Answer:
To solve this problem, we need to find the dimensions of the rectangle and the radius of the semicircle, and then use those values to calculate the area of the entire shape.
Given information:
The perimeter of the rectangle is 64 cm.
The ratio of the length of AB to the length of BC is 3:1.
Step 1: Find the length of AB and BC.
Let the length of BC be x.
Then, the length of AB = 3x (since AB:BC = 3:1)
The perimeter of the rectangle = 2 × (length + width)
64 = 2 × (x + 3x)
64 = 8x
x = 8 cm
Therefore, the length of BC = 8 cm, and the length of AB = 3 × 8 = 24 cm.
Step 2: Find the radius of the semicircle.
The radius of the semicircle is equal to the width of the rectangle, which is BC = 8 cm.
Step 3: Calculate the area of the rectangle.
Area of the rectangle = length × width
Area of the rectangle = 24 cm × 8 cm = 192 cm²
Step 4: Calculate the area of the semicircle.
Area of the semicircle = (1/2) × π × r²
Area of the semicircle = (1/2) × π × (8 cm)²
Area of the semicircle = 32π cm²
Step 5: Calculate the total area of the shape.
Total area = Area of the rectangle + Area of the semicircle
Total area = 192 cm² + 32π cm²
Total area ≈ 192 + 100.53 = 292.53 cm²
Rounding to 3 significant figures, the area of the shape is 293 cm².