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DIA [1.3K]
2 years ago
12

You are researching the penguin population in the Falkland Islands. There are 1000 penguins in the beginning of the first year:

500 are male, and 500 are female. Each couple makes one chick every spring (assuming every penguin has a partner). Chicks that are one year old also reproduce. However, 20% of all penguins, both adults and chicks, die each year due to natural causes. You need to create a formula to determine the population of the penguins (including chicks) at the end of the 7th year.
What is the formula to determine the population of the penguins at the end of the 7th year?
Mathematics
1 answer:
kolbaska11 [484]2 years ago
8 0

Answer:

The formula to determine the population of penguins at the end of the 7th year is:

P_7=1.3^7\cdot P_0=6.275\cdot 1000=6275

Step-by-step explanation:

With this information, we know that the initial population at end of year 0 is 1000 penguins.

The first year will born 500 chicks (50% of the population) and also 20% of the total population will die.

We can then model the population for the end of year 1 as:

P_1=P_0+0.5P_0-0.2P_0=1.3P_0

As this dynamic will continue with the years, we can generalize as:

P_2=1.3P_1=1.3\cdot (1.3P_0)=1.3^2P_0\\\\P_n=1.3^nP_0

Then, the value of the population at the end of the 7th year should be:

P_7=1.3^7\cdot P_0=6.275\cdot 1000=6275

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. Use Lagrange multipliers to find the maximum and minimum values of the function, f, subject to the given constraint, g. (Place
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Answer:

Minimum value of f(x, y, z) = (1/3)

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f(x, y, z) = x⁴ + y⁴ + z⁴

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g(x, y, z) = x² + y² + z² = 1

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x² + y² + z² - 1 = 0

Using Lagrange multiplier, we then write the equation in Lagrange form

Lagrange function = Function - λ(constraint)

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L(x,y,z) = x⁴ + y⁴ + z⁴ - λ(x² + y² + z² - 1)

We then take the partial derivatives of the Lagrange function with respect to x, y, z and λ. Because these are turning points, each of the partial derivatives is equal to 0.

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Hence, at the point where the box has maximum and minimal area,

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And

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Putting these into the constraint equation or the solution of the fourth partial derivative,

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Inserting these into the function f(x,y,z)

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f(x, y, z) = 3 × (1/9) = (1/3).

We know this point is a minimum point because when the values of x, y and z at turning points are inserted into the second derivatives, all the answers are positive! Indicating that this points obtained are

S = (1/3)

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