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?
Can you please show your work.
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To determine the answers to these questions, we need to understand the properties of an ellipse and how to calculate the distance from the ends of the board and the length of the string.
An ellipse is defined as a set of points in a plane such that the sum of the distances from any point on the ellipse to two fixed points (called foci) is constant.
Since we have a 3ft by 4ft piece of plywood, the foci will be located at the midpoints of the shorter side (3ft) of the rectangle. Thus, the foci will be 1.5ft from each end of the board.
A) How far from the ends of the board should the string be attached?
The string should be attached at a distance equal to the distance from the end of the board to the foci. In this case, the foci are located 1.5ft from each end, so the string should be attached 1.5ft from each end.
B) How long should the string be?
To determine the length of the string, we need to calculate the distance from the foci to any point on the ellipse. Since the ellipse is being cut from a 3ft by 4ft piece of plywood, the longer side of the rectangle will be the major axis of the ellipse.
The formula to calculate the length of the string (l) of an ellipse is given by:
l = 2a + 2ε,
where a is the length of the semi-major axis (half of the length of the major axis) and ε is the distance from the center of the ellipse to the foci.
In this case, the semi-major axis (a) is equal to half of 4ft, which is 2ft. The distance from the center of the ellipse to the foci (ε) is 1.5ft.
So, plugging the values into the formula, we get:
l = 2(2ft) + 2(1.5ft) = 4ft + 3ft = 7ft.
Therefore, the length of the string should be 7ft.
By following these calculations, you can determine that the distance from the ends of the board to attach the string is 1.5ft, and the length of the string required is 7ft.