Question

The rate of transmission in a telegraph cable is observed to be proportional to x^22ln(1/x) where x is the ratio of the radius of the core to the thickness of the insulation (0

Answer 1

Answer:

x = 0.606

Step-by-step explanation:

Data provided in the question:

rate of transmission is proportional to $x^{2}ln(\frac{1}{x})$

or

rate of transmission, R = $C[x^{2}ln(\frac{1}{x})]$

here, C is the proportionality constant

Now,

for point of maxima

differentiating the function with respect to 'x'

R' = $\frac{d(C[x^{2}ln(\frac{1}{x})])}{dx}$

using the product rule, we get\

R' = $C[-2x^{1}\ln\left(x\right)-x^{1}]$

or

R' =  $C[-x^{1}\left(2\ln\left(x\right)+1\right)$

Now,

R' = 0         [for point of maxima]

OR

$C[-2x^{1}\ln\left(x\right)-x^{1}]$ = 0

or

-x²¹ = 0 or  $\left(2\ln\left(x\right)+1\right)$ = 0

or

ln(x) = $\frac{-1}{2}$

or

x = 0.606

Since,

0 < x < 1

Hence,

accepted value of x = 0.606