a contractor plans to construct a semielliptical fireplace arch with an opening that is 2 feet high at the center and 6 feet wide along the base (see figure). the contractor draws the outline of the ellipse on the wall using tacks as described on page 704. determine the required positions of the tacks and the length of the string.



Answer :

The foci occur 2.23 feet above the center of the semi-elliptical arch.

Consider, it is a semi-elliptical arc this would make its width equal to 2a and its length equal to b. In order to find where the foci are placed, get the value of c.

The arch will be 6 feet wide so 2a=6. The arch will be 2 feet high so b=2. Using the equation

                                         [tex]c^{2} = a^{2} - b^{2}[/tex]

2a =6, a = [tex]\frac{6}{2}[/tex], a = 3

                                     ⇒  [tex]c^{2} = (3)^{2} - (2)^{2}[/tex]  

                                     ⇒   [tex]c^{2}[/tex] = 9 -4

                                     ⇒   [tex]c^{2}[/tex]  = 5

                                     ⇒    c  = [tex]\sqrt{5}[/tex] = 2.23

This means the foci occur 2.23 feet above the center of the semi-elliptical arch.

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