A circle of radius r is centered at the origin and a rectangle is inscribed. The area of the rectangle as a function is given by what? Can you also explain the processes behind finding the answer it helps me understand.
Let the rectangle have a base length of 2x and a height of 2y.
The the 4 vertices of the rectangle are:
(x,y) in I, (-x,y) in II, (-x,-y) in III and (x,-y) in quadrant IV
the area of the rectangle is 4xy but
x^2 + y^2 = r^2 , where r is the radius
from which y = √(r^2 - x^2)
Area of rectangle = 4x√(r^2 -x^2)
To find the area of the rectangle inscribed in a circle of radius r, we can use the following steps:
1. Draw a diagram: Draw a circle with center at the origin and radius r. Inside the circle, draw a rectangle with sides parallel to the coordinate axes.
2. Understand the properties: In a rectangle that is inscribed in a circle, the diagonals of the rectangle are equal in length and coincide with the diameters of the circle. Additionally, the rectangle's vertices touch the circle's circumference.
3. Determine the dimensions of the rectangle: Let's denote the length and width of the rectangle as L and W, respectively. Since the rectangle's vertices touch the circle's circumference, the length L will be twice the radius (2r), and the width W will be twice the radius as well (2r).
4. Calculate the area: The area of a rectangle is given by the formula A = length × width. Therefore, the area of the rectangle inscribed in the circle is:
A = L × W
= (2r) × (2r)
= 4r^2
Thus, the area of the rectangle as a function is given by A = 4r^2.