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balu736 [363]
3 years ago
8

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
Mathematics
1 answer:
sergij07 [2.7K]3 years ago
7 0

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.

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40 tickets were sold for a concert, some at 75 cents and the rest at $1.25. if the total raised was $42, how many had the cheape
Murljashka [212]

Answer:

34

Step-by-step explanation:

$1.25 = 125 cents.

$42 = 4200 cents

Tickets sold at 75 cents = x

Tickets sold at 125 cents = y

x + y = 40

75x + 125y = 4200

Multiply the first equation by 75

75x + 75y = 3000

75x + 125y = 4200

Subtract the the second equation from the first.

75x + 75y = 3000

- 75x + 125y = 4200

-------------------------------

0 - 150y = - 1200

Divide both sides by - 150

-150y/-150 = -1200/-150

y = 8

Substitute y = 8 into the first equation

x + y = 42

x + 8 = 42

x = 42 - 8

x = 34

34 tickets were sold for 75 cents

8 tickets were sold for $1.25

3 0
2 years ago
List the factors of 48 and 54
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3 years ago
Write real world problem that involves a positive and a negative value.then graph the values used in the problem on a number lin
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Consider the equation 5x^2-10x+c=0. What values of c result in the equation having a complex solutions? Represent your answer on
stiv31 [10]

Answer:

When we have a quadratic equation:

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There is something called the determinant, and this is:

D = b^2 - 4*a*c

If D < 0, then the we will have complex solutions.

In our case, we have

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Then the determinant is:

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And we want this to be smaller than zero, then let's find the value of c such that the determinant is exactly zero:

D = 0 = 100 - 4*5*c

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As c is multiplicating the negative term in the equation, if c increases, then we will have that D < 0.

This means that c must be larger than 5 if we want to have complex solutions,

c > 5.

I can not represent this in your number line, but this would be represented with a white dot in the five, that extends infinitely to the right, something like the image below:

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Step-by-step explanation:

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