SOLUTION: Hi will somebody be so kind as to help with this particular problem, really finding it difficult In an underground telephone cable the ratio of the radius of the core t

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Question 377581: Hi
will somebody be so kind as to help with this particular problem, really finding it difficult

In an underground telephone cable the ratio of the radius of the core to the thickness of the protective sheath
is denoted by x. The speed v at which a signal is transmitted is given by
v = Kx^2 ln(1/x)
where K is some positive constant. Show that
dv/dx= Kx [2 ln(1/x)- 1]
and hence show that the maximum speed of transmission is K/(2e).
Thank you very much!
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
v+=+Kx%5E2+ln%281%2Fx%29
Use the chain rule and the product rule,
dv%2Fdx=K%28x%5E2%2A%281%2F%281%2Fx%29%29%28-x%5E%28-2%29%29%2Bln%281%2Fx%29%2A2x%29
dv%2Fdx=K%28-x%2B2x%2Aln%281%2Fx%29%29
dv%2Fdx=K%282x%2Aln%281%2Fx%29-x%29
highlight_green%28dv%2Fdx=Kx%282%2Aln%281%2Fx%29-1%29%29
.
.
.
The speed is maximum when dv%2Fdx=0.
2ln%281%2Fx%29-1=0
2ln%281%2Fx%29=1
ln%281%2Fx%29=1%2F2
1%2Fx=e%5E%281%2F2%29
highlight%28x=e%5E%28-1%2F2%29%29
.
.
v+=+Kx%5E2+ln%281%2Fx%29
v+=+K%28e%5E%28-1%29%29+ln%28e%5E%281%2F2%29%29
v+=+K%28e%5E%28-1%29%29%2A%281%2F2%29
highlight_green%28v+=+K%2F%282e%29%29