Question

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:

• A. \( \frac{1}{2} \)
• B. \( \frac{2}{3} \)
• C. \( \frac{3}{\sqrt{e}} \)
• D. \( \frac{1}{\sqrt{e}} \)

Hint:

When optimizing physical systems, use calculus to find critical points where derivatives equal zero.

Answer 1

The Correct Option is D

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 \( x = \frac{1}{\sqrt{e}} \).