a rectangular board is 2 by 12 units. How far from the short sides of the board will the foci be located to determine the largest elliptical tabletop?
a rectangular board is 2 by 12 units. How far from the short sides of the board will the foci be located to determine the largest elliptical tabletop
Clearly, the largest elliptical tabletop will have a semi-major axis a = 6 and a semi-minor axis b =1.
The distance of the foci from the center of the ellipse derives from c = sqrt(a^2 - b^2 = sqrt(36 - 1) = 5.916 making the distance of both foci from each end .0839 units.
To determine the largest elliptical tabletop, we first need to find the semi-major axis (a) and semi-minor axis (b) of the ellipse that can fit within the rectangular board.
Since the board is 2 units by 12 units, the longer side of the board will be the major axis of the ellipse, and the shorter side will be the minor axis. Therefore, a = 12/2 = 6 units and b = 2/2 = 1 unit.
The distance between the center of the ellipse and each focus is denoted by c. To find c, we use the equation c = √(a^2 - b^2). In this case, c = √(6^2 - 1^2) = √(36 - 1) = √35 ≈ 5.916 units.
Since the distance from the center of the ellipse to each focus is c, we divide c by 2 to find the distance of each focus from the short sides of the board. Therefore, each focus will be located approximately 5.916/2 = 2.958 units away from the short sides of the board.
So, the foci will be located approximately 2.958 units away from the short sides of the rectangular board to determine the largest elliptical tabletop.