Answer:
16 square inches
Explanation:
You want the largest rectangle that can be inscribed in a semicircle with radius 4 inches.
Area
The equation of a circle centered at the origin with radius 4 is ...
x² +y² = 16
The height (y) of a rectangle whose corner is on the circle will be ...
y = √(16 -x²)
The width of such a rectangle is 2x, so the area of it is ...
A = WH = (2x)√(16 -x²)
Maximum
The maximum area will be had when the derivative of the area with respect to x is zero.
A' = 2√(16 -x²) - 2x²/√(16 -x²) = 0
Multiplying by √(16 -x²)/2 gives ...
16 -x² -x² = 0
16 = 2x²
x = √(16/2) = 2√2
The corresponding area of the rectangle is ...
A = 2(2√2)√(16 -(2√2)²) = 16
The largest rectangle has an area of 16 square inches.
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Additional comment
As is often the case with rectangle optimization problems, the largest rectangle that can be inscribed in the quarter circle in each quadrant is a square. It has a diagonal of 4 inches, so an area of 4²/2 = 8 square inches.
The attachment shows the cubic curve representing the area of that square as a function of its width. As above, the rectangle area is double this value.