Evaluate the gradient of f(x, y, z) = log(x2 + y2 + z2) at (1, 0, 1)

The average value of both f(x) and g(x) on 0≤x≤2 is 1/4. The average value of (f.g) on 0≤x≤2 is 0 and the statement avg(f) . avg(g) = avg(f . g) is false.

avg value of a function = 1/(b - a) ∫f(x) with limts a and b. Here a and b are the constraints under which the average has to be calculated.

a)

Hence average value of f(x) on 0≤x≤2

= 1/2 ∫f(x)  with 0 and 2 as intervals

Since integration is basically the area under the curve we get

avg of f(x) = 1/2[area under curve from 0 to 1 + Area under the curve from 1 to 2]

Since from 0 to 1, the figure is a triangle and beyond that a straight line we get

1/2[1/2 X 1 X 1 + 0]

= 1/4 sq units

b)

Similarly, the average value of g(x) on 0 ≤ x ≤ 2

1/2[area from 0 to 1 + area from 1 to 2]

= 1/2[0 + 1/2]

= 1/4 sq units

c) here we need to find the average of f(x) . g(x)

Here we see that f(x) is 0 from x = 1 to x = 1 while g(x) id 0 from x = 0 from x = 1

Hence we see that on multiplying the functions, the resultant has to be a 0 function.

Hence its average value will be 0.

d)

Average(f) . Average(g) = 1/4 X 1/4

= 1/16

Average of (f.g) = 0

Hence the statement is false

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Complete Question

The figure attached to the left is a graph of f(x), and below to the right is g(x).

The figure below to the left is a graph of f(x), a

(a) What is the average value of f(x) on 0≤x≤2? avg value = __________

(b) What is the average value of g(x) on 0≤x≤2? avg value = __________

(c) What is the average value of f(x) . g(x) on 0≤x≤2? avg value = _________

(d) Is the following statement true?

Average(f) . Average(g)=Average(f.g)

A. Yes

B. No

The Triangle Similar to ΔABC is ΔDEF in which m <E = 80 and m <F = 35.

The Correct option is (A).

According to the triangle's "angle sum property," a triangle's interior angles add up to 180°.

If two triangles have an equal number of corresponding sides and an equal number of corresponding angles, then they are comparable.

Given:

In  ABC, m<A= 65 and m<B= 80

Now, using Angle sum Property

m<A + m <B + m <C = 180

65+ 80 + m<C= 180

m <C = 180 - 145

m <C = 35

So, the similar Triangle will be  DEF in which m <E = 80 and m <F = 35.

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Answer:

-15

Step-by-step explanation:

That's the answer because

-15+4=-11

-15+4=-11-11<-7

For given inequality, graph number 3 is correct.

In mathematics, "inequality" refers to a relationship between two expressions or values that is not equal to each other. Therefore, inequality results from a lack of balance.

Make use of the following steps to solve an inequality: Step 1: Remove fractions by multiplying all terms by the fractions' lowest common denominator. Step 2 Combine like terms on both sides of the inequality to simplify. Step 3 Obtain the unknown on one side and the numbers on the other by adding or subtracting quantities.

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The answer for it is r = -4 :))

The variable in 2x + 3 = 7 is x because rest 2, 3 and 7 are constant terms only x is the variable.

The meaning of such an equations in algebra is just a mathematical expression stating the equality between two mathematical terms. For instance, 6x + 5 = 11 is an equation where 6x + 5 and 11 are two expressions separated by the 'equal' sign.

Identity equations and conditional equations are the two types of equations. For just any value of both the variables, an equality remains true. Only such variables have values that make a condition expression true. Two expressions joined by an equals sign ("=") form an equation.

The variable in 2x + 3 = 7 is x because rest 2, 3 and 7 are constant terms.

if we solve the above equation, we get

2x + 3 = 7

or, 2x = 7 - 3

or, 2x = 4

or, x = 2

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The probability that the sample mean is between 1300 and 1400 is approximately 1.34%.

We can use the Central Limit Theorem and the standard normal distribution to calculate the probability that the sample mean is between 1300 and 1400. The Central Limit Theorem states that as the sample size increases, the distribution of the sample means will approach a normal distribution, regardless of the distribution of the population.

Since we have a sample of 80 observations, we can assume that the sample mean follows a normal distribution with a mean of 1460 (the population mean) and a standard deviation of:

Standard deviation of the sample mean

= population standard deviation ÷ [tex]\sqrt {sample size[/tex]

[tex]= 270 \div \sqrt{80} = 27[/tex]

So the standard deviation of the sample mean is 27.

We can now standardize the interval between 1300 and 1400 by subtracting the mean and dividing it by the standard deviation of the sample mean.

[tex]Z_1[/tex] = (1300 - 1460) / 27 = -2.59

[tex]Z_2[/tex] = (1400 - 1460) / 27 = -2.22

We need to find the area under the standard normal distribution curve between Z1 and Z2.

Using a standard normal table or a calculator with the standard normal distribution function, we can find the probability that a random variable from the standard normal distribution is between -2.59 and -2.22. This probability is approximately 0.0134 or 1.34%.

Therefore, the probability that the sample mean is between 1300 and 1400 is approximately 1.34%.

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The volume of the triangular prism is 120 cm cube.

A triangular prism  is a three-sided prism.  The volume of a triangular prism can be represented as follows:

volume of a triangular prism = 1 / 2 bhl

where

Therefore,

b = 10 cm

h = 12 cm

l = 6 cm

Therefore,

volume of a triangular prism = 1 / 2 × 10 × 12 × 6

volume of a triangular prism = 1 / 2 × 120 × 6

volume of a triangular prism = 60 × 6

volume of a triangular prism = 120 cm³

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The required two numbers are 9 and 41.

The equation is defined as a mathematical statement that has a minimum of two terms containing variables or numbers that are equal.

Let x be the lesser number, and y be the greater number.

From the question, we know that:

x + y = 50 (Equation 1: The sum of the two numbers is 50)

y = 5x - 4 (Equation 2: The greater number is four less than five times the lesser number.)

The equation is to solve for one of the variables, and then substitute that value into the other equation to find the other variable.

Then, two numbers are x = 9 and y = 41.

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Answer:128 cm^2

Step-by-step explanation:

On solving the provided question, we can say that the values of the equation here will be y +1 = 8X 8y = 16 => y = 9.455

An equation is a formula in mathematics that joins two statements with the equal symbol = to represent equality. The definition of an equation in algebra is a mathematical statement proving the equality of two mathematical expressions. In the equation 3x + 5 = 14, for instance, the terms 3x + 5 and 14 are separated by an equal sign. The link between two phrases on either side of a letter is expressed mathematically. There is often only one variable, which is also the symbol. instance: 2x - 4 Equals 2.

y+1= -4(x - 2) 2x8y = 16​

x = 0

y +1 = 8X 8y = 16

y = 9.455

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When selecting four coins from a jar containing quarters, loonies, and toonies, there are a total of 27 unique combinations.

This can be calculated by using the formula for combination, which is nCr.

In this case,

n is the total number of coins in the jar (3) and

r is the number of coins selected (4).

Therefore, 3/(4(3-4)) = 3/0 = 3/1 = 3,

which is the number of possible combinations in the jar for the quarters, loonies, and toonies. When you multiply 3 by 3, you get 9, and when you multiply 9 by 3 again, you get 27.

Therefore, there are 27 unique coin combinations when selecting four coins from the jar.

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An expression that describes the team’s total points will be 2w + t.

It is important to note that an expression is simply used to show the relationship between the variables that are provided or the data given regarding an information. In this case, it is vital to note that they have at least two terms which have to be related by through an operator

In this case, the soccer team earns 2 points for a win and 1 point for a tie.

Let w represent the number of wins and t represent the number of ties.

The expression that describes the team’s total points will be:

= 2(w) + t

= 2w + t

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The values for figure is  x = 2, y = 9,

GH = 8, GJ = 7,

FE = 3, GE = 7​.

A closed quadrilateral has four sides, four vertices, and four angles. It is a form of polygon. In order to create it, four non-collinear points are joined. Quadrilaterals always have a total internal angle of 360 degrees.

The Latin words quadra, which means four, and latus, which means sides, are combined to form the English word quadrilateral. A quadrilateral does not always have to have equal lengths on each of its four sides.

Given figure shows

GH parallel to EG

GH = EG

GH = 3y - 19

EG = 8

3y - 19 = 8

3y = 27

y = 9

GH = 3y - 19 = 3(9) - 19 = 8

and GJ parallel  to JE

GJ = JE

GJ = 3x + 1

JE = 5x - 3

3x + 1 = 5x - 3

5x - 3x = 1 + 3

2x = 4

x = 2

values are,

GJ = 3x + 1 = 3(2) + 1 = 7

JE = 5x - 3 = 5(2) - 3 = 7

FE = 6x - y

substitute values,

x = 2, y = 9

FE = 6x - y = 6(2) - 9 = 12 - 9 = 3

Hence x = 2, y = 9

GH = 8, GJ = 7,

FE = 3, GE = 7​.

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The process of solving the 3 variable with 2 equation is explained below.

The term equation is defined as a relationship between two terms on either side of an equal sign.

Here we have the following steps to solve the 3 variable for two equation is solved accordingly.

First of all we have to take any two equations and solve it for one variable.

And then again we have to take another two pair of equations and solve for the same variable.

Now, we have a system of two equations with two unknown variables.

Then we have to solve the equation by using the appropriate arithmetic operations.

and then substitute it accordingly until all the value of the variable are solved.

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The park should be expanded based on the information as the majority want the expansion.

From the information, on the small town surveyed 250 adults, 180 of whom were parents, about whether a park should be expanded.

Also, the results showed that 150 of the adults who were parents and 40 of the adults who were not parents thought the park should be expanded.

Based on the above the number of people that wants the expansion will be:

= (150 + 40) / 250 × 100

= 190 / 250 × 100

= 76%

It should be expanded.

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The line x= -4 is a vertical line passing through x-axis.

It is a distance of 4 units from the x-axis and is parallel to the y-axis.

Vertical Line in a coordinate plane refers to a line that is perpendicular to the Y-axis. From top to bottom and from bottom to top, it is a straight line. The x-coordinate of every point along this line will be the same. The points of vertical lines include, for instance, (2,0), (3,0), (-4,0), etc. Similar to this, a horizontal line is a line that runs parallel to the x-axis from left to right.

There is no slope to the vertical lines. It follows the y-axis parallel, hence the slope is not known. As a result, the equation for the vertical line that at any point, let's say 'a', crosses the x-axis is:

x = a

where x is a point on the line's coordinate.

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Answer:

-3x + 549

Step-by-step explanation:

The order of operations is PEMDAS(parenthesis, exponents, multiplication, division, addition, subtraction)

So, according to PEMDAS, the first thing that we would do is solve the parenthesis. Since we can't solve the parenthesis because they aren't like terms, we're going to use distributive property to solve it instead.

-3(x+8) = -3(x) + -3(8) = -3x -24

Now we have: -3x - 24 ^ 2 -3

Our next step is to solve the exponent

24 ^ 2 or 24 x 24 = 576

Now we have: -3x - 24 + 576 -3

Our next step would be to add and subtract the like terms

-24 + 576 = 552 and 552 - 3 = 549

Now we have: -3x + 549

This would be the simplest form of the equation because -3x and 549 are not like terms

Answer:

2 is the highest degree because it's the only exponent

Answer: (-4, -6)(9, 2):

y=8/13x - 46/13 (Point slope form: y + 6 = 8/13 *(x+4)

Answer:

2 pounds of cheese

Step-by-step explanation:

First, we have to find out how much did she pay by taking

$5.00 - $1.40 = $3.60

So, she paid $3.60 in total. Then we find how many pounds of cheese she bought by taking

3.60 / 1.80 = 2 pounds of cheese

So, she bought 2 pounds of cheese.

Answer:

19%

Step-by-step explanation:

First, we have to find how much did he spend by taking

$575.00 - $465.75 = $109.25

Then we find the percentage decrease by taking

109.25 divided by 575, then times 100% = 19%

So,  the percentage decrease of the amount of available credit in Dylan's account is 19%

Identification if a polynomial is a sum or difference of two cubes can be done by factorizing the provided polynomial.

A polynomial is a mathematical expression that is a sum of one or more terms. Each term is a constant or a variable raised to a non-negative integer power and multiplied by a coefficient. The degree of a polynomial is the highest degree of the terms in the expression. A polynomial with one variable is called a univariate polynomial, and a polynomial with more than one variable is called a multivariate polynomial. Polynomials are used in various areas of mathematics, physics, engineering, computer science, and other fields. They can be used to model a wide range of phenomena and to solve problems that involve polynomial equations or inequalities.

Factorization, also known as factoring, is the process of breaking down a mathematical expression into a product of simpler factors. The goal is to express a given expression as a product of simpler expressions, which can be useful in solving equations and understanding the structure of the expression.

Factor the polynomial completely into its irreducible factors.

Look for terms that are of the form (x - a)^3 or (x + a)^3, where a is a constant. These are the cubes that you are looking for.

If there are two such terms, check if they can be combined into a single term by adding or subtracting them. If they can, then the polynomial is a sum or difference of two cubes.

If there are more than two such terms, check if any two of them can be combined in the same way. If they can, then the polynomial is a sum or difference of two cubes.

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Therefore ,the answer to the above trigonometry issue is therefore that the river is 70 feet wide.

The study of the connections between the angles and sides of triangles is known as trigonometry. Due to the fact that each straight-sided figure may be reduced to a collection of triangles, trigonometry is commonly utilized in geometry. There are a mind-boggling number of connections between trigonometry and other branches of mathematics, especially algebra, real or complex, integrals, logarithms, and infinite series.

Here,

Figure depicts a right triangle which, in relation to angle, its next side has length d or its wrong side is equal to the river's width, y. As a result,

tanθ= d/ y

=> y=dtanθ

=> y = (100m)tan(35.00)=70ft

The river measures 70 feet in width.

As a result, the trigonometry problem's answer indicates that the river's width is 70 feet.

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The probability of getting 2 blue is 1/9.

You are choosing replacement marbles. The trials (marble selections) are independent from one another and follow the binomial distribution.

1/3 of the time, a blue marble will be the first one chosen.

(This is due to the fact that 4 out of 12 marbles are blue, therefore 4/12 equals 1/3.

A blue marble is 1/3 more likely to be chosen the second time.

The two probabilities can be multiplied to obtain the following results because the marble selections are independent:

The likelihood of receiving two blues is (1/3)2 =1/9.

Therefore, there is a 1/9 chance of obtaining two blues.

Simply put, probability is the likelihood that something will occur. When we don't know how an event will turn out, we can discuss the likelihood or likelihood of several outcomes. Statistics is the study of events that follow a probability distribution.

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As per the given conditions in the question, the amount of money that the second son was left with by the man is 2,40,000.

To add two fractions with different denominators, you must first find a common denominator. A common denominator is a number that is divisible by both denominators of the fractions. Once you have a common denominator, you can add the numerators of the fractions together and place the sum over the common denominator. Then simplify the fraction if possible.

The reciprocal of a fraction is another fraction whose numerator and denominator are flipped. For example, the reciprocal of the fraction 2/3 is 3/2. The product of a fraction and its reciprocal is always equal to 1. To find the reciprocal of a fraction, simply swap the numerator and denominator.

Considering the conditions given in the question,let us suppose that the third son was left with x part of the money,

so we can use the equation,

part of money left for fist son + part of money left for second son + part of money left for third son = total part of money

or, 1/2 + 3/8 + x = 1

therefore, x = 1/8

Also we know that the third son was left 80,000 ,

so 1/8 of total money = 80,000

therefore, total money = 80000 * 8 =6,40,000

Now the amount left for the second son is , 3/8 of total money

which is equal to , 3/8 of 6,40,000 = 2,40,000

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The length of y using sine rule is 6664.2 inches

The sine rule states that if a, b and c are the lengths of the sides of a triangle, and A, B and C are the angles in the triangle; with A opposite a, etc., then a/sinA=b/sinB=c/sinC.

in this case w/sinW = y/sinY

Y = 180-(55+42)

Y = 180-(97)

Y = 83°

Therefore 470/sin55 = y/sin83

470× sin83 = y sin55

= 466.50 = 0.707y

divide both sides by 0.707

y = 466.50/0.707

y = 6664.2 inches

therefore the length of y is 6664.2 inches

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The fraction of the voters who voted in favor of the measure will be 1/5.

Algebra is the study of abstract symbols, while logic is the manipulation of all those ideas.

The acronym PEMDAS stands for Parenthesis, Exponent, Multiplication, Division, Addition, and Subtraction. This approach is used to answer the problem correctly and completely.

Ricardo's town recently voted on a new education funding package. Of the voters who voted for the bill, 20% voted for it.

The fraction of the voters who voted in favor of the measure will be given as,

⇒ 20%

⇒ 20 / 100

⇒ 1 / 5

The fraction of the voters who voted in favor of the measure will be 1/5.

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The mean, variance, and standard deviation for the probability distribution given are A. Mean = 5.5; B. Variance = 13.875; C. Standard Deviation = 3.722.

The mean, variance and standard deviation of the probability distribution can be calculated using the formula. The mean is calculated by summing up the product of all the values and their respective probabilities and then dividing it by the total probability. In this case, the mean is calculated as:

(1*0.587 + 3*0.115 + 9*0.21 + 12*0.088)/(0.587 + 0.115 + 0.21 + 0.088) = 5.5.

The variance is calculated by summing up the product of the squares of the values and their respective probabilities and then dividing it by the total probability. In this case, the variance is calculated as:

(1^2*0.587 + 3^2*0.115 + 9^2*0.21 + 12^2*0.088)/(0.587 + 0.115 + 0.21 + 0.088) = 13.875.

The standard deviation is the square root of the variance and it is calculated as √13.875 = 3.722.

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a) Since data were collected for "score quantitative" variable(s), the correct graph to make is a histogram

b) For this one the class width is given to be the following: class with = 10 .  See the attachment for table

c)  See the attachment for the graph

What is histogram ?

A histogram is a collection of rectangles with bases and the gaps between class boundaries. All of the rectangles are contiguous, and each rectangle bar represents some kind of data. Rectangles have heights that are proportionate to the corresponding frequencies for both related and dissimilar classes.

a) Since data were collected for "score quantitative" variable(s), the correct graph to make is a histogram

b) For this one the class width is given to be the following: class with = 10 .  See the attachment for table

c)  See the attachment for the graph

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