Answer:
(1)To wrap a rope exactly once around the Earth's equator, we need to calculate the circumference of the Earth. The circumference of the Earth at the equator is approximately 40,075 kilometers or 24,901 miles. Therefore, we would need a rope that is at least 40,075 kilometers or 24,901 miles long to wrap around the Earth's equator exactly once.
(2)If we lengthen the rope and suspend it an equal distance away from Earth at all points around the equator, we would need to add enough length to the rope to allow a person to walk underneath it. The height of an average person is approximately 1.7 meters or 5.6 feet. To calculate the length of the added rope, we can use the Pythagorean theorem. The distance from the center of the Earth to the rope is the radius of the Earth plus the height of the person, so we have:
r + h = 6371 km + 1.7 m = 6371.0017 km
Using the Pythagorean theorem, the length of the added rope would be:
l = 2√(r^2 - (r+h)^2) = 2√(6371.0017^2 - 6371^2) = 22.6 meters or 74.2 feet.
Therefore, we would need to add 22.6 meters or 74.2 feet of rope to allow a person to walk underneath the suspended rope.
(3)To wrap a rope around the circumference of a hula hoop with a radius of 70 centimeters, we need to calculate the circumference of the hula hoop. The circumference of a circle is given by the formula 2πr, where r is the radius. Therefore, the circumference of the hula hoop is:
C = 2πr = 2π(70 cm) ≈ 439.8 cm or 4.398 meters.
So, we would need a rope that is at least 4.398 meters long to wrap around the circumference of the hula hoop exactly once.
(4)If we lay the hula hoop and the rope on the floor with the rope arranged in a circle with the same center as the hula hoop, we would need to add enough length to the rope to allow a person to lie down on the floor with their toes pointing towards the hula hoop and their head towards the rope, but touching neither. Let's assume that the person's height is h meters. We need to find the radius of the circle that is formed by the rope plus the added length, such that the person can lie down without touching the hula hoop or the rope.
Using the Pythagorean theorem, we have:
r^2 = (r - 0.7)^2 + h^2
Simplifying and solving for r, we get:
r = √(h^2 + 0.49) + 0.7
The circumference of the circle formed by the rope and the added length is:
C = 2πr = 2π(√(h^2 + 0.49) + 0.7)
To find the length of the added rope, we subtract the circumference of the hula hoop from the circumference of the circle formed by the rope and the added length:
l = C - 2π(0.7) = 2π(√(h^2 + 0.49) + 0.7) - 1.4π
Simplifying, we get:
l = 2π(√(h^2 + 0.49) - 0.3)
Explanation: