Final answer:
The value of the shaded area in the circle with a radius of 3.8 cm and an included angle of 80° is approximately 10 cm², which is calculated using the area of a sector formula, but none of the given options match this result.
Step-by-step explanation:
The value of the shaded area in the circle with a radius of 3.8 cm and an included angle of 80° can be found by calculating the area of the sector formed by the 80° angle. The area of a sector (A) is a fraction of the area of the entire circle, proportional to the angle of the sector out of 360°. So, the formula to calculate the area of a sector is A = (θ/360) × πr², where θ is the central angle in degrees and r is the radius.
For a circle with a radius (r) of 3.8 cm and an angle (θ) of 80°, we use the formula:
A = (80°/360°) × π × (3.8 cm)²
Performing the calculation:
A = (80/360) × π × 14.44 cm²
A = (2/9) × π × 14.44 cm²
A = (2/9) × 3.1415927 × 14.44 cm²
A ≈ 10.05 cm²
Since the radius is given to two significant figures, the answer should also be in two significant figures:
A ≈ 10 cm²
However, looking at the multiple choice options provided in the question, it seems none of the answers match the calculated value of 10 cm². Re-checking the provided answers, option (b) 15.19 cm² seems to be the closest. This discrepancy could be due to rounding or an error in the provided options. Therefore, with provided options, the best choice would be but none of them are correct based on our calculation.