SOLUTION: A racetrack is in the shape of an ellipse 80 feet long and 40 feet wide. What is the width 10 feet from a vertex?

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Question 505613: A racetrack is in the shape of an ellipse 80 feet long and 40 feet wide. What is the width 10 feet from a vertex?
Answer by Edwin McCravy(20059) About Me  (Show Source):
You can put this solution on YOUR website!
A racetrack is in the shape of an ellipse 80 feet long and 40 feet wide. What is the width 10 feet from a vertex?
 

The equation of that ellipse is

2a = Major axis = 80, so a = semi-minor axis = 40
2b = Minor axis = 40, so a = semi-minor axis = 20
 
Equation is of the form:

x%5E2%2Fa%5E2+%2B+y%5E2%2Fb%5E2=1

or

x%5E2%2F40%5E2+%2B+y%5E2%2F20%5E2+=+1

x%5E2%2F1600+%2B+y%5E2%2F400+=+1


To find the width 10 feet from the vertex, we want to
know the length of this green line:


 
We want to know the value of y when x = 30, so we substitute
30 for x in

x%5E2%2F1600+%2B+y%5E2%2F400+=+1

30%5E2%2F1600+%2B+y%5E2%2F400+=+1

900%2F1600+%2B+y%5E2%2F400+=+1

9%2F16+%2B+y%5E2%2F400+=+1

Multiply through by 1600

900+%2B+4y%5E2+=+1600

4y%5E2+=+700

y%5E2=175

y+=+%22%22+%2B-+sqrt%28175%29

y+=+%22%22+%2B-+sqrt%2825%2A7%29

y+=+%22%22+%2B-+5sqrt%287%29

So the endpoints of that green line are

%28matrix%281%2C3%2C30%2C%22%2C%22%2C5sqrt%287%29%29%29 and %28matrix%281%2C3%2C30%2C%22%2C%22%2C-5sqrt%287%29%29%29

and its length is 10sqrt%287%29 or about 26.46 feet.

Edwin