Find the derivative of the function at P0 in the direction of A.
​f(x,y,z)=xy+yz+zx, (1-,1,-2)
A= 3i+2j-6k

√13 is the absolute value of |-3 + 2i| .

Complex numbers are the numbers that are expressed in the form of a+ib where, a, b are real numbers and 'i' is an imaginary number.

For a complex number z = x + iy,

The absolute value |z| as being the distance from z to 0 in the complex plane C

Complex numbers are the numbers that are expressed in the form of a+ib

The complex number here is minus three plus two i.

|-3 + 2i|

Absolute value=√(-3)²+(2)²

=√9+4

=√13

Hence, √13 is the absolute value of |-3 + 2i| .

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On solving the provided question, from the information we can say that this is a example of Research and development (R&D).

Research and development (R&D) refers to the knowledge a business accumulates to create new items or discover innovative ways to enhance current goods and services. Before using their products or processes for commercial purposes, large corporations may have their own R&D teams that test and improve them.

Research and development (R&D) refers to the processes businesses use to innovate and launch new goods and services. It is frequently the initial stage of the development process. Typically, the objective is to expand the company's revenue while introducing a new good or service to the market.

This is an example of  Research and development (R&D)

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Answer:

The answer is 3 ft

Step-by-step explanation:

Now, we know that

A = l × w

30 = (x) × (3x + 1)

30 = 3x² + x

3x² + x = 30

3x² + x - 30 = 0

3x² - 9x + 10x - 30 = 0

3x(x - 3) + 10(x - 3) = 0

(x - 3) (3x + 10) = 0

x = 3 ft

Thus, The value of x is 3

For Verification Only ;

30 = (x) × (3x + 1)

30 = (3) × (3*3 + 1)

30 = 3 × 10

30 = 30

Hence, L.H.S = R.H.S

Answer:

A: 74.9

B: 51.4

C: 53.7

Sorry I don't have enough time for the rest

Step-by-step explanation:

all angles should add up to 180

The probability is it that she actually uses performance-enhancing drugs is 0.5975.

Given that,

The national flufferball association makes the decision to put in place a drug testing programme to check its athletes for the use of illicit performance-enhancing substances. In reality, 3% of professional flufferball players abuse performance-enhancing substances. A drug test has a 2% false positive rate and a 4% false negative rate. Alternatively, a person who abstains from drug use has a 0.02 percent chance of testing positive. With a 0.04 probability, a person who uses the medications will test negative. A player is tested at random, and the results are positive.

We have to find how probable is it that she actually uses performance-enhancing drugs.

We know that,

The provided data is

P(D)=0.03

P(D')=1-0.03=0.97

P(T|D')= 0.02

P(T'|D)=0.04

P(T|D)=1-P(T'|D)=1-0.04=0.96

The probability that she really takes the drugs is,

P(D|T)=P(D)P(T|D)/P(D)P(T|D)+P(D')P(T|D')

P(D|T)=(0.03×0.96)/(0.03×0.96)+(0.02×0.97)

P(D|T)=0.5975

Therefore, The probability is it that she actually uses performance-enhancing drugs is 0.5975.

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The evaluation of the given fraction expression is 1 11/16. The correct option is the last option

From the question, we are to evaluate the given expression.

The given expression is

[tex]12 \div (2+\frac{2}{3})^{2}[/tex]

Evaluating the expression

[tex]12 \div (2+\frac{2}{3})^{2}[/tex]

[tex]12 \div (\frac{8}{3})^{2}[/tex]

[tex]12 \div (\frac{8^{2}}{3^{2}})[/tex]

[tex]12 \div \frac{64}{9}[/tex]

= [tex]12 \times \frac{9}{64}[/tex]

= [tex]\frac{108}{64}[/tex]

= [tex]\frac{27 \times 4}{16 \times 4}[/tex]

= [tex]\frac{27}{16}[/tex]

= [tex]1\frac{11}{16}[/tex]

Hence, the fraction is 1 11/16

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Answer:

7/8

Step-by-step explanation:

2/3 * 1 5/16 =

2/3 * (1 + 5/16)=

2/3 * (16/16 + 5/16)=  -> common denominator numbers are easy to simplify

2/3 * (16+5)/16 =

2/3 * 21/16=

(2*21)/(3*16)=

42/48=

21/24=7/8

The circumference around the flower bed in yards is D) 8×π.

The circumference of a circle is defined as the linear distance around it.

The formula for the circumference of a circle is [tex]2\pi r[/tex]. Where r is the radius of the circle.

It is given that the radius of the flower bed is 4 yards.

The flower bed is in the shape of a circle.

The circumference of the flower bed is 2×π×r.

Substitute the value of r in the equation.

Therefore the circumference around the flower bed is 8×π.

The correct option is D) 8×π.

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The zeros of the expression 3x² + 5x - 2 are 1/3 and -2

A quadratic equation can simply be defined as an algebraic equation having the second degree of x.

From the information given, we have that;

3x² + 5x - 2

Equate the expression to zero

3x² + 5x - 2 = 0

Find the pair factors of -6 that sum up to + 5, we get;

+ 6 and -1

Substitute the values into the equation;

3x² + 6x - x - 2 = 0

Enclose in brackets

(3x² + 6x) - ( x - 2) = 0

Factor the common terms

3x( x + 2) - 1( x + 2)

Then,

3x - 1 = 0

x = 1/3

x +2 = 0

x = -2

Hence, the values are 1/3 and -2

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The equation of the line in slope intercept form of  y is  -2x + 19

When you know the slope of the line to be examined and the given point is also the y intercept, you can use the slope intercept formula, y = mx + b. The y value of the y intercept point is denoted by the symbol b in the formula.

Y = mx + b, where m denotes the slope and b the y-intercept, is how the equation of the line is expressed in the slope-intercept form. We can see that the y-intercept of the line in our equation, y = 7 x + 4, is 4.

To find this, we can use the point and the slope in point-slope form. The we solve for y.

y - y1 = m(x - x1)

y - 7 = -2(x - 6)

y - 7 = -2x + 12

y = -2x + 19

The equation would be y = -2x + 19

The complete question is : Write an equation in slope- intercept form y=mx=b of the line through point p(6,7) with slope -2

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The answer of the given scientific multiplication is,  8.84 x 10^-1.

What is scientific multiplication?

Multiply the decimal numbers to multiply the numbers in scientific notation. Add the 10 power exponents after that. In scientific notation, place the new power of 10 with the decimal.

Given:

(6.8 x 10^2) x (1.3 x 10^-3)

We have to solve this using scientific multiplication.

First to multiply the decimals,

6.8 x 1.3 = 8.84

Now to solve

10^2 x 10^-3 = 10^(2-3) = 10^-1

⇒  (6.8 x 10^2) x (1.3 x 10^-3) = 8.84 x 10^-1

Hence, the answer of the given scientific multiplication is,  8.84 x 10^-1.

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Prior to performing the clustering analysis, we choose the value for k. In a cluster analysis, the distance between the clusters should be minimized.

K-means clustering is an unsupervised machine learning technique used to identify clusters of data points. Before performing the clustering analysis, we must first choose the value of k (the number of clusters we wish to identify). The cluster analysis will then attempt to find the k clusters by minimizing the distances between the data points in each cluster. Once the clusters have been identified, the distance between them should be minimized to ensure that the clusters are as distinct as possible.

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The lowest temperature in January is -10°C if the lowest temperature in January is less than -6°C.

A scaled way of measuring how hot or how chilly something is a living body's normal level of heat production; see thermometer. a condition of extreme body heat with an average oral temperature of 98.6°F. The ideas of hotness and coolness are expressed numerically as a physical value called temperature.

The calibration of thermometers involves employing a variety of temperature scales with clearly defined reference points and thermometric materials. The Celsius scale (°C), formerly known as centigrade, with the unit symbol °C, is the most commonly used scale, followed by the Fahrenheit scale (°F) and the Kelvin scale (K), which is mostly used for scientific purposes. One of the seven base units in the International System of Units is the kelvin (SI).

Given,

Less than 6° was the lowest temperature recorded in January.

The lowest temperature in January will be=?

It is -10°C.

Therefore, the lowest temperature in January is -10°C.

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Simple probability is the calculation of an outcome or the chance of an event ever happening.

18.65% probability that the person is from Washington

Two events, A and B.

[tex]P(B/A)=\frac{P(B)*P(A/B)}{P(A)}[/tex]

In which P(B|A) is the probability of B happening when A has happened and P(A|B) is the probability of A happening when B has happened.

In this question:

Event A: Arrived by train.

Event B: From Washington.

15% are from Washington

This means that [tex]P(B)=0.15[/tex]

12% of the attendees from Washington came to the conference by train.

This means that [tex]P(A/B)=0.12[/tex]

Probability of arriving by train:

6% of 60%(from California), 17% of 25%(from Oregon) or 12% of 15%(from Washington). So

[tex]P(A)=0.06*0.6+0.17*0.25+0.12*0.15=0.0965[/tex]

If an attendee is selected at random and found to have arrived by train,

[tex]P(B/A)=\frac{0.15+0.12}{0.0965} =0.1865[/tex]

=0.1865

18.65% probability that the person is from Washington

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The y-intercept in Duncan's hike situation as described in the task content is; 1000 feet.

It follows from the task content that the y-intercept of Duncan's hike scenario is to be determined.

In context, the y-intercept is otherwise termed; the starting value.

According to the task content; Duncan begins to hike a trail that is at an elevation of 1000 feet.

On this note, Duncan's starting elevation is; 1000 feet and consequently, the y-intercept in this scenario is; 1000 feet.

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Answer:

100 is the answer

Step-by-step explanation:

The probability to determine the time needed to complete the project, we simply added the time required for activities 1 and 2.

1 (1): the chance that a given event will occur

 (2): the ratio of the number of outcomes in an exhaustive set of equally     likely outcomes that produce a given event to the total number of possible outcomes a branch of mathematics concerned with the study of probabilities

2 : something (such as an event or circumstance) that is probable

3: the quality or state of being probable

4: a logical relation between statements such that evidence confirming one confirms the other to some degree

The best-case scenario's calculation of the project's completion time is displayed below;

= Week spent on Activities A1 +  Week spent on Activities A2.

= 5 weeks plus 7 weeks.

= 12

To determine the time needed to complete the project, we simply added the time required for activities 1 and 2.

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Answer:

85.7142857%

Step-by-step explanation:

To find the percentage, take the first quantity and divide by the second quantity

480/560 =.857142857

Then multiply by 100 %

857142857 * 100% = 85.7142857%

The original cost of each sports drink that Jenny bought for her soccer team was $1.20.

Number of sports drink that Jenny bought for her soccer team

= 22

The amount of discount that Jenny got on each drink as they were on sale

= $0.35

Amount that Jenny paid for 22 sports drinks

= $18.70

Let the original cost of each sports drink be $x

Amount that Jenny paid for each drink

= $(x-0.35)

Therefore, amount that Jenny paid for 22 sports drink

= 22× (x-35) = 18.70

(x-0.35) = 18.70/22

x = 0.85 + 0.35

x = $1.20

The question is incomplete, the complete question is:

Jenny buys 22 sports drinks for her soccer team, they are on sale for $0.35 off per drink. Jenny paid $18.70 for 22 sports drink. How much does each drink cost originally?

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Answer:

y = 1/4 x - 1

Step-by-step explanation:

y = mx + b

m = slope = (-2 - 0)/(-4 - 4) = -2/(-8) = 1/4

y = 1/4 x + b

0 = 1/4 × 4 + b

0 = 1 + b

b = -1

y = 1/4 x - 1

The solution to the given system of equations is;

x = -4 and y = 6

We are given the simultaneous equations as;

y = -3x - 6   ------(1)

x - 4y = -28   ------(2)

For eq 1, we can get the coordinates as;

When x = -2, y = -3(-2) - 6 = 0

When x = -1, y = -3(-1) - 6 = -3

When x = 0, y = -3(0) - 6 = -6

When x = 1, y = -3(1) - 6 = -9

When x = 2, y = -3(2) - 6 = -12

For eq 2, we have;

When x = -2, y = (-2 + 28)/4 = 26/4

When x = -1, y = (-1 + 28)/4 = 27/4

When x = 0, y = (0 + 28)/4 = 7

When x = 1, y = (1 + 28)/4 = 29/4

When x = 2, y = (2 + 28)/4 = 7.5

From the graph attached, the solution is (-4, 6)

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dimensions of the poster with margin are L  =  12 in and h  = 24 in

The shape/polygon of a rectangle is two dimensional, having four sides, four vertices, and four right angles. The rectangle's two opposing sides are equal and parallel to one another. The space a rectangle occupies is known as its area. The area of a rectangle can also be defined as the region inside its border.

We utilise the unit squares to calculate a rectangle's area. Rectangle ABCD should be divided into unit squares. The total number of unit squares that make up a rectangle ABCD is its area.

Rectangle area equals length times width.

Let call length of printed area of the poster be " x "  and  height of printed area of the poster be " y ".

Area of the poster = length and height

200 = x*y  

y = 200/x

We also know that dimensions of the poster with margin is:

L  =  x + 2   in  and     H  =  y + 4 in

Therefore area of the poster is:

A(p)  = ( x + 2 ) * ( y + 4 )

And area as function of x is:

A(x)  =   ( x + 2 ) * ( 200/x + 4 )

A(x)  =  200 + 4*x + 400 /x  + 8

Taking derivatives on both sides of the equation we have:

A´(x)  =  4 - 400/x²

By taking A´(x)  = 0

4 - 400/x² = 0 ⇒ 4*x² - 400 = 0

x² = 400 / 4

x² = 100

x = 10 in

and y = 200/x ⇒ y = 20

The second derivative A´´(x) = 400/x4 which is > 0

there is a minimum for the function at the point x = 10

As x and y are dimensions of the printing area of the poster, dimensions of the poster with margin are

L  = x  +  2   =   10  + 2  =  12 in   and

h = y  +  4   =   20  + 4  =  24  in

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The derivative of 1/logx is With the chain rule.

1log(x)=log(x)−1 is ,= -1xlog(x)2 .

The by-product of logₐ x (log x with base a) is 1/(x ln a). Here, the thrilling issue is that we have "ln" withinside the by-product of "log x". Note that "ln" is referred to as the logarithm (or) it's miles a logarithm with base "e".

The by-product of 1/log x is -1/x(log x)^2. Note that 1/logx is the reciprocal of log.

[tex]With the chain rule. \\

1log(x)=log(x)−1 \\

So you follow the chain rule to the electricity after which to the log: \\

ddxlog(x)−1=−1⋅log(x) \\ \\−2⋅ddxlog(x)

\\ =−1log(x)21x

\\ =−1xlog(x)2 .[/tex]

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The person is 8.325 miles far from her home and she should walk in the direction of 64.86° North Of East to head directly home .

In the question ,

from the figure ,

we see that point O is  home of the person ,

it is given that person leaves the home and walks 4 miles west to point A , then 5 miles in south west direction to the point B .

So ,distance OA = 4 miles

AB = 5 miles .

In triangle OAB ,

angle OAB = 45+90 = 135°

Using Cosine Rule ,

OB² = OA² + AB² - 2*OA*AB*Cos(135)°

= 16 + 25 + 28.2843 = 69.2843

So ,OB = √69.2843  = 8.325 miles .

Now using Sine Rule ,

Sinθ/OA = Sin(135°)/OB

Sinθ = 4*Sin(135°)/8.324

Sinθ = 0.339792

So , θ = 19.86°

hence the direction from North of East is

= 45° + θ

= 45° + 19.86°

= 64.86° North Of East .

Therefore , the distance is 8.325 miles and the direction is 64.86° North Of East .

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A) D = 1/2[C]  = 1/2[-16]

B) Elevation of city D is -8 feet.

Answer is negative due to -8 feet below sea level.

What is elevation?

The elevation is the height above sea level. Typically, elevations are expressed in feet or meters. Maps can depict them using contour lines, which link points of the same elevation, color bands, or numbers that indicate the precise elevations of specific points on the surface of the Earth.

The elevation of city A is 60 feet below sea level
⇒ A = -60
The elevation of city C is 16 feet below sea level
⇒ C = -16                        

The elevation of city D is 1 over 2 of the elevation of city C.
⇒  D = 1/2[C]

Part A: Write an expression to represent the elevation, in feet, of city D.
⇒ D = 1/2[C]  = 1/2[-16]

Part B: Solve the expression and explain why the answer is negative or positive.              
⇒  D = 1/2[-16]
     D = -8
As a result Elevation of city D is -8 feet.
Here founded answer is negative due to -8 feet below sea level.

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The equation (C) (√2x + √3)² + x² = 3x² - 5x is not a quadratic equation.

A quadratic equation, or second-order polynomial equation with a single variable, is x ax2 + bx + c = 0.

Since it is a second-order polynomial equation, which is ensured by the algebraic 's theorem, it must have at least one answer.

The solution might be simple or difficult.

(A) 2(x - 1)² = 4x² - 2x + 1

Algebraic identity:

(a - b)² = a² - 2ab + b²

2(x² + 1 - 2x) = 4x² - 2x + 1

2x² + 2 - 4x = 4x² - 2x + 1

4x² - 2x² - 2x + 4x + 1 - 2 = 0

2x² + 2x - 1 = 0

The equation has a degree of two.

As a result, the equation 2(x - 1)² = 4x² - 2x + 1 is quadratic.

(B) 2x - x² = x² + 5

x² + x² + 5 - 2x = 0

2x² - 2x + 5 = 0

The equation has a degree of two.

As a result, the equation 2x - x² = x² + 5 is quadratic.

(C) (√2x + √3)² + x² = 3x² - 5x

Algebraic identity is used:

(a + b)² = a² + 2ab + b²

2x² + 3 + 2√6x + x² = 3x² - 5x

3x² - 3x² + 2√6x + 5x + 3 = 0

5x + 2√6x + 3 = 0

The equation has a degree of 1.

As a result, the equation (√2x + √3)² + x² = 3x² - 5x  is not quadratic.

(We got our answer so we don't need to solve further)

Therefore, the equation (C) (√2x + √3)² + x² = 3x² - 5x is not a quadratic equation.

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For the parabola y = 2x² + 3x + 1,

a. i.  The x-intercepts are x = -1/2 and x = -1

a. ii. The y-intercept is y = 1

b. i. The line of symmetry is at x = -3/4

b. ii. The equation of the line of symmetry is x = -3/4

c.  The vertex is at (-3/4, -1/8)

A parabola is part of a conic section which is a curve.

Given the parabola y = 2x² + 3x + 1, we find the x- intercept when y = 0.

So, y = 2x² + 3x + 1

2x² + 3x + 1 = 0

2x² + 2x + x + 1 = 0

Factorizing, we have

2x(x + 1) + (x + 1) = 0

(2x + 1)(x + 1) = 0

2x + 1 = 0 or x + 1 = 0

2x = -1 or x = -1

x = -1/2 or x = -1

So, the x-intercepts are x = -1/2 and x = -1

Given the parabola y = 2x² + 3x + 1, we find the y- intercept when x = 0.

So, y = 2x² + 3x + 1

So, y = 2(0)² + 3(0) + 1

y = 0 + 0 + 1

y = 1

So, the y-intercept is y = 1

The line of symmetry passes through the vertex of the parabola where dy/dx = 0.

So, y = 2x² + 3x + 1

dy/dx = d(2x² + 3x + 1)/dx

= d(2x²)/dx + d3x/dx + d1/dx

= 4x + 3 + 0

= 4x + 3

So, when dy/dx = 0,we have

4x + 3 = 0

4x = -3

x = -3/4

So, the line of symmetry is at x = -3/4

The equation of the line of symmetry is x = -3/4

Since the vertex of the parabola is a t x = -3/4, substituting this into the equation, we have

y = 2x² + 3x + 1

y = 2(-3/4)² + 3(-3/4) + 1

y = 2(9/16) - 9/4 + 1

y = 9/8 - 9/4 + 1

y = (9 - 18 + 8)/8

y = (17 - 18)/8

y = -1/8

So, the vertex is at (-3/4, -1/8)

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The value of the variable BC will be equal to 20.

A line segment in geometry is a section of a line that has two clearly defined endpoints and contains every point on the line that lies within its confines.

Given that B is between A and C. AB=19, BC = 4y, AC = 8y-1.

Here If B is between the segment AC the value of BC will be calculated as,

BC = AC - AB

Substitute the values in the above equation and solve for the value of y,

4y = 8y - 1 - 19

4y = 8y - 20

8y - 4y = 20

4y = 20

y = 5

The value of segment BC will be.

BC = 4y

BC = 4 x 5

BC = 20

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