Answer to Question #151085 in Calculus for Angelo

Question

In a telegraph cable, the measure of the speed of the signal is proportional to 𝑥^2 𝑙𝑛 (1/𝑥), where

𝑥 is the ratio of the measure of the radius of the core of the cable to the measure of the thickness of the cable’s winding. Find the value of 𝑙𝑛𝑥 for which the speed of the signal is greatest.

Answer 1

I presume the function is:

t(x)=x²ln(x). You are looking for a maximum of t(x). This will happen if t'(x) = 0. Thus calculate the first derivative. Use the product rule:

t'(x)=ln(x)⋅2x+x²1/x=x(2ln(x)+1)

Now t'(x) =0. Solve for x.

x(2ln(x)+1)=0.

x=0 this is not allowed or ln(x)=-1/2.

The answer is: ln(x)=-1/2