The rate of transmission in a telegraph cable is observed to be proportional to $$x^2 \ln(1/x)$$ where $x$ is the ratio of the radius of the core to the thickness of the insulation $(0 < x < 1)$. What value of $x$ gives the maximum rate of transmission?
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Hint: Think about what does 'proportional' means in this case. Essentially, this means that the rate of transmission is a constant multiple of the expression given. Write the rate, $R$, this way, differentiate and apply your knowledge of rates of change and maxima and minima to solve for $x$.