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MATHEMATICS - CASE STUDY SOLUTION

   

Recall that when a curve is given in polar coordinates r = f(θ) for a ≤ θ ≤ b, the arc length of a polar curve is determined as follows:

     

For the given function r = 100cos2θ for 0 ≤ θ ≤ π/2, the derivative dr/dθ is given by

     

The arc length of the ski track is determined as:

     

     
n θ f(θ)
0 0 1.00
1 π/20 1.13
2 π/10 1.43
3 3π/20 1.72
4 π/5 1.93
5 π/4 2.00
6 3π/10 1.93
7 7π/20 1.72
8 2π/5 1.43
9 9π/20 1.13
10 π/2 1.00
 

Since there is no obvious technique to perform the integration, Simpson's Rule is used. According to the Simpson's Rule, an integration can be approximated as,

 

where n is even and Δθ = (b - a)/n.

In this case, , n = 10, and Δθ = (π/2)/10 = π/20.

The values of f(θ) subject to different values of θ are summarized in the table on the left.

The integration can then be evaluated to give the length of the ski track as