The figure above shows the design of a theatre stage which is in the shape of a semicircle attached to a rectangle - English General - NSC Mathematics - Question 10 - 2017 - Paper 1

To find the area $A$ of the stage, we need the area of both the rectangle and the semicircle:

A(r) = ext{length} imes ext{breadth} + rac{1}{2} ext{area of circle}

The length of the rectangle can be defined as $2r$, thus the area becomes:

A(r) = (2r)(30 - r - rac{1}{2}r ext{π}) + rac{1}{2}( ext{π}r^2)

This simplifies and can be set for maximization:

A(r) = 60r - 2r^2 - rac{1}{r}r^2 ext{π}

To find the maximum area, we take the derivative and set it to 0:

$A'(r) = 60 - 4r - ext{π}r = 0$

Solving for $r$ gives:

r = rac{60}{4 + ext{π}}

Calculating this:

$r ext{ approximately equals } 8.40 ext{ m}$