A rectangle is inscribed in a circle of radius 4 centered around the origin. Write a function A(x) for the area of the rectangle.
==> I have no idea how to do this. Can someone help me please? Thanks!!
The circle has a radius of r=4, centred around the origin. Therefore any point (x,y) on the circumference can be obtained by the relation
r²=x²+y².......(1)
by the Pythagorean theorem.
The four corners of the rectangle have to lie on the circumference of the circle.
In fact, if one corner is x,y (in the first quadrant, therefore x≥0, y≥0), the the other points are (-x,y), (x,-y) and (-x,-y).
The width and breath are therefore
2x and 2y, and the area, A(x,y)=2x*2y=4xy........(2)
Using equation (1) above, it is possible to express y in terms of x and r.
Substitute and eliminate y from equation (2) to get:
A(x) = 4x.(..y...)
Can you take it from here?
Certainly! To find the area of the rectangle, we need to determine the dimensions of the rectangle first.
Since the rectangle is inscribed in a circle of radius 4, its diagonal is equal to the diameter of the circle, which is 2 times the radius. So, the diagonal of the rectangle is 2 * 4 = 8.
Let's assume the length of the rectangle is along the x-axis and its width is along the y-axis. By using the Pythagorean theorem, we can find the dimensions of the rectangle as follows:
The formula for the Pythagorean theorem is: hypotenuse^2 = base^2 + height^2.
The diagonal of the rectangle is the hypotenuse, the base is the length of the rectangle along the x-axis, and the height is the width of the rectangle along the y-axis.
Plugging in the values, we have:
8^2 = (base)^2 + (height)^2.
64 = (base)^2 + (height)^2.
Since the rectangle is inscribed in the circle, the base and height represent the x and y coordinates of a point on the circle. Thus, we have the equation:
x^2 + y^2 = (base)^2 + (height)^2.
Since the circle is centered at the origin, we can rewrite the equation as:
x^2 + y^2 = 64.
Now, let's solve this equation for the base and height.
We can rearrange the equation to isolate the base:
x^2 = 64 - y^2.
Taking the square root of both sides gives us:
x = sqrt(64 - y^2).
Now, we can write the function A(x) for the area of the rectangle:
A(x) = x * y.
Substituting the x value we obtained above into this equation:
A(x) = (sqrt(64 - y^2)) * y.
And that's it! The function A(x) represents the area of the rectangle inscribed in a circle of radius 4 centered around the origin.