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2 years ago

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Mathematics

1 answer:

TEA [102]2 years ago

7 0

Answer:

24 kilograms of catfish

Step-by-step explanation:

Take the percentage of catfish as a decimal or fraction and multiply it by the total amount of fish caught. The percent of catfish is 40% which is the same thing as 0.4. Multiply that by the total 60 kg of fish to get 24 kg of catfish.

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Use Lagrange multipliers to find the maximum and minimum values of the function subject to the given constraint. (If an answer d

andrey2020 [161]

Answer:

The maximum value is 1/27 and the minimum value is 0.

Step-by-step explanation:

Note that the given function is equal to $(xyz)^2$ then it means that it is positive i.e $f(x,y,z)\geq 0$ .

Consider the function $F(x,y,z,\lambda)=x^2y^2z^2-\lambda (x^2+y^2+z^2-1)$

We want that the gradient of this function is equal to zero. That is (the calculations in between are omitted)

$\frac{\partial F}{\partial x} = 2x(y^2z^2 - \lambda)=0$

$\frac{\partial F}{\partial y} = 2y(x^2z^2 - \lambda)=0$

$\frac{\partial F}{\partial z} = 2z(x^2y^2 - \lambda)=0$

$\frac{\partial F}{\partial \lambda} = (x^2+y^2+z^2-1)=0$

Note that the last equation is our restriction. The restriction guarantees us that at least one of the variables is non-zero. We've got 3 options, either 1, 2 or none of them are zero.

If any of them is zero, we have that the value of the original function is 0. We just need to check that there exists a value for lambda.

Suppose that x is zero. Then, from the second and third equation we have that

$-2y\lambda = -2z\lambda$ . If lambda is not zero, then y =z. But, since $-2y\lambda=0$ and lambda is not zero, this implies that x=y=z=0 which is not possible. This proofs that if one of the variables is 0, then lambda is zero. So, having one or two variables equal to zero are feasible solutions for the problem.

Suppose that only x is zero, then we have the solution set $y^2+z^2=1$ .

If both x,y are zero, then we have the solution set $z^2=1$ . We can find the different solution sets by choosing the variables that are set to zero.

NOw, suppose that none of the variables are zero.

From the first and second equation we have that

$\lambda = y^2z^2 = x^2z^2$ which implies $x^2=y^2$

Also, from the first and third equation we have that

$\lambda = y^2z^2 = x^2y^2$ which implies $x^2=z^2$

So, in this case, replacing this in the restriction we have $3z^2=1$ , which gives as another solution set. On this set, we have $x^2=y^2=z^2=\frac{1}{3}$ . Over this solution set, we have that the value of our function is $\frac{1}{3^3}= \frac{1}{27}$

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3 years ago

A/an _______ is a mathematical statement that calculates a value.

VashaNatasha [74]

A <u>formula</u> is a mathematical statement that calculates a value and each formula must always begin with an equal sign. A formula is used to express the mathematical relationship, information and rules symbolically and concisely. It basically summarizes the solution to a problem.

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2 years ago

Read 2 more answers

Assume that random guesses are made for nine multiple choice questions on an SAT test, so that there are nequals9 trials, each

Marina86 [1]

Answer: Find the probability that the number x of correct answers is fewer than 4 = 0.6087

Step-by-step explanation: Please find the attached files for the solution