Final answer:
The standard form of the equation of an ellipse with a 28 feet length and 18 feet width with the center (0,0) is (x^2/196) + (y^2/81) = 1, where x is the horizontal coordinate and y is the vertical coordinate.
Step-by-step explanation:
The student is interested in finding the standard form of the equation of an ellipse to represent the outline of a room. The room is said to be in the shape of an ellipse with a given length of 28 feet and a width of 18 feet while the center of the ellipse is at the point (0,0). An ellipse is a conic section and can be represented by a standard equation in Cartesian coordinates where the center of the ellipse is at the origin (0,0).
The standard form of the equation for an ellipse that is horizontally oriented (the major axis is along the x-axis) is given by:
(x^2/a^2) + (y^2/b^2) = 1
where a is the semi-major axis and b is the semi-minor axis. Since the length of the ellipse is its major axis and the width is its minor axis, we divide these by 2 to get the semi-major and semi-minor axes, respectively:
Now, we can write down the standard form equation for the ellipse:
(x^2/14^2) + (y^2/9^2) = 1
or, simplifying the denominators:
(x^2/196) + (y^2/81) = 1
This is the equation of the ellipse that would represent the room in standard form.