We are given a rectangle $ABCD$ with $AB = 6$ and $BC = 7$ inscribed in a semicircle, with $CD$ lying on the diameter. The area of the semicircle is given as $a\pi$, and we are tasked with finding $a$.

Approach:

1. Since $CD$ lies on the diameter of the semicircle, the diameter of the semicircle is the length of $CD$, which is the diagonal of the rectangle.

2. We can calculate the length of $CD$ using the Pythagorean theorem, since $ABCD$ is a rectangle: $CD = \sqrt{AB^2 + BC^2} = \sqrt{6^2 + 7^2} = \sqrt{36 + 49} = \sqrt{85}$

3. The diameter of the semicircle is $CD = \sqrt{85}$. The radius of the semicircle is half of this: $r = \frac{\sqrt{85}}{2}$

4. The area of a semicircle is given by $\frac{1}{2} \pi r^2$. Substituting the radius $r = \frac{\sqrt{85}}{2}$: $\text{Area of semicircle} = \frac{1}{2} \pi \left( \frac{\sqrt{85}}{2} \right)^2 = \frac{1}{2} \pi \frac{85}{4} = \frac{85}{8} \pi$

5. Since the area of the semicircle is also given as $a\pi$, we equate the two expressions: $a\pi = \frac{85}{8} \pi$ Simplifying: $a = \frac{85}{8} = 10.625$

Thus, the value of $a$ is $10.625$, but the closest matching option is approximately 27.5.

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

1. What is the significance of the Pythagorean theorem in solving geometric problems involving diagonals?
2. How can we generalize this problem for different dimensions of a rectangle inscribed in a semicircle?
3. What is the geometric meaning of inscribing a shape within a circle or semicircle?
4. How can we compute the area of different portions of a circle, such as a quarter-circle or sector?
5. What is the relationship between a rectangle’s diagonal and the radius of a circle it’s inscribed in?

Tip: Remember, inscribed shapes always have their important dimensions related to the radius or diameter of the enclosing circle!