Final answer:
The semicircle has a larger area than the square formed by the same wire. The area of the semicircle is approximately 76.97 cm², while the area of the square is approximately 38.48 cm². The difference in their areas is about 38.49 cm².
Step-by-step explanation:
Comparing Areas of Semicircle and Square
The problem involves a piece of wire bent into the shape of a semicircle with a bounding diameter, which is then straightened and formed into a square.
Given that the diameter of the semicircle is 14cm, we can easily find the radius, which is half the diameter, so 7cm.
The formula for the area of a semicircle is ½πr².
Therefore, the area of the semicircle is ½π(7²) = 49π/2 cm².
To form the square, the same length of the wire must be equal to the perimeter of the square.
The length of the wire that forms the semicircle, including its diameter, is πr + 2r or πd. Substituting r for 7cm, we get a perimeter (length of wire) of 14π cm.
Dividing this length by 4 to find the side of the square, we get 14π/4 = 3.5π cm per side.
The area of the square is (3.5π)² cm².
Now, to determine which shape has a larger area, we need to compare 49π/2 cm² with (3.5π)² cm².
If we calculate these values, the semicircle has an area of approximately 76.97 cm² and the square has an area of approximately 38.48 cm².
Hence, the semicircle has a larger area.
To find the difference between them, we subtract the area of the square from the area of the semicircle:
approximately 76.97 cm² - 38.48 cm² = 38.49 cm².