# Math Help - ellipse word problem

1. ## ellipse word problem

this is the word problem: a stadium is to be built with stands in the form of two concentric ellipses. The outer ellipse is to be 240 yds long and 200 yds wide. The inner ellipse is to be 200 yds long and 100 yds wide. The football field of standard dimensions is to be laid out in the center of the inner ellipse.

I sketched this ellipse with the outer coordinate (240,200) and the inner coordinate (200, 100). I found the equations...outer: (x^2/120^2)+(y^2/100^2)= 1 AND inner: (x^2/100^2)+(y^2/50^2)= 1. I found those numbers because the outer a is 120 and outer b is 100. also the inner a is 100 and inner b is 50.

With all this information I found the inner area and outer area.
Inner: \pi (100)(50)= 15707.96 AND outer: \pi (120)(100)= 37699.11

Next I have to find the clearance will there be between the field and the inner ellipse in the direction of the end line. Im not sure where to begin at this step..

2. By football field I assume you mean American football. The field has dimensions 120yds by 53.3 yards. Since you need the distance from the end lines center a rectangle with these dimensions at the origin. The end lines should be at -60yds and 60yrds. I hope this helps.

3. wouldnt that mean that the endlines would be at -120 and 120?
am I trying to find the distance between the endlines and the inner ellipse?

4. Originally Posted by kassums17
wouldnt that mean that the endlines would be at -120 and 120?
am I trying to find the distance between the endlines and the inner ellipse?
If what you said is true that would make the field

$120 -(-120)=240\text{yrds}$ long twice as long as it needs to be. Draw the picture of the field on you graph to help you visualize it.

5. thank you. the next thing I have to find with this problem is what the area of the stands is? how would I approach solving this?

6. Originally Posted by kassums17
thank you. the next thing I have to find with this problem is what the area of the stands is? how would I approach solving this?
Again look at your diagram. What is the area you want? It is inside the outer ellipse but outside the inner ellipse. You know both of these area's from your first post. How can you find it

7. is there some kind of formula i can use?
maybe my diagram isnt accurate because its hard to depict the coordinates

8. Originally Posted by kassums17
is there some kind of formula i can use?
maybe my diagram isnt accurate because its hard to depict the coordinates
Alright. So in your first post you found the area of the outer ellipse and the inner ellipse correct?

The area of the outer ellipse is too big because we don't want the area inside the inner ellipse. So how much extra area does the outer ellipse have?
Think on this. You have all of the information that you need to answer the question.