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MATHEMATICS - CASE STUDY SOLUTION


Solution Diagram

n
t
f(t)
0
0
5.00
1
π/5
4.68
2
2π/5
4.11
3
3π/5
4.11
4
4π/5
4.68
5
π
5.00
6
6π/5
4.68
7
7π/5
4.11
8
8π/5
4.11
9
9π/5
4.68
10
5
 

Recall that when a curve is given using the following parametric equations

     x = f(t) and y = g(t) for a ≤ t ≤ b,

the arc length of the curve is defined as

     

The ellipse describing the planet's orbit is

     x = 4 + 4cos(t) and y = 5 + 5sin(t)
     for 0 ≤ t ≤ 2π

The first step is to evaluate the derivatives of the integral as,

     

The arc length of the ellipse is given by:

     

Since there is no obvious technique to perform the closed form integration, a numerical method called the Simpson's Rule is used. According to the Simpson's Rule, an integration can be approximated as follows:

 

where n is even and Δt = (b - a)/n

In this case, let , n = 10, and Δt = (2π)/10 = π/5

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

The integration can then be evaluated to give the distance of a complete orbit as