what is definition of integral?

To find the volume of a solid obtained by rotating a region bounded by curves around an axis, we can use the disk or washer method.

Suppose we want to find the volume of a solid obtained by rotating the region bounded by the curves y = f(x), y = g(x), and the lines x = a and x = b around the x-axis.

We can divide the region into vertical slices of width Δx, and rotate each slice around the x-axis to obtain a disk or washer. The volume of each disk or washer can be approximated as the area of the base times the thickness, or π(R² - r²) Δx, where R and r are the outer and inner radii of the disk or washer, respectively.

To find the exact volume, we take the limit as Δx approaches zero and sum up the volumes of all the disks or washers using integration:

V = ∫[a, b] π(R² - r²) dx

where R = f(x) and r = g(x).

Alternatively, if we want to rotate the region around the y-axis, we can divide the region into horizontal slices of width Δy, and rotate each slice around the y-axis to obtain a disk or washer. The volume of each disk or washer can be approximated as π(R² - r²)Δy, where R and r are the outer and inner radii of the disk or washer, respectively.

The exact volume can be found using integration as:

V = ∫[c ,d] π(R² - r²) dy

where R = f(y) and r = g(y), and c and d are the y-coordinates of the endpoints of the region.

In either case, it is important to sketch the region and identify the appropriate radii before setting up the integral.

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A p-value two-tailed calculator is a statistical tool used to calculate the p-value for a two-tailed test of a population mean or proportion.

A p-value two-tailed calculator is an online tool used in statistical analysis to determine the probability value of a two-tailed hypothesis test. This calculator helps researchers and analysts to determine if there is a significant difference between two population means or proportions. It calculates the p-value, which is the probability of obtaining results as extreme or more extreme than the observed results under the null hypothesis.

Users input sample data, population parameters, significance level, and type of test to obtain the calculated p-value. This tool is essential in hypothesis testing in fields such as psychology, medicine, and economics.

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Rounding to the nearest tenth, we get Average rate of change ≈ -27.5

In mathematics, an average is a measure of central tendency that represents the typical or common value of a set of data. There are several types of averages, including:

Arithmetic mean: The arithmetic mean is the most commonly used measure of average. It is calculated by adding up all the values in a dataset and dividing by the total number of values. For example, the arithmetic mean of the numbers 1, 2, 3, 4, and 5 is (1 + 2 + 3 + 4 + 5) / 5 = 3.

Median: The median is the middle value in a dataset when the values are arranged in numerical order. If there is an even number of values, the median is the average of the two middle values. For example, the median of the numbers 1, 3, 4, 6, and 7 is 4.

Mode: The mode is the value that appears most frequently in a dataset. If there are multiple values that appear with the same frequency, the dataset is said to have multiple modes. For example, the mode of the numbers 1, 2, 2, 3, and 4 is 2.

Averages are used in a wide range of applications, including statistics, finance, and science. They can help summarize data, identify trends, and make comparisons between different sets of data. However, it is important to note that averages can be affected by outliers or extreme values in a dataset, so it is important to consider other measures of variability and dispersion when analyzing data.

The average rate of change of the cost from the 6th month to the 8th month is given by the expression:

(Amount of change in cost) / (Amount of change in month)

The amount of change in cost from the 6th to the 8th month is:

c(8) - c(6) = 100 - 155 = -55

The amount of change in month from the 6th to the 8th month is:

8 - 6 = 2

Therefore, the average rate of change of the cost from the 6th month to the 8th month is:

(Amount of change in cost) / (Amount of change in month) = -55 / 2 ≈ -27.5

Rounding to the nearest tenth, we get:

Average rate of change ≈ -27.5

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At 95% confidence, the margin of error for the population mean is:

3.920

The margin of error for the population mean can be calculated using the formula:

margin of error = (z-score)*(standard deviation/√(sample size))

Where z-score is the value that corresponds to the desired confidence level, standard deviation is the square root of the variance, and sample size is the number of items in the sample.

Given that the variance is 900, the standard deviation is √(900) = 30. The sample size is 225, and the z-score for a 95% confidence level is 1.96.

Plugging these values into the formula, we get:

margin of error = (1.96)*(30/√(225))

margin of error = (1.96)*(30/15)

margin of error = (1.96)*(2)

margin of error = 3.92

Rounding to three decimal places, the margin of error is 3.920.

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As per the information provided, a = 4√3/3 + 4, b = 4 - 4√3/3 the answer can be calculated with optimization method. it will be as follows:

Sum of a and b is 8, we get

a+b=8

b=8−a

Now, let x=a−b

Then we get,

[tex]x=a−(8−a) \\ x=2a−8 \\ x+8=2 \\ 1 \div 2x+4=a

[/tex]

we use this to answer to solve for b

[tex]b=8−a \\ =8−(1 \div 2x+4) \\ =4−1 \div 2x

[/tex]

Now we use the product of two numbers and its difference. This can be expressed as:

[tex]a⋅b⋅x=(1 \div 2x+4)(4−1 \div 2x) \\ x=2 {x}^{2} − \frac{1}{4} {x}^{3} +16x−2x^{2} \\ =−14x^{3} +16x

[/tex]

Thus, this expression that we need to maximize. Take the derivative, set it equal to zero, and solve for x

[tex]−3 \div 4x ^{2} +16=0 \\ 16 =3 \div 4 x ^{2} \\ 643=x \\ 28√3 \div 3=x[/tex]

Now for us to check that this is a maximum, we have to note that the second derivative is

[tex]−3 \div 2x

\\ At \\

x=8√3 \div 3

[/tex]

the second derivative is −4√3. Since this number is negative, the point is a maximum.

Now we must find the values of a and b for this x. We have to use the relationship

[tex]a=1 \div 2x+4[/tex]

[tex]a=1 \div 2 \times 8√3 \div 3+4 \\ =4√3 \div 3+4[/tex]

now we use the relationship b=8−a

[tex]b=8−(4√3 \div 3+4) \\ =4−4√3 \div 3[/tex]

The first step in determining a function's maximum or minimum value is differentiating it. Then, set this derivative to zero and conduct the computation.

x. This will reveal the location of a function's maximum or minimum, but it won't reveal which.

Take the second derivative to get more details. A local maximum occurs when both the first and second derivatives are negative. You have a local minimum when both the first derivative and the second derivative are zero.

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So the answer is: the first arc should pass through point B (not A). All other elements in the diagram appear to be correct.

The elements or members of a set are the parts that make it up. To divide the components of a set, commas and two curly (idle) braces are typically employed. Capital letters are always used to print the set's name. The set "A" in this context consists of the letters v, w, x, y, and z.

Based on the given diagram and instructions, it appears that the error in the construction is that the first arc is centered at point C instead of point A. The correct procedure for constructing a line parallel to line AB and passing through point P is as follows:

1.Draw a line AB and mark point P not on line AB.

2.From point P, draw an arc that intersects line AB at points D and E.

3.Draw another arc, centered at point D, with the same radius as the first arc. This arc should intersect the first arc at point F.

4.Draw a line through point P and point F. This line is parallel to line AB and passes through point P.

So the answer is: the first arc should pass through point B (not A). All other elements in the diagram appear to be correct.

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The third arc should be drawn centred at F.

What is an arc?

In geometry, an arc is a curved segment of a circle or any curved line. It is a part of a curve that is bounded by two distinct endpoints and is characterized by its length, its curvature, and its central angle.

Constructing a parallel line to line AB and passing through point P.

To find:

The error occurred in the construction.

The procedure is,

The first arc must be drawn centred at C.

The second must be drawn centred at P.

Finally, the third arc must be drawn centred at F.

But here, the third arc is drawn centred at D. This is wrong.

The correct step is that the third arc must be centred at F.

Hence, The third arc should be drawn centred at F.

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The required height of the triangle whose area and base are given is 9 yd.

The area of a triangle in a two-dimensional plane is the space that it completely encloses. A triangle is a closed shape with three sides and three vertices, as is pretty obvious.The area of a triangle is the total space occupied by its three sides. The general formula for determining the triangle's area is half of the product of its base and height.

In the given question, we have to find the height of the given triangle ,

we know that ,

Area of triangle = base x height x 1/2.

63.9 = 1/2 × 14.2 × height

Height = (63.9 × 2) ÷ 14.2

Height = 127.8 ÷ 14.2

Height = 9 yd

Therefore , the height of the triangle is 9 yd .

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Yes, in solid geometry a square is a rhombus. A rhombus has all four sides equal just like a square for why it is also known as a titled square.

A rhombus can be thought of as a special parallelogram since it is a quadrilateral with two pairs of parallel sides. Similar to a square, a rhombus likewise has equal sides on each of its four sides. As a result, it's frequently referred to as a tilted square.

According to definition, a square is a regular quadrilateral with four equal sides and four equal 90-degree angles.

Rhombus also has four equal sides.

Therefore, a square is a rhombus with all of its angles being right angles.

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The complete question is:

Given. a square proof it’s a rhombus.

The conversion of mass unit lbs, 163 lbs to other mass unit kilogram is equals to the 73.93555631 kg.

Kilogram (kg) and pound (lbs) are used to measure the mass of the substance/objects. Kilogram is used for mass in the International System of Units, whereas pound is used in the imperial and US customary systems. To convert lbs to kg, multiply the particular lbs value by 0.45359237 kg. For example, to convert x lbs to kilogram, multiply the x lbs by 0.45359237 kg. So,x lbs = x × 0.45359237 kg

x lbs = 0.45359237x kg, if x = 6

Therefore, 6 lbs is approximately equal to 2.72155422 kg. It is very important skill to convert the lbs to kilograms in mass measurment. Now, we have 163 lbs. We know that one lbs = 0.45359237 kg

Therefore, 163 lbs = 163 x 0.45359237 kg

163 lbs = 73.93555631 kg.

Hence, 163 lbs is equal to 73.93555631 kg.

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The value of x in the triangle is 3.4 cm

From the question, we have the following parameters that can be used in our computation:

The triangle

Using the above as a guide, we have the following:

tan(43) = 3.2/x

Make x the subject of the formula

So, we have the following representation

x = 3.2/tan(43)

Evaluate

x = 3.4

Hence, the solution is 3.4

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A) 500

B) 54 minutes

C) 68cents and 60 cents

D)

E) $40.95

A proportion is a statement that says that two ratios are equal. They can be used in many everyday situations like comparing sizes, cooking, calculating percentages, and more.

Proportions can be written as equivalent fractions or as equal ratios.

A) 42% = 210

100% = (210 x 100)/42

= 500 people

B) 135 berries = 15 minutes

    486 berries = ??minutes

= (486 x 15)/135

= 54 minutes

C) unit price = $6.75/10 = 0.675cents or 68cents

    ii) unit price = $7.25/12 = 0.604cents or 60cents

D) Inconclusive question.

E) 40% of 65 = $26

65 - 26 = $39

Tax = 5% = $1.95

Total = $39 + 1.95

Final Price of jacket = $40.95

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The table mentioned in the question is not provided. However, we can use the following formula to find the probability of two independent events occurring together:

P(A and B) = P(A) x P(B)

Here, A represents the event of the coin landing heads up, and B represents the event of the pointer landing on either 1, 2, or 4.

Assuming that the spinner has equal sections, the probability of the pointer landing on either 1, 2, or 4 is 3/8 (since there are three out of eight possible sections on the spinner).

The probability of flipping a fair coin and getting heads is 1/2.

Therefore, the probability of the coin landing heads up and the pointer landing on either 1, 2, or 4 is:

P(A and B) = P(A) x P(B) = (1/2) x (3/8) = 3/16

So, the probability of the coin landing heads up and the pointer landing on either 1, 2, or 4 is 3/16.

The probability of the coin landing heads up and the pointer landing on either 1, 2, or 4 is 3/16.

Probability denotes the possibility of the outcome of any random event. The meaning of this term is to check the extent to which any event is likely to happen.

Given that, a spinner is spined and coin is flipped,

P(A and B) = P(A) x P(B)

Here, A represents the event of the coin landing heads up, and B represents the event of the pointer landing on either 1, 2, or 4.

Assuming that the spinner has equal sections, the probability of the pointer landing on either 1, 2, or 4 is 3/8 (since there are three out of eight possible sections on the spinner).

The probability of flipping a fair coin and getting heads is 1/2.

Therefore, the probability of the coin landing heads up and the pointer landing on either 1, 2, or 4 is:

P(A and B) = P(A) x P(B) = (1/2) x (3/8) = 3/16

So, the probability of the coin landing heads up and the pointer landing on either 1, 2, or 4 is 3/16.

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The values of  x and y are

Figure (a)

From the question, we have the following parameters that can be used in our computation:

Parallel and congruent lines

This means that

5y - 8 = 2y + 1

So, we have

3y = 9

Divide

y = 3

Also, we have

1/5x + 3 = 4x - 35

So, we have

x + 15 = 20x - 175

This gives

19x = 190

x = 10

Figure (a)

Here, we have

1/3y - 6 = 66 - 2/3y

This gives

y - 18 = 198 - 2y

Evaluate

3y = 216

Divide

y = 72

For x, we have

1/4x + 5 = 1/2x - 7

So, we have

1/2x =+ 12

Evaluate

x = 24

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The two triangles are similar by the Angle- Angle and also SIde -Side postulates.

A postulate is defined as a statement that is assumed to be true without any proof.

A theorem is defined as a true statement and is supported by a proof (or can be proven).

Here, two triangles can be said to be similar through many postulates :

1. △ABC and △DEF are similar if

angle ABC is equal to angle DEF; and

angle BCA is equal to angle EFD

(or any other two corresponding angles are same)

This can be said that △ABC and △DEF are similar with Angle- Angle (AA) postulate.

2.  △ABC and △DEF are similar if

side AB/ BC is proportional to side DE/ EF;

(or ratio of two corresponding sides are similar)

This can be said that △ABC and △DEF are similar with Side Side (SS) postulate.

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The scalar multiples of u that have a direction opposite that of u is -54  i + 21 j .

Initial Point of vector , u= (-7,2)= -7 i +  2 j

Terminal point of Vector , u= (11, -5)= 11 i -5 j

Vector , u= Terminal point - Initial Point

               = 11 i - 5 j - (-7 i+  2 j)

                = 11  i - 5 j + 7 i - 2 j

                = 18 i - 7 j

⇒Vector, v= 3 ×Direction Opposite to that of vector, u

             = -3 u

            = -3 × (18 i - 7 j )

            = 3× (-18 i + 7 j)

             = -54  i + 21 j

Hence, The scalar multiples of u that have a direction opposite that of u is -54  i + 21 j .

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Number of significant figures are in the measured number 0.00208 m is option (C) three

The significant figures in a measured number are the digits that carry meaning or contribute to the precision of the measurement. In general, significant figures include all non-zero digits and any zeros between non-zero digits.

In this case, the measured number is 0.00208 m. The first two zeros in this number are not significant because they are only placeholders to indicate the decimal point's position. The first significant digit is 2, which is followed by two more significant digits, 0 and 8. Therefore, there are three significant figures in the measured number.

Therefore, the correct option is (C) three

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The perimeter of the rectangle, in terms of x, is [tex]9x + 4 + 2/(2x+1).[/tex]

What is area of a rectangle?

The formula for the area of a rectangle is multiply length x width.

We know that the area of a rectangle is equal to the product of its length and b. In this case, the area is given by the expression [tex]2x^2+5x+2[/tex], and one side has a length of (2x+1) units.

To find the width of the rectangle, we can divide the area by the length:

Width = Area / Length

Width = ([tex]2x^2+5x+2) / (2x+1)[/tex]

The perimeter of the rectangle is the sum of the lengths of all four sides. Since we know one side has a length of (2x+1), the other side must have a length of:

Width = [tex](2x^2+5x+2) / (2x+1)[/tex]

Other side = Area / Length = ([tex]2x^2+5x+2) / (2x+1)[/tex]

Therefore, the perimeter of the rectangle is:

Perimeter = 2(Length + Width)

= [tex]2[(2x+1) + (2x^2+5x+2)/(2x+1)][/tex]

Simplifying this expression, we get:

Perimeter = [tex]4x + 2 + 5x + 2 + 2/(2x+1)[/tex]

                [tex]= 9x + 4 + 2/(2x+1)[/tex]

So the perimeter of the rectangle, in terms of x, is   [tex]9x + 4 + 2/(2x+1).[/tex]

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We can calculate this using this equation:

[tex]V =\dfrac{4}{3}\pi r^3[/tex]

[tex]V =\dfrac{4}{3}\pi r^3[/tex]

We're given that the radius is 9 cm:

[tex]V =\dfrac{4}{3}\pi (9)^3\\\\V=3053.63[/tex]

V = 3053.63 cm³

Therefore, the area of the quarter circle of radius 8 cm is 16π square centimeters (or approximately 50.2655 square centimeters if we use a decimal approximation for π).

A circle is a two-dimensional shape that is defined as the set of all points that are a fixed distance (called the radius) from a central point (called the center) in a plane. It is one of the most basic geometric shapes and is often used in a variety of mathematical and scientific contexts. One important thing to note about circles is that they are symmetrical: any line passing through the center of a circle divides the circle into two halves that are mirror images of each other. Circles are commonly used in a variety of contexts, including geometry, trigonometry, physics, engineering, and many other fields. They are also used in everyday life, such as in the design of wheels and other circular objects, the calculation of the areas of circular fields or gardens, and the measurement of circular objects like plates and bowls.

Here,

The formula for the area of a quarter circle is given by:

A = (1/4)πr²

where A is the area of the quarter circle, r is the radius of the circle, and π is the mathematical constant pi (approximately 3.14159).

In this case, the radius of the quarter circle is 8 cm, so we can substitute this value into the formula and simplify as follows:

A = (1/4)π(8 cm)²

A = (1/4)π(64 cm²)

A = 16π cm²

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On factorizing each equation we get

a) x² + 6x + 9 =  (x + 3) (x + 3)  

b) x² - 6x + 8 = (x - 4) (x - 2)  

c) x² - 4x - 32 = (x - 8) (x + 4)  

d) x² + x - 42 = (x + 7) (x - 6)  

Factorization is the process of finding the factors of an expression or number. In algebra, factorization involves expressing an algebraic expression as a product of its factors.  

To factorize the following split the middle term and group out the common factors as given below.

Here we have four equations that can be factorized as follows

a) x² + 6x + 9  

=> x² +3x + 3x + 9         [ Split middles term ]

=> x(x + 3) +3(x + 3)      [ Grouping the terms ]

=> (x + 3) (x + 3)  

∴  a) x² + 6x + 9 =  (x + 3) (x + 3)  

b) x² - 6x + 8

=> x² - 4x - 2x + 8               [ Split middles term ]

=> x(x - 4) -2(x - 4)              [ Grouping the terms ]  

=> (x - 4) (x - 2)

∴ x² - 6x + 8 = (x - 4) (x - 2)  

c) x² - 4x - 32  

=> x² - 8x + 4x - 32         [ Split middles term ]  

=> x(x - 8) + 4(x - 8)         [ Grouping the terms ]    

=> (x - 8) (x + 4)

∴ x² - 4x - 32 = (x - 8) (x + 4)  

d) x² + x - 42  

=> x² + 7x - 6x - 42             [ Split middles term ]  

=> x(x + 7) - 6(x + 7)            [ Grouping the terms ]    

=> (x + 7) (x - 6)

∴ x² + x - 42 = (x + 7) (x - 6)  

Therefore,

On factorizing each equation we get

a) x² + 6x + 9 =  (x + 3) (x + 3)  

b) x² - 6x + 8 = (x - 4) (x - 2)  

c) x² - 4x - 32 = (x - 8) (x + 4)  

d) x² + x - 42 = (x + 7) (x - 6)  

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The height of the tree is given as follows:

44.4 feet.

The three trigonometric ratios are defined as follows:

For the height, we have that:

Hence the height is obtained as follows:

tan(32º) = h/71.

h = 71 x tangent of 32 degrees

h = 44.4 feet.

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A function to represent the total charges for your hotel and car rental is Option C.

In mathematics, a function is a relation between two sets of values, where each input value from the first set (called the domain) corresponds to exactly one output value in the second set (called the range). The output value of a function is determined by its input value and any relevant rules or formulas.

Formally, a function can be defined as follows: Let X and Y be two non-empty sets. A function f from X to Y is a rule or formula that assigns to each element x in X a unique element y in Y, denoted by f(x), such that for any two elements x1 and x2 in X, if x1 = x2, then f(x1) = f(x2). The set X is called the domain of the function, and the set Y is called the codomain or range of the function.

Functions can be represented graphically, algebraically, or in tabular form. The graph of a function shows how the output value of the function varies with the input value, and can be used to visualize the behavior of the function. The algebraic representation of a function can be given by a formula or equation that expresses the output value in terms of the input value. A tabular representation of a function shows the input-output pairs in a table.

Functions are used in many areas of mathematics, science, engineering, and economics to model and describe relationships between variables. They are also used in practical applications, such as in computer programming, where they are used to define algorithms and data structures.

The total charges for the hotel and car rental can be found by adding the charges for the hotel and the car rental. Therefore, the function that represents the total charges can be found by adding the two given functions:

p(x) = 135x + (45 + 79x)

p(x) = 135x + 45 + 79x

p(x) = 214x + 45

Therefore, the function that represents the total charges for the hotel and car rental is p(x) = 214x + 45.

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The probability that someone would have done at least as well as the man with no ESP is 0.077, or 7.7%.

The probability that someone would have done at least as well as the man with no ESP is calculated by the binomial probability formula. This formula takes into account the number of trials, the number of successes, and the probability of success in each trial. In this case, the number of trials is 22, the number of successes is 16, and the probability of success in each trial is 50% (0.5). The binomial probability formula states that the probability of success given this information is 0.077. This means that the probability that someone would have done at least as well as the man with no ESP is 0.077, or 7.7%.

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Answer:  430000

Step-by-step explanation: you have to do pemdas

Answer: 430,000. /four hundred thirty thousand

Step-by-step explanation:

The value of 4.3 x 10^5 is 430,000.

To understand this, the notation 4.3 x 10^5 represents a number in scientific notation, where 4.3 is the coefficient or significand, and 10^5 is the exponent. The exponent of 10 tells us how many places to move the decimal point to the right. In this case, the exponent 5 means we need to move the decimal point five places to the right.

Starting with 4.3, we move the decimal point five places to the right to get 430,000. So, 4.3 x 10^5 is equivalent to 430,000.

The letter "F" has one reflectional symmetry. This symmetry can be seen by drawing a line that divides the letter "F" into two equal parts, such that the left and right sides are mirror images of each other.

This line of symmetry would run vertically down the middle of the letter "F", passing through the crossbar. The letter "F" has one reflectional symmetry. A reflectional symmetry is a type of symmetry where one half of an object is a mirror image of the other half. In the case of the letter "F", a line can be drawn vertically down the middle of the letter, passing through the crossbar, such that the left and right sides of the letter are mirror images of each other. This line is called a line of symmetry. Therefore, the letter "F" has one reflectional symmetry.

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The figure which is missing in the question is attached here-with the answer

Answer:

................ ⅓........

The measures of the other two sides of LMN are approximately 10.67cm and 16 centimeter.

We can start by using the fact that similar triangles have proportional side lengths.

Let x be the length of the side of LMN that is similar to EF. Then, we can set up the following proportion:

 [tex]\frac{LMN}{EF}=\frac{x}{15}[/tex]

We can cross-multiply to get:

[tex]LMN=\frac{x}{15} \times EF[/tex]

Next, we can use the fact that EFD is similar to LMN to set up another proportion:

[tex]\frac{LMN}{EFD}=\frac{x}{6}[/tex]

We can cross-multiply to get:

[tex]LMN=\frac{x}{6} \times EFD[/tex]

Now we can substitute the expressions we found for LMN in terms of x into both sides of the equation:

[tex]\frac{x}{15} \times EF[/tex]=   [tex]\frac{x}{6} \times EFD[/tex]

Substituting the given values, we get:

[tex]\frac{x}{15} \times 15[/tex] =   [tex]\frac{x}{6} \times 8[/tex]

Simplifying, we get:

x = 20

The length of the other two sides of LMN are:

 [tex]\frac{20}{15} \times 8[/tex]= 10.67cm and  [tex]\frac{20}{15} \times 12[/tex]

= 16cm

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The number of terms in the expression, 6 + 2 x - 4 y + 5 z is 4 terms.

In the expression 6 + 2x - 4y + 5z, the number of terms is four, not the number of signs. The terms in this expression are:

Each term is separated by an operator (either addition or subtraction), which is represented by a sign. Therefore, the expression contains three addition signs and one subtraction sign.

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The full question is:

How many terms are in the following expression 6 + 2 x - 4 y + 5 z

The values are 45, 4.5, 0.45, and 0.045.

Numbers are representations of quantity or quality.

We have many types of numbers such as,

- Decimal numbers

- Real numbers

- Rational numbers

- Irrational numbers

- Complex numbers

- Even numbers

- Odd numbers

- Whole numbers

We have,

We need to multiply the numbers without using a calculator.

The steps to multiply two decimal numbers without using a calculator.

Step 1:

Ignore the decimal points and multiply the two numbers as if they were whole numbers.

Step 2:

Count the total number of digits to the right of the decimal points in both numbers.

Step 3:

Add the number of digits from step 2, and place the decimal point in the product so that it has that many digits to the right.

Now,

3 x 15 =45

0.3 x 15 = 4.5

3 x 0.015 = 0.045

0.03 x 15 = 0.45

3 x 1.5 = 4.5

3 x 0.15 = 0.45

0.3 x 1.5 = 0.45

0.3 x 0.15 = 0.045

Thus,

The values are 45, 4.5, 0.45, and 0.045.

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The complete question.

Multiply the two decimal numbers without using a calculator,

3 x 15

0.3 x 15

3 x 0.015

0.03 x 15

3 x 1.5

3 x 0.15

0.3 x 1.5

0.3 x 0.15