１feet & 13.4 feet

Imagine that you are in an American football field. The distance from one end of the field to the other is 360 feet. Now, you have a rope which is as long as the distance between two ends of the field, 360 feet. You place this rope on the ground, letting two ends of the rope attach to the boundaries of the field. Next, you walk to the center of the field, grabbing the rope and lift it up. Here's the question, how high can you raise the rope with two ends of rope attaching to the boundary of the field? The answer is zero. Since the rope is as long as the distance between two ends of the field,  two ends of the rope will be dis-attached to the boundaries of the field as long as the rope is not entirely laid on the ground. This isn't too hard to imagine. However, things will be different when we lengthen the rope.

You may wander how significant the change is if we add extra length to the rope. Suppose that we add one extra feet to the rope, making two ends of the rope exceed each boundary of the field for 0.5 feet. Next, let's move to the center of the field, and lift the rope as high as possible until its ends attach perfectly to the boundaries of the field. If we measure the height of the center of the rope, we will find that the center of the rope is about 13.4 feet above the ground. This result is astonishing and seems impossible. However, it is totally reasonable and explainable with a mathematics equation.

Explanation: To understand how the equation works, we have to divide the football field into half. When we look at one of two parts, we  imagine that the rope and the ground form a right triangle together. According to the Pythagorean theorem, the square of hypotenuse(the side opposite to the right angle) is equal to the sum of the squares of the other two sides.  In this case, one side of the right triangle is 180 feet(the distance from the center of the field to one end of the field), and the hypotenuse of the triangle is 180.5 feet (half of the length of the rope). To know the length of the other side (the height of the center of the rope), we can use the equation: 180²+x²=180.5² 

If we solve X, we will know that the height of the center of the rope is about 13.4 feet.(See the picture below)

The result is amazing and unquestionable. The capacity becomes 13 feet only by adding one feet of extra length. That is the beauty of math.