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ose90

Member

Registered: 2008-09-20

Posts: 3

Application of maxima

In the diagram above, Ox and Oy are the rectangular axes and the straight line PAQ passes through the fixed point A ( 4,1 ). The angle QPO= Theta and 0 < Theta < pi/2

Let T = theta ( Sorry , I do not know how to insert the symbol, I tried &Theta but no luck )

a) Show thatOP + OQ = 5 + 4 tan T + cot T and show that as T changes, the least value of OP + OQ is 9.

b) Show that PQ = 4 sec T + cosec T and find, correct to two significant figures, the minimum value of Q when T changes.

luca-deltodesco

Member

Registered: 2006-05-05

Posts: 1,470

Re: Application of maxima

OP = 4 + cot θ  -> look at small triangle tan θ = 1/(OP-4)
OQ = 1 + 4.tan θ -> look at big triangle, tan θ = OQ-1

hence, OP + OQ = 5 + 4.tan θ + cot θ = f(θ)

since this is positive, we know that the value of 9 is a minimum.

b) Show that PQ = 4 sec T + cosec T and find, correct to two significant figures, the minimum value of Q when T changes.

i assume you mean, minimum of PQ.

PQ^2 = OP^2 + OQ^2  (pythagoras)

if you want, go ahead and find second derivitive to prove that the following is a maximum

Last edited by luca-deltodesco (2008-09-22 03:21:16)

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