find best route for running power cable to island

[Nigell]

I need help with the following questions, answering any of them would be much appreciate, as you can see they are very hard and i doubt anyone could solve the latter ones. Question 1 and 2 are easy, what i would like to see is if anyone can solve questions 3,4. Dont worry about question 5 So go on take up the challenge.
Some of what i have done and what others have contributed can be found via this link:
http://www.freemathhelp.com/forum/viewtopic.php?p=59815

Question 3

a) Write down expressions for C₁ and C₂ in terms of y, and find $\displaystyle \frac{dC_1}{dy}$ and $\displaystyle \frac{dC_2}{dy}$ in terms of y.

b) Express $\displaystyle \frac{dC_1}{dy}$ and $\displaystyle \frac{dC_2}{dy}$ in terms of $\displaystyle \theta$ and $\displaystyle \phi$ respectively.

c) Hence express $\displaystyle \frac{dC}{dy}$ in terms of $\displaystyle \theta$ and $\displaystyle \phi$, and show that the minimum cost occurs when $\displaystyle a\sin{\theta}\,=\,b\sin{\phi}$ where a and b are constants. Give an interpretation of these constants.

d) Show that $\displaystyle \theta$ and $\displaystyle \phi$ are also related by the equation $\displaystyle c\tan{\theta}\,+\,d\tan{\phi}\,=\,e$ where c, d, and e are constants.

To find the values of $\displaystyle \theta$ and $\displaystyle \phi$ corresponding to the minimum cost, we must solve the equations in Question 3 c and d simultaneously. Since they are nonlinear, one approach is to use a suitable technology approach.

Question 4

a) Use the equation in Question 3 b to express $\displaystyle \phi$ in terms of $\displaystyle \theta$ and substitute this expression in the equation in Question 3 c to obtain an equation in $\displaystyle \theta$.

b) Use a suitable technology approach to find approximately the value of $\displaystyle \theta$ that minimizes C, and hence find:

. . . .i. the corresponding value of $\displaystyle \phi$
. . . .ii. the location of Y to the nearest metre along the shoreline
. . . .iii. the best route for running the power cable to the island

Component 3

Question 5

Investigate possible effects in a change in the cost of running the cable underwater and/or on land.

 

[stapel]

What have you tried? How far have you gotten?

Please be specific. Thank you.

Eliz.

P.S. The third of your images has been replaced with text, for ease of readability. If the first graphic could be enlarged a bit and the second reduced (at least in width, perhaps by typing out the textual portions), that would be very helpful. Thank you.

 

[Nigell]

What i have done on it so far and what others have done on it can be find via this link.
http://www.freemathhelp.com/forum/viewtopic.php?p=59815
But that is only the first part of the equation.