1. **State the problem:** We have a semicircle with diameter $BC = 16$ cm and a triangle $ABC$ inside it. The angle at $A$ is $0.68$ radians. We need to find the area of the shaded region, which is the area of the semicircle sector minus the area of triangle $ABC$. 2. **Find the radius of the semicircle:** $$r = \frac{16}{2} = 8 \text{ cm}$$ 3. **Calculate the area of the semicircle:** The area of a full circle is $\pi r^2$, so the semicircle area is: $$\text{Area}_{semicircle} = \frac{1}{2} \times \pi \times 8^2 = \frac{1}{2} \times 3.142 \times 64 = 100.544 \text{ cm}^2$$ 4. **Calculate the area of the sector formed by angle $0.68$ radians:** The area of a sector is given by: $$\text{Area}_{sector} = \frac{1}{2} r^2 \theta = \frac{1}{2} \times 8^2 \times 0.68 = \frac{1}{2} \times 64 \times 0.68 = 21.76 \text{ cm}^2$$ 5. **Calculate the area of triangle $ABC$ using the formula:** $$\text{Area}_{triangle} = \frac{1}{2} r^2 \sin(\theta) = \frac{1}{2} \times 8^2 \times \sin(0.68) = 32 \times \sin(0.68)$$ Calculate $\sin(0.68)$: $$\sin(0.68) \approx 0.6293$$ So, $$\text{Area}_{triangle} = 32 \times 0.6293 = 20.1376 \text{ cm}^2$$ 6. **Calculate the shaded area:** The shaded area is the sector area minus the triangle area: $$\text{Shaded area} = 21.76 - 20.1376 = 1.6224 \text{ cm}^2$$ **Final answer:** $$\boxed{1.62 \text{ cm}^2}$$