When the size of a three-dimensional structure is doubled, the volume increases by 23, or eight times, but the weight merely doubles. When its size is tripled, volume goes up by 33 or 27 times. Again, the weight only triples. A modest increase in size therefore results in a huge increase in volume, without a concurrent weight penalty.
Assume the propellant-carrying capacity of a spaceplane is represented by two volumes, a right circular cylinder and two hollow triangular wings. The two triangular wings can be mathematically combined to create a single rectangular volume that can be calculated easily. The formula will have two terms, one for the cylinder and one for the wings. We will need the radius and length of the fuselage, as well as the length, width, and height of the wing.
This simplified formula reveals that if the dimensions (length l, width w, height h, and radius r) are increased, the volume will once again go up by the cube of that increase. The first term guarantees this result because length and the square of the radius are multiplied together, and the second term ensures this result because length, width, and height are all multiplied together.
The unmistakable conclusion is that if a vehicle, say, four times the size of the X-15 had been constructed, it would have had a propellant capacity of 4 cubed, or 64 times as much. Of course, the fully fueled vehicle would also have weighed 64 times as much, making it 64 times more difficult to accelerate into space.
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