Another way to write 12 x 10 to the 3rd power

The extreme values of a function are the minimum and the maximum values of the function.

The extreme values are: -8 and 131.91

The given parameters are:

[tex]\mathbf{F(x,y) = x^2 + y^2 + 4x - 4y}[/tex]

[tex]\mathbf{x^2 + y^2 \le 81}[/tex]

Find the gradient of F(x,y)

[tex]\mathbf{f_x(x) = 2x + 4}[/tex]

[tex]\mathbf{f_y(y) = 2y - 4}[/tex]

Set to 0, to solve for x and y

[tex]\mathbf{2x + 4 = 0}[/tex]

[tex]\mathbf{2x = -4}[/tex]

[tex]\mathbf{x = -2}[/tex]

[tex]\mathbf{2y - 4 = 0}[/tex]

[tex]\mathbf{2y = 4}[/tex]

[tex]\mathbf{y = 2}[/tex]

So, the critical point is:

[tex]\mathbf{(x,y) = (-2,2)}[/tex]

Also, we have: [tex]\mathbf{x^2 + y^2 \le 81}[/tex]

Calculate the gradients

[tex]\mathbf{g_x = 2x}[/tex]

[tex]\mathbf{g_y = 2y}[/tex]

Equate the gradients as follows:

[tex]\mathbf{f_x = \lambda \cdot g_x}[/tex]

[tex]\mathbf{f_y = \lambda \cdot g_y}[/tex]

So, we have:

[tex]\mathbf{2x + 4 = \lambda \cdot 2x}[/tex]

[tex]\mathbf{2y - 4 = \lambda \cdot 2y}[/tex]

Make [tex]\mathbf{\lambda}[/tex] the subject, in the above equations

[tex]\mathbf{\lambda = \frac{x + 2}{x}}[/tex]

[tex]\mathbf{\lambda = \frac{y - 2}{y}}[/tex]

Equate the above equations, so we have:

[tex]\mathbf{\frac{x + 2}{x} = \frac{y - 2}{y} }[/tex]

Cross multiply

[tex]\mathbf{xy + 2y = xy - 2x}[/tex]

Subtract xy from both sides

[tex]\mathbf{2y =- 2x}[/tex]

Divide both sides by 2

[tex]\mathbf{y =- x}[/tex]

Substitute [tex]\mathbf{y =- x}[/tex] in [tex]\mathbf{x^2 + y^2 = 81}[/tex]

[tex]\mathbf{x^2 + (-x)^2 = 81}[/tex]

[tex]\mathbf{x^2 + x^2 = 81}[/tex]

[tex]\mathbf{2x^2 = 81}[/tex]

Divide both sides by 2

[tex]\mathbf{x^2 = \frac{81}{2}}[/tex]

Take square roots

[tex]\mathbf{x = \±\frac{9}{\sqrt2}}[/tex]

Rationalize

[tex]\mathbf{x = \±\frac{9\sqrt2}{2}}[/tex]

Recall that: [tex]\mathbf{y =- x}[/tex]

So, we have:

[tex]\mathbf{y = \±\frac{9\sqrt2}{2}}[/tex]

The ordered pairs are:

[tex]\mathbf{(x,y) = \{(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}),(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2})\}}[/tex]

Hence, the points are:

[tex]\mathbf{(x,y) = \{(-2,2),(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}),(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2})\}}[/tex]

Substitute these values in [tex]\mathbf{F(x,y) = x^2 + y^2 + 4x - 4y}[/tex]

[tex]\mathbf{F(2,2) = (-2)^2 + 2^2 + 4(-2) - 4(2) =-8}[/tex]

[tex]\mathbf{F(\frac{9\sqrt2}{2},-\frac{9\sqrt2}{2}) = (\frac{9\sqrt2}{2})^2 + (-\frac{9\sqrt2}{2})^2 + 4(\frac{9\sqrt2}{2}) - 4(-\frac{9\sqrt2}{2}) =131.91}[/tex]

[tex]\mathbf{F(-\frac{9\sqrt2}{2},\frac{9\sqrt2}{2}) = (-\frac{9\sqrt2}{2})^2 + (\frac{9\sqrt2}{2})^2 + 4(-\frac{9\sqrt2}{2}) - 4(\frac{9\sqrt2}{2}) =30.09}[/tex]

Hence, the extreme values of the function are: -8 and 131.91

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Answer:

4.16

Step-by-step explanation:

Sence Lin estimated $20 and her actucal bill was $24.16. To find the difference, we subtract 20 from 24.16.

[tex]24.16-20=4.16[/tex]

Answer:

Fraternity of 32 members with 8 taking neither math or physics

Of the 24, 5 took both. 24 - 5 = 19 taking one. 18 taking just Math.

Just 1 took only Physics.

Step-by-step explanation:

Answer:

0.30j + 1.19 = 2.99

Step-by-step explanation: Hope this helps! :D

There are a total of 26 letters that can be made to follow. There are 4 independent events here, namely:

First: Starts with K and followed by two additional letters = 1 * 26 * 26 = 676

Second: Starts with K and followed by three additional letters = 1 * 26 * 26 * 26 = 17,576

Third: Starts with W and followed by two additional letters = 1 * 26 * 26 = 676

Fourth: Starts with W and followed by three additional letters = 1 * 26 * 26 * 26 = 17,576

The total possibilities that can be made from here is:

Total possibilities = 676 + 17,576 + 676 + 17,576

Total possibilities = 36,504

By subtracting the number of blue pens from the number of black pens, we find that there are 183,891 more black pens than blue pens.

To find out how many more black pens there are than blue pens, we need to subtract the number of blue pens from the number of black pens. The student provided the numbers as 365,493 blue pens and 549,384 black pens.

Here is the calculation:

There are 183,891 more black pens than blue pens.

density because everything else would fall under the category of chemical properties instead of physical properties and the other 3 properties would result in a chemical change while density would not

Answer:

5400

Step-by-step explanation:

Convert 5.4 ⋅ 103 to standard form.

[tex]5.4 \ times \ 10^3[/tex]

for 10^3 we add 3 zeros

10^3= 10 times 10 times 10=1000

[tex]5.4 \ times \ 10^3[/tex]

Replace 10^3 by 1000. then multiply with the decimal number

[tex]5.4 \ times \ 1000[/tex]

When we multiply by 1000 we move the decimal point 3 places to the right

5400

The equation y = (1/2)x - 2.

A relation between a collection of inputs and outputs is known as a function. A function is a connection between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain.

We have points from the graph (6,1) and (8,2).

The slope of the equation is;

m = (2-1) / (8-6)

m = 1/2

And the y-intercept;

We have the slope-intercept form of the equation,

y = mx + b

y = x/2 + b

To find the y-intercept, we substitute the point (6,1)

b = -2

Therefore, we have the equation of the linear function y = (1/2)x - 2.

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A child can correctly explain that 3 x 2 represents three sets of two. This child is most likely in Piaget's Concrete Operational stage option (C) is correct.

It is defined as the operation in which we do the addition of numbers, subtraction, multiplication, and division. It has basic four operators that is +, -, ×, and ÷.

As we know,

A young toddler may properly identify three sets of two as being represented by 3 x 2.

This child is most likely in Piaget's Concrete Operational stage.

Thus, a child can correctly explain that 3 x 2 represents three sets of two. This child is most likely in Piaget's Concrete Operational stage option (C) is correct.

The complete question is:

A child can correctly explain that 3 x 2 represents three sets of two. This child is most likely in Piaget's _____________ stage.

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The equivalent ratios could be used to get the number of counselors who will be 13.

Division means the separation of something into different parts, sharing of something among different people, places, etc.

At a sports camp, there is one counselor for every 12 campers.

If there are 156 campers attending the camp.

The equivalent ratios could be used to get the number of counselors who will be

⇒ 156 / 12

⇒ 13

The equivalent ratios could be used to get the number of counselors who will be 13.

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Answer:

Option A: Exponential growth

Step-by-step explanation:

If  for values of x values of y increases and forms a geometric sequence then it would be an exponential growth function because geometric sequence is an exponential function and since, it is increasing hence, an exponential growth.

Option B is incorrect because  because y is not decreasing

Option C and D are incorrect because geometric sequence can never be linear since, it gives common ratio.

Final answer:

The function described, where y-values form an increasing geometric sequence as x-values increase, is characterized as Exponential growth. This is consistent with an exponential curve where the rate of growth is proportional to the current value, which aligns with the behavior of a geometric sequence.

Explanation:

For the given scenario where the y-values of a function form a geometric sequence that increases for the x-values 1, 2, 3, and so on, the type of function is described as Exponential growth. This is because in a geometric sequence each term after the first is found by multiplying the previous term by a constant called the common ratio, which is analogous to the exponential function where the growth rate of the value is proportional to its current value. The more the x-value increases, the higher and faster the y-value grows, following the pattern of an exponential growth curve.

Answer A. Exponential growth fits the description as it involves an increasing geometric sequence. Answer B. Decreasing linear refers to a linear function that decreases with each increment in x, which does not describe the function in question. Similarly, Answer C. Increasing linear describes a function that increases at a constant rate, not geometrically. Answer D. Exponential decay would imply the y-values decrease as x increases, which is also not the case described.

In 22 gross of nails, there are 264 dozens as one gross is equal to 12 dozen.

To find out how many dozens are there in 22 gross of nails, we need to determine the number of dozens in one gross and then multiply it by 22.

One gross is equal to 12 dozen.

So, to find the number of dozens in 22 gross of nails.

we can multiply 12 (number of dozens in one gross) by 22:

12 x 22

= 264

Therefore, there are 264 dozens in 22 gross of nails.

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The symbol x is the sample mean, and the Greek symbol μ is the population mean.

The symbol x is the sample mean, and the Greek symbol μ is the population mean. Calculating the Standard Deviation: If x is a number, then the difference x - mean is called its deviation. In a data set, there are as many deviations as there are items in the data set. The deviations are used to calculate the standard deviation. If the numbers belong to a population, in symbols, a deviation is x - μ. For sample data, in symbols, a deviation is x - x.