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.
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Here are the calculations for the shaded areas in Fig. 14.33, using = (22)/(7). All dimensions are in centimeters. a) Shape a: A square with a circular hole. Step 1: Calculate the area of the square. The side length of the square is 10 cm. Area of square = (10 cm)^2 = 100 cm^2 Step 2: Calculate the area of the circle. The diameter of the circle is 7 cm, so the radius is r = (7)/(2) cm. Area of circle = r^2 = (22)/(7) × ((7)/(2) cm)^2 = (22)/(7) × (49)/(4) cm^2 = (11 × 7)/(2) cm^2 = 38.5 cm^2 Step 3: Calculate the shaded area. Shaded area = Area of square - Area of circle = 100 cm^2 - 38.5 cm^2 = 61.5 cm^2 b) Shape b: An annulus (ring). Step 1: Calculate the area of the outer circle. The outer diameter is 14 cm, so the outer radius is R = (14)/(2) = 7 cm. Area of outer circle = R^2 = (22)/(7) × (7 cm)^2 = (22)/(7) × 49 cm^2 = 22 × 7 cm^2 = 154 cm^2 Step 2: Calculate the area of the inner circle. The inner diameter is 7 cm, so the inner radius is r = (7)/(2) cm. Area of inner circle = r^2 = (22)/(7) × ((7)/(2) cm)^2 = (22)/(7) × (49)/(4) cm^2 = (11 × 7)/(2) cm^2 = 38.5 cm^2 Step 3: Calculate the shaded area. Shaded area = Area of outer circle - Area of inner circle = 154 cm^2 - 38.5 cm^2 = 115.5 cm^2 c) Shape c: A quarter circle. Step 1: Calculate the area of the quarter circle. The radius of the quarter circle is r = 14 cm. Shaded area = (1)/(4) r^2 = (1)/(4) × (22)/(7) × (14 cm)^2 = (1)/(4) × (22)/(7) × 196 cm^2 = (1)/(4) × 22 × 28 cm^2 = 22 × 7 cm^2 = 154 cm^2 d) Shape d: Two semicircles joined. Step 1: Calculate the area of the two semicircles. Each semicircle has a diameter of 7 cm, so the radius is r = (7)/(2) cm. Two semicircles form one full circle. Shaded area = r^2 = (22)/(7) × ((7)/(2) cm)^2 = (22)/(7) × (49)/(4) cm^2 = (11 × 7)/(2) cm^2 = 38.5 cm^2 e) Shape e: A rectangle with two semicircles removed. Step 1: Calculate the area of the rectangle. The width of the rectangle is 14 cm and the height is 20 cm. Area of rectangle = 14 cm × 20 cm = 280 cm^2 Step 2: Calculate the area of the two semicircles. Each semicircle has a diameter of 14 cm, so the radius is r = (14)/(2) = 7 cm. Two semicircles form one full circle. Area of two semicircles = r^2 = (22)/(7) × (7 cm)^2 = (22)/(7) × 49 cm^2 = 22 × 7 cm^2 = 154 cm^2 Step 3: Calculate the shaded area. Shaded area = Area of rectangle - Area of two semicircles = 280 cm^2 - 154 cm^2 = 126 cm^2 f) Shape f: A rectangle with a semicircle removed. Step 1: Calculate the area of the rectangle. The width of the rectangle is 5 cm and the height is 9 cm. ✂️ _That answer was long and got cut off. Reply continue and I'll finish it._