(a) The radius of the small sphere is 2 centimeters.
(b) The measure of the angle XAB is approximately 36.870°.
(c) The area of the shaded region is approximately 3.107 square centimeters.
How to analyze a geometric system
Herein we have a geometric system formed by three big semicircles and a small circle, whose geometric diagram is attached below. (a) The relationship between the semicircles and the circle is represented by two formulae:
8² + h² = (8 + r)² (1)
h + r = 8 (2)
Where r is the radius of the small circle.
Then, we solve the system:
8² + (8 - r)² = (8 + r)²
64 + 64 - 16 · r + r² = 64 + 16 · r + r²
32 · r = 64
r = 2
The radius of the small sphere is 2 centimeters.
(b) The measure of the angle XAB is found by trigonometric relations:
cos m ∠ XAB = OA / AX
cos m ∠ XAB = 8 / 10
m ∠ XAB ≈ 36.870°
The measure of the angle XAB is approximately 36.870°.
(c) The area of the shaded region is the area of the triangle minus the area of the three circular sections:
A = 0.5 · (16 cm) · (10 cm) · sin 36.870° - (36.870° /180) · π · (8 cm)² - 0.5 · (106.260° / 180) · π · (2 cm)²
A ≈ 3.107 cm²
The area of the shaded region is approximately 3.107 square centimeters.
To learn more on triangles: https://brainly.com/question/2773823
#SPJ1
