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from my point of view, we come up with some thing like this (pardon my crap skills with MSpaint):



What we need to figure out is how much the Earth curves. I propose that we find the length of the bottom arc by using the radii, which we know, and by using some math thing that I forgot.

Also, once we find the lenght of the bottom side/arc, we can find how wide the triangle will have to be and how tall.

I really don't have the math skills to do any of this, but I know that some of you do, and it seems much simpler than finding all of those variables etc.

what if the cable passes directly over a skyscraper???
the cable will then have to be longer than the earth's circumference to reach around the earth.

Kudos to germanboyno2 for a very original solution to this problem, although his answer is off by about 10%. This is actually rather impressive. As mentioned previously, the problem does require a bit of calculus.

Consider the triangle which is half the kite in his construction. This will be a right triangle. In other words, pull up the cable at a single point, which is possible until the raised segment is a tangent. Thus, we have three sides of a triangle defined by the pulled-up point, the point of tangency, and the center of the Earth: two sides are R and R+h, making some angle φ, and the third side is 1/2 meter more than the circular arc inside the triangle formed by the Earth's surface, i.e., Rφ+1/2. From basic trigonometry, tan φ = φ+1/(2R) and h = R(sec φ - 1). The problem is: what is φ if R = 6.3710e6m (Earth radius)?

Fortunately, we can expect φ to be small. Using the Maclaurin series for tangent, we have tan φ ~ φ + φ³/3 + O(φ^5) ~ φ + 1/(2R), or φ ~ [3/(2R)]^(1/3) = 6.1749e-3 (one can check this to be a very good approximation to a solution of tan φ = φ + 1/(2R)). This immediately gives h = 121.5m. If your calculator loses precision when handling such small numbers (sec φ is almost 1), one can apply the Maclaurin series for secant: h = R[sec φ - 1] ~ (R/2)φ² = 121.5m.

One can indeed stack some elephants with just a meter of slack.