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C-102 Appendix C<br />
MINI PROJECT<br />
The Circumference of an Ellipse<br />
As you know, there is a simple expression for the circumference of a circle of<br />
radius a, namely, 2pa. However, there is no similar type of elementary expression<br />
for the circumference of an ellipse. (The circumference of an ellipse can<br />
be computed to as many decimal places as required using the methods of calculus.)<br />
Nevertheless, there are some interesting elementary formulas that allow<br />
us to approximate the circumference of an ellipse quite closely. Four such formulas<br />
follow, along with the names of their discoverers and approximate dates<br />
of discovery. Each formula yields an approximate value for the circumference<br />
of the ellipse (x 2 a 2 ) (y 2 b 2 ) 1.<br />
Discoverer Date Formula<br />
Giuseppe Peano 1887<br />
C 1 p c a b 1 2 11a 1b22 d<br />
Scrinivasa Ramanujan 1914<br />
C 2 p33(a b) 1(a 3b)(3a b)4<br />
Roger A. Johnson 1930<br />
C 3 p 2 3a b 22(a2 b 2 )4<br />
Roger Maertens 2000 C 4 4(a y b y ) 1y , where y ln 2<br />
ln(p2)<br />
y<br />
Approximation<br />
to<br />
Circumference<br />
C 1<br />
C 2<br />
C 3<br />
C 4<br />
Percentage<br />
Error<br />
x<br />
(a) In the figure on the left, the red ellipse has an eccentricity of 0.9, and the<br />
blue ellipse an eccentricity of 0.5. The outer circle has a radius of 10,<br />
which is equal to the semimajor axis of each ellipse. As preparation for<br />
parts (b) and (c), determine the equation of each ellipse.<br />
(b) For the red ellipse, use the approximation formulas given above to complete<br />
the table at left. Round the values of C 1 , C 2 , C 3 , and C 4 to six decimal<br />
places. Round the percentage errors to two significant digits. In computing<br />
the percentage errors, use the fact that the actual circumference, rounded to<br />
six decimal places, is 23.433941. Which of the four approximations for the<br />
circumference of this ellipse is the best? Which is worst? How does the circumference<br />
of this ellipse compare to that of the black circle in the figure?<br />
That is, using the given six-place value for circumference, compute the<br />
ratio of the circumference of the ellipse to the circumference of the circle.<br />
(c) Follow part (b) for the blue ellipse. The actual circumference here,<br />
rounded to six decimal places, is 29.349244. Compare your results for percentage<br />
errors to those in part (b), and summarize your observations<br />
(using complete sentences).<br />
(d) What is the circumference of the circle in the given figure? What value<br />
does each approximation formula yield for the circumference of the circle?
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