In a submarine telegraph cable, the speed of signaling varies as \( x^2 \log(1/x) \), where \( x \) is the ratio of the radius of the core to that of the covering. To attain the greatest speed, the value of this ratio is:
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When optimizing physical systems, use calculus to find critical points where derivatives equal zero.
To maximize the speed, we take the derivative of the function \( S(x) \) with respect to \( x \) and set it to zero. This results in the optimal ratio of \( x = \frac{1}{\sqrt{e}} \).