The perimeter of a square is equal to twice the perimeter of a rectangle whose length is 8 cm and breadth is 7 cm. If the diameter of a semicircle is equal to the side of this square, find the circumference of the semicircle (use π ≈ 3.14).

Introduction / Context:
This problem connects the perimeter of a rectangle, the perimeter of a square, and the circumference of a semicircle. It tests your ability to link several geometric formulas and carefully follow through multiple steps, including working with a semicircle where the formula includes both the curved part and the diameter.

Given Data / Assumptions:

• Rectangle length = 8 cm
• Rectangle breadth = 7 cm
• Perimeter of square = 2 * perimeter of rectangle
• Diameter of semicircle = side of square
• Use π ≈ 3.14
• Circumference of semicircle = π * r + diameter

Concept / Approach:
First, find the perimeter of the rectangle. Then calculate the perimeter of the square, which is twice that value, and use it to find the side of the square. That side becomes the diameter of the semicircle. From the diameter, compute the radius and then use the formula for the total boundary length of a semicircle, which consists of half the circular arc plus the diameter.

Step-by-Step Solution:
Perimeter of rectangle P_rect = 2 * (length + breadth) = 2 * (8 + 7) = 2 * 15 = 30 cm Perimeter of square P_sq = 2 * P_rect = 2 * 30 = 60 cm Side of square s = P_sq / 4 = 60 / 4 = 15 cm Diameter of semicircle D = 15 cm Radius r = D / 2 = 15 / 2 = 7.5 cm Circumference of semicircle = (1/2) * (2 * π * r) + D = π * r + D Substitute values: = 3.14 * 7.5 + 15 3.14 * 7.5 = 23.55 Total = 23.55 + 15 = 38.55 cm ≈ 38.57 cm

Verification / Alternative check:
The key step is recognising that a semicircle boundary includes the straight diameter plus the curved half circle. If you mistakenly used only π * r without adding the diameter, you would get 23.55 cm which does not match any good option. Including the diameter gives a result very close to 38.57 cm, which is the listed option that best matches 38.55 cm.

Why Other Options Are Wrong:
55.12 cm and 42.51 cm: These values are larger than the correct circumference and arise from incorrect formulas or use of full circle instead of semicircle. 22.54 cm: Too small, corresponding roughly to π * r without diameter. 34.10 cm: Does not correspond to any correct computation from the given data.

Common Pitfalls:
Forgetting to double the rectangle perimeter when finding the square perimeter. Using radius instead of diameter for the relationship to the square side. Omitting the diameter term in the semicircle circumference formula.

Final Answer:
The circumference of the semicircle is approximately 38.57 cm