A carpenter is cutting a 3ft by 4ft elliptical sign from a 3ft by 4ft piece of plywood. The ellipse will be drawn using a string attached to the board at the foci of the ellipse.
A)How far from the ends of the board should the string be attached?
B)How long should the string be?
Ignore the final '=2s '
The answer got sent before I could finish.
2c = 2.646 ft is the string length
The semimajor axis is a = 2 ft.
The semiminor axis is b = 1.5 ft.
The equation of the ellipse is
x^2/a^2 + y^2/b^2 = 1
I have assumed (0,0) is the center of the rectangle.
The foci are at y = 0 and x = +/- c, where
c = sqrt[a^2 - b^2] = sqrt1.75
= 1.323 ft
The distance of the foci from the closest end of the board is 4 - 1.323 = 2.677 ft
String length = 2c = 2*s
Slight mistake in the first answer...The total distance the foci are apart from each other is 2.6457 ft. BUT the distance of the foci from the closest end of the board is 2 - 1.32288 = .67712 ft. Since c^2= a^2 - b^2, and 2a=4, and a=2, therefore a-c is equal to 2-1.32288 which is only .67712 ft. Also, string length is equal to 2c + 2(a-c)=2c + 2a - 2c which is only equal to 2a. So, the string length is 4 ft.
To determine how far from the ends of the board the string should be attached (question A), we need to find the locations of the foci of the ellipse.
An ellipse can be defined as the set of all points where the sum of the distances to two fixed points, called the foci, is constant. In this case, we can assume that the width of the ellipse will be constant at 3ft (since it is cut from a 3ft board) and the length will be constant at 4ft (since it is also cut from a 4ft board).
Since the width and length of the ellipse match the dimensions of the board, we can conclude that the foci of the ellipse will lie exactly on the ends of the board.
Therefore, the string should be attached at the ends of the board, which is 3ft away from each end.
To determine the length of the string (question B), we need to calculate the distance between the foci of the ellipse.
The distance between the foci of an ellipse can be calculated using the formula:
c = sqrt(a^2 - b^2)
where c is the distance between the foci, a is the semi-major axis, and b is the semi-minor axis.
In this case, the semi-major axis is half of the length of the board, which is 4ft/2 = 2ft.
The semi-minor axis is half of the width of the board, which is 3ft/2 = 1.5ft.
Using the formula, we can calculate the distance between the foci:
c = sqrt(2^2 - 1.5^2) = sqrt(4 - 2.25) = sqrt(1.75) ≈ 1.32ft.
Therefore, the length of the string should be approximately 1.32ft.