A racetrack is in the shape of an ellipse, 100 feet long and 50 feet wide. Write the standard form of the elliptical track.

Question

<!--[if !supportLists]-->1.     <!--[endif]-->A racetrack is in the shape of an ellipse, 100 feet long and 50 feet wide. 

a.	Write the standard form of the elliptical track.
b.	What is the width of the line z, which is 10 feet in from the right vertex, assuming your new ellipse is proportional?
c.	Identify both endpoints of the new width.

Answer 1

(a) The given information establishes the major and minor axes of the ellipse with its origin at (0,0).

The semi-axes are therefore 50ft and 25ft and the equation in standard form is:

x²/50²+y²/25²=1, that is: x²/2500+y²/625=1.

(b) and (c) 10ft in from the right vertex is the point (40,0) because the vertex is at (50,0).

Plugging in x=40 into the equation of the ellipse:

1600/2500+y²/625=1,

1600+4y²=2500,

4y²=900, y²=225, y=±15ft, making (40,15) and (40,-15) the endpoints of z. Therefore its length is 30ft.