**Hello. This is a crazy modelling i found in my maths work. i've been making assumptions and they are crap >_< plz help:**
A function f(x) is considered to be “smooth” in the interval [a,b] if f'(x) is continuous throughout the interval.

The arc length of a smooth function in the interval (that is, the length of the curve) is given by:

Totally Twak (TT) Industries manufactures exercise equipment. The company wants its new model of exercise bike to feature a “hill profile” that simulates riding the bike over two mountains (one small and one large). The approximate shape of the mountains involved is shown below:

When the profile is chosen, the bike alters the resistance to simulate riding uphill (greater resistance) and downhill (less resistance). Although the starting and finishing points are to be only 15 kilometres apart horizontally, the distance that is to be “ridden” is 20 kilometres because of the mountains.

TT hires you to design the profile (but not to calculate the sizes of the resistances). Your task is to find intervals of a minimum of 2 functions that can be placed together to closely match the shape of the hills. Both x and y for the functions are to be measured in kilometres. The following conditions must be met:

• the total arc length must be as close to 20 kilometres as possible (the range 19.5<=L<=20.5 is acceptable).

• at the point(s) where the intervals join up, the derivatives of the functions must not differ significantly so that a “smooth” ride is guaranteed.

Checklist

(i) The equations of the functions and the intervals that have been chosen.

(ii) A graph that clearly shows the profile.

(iii) Verification that the derivatives at the “join” point(s) do not differ greatly.

(iv) Verification that the total arc length lies in the required range. (All definite integrals may be performed on your graphics calculator.)

(v) A detailed explanation of how you approached the exercise (how you chose your functions, etcetera).