Opimization?

Question

Find the rectangle of largest area that can be inscribed in a semicircle of diameter 73, assuming that one side of the rectangle lies on the diameter of the semicircle

What is the largest possible area?

Answer 1

Let
r = radius semicircle
2x = width rectangle
y = height rectangle
A = area rectangle

If we look at one-half of the rectangle (let's call it the right half), then we can draw a radius line from the center of the circle to the upper right corner of the rectangle. We get a right triangle with

x = base
r = hypotenuse
y = height

y = √(r² - x²)

Now we can calculate the area.

A = 2xy = 2x√(r² - x²)

Take the derivative and set it equal to zero to find the critical point(s).

dA/dx = 2√(r² - x²) - 2x²/√(r² - x²) = 0
2√(r² - x²) = 2x²/√(r² - x²)
√(r² - x²) = x²/√(r² - x²)
r² - x² = x²
r² = 2x²
x² = r²/2
x = r/√2

Now solve for the largest area.

A = 2x√(r² - x²)
A = 2(r/√2)√(r² - r²/2)
A = (r√2)√(r²/2) = (r√2)(r/√2)
A = r²

Plug in the value for r.

r = 73/2

A = (73/2)² = 1,332.25