IBM Research | Ponder This | May 1998 solutions
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May 1998

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First, our assumptions:
  • The radius of the Earth denoted r, is 6,400 km.
  • The additional belt length, denoted d is 6 meters, or .006 kilometers.
If we pull the belt up at a point, we get a height h meters above the earth's surface. To determine this height , we begin by constructing a sketch of the earth with its center at c. The belt is tight against the Earth for most of its circumference, but at some point p leaves the Earth's surface, traveling up to the apex s, and returning to the Earth's surface at point q.

  • Draw the line segments cs and side cp, and let be the angle formed between them, expressed in radians. Also, let t denote the point at which the line segment cs intersects the earth. The arc that lies on the earth surface connecting p and t has length . (This is because the circumference of a circle with radius r is and an arc that spans radians spans of the entire circle.)
  • Next draw the line ps connecting p and s. This line is formed by using the portion of the belt that would lie on the arc connecting p and t plus one half of the slack in the belt; thus its length is
The length cp, is r, the radius of the earth

The line segment cs, has length r + h (radius of the earth plus the unknown height). Since the line segment ps is tangent to the Earth's surface at point p, the three lines form a right triangle.

Hide details for View an IBM Researcher's sketchView an IBM Researcher's sketch
View an IBM Researcher's sketch

We can use high school trigonometry (yes, it's really good for something) to compute the other angles and the length of the third side of the triangle.

We first find by solving for in the following equality.

This can then be solved accurately by computer computation using special mathematical software. Our software found that

How did the computer do it?
Hide details for Calculating Theta with a computerCalculating Theta with a computer
Calculating Theta with a computer

How does the computer find the value of ?

Basically the computer is used to find the root of the equation

Finding the root is accomplished by selecting and initial value of , evaulating the function f, and adjusting if the value is not zero. The program systematically searches through values of until a for which f is zero (that is, a root) is found.

This value of theta is then used, together with the trigonometry formula,

and some simple algebra to compute h = 402 meters.

What if I don't have a computer to calculate the value of

Hide details for Calculating Theta without a computerCalculating Theta without a computer

    Calculating Theta without a computer

    Calculating without a computer


    To solve without mathematical software requires some freshman calculus. Since the angle is small for large radius r (proving this is left as an exercise for the reader) the first two terms of the Taylor series at 0 for the tan and cos functions provide a good approximation for tan and cos , respectively.

    Solving we get


    Substituting this value of into the equality

    yields h = 401 meters.

Doubting Thomas? If you don't trust our handiwork, we will prove it.

Hide details for Proving the answerProving the answer
Proving the answer

Prove it with Pythagorus

Since we are working with a right triangle, the simplest way to check our answer is to plug it into the durable and ever-useful Pythagorean Theorem.

Substituting, we arrive at:

The left hand side is 40960000 + 5138.45 = 40965138.45 while the right hand side is 40965133, so the terms agree to 6 significant digits.

As an alternative to using the cosine approximation, the Pythagorean Theorem could have been used to solve for h once the value of theta and the length of the side ps had been determined.

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