This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
To calculate the shaded area, we need to find the area of the circular segment in the circle with center R. The shaded area is the region bounded by the arc AB and the chord AB in the circle with center R. The formula for the area of a circular segment is: Area of segment = Area of sector - Area of triangle For the circle with center R: Radius, r = 35 cm Central angle, = ARB = 65.76^ Step 1: Calculate the area of the sector ARB. The formula for the area of a sector is ()/(360^) × r^2. A_sector = (65.76^)/(360^) × (35 cm)^2 A_sector = (65.76)/(360) × × 1225 A_sector ≈ 0.1826666667 × 1225 × 3.1415926535 A_sector ≈ 703.9510 cm^2 Step 2: Calculate the area of the triangle ARB. The formula for the area of a triangle given two sides and the included angle is (1)/(2)ab C. Here, a=b=r. A_triangle = (1)/(2) r^2 () A_triangle = (1)/(2) (35 cm)^2 (65.76^) A_triangle = (1)/(2) × 1225 × (65.76^) A_triangle = 612.5 × 0.9117601 A_triangle ≈ 558.4561 cm^2 Step 3: Calculate the shaded area (Area of segment). A_shaded = A_sector - A_triangle A_shaded ≈ 703.9510 cm^2 - 558.4561 cm^2 A_shaded ≈ 145.4949 cm^2 Rounding to two decimal places, the shaded area is 145.49 cm^2. The final answer is 145.49 cm^2.