Respuesta :
Answer:
Step-by-step explanation:
The question doesn't seem complete, but the question is likely to ask for 2 things: the equation of the ellipse and it's foci. So I'll respond to that. Also, the ellipse will be at the center since the artist wants the largest.
Firstly, the equation of an ellipse at the center is given by:
(x²/a²) + (y²/b²) = 1 where a > b.
Since the artist wants to make the largest ellipse, then the length of the rectangle will be the length of the ellipse, same applies to the width.
Since the length is 60, then a = 60/2 = 30
Since the width is 40, then b = 40/2 = 20
Therefore, the equation of the ellipse is:
(x²/30²) + (y²/20²) = 1
To find the foci we need to know the center and this can be gotten by
c² = a² – b²
c² = 30² – 20² = (30 + 20)(30 – 20) = (50)(10) = 500
c = √(500) = 10√(5).
The foci, since it's at th center would be:
(–10√(5), 0) and (+10√(5), 0)
The largest ellipse that the artist can make on the considered rectangular wooden piece will have area as: [tex]\dfrac{x^2}{900} + \dfrac{y^2}{400} = 1\\[/tex](assuming its major and minor axes lie on x and y axis respectively)
What is the equation of ellipse if its major and minor axis are given?
Suppose that the major axis is of the length 2a units, and that minor axis is of 2b units, then if major axis is on x-axis and minor axis is on y-axis, then the equation of that ellipse would be:
[tex]\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} =1[/tex]
For this case, as we have rectangular piece of board the maximum ellipse will have it's major axis of length 60 inches, and of minor axis as 40 inches.
Thus, we have:
[tex]a = 60/2 = 30\: \rm inches\\b = 40/2 = 20 \: inches[/tex]
Thus, if the rectangle is placed such that the major axis is on x-axis, and minor axis is on y-axis, then the equation of that ellipse would be:
[tex]\dfrac{x^2}{30^2} + \dfrac{y^2}{20^2} = 1\\\\\dfrac{x^2}{900} + \dfrac{y^2}{400} = 1\\[/tex]
Thus, the largest ellipse that the artist can make on the considered rectangular wooden piece will have area as: [tex]\dfrac{x^2}{900} + \dfrac{y^2}{400} = 1\\[/tex](assuming its major and minor axes lie on x and y axis respectively)
Learn more about ellipse here:
https://brainly.com/question/16868628
