y-k+=+a%28x-h%29%5E2

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.

y-20+=+a%28x-0%29%5E2
0-20+=+a%2830-0%29%5E2
-20+=+900a
a+=+-20%2F900+=+-1%2F45

The equation for the parabola is

y+=+%28-1%2F45%29x%5E2%2B20

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(20059) About Me  (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:

x%5E2%2Fa%5E2%2By%5E2%2Fb%5E2=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%5E2%2F30%5E2%2By%5E2%2F20%5E2=1

x%5E2%2F900%2By%5E2%2F400=1

The parabola's equation is of the form:

y=a%28x-h%29%5E2%2Bk

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

y=a%28x-0%29%5E2%2B20

y=ax%5E2%2B20

It goes through (30,0)

0=a%2830%29%5E2%2B20

0=900a%2B20

-20=900a

%28-20%29%2F%28900%29=a

-1%2F45=a

y=%28-1%2F45%29x%5E2%2B20 <-- equation of parabola
 
We find the ordinate of the ellipse at x=25

x%5E2%2F900%2By%5E2%2F400=1
4x%5E2%2B9y%5E2=3600
Substitute x=25
4%2825%29%5E2%2B9y%5E2=3600
4%28625%29%2B9y%5E2=3600
2500%2B9y%5E2=3600
9y%5E2=1100
y%5E2=1100%2F9
y=sqrt%281100%2F9%29
y=sqrt%281100%29%2Fsqrt%289%29
y=sqrt%28100%2A11%29%2F3
y=10sqrt%2811%29%2F3

We find the ordinate of the parabola at x=25

y=%28-1%2F45%29x%5E2%2B20

y=%28-1%2F45%29%2825%29%5E2%2B20
y=%28-1%2F45%29%28625%29%2B20
y=-625%2F45%2B20
y=-125%2F9%2B20
y=-125%2F9%2B180%2F9
y=55%2F9

We subtract the two y-coordinates:

difference=10sqrt%2811%29%2F3-55%2F9

difference=30sqrt%2811%29%2F9-55%2F9

difference=%2830sqrt%2811%29-55%29%2F9 <--exact difference

4.944304857 <--approximate difference

Edwin