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# Finding the shortest length of a track

Fri 25 Sep, 2015 09:25 pm
A new train track is to be constructed. There is an existing straight train track, 10ā2km long, running in a north easterly direction from Riemann's Pier. The new track has to continue smoothly from this existing track and continue until Newtown's Jetty which is 30 kms due east of Riemann's Pier. Newton's Jetty and Riemann's Pier lie on the edge of the nortern bank of Euler's River which runs straight in course. Due to budgetary constraints the track must be as short as possible. Furthermore the area between th river bank and the tracks both existing and proposed must be exactly 250 square kilometres.

a) On the xy plane model the existing track with the function f(x)=x which strarts at the origin (Riemann's Pier) and continues to the point (10,10). Determine g(x), giving reasons which models the proposed track starting at (10,10) and continues to (30,0) representing Newton's Jetty. The norther bank of Euler's River is of course the x-axis. The continuous function g has the following properties: g(10) = 10, g(30) = 0, it is non-negative, the area between g and the x-axis from x = 10 to x = 30 is 200, and g must be 'smooth' from x = 10 to x = 30.

b) Calculate the length of the proposed section of train track. The arc length for your fucntion g must be found using L = ā«[10,30] ā(1 + [gā(x)]2) dx
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