The surface of a mirror is composed of a rectangular piece..

Hi swerve,

You seem to have doubled the the area of the two semicircular sections.

Here is how I did it:

Given:
Length = 9
Assume w = width;
Diameter = w, therefore, radius = w/2
Area of rectanlge/area of two semicircles = 9/pi

Calculations
Area of the rectangular section = 9*w = 9w
Combined area of two semicircles with radius w/2 = Area of one full circle = Pi* r^2 = Pi * [(w/2)^2] (note that the two is included in the squaring operation), so Area of circle = Pi*w^2/4
Since we know that dividing the area of the rectangle with the combined area of the semi-circles gives us 9/Pi, we can say:
$$\frac{9w}{\left[\frac{\pi\times w^2}{4}\right]}=\frac{9}{\pi}$$
As you can see, the numerator of both expressions is multiplied by 9 and the denominator of both expressions is multiplied by Pi, so we can get rid of them:
$$\frac{1\times w}{\left[\frac{w^2}{4}\right]}=\frac{1}{1}$$
We can now multiply both the numerator and denominator by 4 for the left-hand expression, which gives:
$$\frac{4w}{\left[w^2\right]}=1$$
Then multiply both sides of the equation by w^2:
$$4w\ =\ w^2\ ;\ w^2-4w\ =\ 0\ ;\ w\left(w-4\right)\ =\ 0;\ so\ w\ =\ \left\{0,4\right\}$$
Since we know that w is nonzero (otherwise the shape would not exist), w must equal 4, which is OA D.