(a)

Observe the diagram:

Radius of the semi circle is .

Width of rectangle is .

Length of rectangle .

Pythagorean theorem: .

From the triangle:

Substitute in the above equation.

Substitute in the above equation.

Area of the rectangle is product of length and width of rectangle.

Substitute and .

Area of the rectangle is .

(b)

Area of the rectangle is .

Double angle formula: .

Hence .

(c)

Area of the rectangle is .

Find the largest area of the rectangle differentiate the area equation with respect to .

Double angle formula: .

Equate to zero.

The area of the rectangle is maximum when the angle is .

(d)

Pythagorean theorem: .

Substitute in the above equation.

Area of rectangle is product of length and width of rectangle.

Find the largest dimension of the rectangle differentiate the above equation with respect to .

Equate to zero.

Substitute in the above equation.

The largest dimensions of the rectangle is .

(a) Area of rectangle .

(b) .

(c) The area of the rectangle is maximum when the angle is .

(d) The largest dimensions of the rectangle is .