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's how to solve this problem: Given: Radius of each circle, r = 7 cm. Use = (22)/(7). Part (a): Find the perimeter of the shaded part. Step 1: Identify the components of the perimeter. The perimeter of the shaded part consists of arcs from the circles. There are 4 quarter-circle arcs (from the top-left, top-right, bottom-left, and bottom-right shaded regions) and 2 semi-circle arcs (from the middle-left and middle-right shaded regions). Step 2: Calculate the length of one quarter-circle arc. The circumference of a full circle is 2 r. Length of one quarter-circle arc = (1)/(4) × 2 r = (1)/(2) r. Step 3: Calculate the length of one semi-circle arc. Length of one semi-circle arc = (1)/(2) × 2 r = r. Step 4: Calculate the total perimeter. Total perimeter = 4 × (length of a quarter-circle arc) + 2 × (length of a semi-circle arc) Perimeter = 4 × ((1)/(2) r) + 2 × ( r) Perimeter = 2 r + 2 r = 4 r Substitute the given values for r and : Perimeter = 4 × (22)/(7) × 7 Perimeter = 4 × 22 = 88 cm The perimeter of the shaded part is 88 cm. Part (b): Find the area of the shaded part. Step 1: Identify the components of the area. The shaded part consists of 4 quarter-circles (from the top-left, top-right, bottom-left, and bottom-right shaded regions) and 2 semi-circles (from the middle-left and middle-right shaded regions). Step 2: Calculate the area of one quarter-circle. The area of a full circle is r^2. Area of one quarter-circle = (1)/(4) r^2. Step 3: Calculate the area of one semi-circle. Area of one semi-circle = (1)/(2) r^2. Step 4: Calculate the total area. Total area = 4 × (area of a quarter-circle) + 2 × (area of a semi-circle) Area = 4 × ((1)/(4) r^2) + 2 × ((1)/(2) r^2) Area = r^2 + r^2 = 2 r^2 Substitute the given values for r and : Area = 2 × (22)/(7) × (7)^2 Area = 2 × (22)/(7) × 49 Area = 2 × 22 × (49)/(7) Area = 2 × 22 × 7 Area = 44 × 7 = 308 cm^2 The area of the shaded part is 308 cm^2. Last free one today — make it count tomorrow, or type /upgrade for unlimited.