Question

two poles, one 12 meters tall and one 30 meters tall, stand 40 meters apart. a worker attaches a piece of wire to the top of each pole and anchors the other ends of these wires to a stake in the ground somewhere between the two poles. how far from the 12-meter pole should the worker position the stake to use the least amount of wire?

Answer 1

To use the least amount of wire, the stake should be positioned approximately 27.5 meters from the 12-meter pole.

Step-by-step explanation:

To find the position where the worker should place the stake, we need to use the principle of least action. The principle states that the path taken by a system between two points will minimize the total action, which is a quantity related to the energy spent. In this case, the total action is related to the length of the wires. Since the wires form a triangle with the stake, we can use the Pythagorean theorem to find the length of the wires and minimize it by finding the height of the triangle.

Using the Pythagorean theorem, the height of the triangle is obtained as:

height = √(30² - 12²) = √(900 - 144) = √756 ≈ 27.5 meters

Therefore, the stake should be positioned approximately 27.5 meters from the 12-meter pole to use the least amount of wire.