A fireplace arch is to be constructed in the form of a semi ellipse. The opening is to have a height of 2 feet at the center and a width of 6 feet along the base. The contractor cuts a string of a certain length and nails each end of the string along the base in order to sketch the outline of the semi ellipse.1. What is the total length of the string?

2. How far from the center should the string be nailed into the base?

1. One definition of an ellipse is that it is the locus of points such that the sum of the distances to the two foci is equal to the length of the major axis. Hence, the string can be 6 feet long, the length of the major axis.

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2. The distance from a focus to a co-vertex is the length of the semi-major axis. Using the Pythagorean theorem, we can find the distance from the ellipse center to the focus as ...

center-to-focus = √( (semi-major axis)² - (semi-minor axis)² ) = √(3² -2²)

center-to-focus = √5

The string should be fastened √5 feet, about 26.83 inches, from the center of the base.

oeerivona oeerivona · Answer 2 · 2021-10-07T21:26:48+02:00

The sum of the distances of points on the perimeter of an ellipse to the foci is a constant

1. The total length of the string is 6 feet

2. The distance from where the string is fastened on either side from the center of the base is √5 feet

Reason:

1. An ellipse is the locus of points such that the distances from the point to the foci when added together gives twice the length of the major axis, or 2·a

The length of the major axis, a = 3 feet, therefore, to draw the ellipse, the contractor can place the end of the string at the foci and draw sketch the outline with a length of string of 2 × 3 feet = 6 feet

The total length of the string required = 6 feet

2. The string should nailed to the foci to sketch the ellipse

The distance from the center to the foci, c = [tex]\sqrt{a^2 - b^2}[/tex]

c = [tex]\sqrt{3^2 - 2^2} = \sqrt{5}[/tex]

Therefore, the string should nailed at approximately √5 feet from the center, on either side

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