Let P=(x,y) be the point in quadrant 1 that is a vertex of the rectangle and is on the semicircle of radius r=8.

Question

what is the area of the rectangle as a function of x?

Answer 1

The point is on a circle of radius 8. Therefore, we start with the basic equation of a circle:

x² + y² = 8²

The area of the rectangle, A = x*y, so we can write y = A/x.

Substitute this into the equation of the circle to get:

x² + (A/x)² = 8²

Solve for A:

x² + (A/x)² = 8²

x² + A²/x² = 8²

x⁴ + A² = 64x²

A = √(-x⁴ + 64x²)

Test this using known values of x and y. From the circle equation, if x = 1, y = √63, and A = √63

Using the equation for area:

A = √(-x⁴ + 64x²)

A = √(-1 + 64) = √63

Note that our equation covers all quadrants, but--since the exponents of x are even--the equation still returns correct values for the area.

There is one problem: if x is larger than 8, the formula returns the square root of a negative number.

To fix this, I think the correct equation is:

A = √(-x⁴ + 64x²), x ≤ 8