Parametric curves, optimization, and 3D force vectors

[Danielll]

Hi guys I'm trying to prepare for an exam coming up (that time again :( ) and I've found a few problems in my textbook I don't really know how to do. I really need to do well on this so if anyone could help me out i'd really appreciate it! :D

Question1>
A curve has parametric equations: x = 2t + sin2t, y = cos2t
a) Show that dy/dx = -tant
b) Find d^2y/dx^2 and d^2x/dy^2 in terms of variable t, and demonstrate that d^2y/dx^2 != (does not equal) (1/d^2x/dy^2)

Question2>
A fruit juice manufacturer wishes to design a carton that has a square face. The carton is to contain 1 litre of joice and is made from a rectangular sheet of waxed cardboard by folding it into a rectangular tube and sealing down the edge and then folding and sealing the top and bottome. To make the carton airtight and robust for handling, an overlap at least 0.5cm is needed.

a) Show that the amount A(h) cm^2 of card used is given by
A(h) = (h + (1000/h^2) + 1)(2h + (2000/h^2) + 0.5)
b) Verify that
A(h) = 2(h + (1000/h^2) + 5/8)^2 - 9/32.
And hence find the value of hm of h which minimizes A(h).
http://members.optusnet.com.au/h0ax/pics/temp/2.jpg


Question3>
A telegraph pole OP has three wires connected to it at P. The other ends of the wires are connected to houses at A,B and C. Axes are set up as shown below. The points relative to these axis, with distances in metre, are OP = 8k, OA = 20j+6k, OB = -i - 18j +10k and OC = -22i + 3j + 7k. The tension in each wire is 900N. Find the total force acting at P. A tie cable at an angle of 45 degrees is connected to P and fixed in the ground. Where should the ground fixing be placed, and what isthe tension requred to ensure a zero horizontal force at P?
http://members.optusnet.com.au/h0ax/pics/temp/1.jpg

Thanks guys.

 

[dextercioby]

What did u do so far?The first looks kinda simple.

Daniel.

 

[p53ud0 dr34m5]

yes, the first one looks fairly simple. what do you know about parametric equations and their derivatives?

 

[Danielll]

i've worked the first one out but still stuck with what to do on the other ones?

can anyone tell me if this is right for question 1??
a)
. . .dy/dt .= .-2sin(2t)
. . .dx/dt .= .2 + 2cos(2t)

. . . . . . . . .dy/dt . . . . . . .-2 sin(2t) . . . . . - 2 sin(t)cos(t) . . . . - sin(t)
dy/dt . = . -------- . = . ------------------ . = . ------------------ . = . --------- . = . - tan(t)
. . . . . . . . .dx/dt . . . . .2[1 + cos(2t)] . . . . . .2 cos2(t) . . . . . . . cos(t)

and for b)
Second derivative = d/dt(dy/dx)/(dx/dt) and d/dt(dy/dx) = (d/dt)(dy/dx) x dt/dx.
So d^2y/dx^2 = (-sec^2t x (1/(2+cos2t)))/(2+cos2t)
and d^2x/dy^2 = -((-sec^2t x (1/(2+cos2t))))/(2+cos2t)/(-tant)^3

is this right or am i totally way off?