Question

The rate of transmission in a telegraph cable is observed to be proportional to x2ln(1/x) where x is the ratio of the radius of the core to the thickness of the insulation (0

Answer 1

Answer:

The value of x that gives the maximum transmission is 1/√e ≅0.607

Step-by-step explanation:

Lets call f the rate function f. Note that f(x) = k * x^2ln(1/x), where k is a positive constant (this is because f is proportional to the other expression). In order to compute the maximum of f in (0,1), we derivate f, using the product rule.

$f'(x) = k*((x^2)'*ln(1/x) + x^2*(ln(1/x)')) = k*(2x\,ln(1/x)+x^2*(\frac{1}{1/x}*(-\frac{1}{x^2})))\\= k * (2x \, ln(1/x)-x)$

We need to equalize f' to 0

• k*(2x ln(1/x) - x) = 0 -------- We send k dividing to the other side
• 2x ln(1/x) - x = 0 -------- Now we take the x and move it to the other side
• 2x ln(1/x) = x -- Now, we send 2x dividing (note that x>0, so we can divide)
• ln(1/x) = x/2x = 1/2 -------  we send the natural logarithm as exp
• 1/x = e^(1/2)
• x = 1/e^(1/2) = 1/√e ≅ 0.607

Thus, the value of x that gives the maximum transmission is 1/√e.