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 C<sub>1</sub> and C<sub>2</sub> 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.
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 C<sub>1</sub> and C<sub>2</sub> 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.
