We are given a rectangle $ABCD$ inscribed in a semicircle, with the side $CD$ lying on the diameter. The sides of the rectangle are $AB = 6$ and $BC = 7$, and the area of the semicircle is $a\pi$. We need to find $a$.

Step 1: Relating the Dimensions of the Rectangle to the Semicircle

Since $CD$ lies on the diameter of the semicircle, the length of $CD$ is the diameter of the semicircle. Using the Pythagorean theorem, we can find the diameter because $ABCD$ is a rectangle, and the diagonal $AC$ serves as the hypotenuse of a right triangle with sides $AB$ and $BC$.

$AC = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85}$

Thus, the diameter of the semicircle is $\sqrt{85}$.

Step 2: Finding the Radius of the Semicircle

The radius of the semicircle is half the diameter:

$r = \frac{\sqrt{85}}{2}$

Step 3: Area of the Semicircle

The area of a semicircle is given by:

$\text{Area of semicircle} = \frac{1}{2} \pi r^2$

Substitute $r = \frac{\sqrt{85}}{2}$:

$\text{Area of semicircle} = \frac{1}{2} \pi \left(\frac{\sqrt{85}}{2}\right)^2 = \frac{1}{2} \pi \frac{85}{4} = \frac{85\pi}{8}$

We are told that the area of the semicircle is $a\pi$, so we equate:

$\frac{85\pi}{8} = a\pi$

Step 4: Solving for $a$

Divide both sides by $\pi$:

$\frac{85}{8} = a$

$a = 10.625$

Thus, the value of $a$ is approximately $10.625$.

However, this value does not match any of the options given. Let's verify if rounding or other assumptions should be revisited. Based on this analysis, none of the listed options align perfectly with the calculated value.

Would you like more clarification, or should I look into an alternative approach? Here are 5 related questions for further exploration:

1. What is the significance of the diagonal in a rectangle inscribed in a semicircle?
2. How does the Pythagorean theorem help in determining the diameter of the semicircle?
3. What is the geometric relationship between a rectangle and a semicircle?
4. How can you derive the area formula for a semicircle?
5. What other properties of inscribed figures can you explore mathematically?

Tip: Always verify if the precision required for a solution matches the given options.