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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You're on a roll — here are the perimeters of the shapes: 1) This shape is a semicircle. Step 1: Calculate the length of the curved part. The diameter (d) is 14 cm. The curved part is half the circumference of a circle. The formula for the circumference of a circle is C = d. We are given = (22)/(7). Length of curved part = (1)/(2) d = (1)/(2) × (22)/(7) × 14 cm Length of curved part = 11 × 2 cm = 22 cm Step 2: Calculate the total perimeter. The perimeter is the sum of the straight side (diameter) and the curved part. Perimeter = 14 cm + 22 cm Perimeter = 36 cm The perimeter of shape 1 is 36 cm. 2) This shape consists of a quarter circle, a semicircle, and a straight line segment. Step 1: Identify the dimensions. The straight line segment is 7 cm. This is the radius of the quarter circle (r_1 = 7 cm) and also the diameter of the semicircle (d_2 = 7 cm). Step 2: Calculate the arc length of the quarter circle. Arc length of quarter circle = (1)/(4) (2 r_1) = (1)/(2) r_1 = (1)/(2) × (22)/(7) × 7 cm Arc length of quarter circle = 11 cm Step 3: Calculate the arc length of the semicircle. Arc length of semicircle = (1)/(2) ( d_2) = (1)/(2) × (22)/(7) × 7 cm Arc length of semicircle = 11 cm Step 4: Calculate the total perimeter. Perimeter = (arc length of quarter circle) + (arc length of semicircle) + (straight line segment) Perimeter = 11 cm + 11 cm + 7 cm Perimeter = 29 cm The perimeter of shape 2 is 29 cm. 3) This shape is a rectangle with a semicircle cut out from the top. Step 1: Identify the lengths of the straight sides. The bottom side is 9 cm. The two vertical sides are 5 cm each. The two small straight segments on the top are 1 cm each. Step 2: Determine the diameter of the inner semicircle. The total length of the top edge of the rectangle is 9 cm. The two small straight segments are 1 cm each. So, the diameter of the semicircle (d) is 9 cm - 1 cm - 1 cm = 7 cm. Step 3: Calculate the arc length of the inner semicircle. Arc length of semicircle = (1)/(2) d = (1)/(2) × (22)/(7) × 7 cm Arc length of semicircle = 11 cm Step 4: Calculate the total perimeter. Perimeter = (bottom side) + (two vertical sides) + (two top straight segments) + (arc length of inner semicircle) Perimeter = 9 cm + (2 × 5 cm) + (2 × 1 cm) + 11 cm Perimeter = 9 cm + 10 cm + 2 cm + 11 cm Perimeter = 32 cm The perimeter of shape 3 is 32 cm. 4) This shape is composed of four semicircles. Step 1: Identify the diameter of each semicircle. The dashed lines indicate a central square with sides of 7 cm. Each semicircle has a diameter (d) equal to the side of this square, so d = 7 cm. Step 2: Calculate the arc length of one semicircle. Arc length of one semicircle = (1)/(2) d = (1)/(2) × (22)/(7) × 7 cm Arc length of one semicircle = 11 cm Step 3: Calculate the total perimeter. Since there are four such semicircles, the total perimeter is four times the arc length of one semicircle. Perimeter = 4 × 11 cm Perimeter = 44 cm The perimeter of shape 4 is 44 cm. 5) This shape is a rectangle with two semicircles on its ends. Step 1: Identify the dimensions. The '7' indicates the length of the straight rectangular part is 7 cm. The vertical dashed lines show that the diameter of each semicircle (d) is also 7 cm. Step 2: Calculate the total perimeter. The perimeter consists of two straight sides and the circumference of one full circle (formed by the two semicircles). Perimeter = (2 × length of straight side) + ( d) Perimeter = (2 × 7