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
Okay, Nmk, let's calculate the shaded areas for each figure. We'll use = 3.1. a) Step 1: Calculate the area of the outer rectangle. The length is 10 cm and the width is 6 cm. Area_outer = 10 cm × 6 cm = 60 cm^2 Step 2: Calculate the area of the inner (unshaded) rectangle. The length is 6 cm and the width is 4 cm. Area_inner = 6 cm × 4 cm = 24 cm^2 Step 3: Calculate the shaded area. The shaded area is the area of the outer rectangle minus the area of the inner rectangle. Area_shaded = Area_outer - Area_inner = 60 cm^2 - 24 cm^2 = 36 cm^2 b) Step 1: Calculate the area of the rectangular part. The length is 12 cm and the width is 12 cm. Area_rectangle = 12 cm × 12 cm = 144 cm^2 Step 2: Calculate the area of the semicircular part. The diameter of the semicircle is 12 cm, so the radius is r = (12)/(2) = 6 cm. Area_semicircle = (1)/(2) r^2 = (1)/(2) × 3.1 × (6 cm)^2 Area_semicircle = (1)/(2) × 3.1 × 36 cm^2 = 3.1 × 18 cm^2 = 55.8 cm^2 Step 3: Calculate the total shaded area. The total shaded area is the sum of the rectangle's area and the semicircle's area. Area_total shaded = Area_rectangle + Area_semicircle = 144 cm^2 + 55.8 cm^2 = 199.8 cm^2 c) Step 1: Calculate the area of the square. The side length of the square is 4 cm + 4 cm = 8 cm. Area_square = (8 cm)^2 = 64 cm^2 Step 2: Calculate the area of the unshaded quarter circle. The radius of the quarter circle is 4 cm. Area_quarter circle = (1)/(4) r^2 = (1)/(4) × 3.1 × (4 cm)^2 Area_quarter circle = (1)/(4) × 3.1 × 16 cm^2 = 3.1 × 4 cm^2 = 12.4 cm^2 Step 3: Calculate the shaded area. The shaded area is the area of the square minus the area of the quarter circle. Area_shaded = Area_square - Area_quarter circle = 64 cm^2 - 12.4 cm^2 = 51.6 cm^2 d) Step 1: Calculate the area of the outer circle. The diameter of the outer circle is 20 cm, so the radius is R = (20)/(2) = 10 cm. Area_outer = R^2 = 3.1 × (10 cm)^2 Area_outer = 3.1 × 100 cm^2 = 310 cm^2 Step 2: Calculate the area of the inner (unshaded) circle. The diameter of the inner circle is 10 cm, so the radius is r = (10)/(2) = 5 cm. Area_inner = r^2 = 3.1 × (5 cm)^2 Area_inner = 3.1 × 25 cm^2 = 77.5 cm^2 Step 3: Calculate the shaded area. The shaded area is the area of the outer circle minus the area of the inner circle. Area_shaded = Area_outer - Area_inner = 310 cm^2 - 77.5 cm^2 = 232.5 cm^2 That's 2 down. 3 left today — send the next one.