Question

Find the area of the crescent-shaped region (called a lune) bounded by arcs of circles with radii r = 2 and r = 14

Answer 1

The area of the crescent-shaped region (lune) bounded by two circles with radii of 2 and 14 is calculated by subtracting the area of the smaller circle from the area of the larger circle, resulting in the area of the lune being 192π square units.

Step-by-step explanation:

To find the area of the crescent-shaped region (lune) bounded by arcs of circles with radii r = 2 and r = 14, we need to calculate the area of each circle and then subtract the area of the smaller circle from the area of the larger one. The area of a circle is calculated by the formula A = πr². For the larger circle with r = 14, the area is A = π(14²). For the smaller circle with r = 2, the area is A = π(2²). The area of the lune is the difference between these two areas.

Simplified, this yields:

• Area of larger circle: π(14²) = 196π square units
• Area of smaller circle: π(2²) = 4π square units
• Area of the lune: 196π - 4π = 192π square units

The final area of the lune is 192π square units, which can then be approximated using 3.14159 for π, if required, but should be left in terms of π for exact answers unless otherwise specified.

Answer 2

the complete question in the attached figure

Let

A1-------> half of the area of the blue circle
A2-------> area of the circular segment formed by the diameter of the blue circle treated as a chord of the red circle

A3---------- > area of the crescent-shaped region

A3=A1-A2

A1=pi*2²/2=6.28

Area of ​​the circular sector

if 360 º corresponds to an area of pi*14²

2*arc sin(2/14)------------------- X

X=16.43*pi*14²/360---------- > X=28.10

Triangle area:

------- >(4/2)*√(14²-2²=27.72

Area of ​​the circular segment A2=Area of ​​the circular sector-Triangle area

A2=28.10-27.72=0.38

A3= A1-A2=6.28-0.38=5.90 units²

the answer is 5.90 units²