Question

Skip designs tracks for amusement park rides. For a new design, the track will be elliptical. If the ellipse is placed on a large coordinate grid with its center at (0, 0), the equation x^2/2500 + y^2/8100 = 1 models the path of the track. The units are given in yards. How long is the major axis of the track? Explain how you found the distance.show the steps

Answer 1

Step 1

Given:

center (0,0)

the equation given should have been:

`(x^2)/(2500)+(y^2)/(8100)=1`

We need to identify the larger denominator. If it is under x, the ellipse is horizontal. If it is under y, the ellipse is vertical. 8100 is the larger denominator and is under y, therefore, the ellipse is vertical

Step 2

The general equation of an ellipse is given as;

`((x-h)^2)/(b^2)+((y-k)^2)/(a^2)=1`

h and k are the center values which are both 0.

a = length of the semi-major axis

b = length of the semi-minor axis

The given equation is;

`(x^2)/(2500)+(y^2)/(8100)=1`

which is equivalent to;

`(x^2)/(50^2)+(y^2)/(90^2)=1`
`\begin{gathered} a^2=90^2 \\ \sqrt[]{a^2}=\sqrt[]{90^2} \\ a=90\text{ yard},\text{ the semi-major ax}is \end{gathered}`

The length of the major axis will thus be; 90x2=180 yards

Answer; 180 yards