(a)

Observe the diagram. \ \

Radius of the semi circle is .

Width of rectangle is .

Length of rectangle .

From Pythagorean theorem  : .

From 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 .

To 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)

From Pythagorean theorem : .

Substitute in the above equation. \ \

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

To 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 .