Show that the area of the shape can be written as $A=200r-r^2 (2+ \pi/2)$

Question

A $\rm200\,m\,$ fence is to placed around a lawn of this shape. We know that $x$ in terms of $r$ :

$$x=100-\dfrac{(2+\pi)r}2$$

How do I show that the area of the lawn, $A$, can be written as:

$$A= 200r-r^2\left(2+\left(\dfracπ2\right)\right) $$

Answer 1

The area is equal to: $$\eqalign{&x\cdot 2r+\text{half the area of the circle with radius $r$.} \\ &=2r\left[100-\dfrac{(2+\pi)r}2\right]+\dfrac12(\pi r^2).}$$ Then you just have to simplify everything as much as possible and voilà!