A wire in the shape of a semicircle with a radius of 5 meters is being pushed along the ground. If it is pushed at a constant rate of 2 meters per second, find the rate at which the enclosed area between the wire and the ground is changing.
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Let's assume the wire starts at point A on the ground and is being pushed clockwise. Let O be the center of the semicircle.
When the wire is at point A, the enclosed area is 0. As the wire moves, it sweeps out a sector of a circle, which represents the change in the enclosed area. Let's call this sector AOB.
The radius, OB, is always 5 meters.
The arc length, AB, is the distance the wire travels along the semicircle, which is equal to the length of the semicircle. The length of a semicircle is (1/2) × circumference of the circle. In this case, the length is (1/2) × (2π)(5) = 5π meters.
The angle AOB is equal to the angle at the center of the circle, which is formed by the wire and the two radii. This angle is always 90 degrees.
The area of the sector, AOB, is (1/2) × OB² × angle AOB. In this case, the area is (1/2) × (5)² × (90) = 112.5 square meters.
Therefore, the rate at which the enclosed area between the wire and the ground is changing is 112.5 square meters per second.
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