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 $x = \frac{1}{\sqrt{e}}$.