Pre- cal question??

Question

A rectangle is inscrbed in an ellipse centered at the origin with vertices at (-4,0), (4,0), (0,-3), and (0,3).
a) What is the standard equation of this ellipse?
b) solve the equation in part a for x
c) If x is the x-coordinate of the vertex of the triangle which lies in the first quadrant, then write an equation for the area of the rectangle in terms of x.
d) What value of x woild miximixe the area of the rectangle
e) What is the maximum area of the inscribed rectangle?

Answer 1

Standard equation:

a) x^2/16 + y^2/9 = 1

b) x=Sqrt((1-y^2/9) * 16)

c) all four triangles are the same area with one side = 4 and one side =3. Since the problem states x as a variable, I will assume y is fixed at 3 and -3

Triangle area = 1/2 * 3 * x

Rectangle area = 4 triangles
=6 * x

substituting x, and setting y=0 since it is a vertex

Sqrt((1-y^2/9) * 16) y=0
x=4
area = 24

d) The rectangle is already minimized at x=4 since the definition of a rectangle requires right angles on all four verticies.

e) Similarly, the area is maximized at 24



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