Three quarter-circle arcs enclose an area as shown.

The largest is part of a unit circle, the smaller two are of a size that varies by a parameter $a$ . The vertical line is $x=a$ . The arc to the left of this line has radius $a$ and to the right radius $1-a$ .

The area of the figure is smallest when $a=0$ and largest when $a=1$ .

When $a$ is increased at a constant rate from $0$ to $1$ , when does the area increase most quickly?

Only when
$a=0$
Always constant
Only when
$a=0.5$
Only when
$a=1$
Only when
$a=0$
or
$a=1$

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The shape is the sum of two arcs less a third. All the arcs are similar so their areas are in proportion to $1^{2}$ , $a^{2}$ , and $(1-a)^{2}$

$1+a^{2}-{1-a}^2=2a$ so the area increases linearly with $a$ . In other words it is $\boxed{\text{Always constant}}$

You can play with the picture via this link: https://www.desmos.com/calculator/pw7c298i0s