Answer the question below

Question

A horse show arena is being laid out in the shape of an ellipse so that it is twice as

long as it is wide. If the focus of the arena is 84 feet from the center, to the nearest

foot, how long is the arena?

Answer 1

Plot the ellipse on the x y axes

foci are (-84, 0) and (84, 0)

center is the origin (0,0)

distance from (-84, 0) to the right end point (a,0) plus (84,0) to the right end point is 84+a + a-84 = 2a. 2a =2(2b), a =2b, a^2=4b^2

a^2 = b^2 +84^2

4b^2 = b^2+ 84^2

3b^2 =84^2

b^2 = 84^2/3

b = 84/sqr3 = about 48.5

2b = minor axis = 97 feet

2a = major axis =4b = 194 feet

the arena is 194 feet long, approximately, rounded to nearest integer