The diameter of the larger circle in the figure shown above is x . What is the area of the shaded region, if the radius of the smaller circle is half of the radius of the larger circle?

The radius of a circle is half its diameter. Given that the diameter of the larger circle is x, the radius of the larger circle = \(\dfrac{x}{2}\)
Using the formula for area of a circle, the area of the larger circle = \(\pi * \left(\dfrac{x}{2}\right)^2 = \dfrac{\pi x^2}{4}\)

Given that the radius of the smaller circle is half of the radius of the larger circle, the radius of the smaller circle = \(\dfrac{x}{4}\)
The area of the smaller circle = \(\pi * \left(\dfrac{x}{4}\right)^2 = \dfrac{\pi x^2}{16}\)

To find the area of the shaded region, we subtract the area of the smaller circle from the area of the larger circle:
\(\dfrac{\pi x^2}{4} - \dfrac{\pi x^2}{16}
= \left(\dfrac{\pi x^2}{4}\right) * \left(\dfrac{4}{4}\right) - \dfrac{\pi x^2}{16}
= \dfrac{4\pi x^2}{16} - \dfrac{\pi x^2}{16}
= \dfrac{3\pi x^2}{16}
\)