, 2024

To calculate the shaded area, we add the area of the rectangle
and the areas of the four semi-circles, and subtract the area of the
larger circle.

Since the rectangle is $6$ by $8$, its area is $6 \times 8 = 48$. The two semi-circles of diameter $6$ together form a complete circle of diameter $6$, or radius $3$. The combined area of these semi-circles is $\pi \times 3^2$ or $9\pi$. The two semi-circles of diameter $8$ together form a complete circle of diameter $8$, or radius $4$. The combined area of these semi-circles is $\pi \times 4^2$ or $16\pi$. Since the larger circle passes through the four vertices of the rectangle, the diagonal of the rectangle is its diameter. (This is because the diagonal subtends an angle of $90\degree$ at each of the other vertices and so is a diameter.) The length of the diagonal is $\$\sqrt{6^2}$ +
8^2} = $\sqrt{100}$ = 10, and so the radius of the larger circle is

5$, and so its area is$$\pi \times$ 5^2 = $25\pi$. Finally, this means that the area of the shaded region is

48 + 9 $\pi$ + 16 $\pi$ - 25 $\pi$$which equals$48$. The closest integer to$48$is$48$.