Let's break down the shape in the image step by step to solve both parts of the problem.

Given Information:

• The shape consists of a quarter circle with two semicircles attached to it.
• The radius of the quarter circle is given as $x$.

a) Perimeter of the shape when $x = 4$:

1. Quarter circle's arc length: The perimeter contribution from the arc of the quarter circle is a quarter of the circumference of a full circle: $\text{Arc length of quarter circle} = \frac{1}{4} \times 2\pi x = \frac{\pi x}{2}$

2. Semicircles: There are two semicircles attached, and each has a radius of $\frac{x}{2}$. The circumference of each semicircle is half of the full circumference of a circle: $\text{Circumference of one semicircle} = \frac{1}{2} \times 2\pi \times \frac{x}{2} = \frac{\pi x}{2}$ Since there are two semicircles, their combined contribution to the perimeter is: $2 \times \frac{\pi x}{2} = \pi x$

3. Total Perimeter: Adding the perimeter contributions from the quarter circle's arc and the two semicircles: $\text{Total Perimeter} = \frac{\pi x}{2} + \pi x = \frac{3\pi x}{2}$ For $x = 4$, the perimeter becomes: $\text{Perimeter} = \frac{3\pi (4)}{2} = 6\pi$

Answer for part (a):

When $x = 4$, the perimeter of the shape is $6\pi$.

b) Expression for the perimeter in terms of $x$ and $\pi$:

From the explanation above, the general expression for the perimeter is: $\text{Perimeter} = \frac{3\pi x}{2}$

Would you like more details or have any further questions? Here are some related questions:

1. How is the formula for the circumference of a circle derived?
2. What is the difference between the arc length and the full circumference of a circle?
3. How can the area of this shape be calculated?
4. What happens to the perimeter if the semicircles' radius changes?
5. Can we apply the same perimeter formula to other shapes composed of circles?

Tip: When solving geometric problems, break complex shapes into simpler parts like arcs and semicircles to easily compute perimeter or area.