Find the area A of the sector shown in the picture
76 degrees
6

Answer: x = 5 and x = 1.

Step-by-step explanation:

To solve fg(x) = gf(x), we need to find the expressions for fg(x) and gf(x) and then set them equal to each other.

fg(x) = f(g(x)) = f(x² + 2) = 2(x² + 2) - 3 = 2x² + 1

gf(x) = g(f(x)) = g(2x - 3) = (2x - 3)² + 2 = 4x² - 12x + 11

Now we set fg(x) equal to gf(x) and solve for x:

2x² + 1 = 4x² - 12x + 11

2x² - 12x + 10 = 0

Dividing both sides by 2 gives:

x² - 6x + 5 = 0

This quadratic equation factors as:

(x - 5)(x - 1) = 0

So the solutions are x = 5 and x = 1.

Therefore, the solutions to fg(x) = gf(x) are x = 5 and x = 1.

When the data is skewed, it is untrue that the mean and standard deviation are better indicators of the centre and spread.

The median is a better tool to use to locate the centre when it is skewed right or left with high or low outliers. The IQR is the most accurate indicator of spread when the median is the centre. When the mean is the centre, the standard deviation should be utilised because it gauges how far a data point is from the mean.

The standard deviation will be greatly overstated in cases when the distribution of the data is highly skewed, making it a poor choice as a measure of variability.

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The probability that, out of 18 independent samples received from one river, just two were contaminated is 0.8438.

In the following 18 samples to be evaluated,

Let X = the number of samples that now the pollutant is present in.

Thus, with p = 0.10 and n = 18, X is a binomial random variable.

Using the binomial theorem:

[tex](^{n} _{r} ) p^{x} q^{n-x}[/tex]

p = 0.10

q = 1 - 0.10 = 0.9

n = 18

The likelihood that only two samples out of 18 obtained in different ways from the same river were polluted

P(x = 2) = [tex](^{18} _{2} ) (0.1)^{2} (0.9)^{18-2}[/tex]

=  [tex](^{18} _{2} ) (0.1)^{2} (0.9)^{16}[/tex]

= 153 x 0.01 x 0.1853

= 0.8438

Thus, the probability that, out of 18 separate samples received from one river, just two were contaminated is 0.8438.

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Therefore , the solution of the given problem of unitary method comes out to be  the attraction sold 943 child tickets and 736 adult tickets on that particular day.

It is possible to accomplish the objective by using previously recognized variables, this common convenience, or all essential components from a prior malleable study that adhered to a specific methodology. If the expression assertion result occurs, it will be able to get in touch with the entity again; if it does not, both crucial systems will undoubtedly miss the statement.

Here,

Assume the attraction sold x tickets for adults and y tickets for kids.

Based on the supplied data, we can construct the following two equations:

=>  x + y = 1679 (equation 1, representing the total number of tickets sold)

=>  35x + 20y = 44620 (equation 2, representing the total revenue generated)

Using the elimination technique, we can find the values of x and y.

When we divide equation 1 by 20, we obtain:

=>  20x + 20y = 33580 (equation 3)

Equation 3 is obtained by subtracting equation 2 to yield:

=>  15x = 11040

=>  x = 736

When we enter x = 736 into equation 1, we obtain:

=>  736 + y = 1679

=> y = 943

As a result, the attraction sold 943 child tickets and 736 adult tickets on that particular day.

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Answer:

m = -1

Step-by-step explanation:

may not be accurate, I haven't done this in a while

Answer:

-1y−y1=m(x−x1)

y−6=−1(x+5)

y−6=−1x+(−1×5)

y−6=−1x+−5

y−6=−1x−5

y=−1x−5+6

y=−1x+1

y=−x+1

m=−1

b=1

Step-by-step explanation: Hope this helps!! Mark me brainliest!

Answer:

y > 0

Step-by-step explanation:

[tex]y = e^{4x}[/tex]

This is an exponential function;

Consider as x → ∞, y → ∞;

If x = 0. y = e⁰ = 1;

As x → -∞, y → 0;

y ranges from 0 to ∞ therefore, i.e. y > 0

The compound interest on ₹20,000 for 3 years at 10% per annum compounded annually is ₹6,620.

To calculate the compound interest on a principal amount of ₹20,000 for 3 years at 10% per annum compounded annually, we can use the formula

A = P(1 + R/100)^n

Where,

A = final amount after n years

P = principal amount

R = annual interest rate

n = number of years

In this case, P = ₹20,000, R = 10%, and n = 3 years.

So, applying the formula

A = 20000(1 + 10/100)^3

= 20000(1.1)^3

= 20000(1.331)

= ₹26,620

The final amount after 3 years is ₹26,620. Therefore, the compound interest is

Compound Interest = Final Amount - Principal Amount

= ₹26,620 - ₹20,000

= ₹6,620

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I have solved the question in general, as the given question is incomplete.

The complete question is:

Find the compound interest on ₹ 20,000 for 3 years at 10% per annum compounded annually.

Every real root of a polynomial with rational coefficients is also a root of a polynomial with integer coefficients.

Suppose that c is a root of the polynomial equation:

[tex]q_n[/tex] × [tex]x^n[/tex] + q_{n-1} × [tex]x^{n-1}[/tex] + ... + q1 × x + q0 = 0

where [tex]q_i[/tex] are rational coefficients. Since c is a root of this polynomial equation, we have:

[tex]q_n[/tex] × [tex]c^n[/tex] + [tex]q_{n-1}[/tex] × [tex]c^{n-1}[/tex] + ... + q1 × c + q0 = 0

Multiplying both sides of the equation by the common denominator of the coefficients [tex]q_i[/tex], we can obtain an equation with integer coefficients. Let d be the least common multiple of the denominators of the coefficients [tex]q_i[/tex]. Then we can write:

d × ([tex]q_n[/tex] × [tex]c^n[/tex] + [tex]q_{n-1}[/tex] × [tex]c^{n-1}[/tex] + ... + q1 × c + q0) = 0

Expanding the left-hand side of the equation, we obtain a polynomial with integer coefficients:

d × [tex]q_n[/tex] × [tex]x^n[/tex] + d × [tex]q_{n-1}[/tex] × [tex]x^{n-1}[/tex] + ... + d × q1 × x + d × q0 = 0

Since c is a root of the original polynomial equation, it is also a root of this polynomial with integer coefficients. Therefore, we have shown that if c is a root of a polynomial with rational coefficients, then it is also a root of a polynomial with integer coefficients.

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The horizontal line at $7.25 on the y-axis of the graph is the one with a slope that most accurately depicts the federal minimum wage of $7.25 per hour as of January 1, 2014.

Although it varies from state to state, the federally mandated minimum wage in the United States is $7.25 per hour. The District of Columbia had the highest minimum wage in the US as of January 1, 2023, at 16.50 dollars per hour.

The variable dearness allowance (VDA) component, which takes into account inflationary trends, such as an increase or fall in the Consumer Price Index (CPI), and, if applicable, the housing rent, are included in the computation of the monthly minimum salary.

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Three youngsters are interviewed at random. Population variation is around [tex]18.67[/tex].

The standard deviation refers to the square base of variance, which is the average of the squares departures from the mean. Both metrics capture distributional variability, although they use different measurement units: The units used to indicate standard deviation are the same as the values' original ones.

In statistics, population standard deviation is a crucial indicator of dispersion. further reading. Statisticians compute variance to determine how order to overcome the drawbacks in a data gathering interact to one another. By calculating the population variance, one may also compute the dispersion in relation to the population means.

we need to first find the population mean [tex]u[/tex]

[tex]u = (2 + 4 + 12)/3 = 6[/tex]

To calculate the population variance

[tex]= [(-4)^{2} + (-2)^{2} + 6^{2} ]/3[/tex]

[tex]= (16 + 4 + 36)/3[/tex]

[tex]= 56/3[/tex]

[tex]= 18.67[/tex] (rounded to two decimal places)

Therefore, the population variance is approximately [tex]18.67[/tex].

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Answer:

b. 5π/6 is not a solution to cosØ=3√2.

The correct option is -C: No, 6 is the geometric mean of 4 and 9, however if the altitude is 6, then the hypotenuse is the geometric mean of the two segments.

An average technique multiplies several values and determines the number's root is known as the geometric mean. You locate the nth root for their product for a collection of n numbers. This descriptive statistic can be used to sum up your data.

Mean Geometric The square root of the product of two numbers is the geometric mean amongst them. The geometric mean of two positive numbers an as well as b is the positive number x as in percentage Cross multiplication results in x² = ab,.

For the given question.

geometric mean of a and b :

From the drawn diagram.

a = 4

b = 9

x = √ab

x = √9*4

x = 6

geometric mean: 6

Applying the altitude rule:

h² = x.y

6² = 9*4

36 = 36

Thus, the geometric mean calculated by friend is correct but the marking on the diagram is wrong.

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D. A statistic is a random variable. A statistic is a figure calculated from a sample, such as the sample mean or the sample standard deviation.

Every statistic is a random variable since samples are chosen at random; these variations cannot be anticipated in advance. It has a mean, a standard deviation, and a probability distribution as a random variable. A statistic's sample distribution is its probability distribution. Generally, sample statistics are calculated to estimate the associated population parameters rather than being ends in themselves. The concepts of mean, standard deviation, and sampling distribution of a sample statistic are introduced in this chapter, with a focus on the sample mean.

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Complete question:

which of the following is true about a sample statistic such as the sample mean or sample proportion?

A. A statistic is constant.

B. A statistic is always known.

C. A statistic is a parameter.

D. A statistic is a random variable.

Answer:

There were 8 bears

Step-by-step explanation:

Letting L = number of lions, T = number of tigers and B = number of bears

L : T = 3 : 2

We can rewrite this as
L/T = 3/2

Cross multiply:
L x 2 = 3 x T

Divide by 3 to get
T = 2/3 L

Since L = 9

T = 2/3 x 9 = 6

In the other ratio we have
T : B = 3 : 4 which we can write as
T/B = 3/4

Cross multiply to get

4T = 3B

B = 4/3 T

Since T = 6, B = 4/3 x 6 = 8

Check
L : T = 9 : 6 = 3: 2 (by dividing both sides of : by 3)
T : B = 6 : 8 = 3:4 (by dividing both sides of : by 2)

After answering the provided question, we can conclude that  This expression process frequently results in bolt diameters that are normally distributed, with minor random variations around the target diameter.

In mathematics, an expression is a collection of symbols, digits, and companies that portray a statistical correlation or formula. An expression can be a single number, a mutable, or a combination of both of them. Addition, subtraction, proliferation, division, and exponentiation are examples of mathematical operators.  Expressions are used extensively in mathematics, including arithmetic, calculus, and geometry. They are used in mathematical formula representation, equation solution, and mathematical relationship simplification.

a) Because of the central limit theorem, the volume of soft drinks in cans is likely to follow a normal distribution. Because soft drink manufacturers typically produce a large number of cans filled with the same volume of liquid, the sample size is large.

b) The diameter of bolts immediately after manufacture is also likely to be distributed normally. Bolt manufacturing typically involves a large number of measurements and adjustments to ensure that the bolts are manufactured to exact specifications. This process frequently results in bolt diameters that are normally distributed, with minor random variations around the target diameter.

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Answer:

3x(x - 4y)

Step-by-step explanation:

3x² - 12xy ← factor out 3x from each term

= 3x(x- 4y)

Therefore , the solution of the given problem of unitary method comes out to be $500.

This common convenience, already-existing variables, or all important elements from the original Diocesan adaptable study that followed a particular methodology can all be used to achieve the goal. Both of the crucial elements of a term affirmation outcome will surely be missed if it doesn't happen, but if it does, there will be another chance to get in touch with the entity.

Here,

Reginald's total moving costs must first be determined in order to determine the moving expense deduction he may claim on his 2022 tax return.

The full cost of relocating is:

=> $450 (gasoline) + $250 (truck rental) + $100 (tow trailer) + $90 (food) + $35 (Starbucks) + $300 (motel lodging) + $175 (storage)

=> $1,400

Reginald's moving expenditures were covered by the government for $900, so the following are his actual moving costs:

=> $1,400 - $900 = $500

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Consider the line segment AB shown the locations for point C makes ABC a right triangle with hypotenuse AB

In a coordinate pair the points are as represented as (x, y).

The point that forms the right triangle is located by tracing the point on the x axis of of the point A and the point on the y axis of the point B.  This is done below

therefore we can say that the point C that forms the right triangle is

C (2, 3)

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If a is a zero of the polynomial P(x), then (x-a) must be a factor of P(x) and [tex](x-a)^m[/tex] be a factor of P(x) when we factor P completely.

The values of x that fulfil the formula f(x) = 0 are the zeros of a polynomial. The polynomial's zeros are the x values for which the function's value, f(x), equals zero in this case. The degree of the equation f(x) = 0 determines how many zeros a polynomial has.

The locations when a polynomial equals 0 overall are known as its zeros. In layman's terms, we may state that a polynomial's zeros are variable values at which the polynomial equals 0. The zeros of a polynomial are often referred to as the equation's roots and are frequently written as,, and. A few techniques for locating polynomial zeros include grouping, factoring, and employing algebraic expressions.

(a) if we have zero at x=a of polynomial P(x)

then,  (x-a)  must be factor of P(x).

(b) if we have zero at x=a of polynomial P(x)

with multiplicity=m

then,  [tex](x-a)^m[/tex]  must be factor of P(x).

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Step-by-step explanation:

4)

based on similar triangles and the common ratio for all pairs of corresponding sides we know

LE/LM = LD/LK = DE/EM

because E and D are the midpoints of the longer sides, all of these ratios are 1/2.

1/2 = DE/8

8/2 = 4 = DE

5)

same principle as for 4)

BQ/BA = BR/BC = QR/AC

again, Q and R are the midpoints, so all these ratios are 1/2.

1/2 = QR/10

QR = 10/2 = 5

Answer:

Step-by-step explanation:

To solve this problem, we need to use the equations of motion for constant acceleration, constant velocity, and constant deceleration. We'll break the problem into several parts and use these equations to find the distance traveled in each part. Then, we'll add up the distances to get the total distance traveled.

First, we need to convert the units of speed from km/h to m/s, since the equations of motion use meters per second. We have:

Initial speed (u) = 0 km/h = 0 m/s

Final speed (v) = 40 km/h = 11.11 m/s

Constant speed = 40 km/h = 11.11 m/s (for 4 minutes)

Final speed before hill = 30 km/h = 8.33 m/s

Speed at top of hill = 0 m/s

Acceleration (a) = (v-u)/t = (11.11-0)/(3*60) = 0.0611 m/s^2

PART 1: ACCELERATION PHASE

Time taken (t) = 3 minutes = 180 seconds

Distance traveled (s) = ut + (1/2)at^2

s = 0 + (1/2)0.0611(180^2) = 331.83 meters

PART 2: CONSTANT SPEED PHASE

Time taken (t) = 4 minutes = 240 seconds

Distance traveled (s) = vt

s = 11.11*240 = 2666.4 meters

PART 3: DECELERATION PHASE

Time taken (t) = 1 minute = 60 seconds

Deceleration (a) = (v-u)/t = (8.33-11.11)/60 = -0.0461 m/s^2 (negative since it's deceleration)

Distance traveled (s) = vt + (1/2)at^2

s = 8.3360 + (1/2)(-0.0461)*(60^2) = 494.7 meters

PART 4: CONSTANT SPEED PHASE

Time taken (t) = 10 minutes = 600 seconds

Distance traveled (s) = vt

s = 8.33*600 = 4998 meters

PART 5: DECELERATION PHASE TO STOP

Time taken (t) = 2 minutes = 120 seconds

Initial speed (u) = 8.33 m/s

Final speed (v) = 0 m/s

Deceleration (a) = (v-u)/t = (0-8.33)/120 = -0.0694 m/s^2

Distance traveled (s) = vt + (1/2)at^2

s = 8.33120 + (1/2)(-0.0694)*(120^2) = 733.3 meters

TOTAL DISTANCE TRAVELED:

Adding up the distances from each part, we get:

Total distance = 331.83 + 2666.4 + 494.7 + 4998 + 733.3 = 9184.23 meters = 9.18 km (rounded to two decimal places)

Therefore, the cyclist has traveled approximately 9.18 km.

The answer of the given question based on solving  13 to the power of 16 is 13 to the power of 16 is 3,947,868,257,259,789. and the simplified expression is 7 + 2ab + br.

A expression is a combination of symbols or values that represents a particular concept or computation.

In mathematics, an expression is a combination of numbers, variables, operators, and/or functions that can be evaluated to produce a numerical result.

an expression is a way to represent an idea, computation, or meaning using a set of symbols or words.

To solve 13 to the power of 16, we can start by multiplying 13 by itself 16 times:

13 × 13 = 169

169 × 13 = 2197

2197 × 13 = 28,561

28,561 × 13 = 371,293

371,293 × 13 = 4,826,389

4,826,389 × 13 = 62,748,857

62,748,857 × 13 = 815,730,721

815,730,721 × 13 = 10,604,807,473

10,604,807,473 × 13 = 137,858,491,849

137,858,491,849 × 13 = 1,792,160,390,737

1,792,160,390,737 × 13 = 23,303,986,079,681

23,303,986,079,681 × 13 = 303,305,489,096,753

303,305,489,096,753 × 13 = 3,947,868,257,259,789

Therefore, 13 to the power of 16 is 3,947,868,257,259,789.

As for the second expression, we can simplify it using the distributive property of multiplication:

2(2 + ab) + b(r + 3) = 4 + 2ab + br + 3b

Simplifying further, we can combine the constant terms:

2(2 + ab) + b(r + 3) = 7 + 2ab + br

So the simplified expression is 7 + 2ab + br.

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In the given scenarios, Paired differences are option A, B, and D, the scenario which shows Differences between 2 independent samples is option C.

Paired differences refer to cases where the difference is measured within the same group of subjects or individuals, such as the difference in time spent with mother and father in a hetero family, or the difference in IQ between first-born and second-born twins.

Differences between 2 independent samples, on the other hand, refer to cases where the difference is measured between two separate groups, such as the difference in the number of books read per year by residents of California and New York.

Recognizing the type of difference is important in determining the appropriate statistical analysis to use in analyzing the data.

The answer option for paired are  A, B, and D and for Differences between 2 independent samples is option C.

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_____The given question is incomplete, the complete question is given below:

Which of the following cases represent paired differences and which are differences between 2 independent samples?

A The difference between the average time a child spends with their mother and the average time the child spends with their father (in a hetero family) [Select]

B The difference between the average IQ of the first-born twin and of the second-born twin [ Select]

C The difference between the average number of books read per year by the residents of the states of California and New York [Select] The difference between the average amount grossed by Paramount Pictures and Warner Bros. movies [Select ]

D The difference between the shares of the republican and the democratic candidate in each year of the US elections. [ Select ]

One pair of functions that satisfies the given condition is:

Exponential function: [tex]f(x) = 1.46^x,[/tex] Quadratic function: [tex]g(x) = x^2[/tex]

In mathematics, an expression is a combination of numbers, variables, and mathematical operations such as addition, subtraction, multiplication, and division. Expressions can also include functions, brackets, and other symbols.

Let's consider the two functions:

Exponential function: [tex]f(x) = a^x, where a > 1[/tex]

Quadratic function: [tex]g(x) = x^2[/tex]

We want to find the pair of functions for which the exponential is consistently growing at a faster rate than the quadratic over the interval 0 ≤ x ≤ 5.

To determine this, we can compare the growth rates of the two functions by looking at their derivatives.

The derivative of the exponential function is:[tex]f'(x) = a^x * ln(a)[/tex]

The derivative of the quadratic function is: [tex]g'(x) = 2x[/tex]

To compare the growth rates of the two functions, we need to compare their derivatives. We want to find the value of x for which the exponential function is growing faster than the quadratic function, i.e., where f'(x) > [tex]g'(x).\\f'(x) > g'(x)\\a^x * ln(a) > 2x[/tex]

Now, we can solve for x:

[tex]a^x * ln(a) > 2xln(a)/2 * a^x > x[/tex]

Since we want to find the pair of functions for which the exponential is consistently growing at a faster rate than the quadratic over the interval 0 ≤ x ≤ 5, we need to find a value of a such that the inequality ln(a)/2 * [tex]a^5 > 5[/tex] is true for all values of a > 1.

We can use a graphing calculator or a numerical solver to find the value of a that satisfies this inequality. One possible solution is a ≈ 1.46.

Therefore, one pair of functions that satisfies the given condition is:

Exponential function: [tex]f(x) = 1.46^x,[/tex] Quadratic function: [tex]g(x) = x^2[/tex]

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The MAD of the portions of popped balloons is 0.046.

The Mean Absolute Deviation (MAD) is a measure of the average distance between each data point and the mean of the data set. It gives an idea of how spread out the data is around the mean.

To find the mean, median, and MAD (mean absolute deviation) of the given data, we first need to find the average value of the portions of popped balloons.

Mean = (16% + 2/25 + 0.06 + 1/50 + 0.12 + 0.04) / 6

Mean = 0.0975

Therefore, the mean of the portions of popped balloons is 0.0975.

To find the median, we need to first arrange the portions in ascending order:

0.04, 0.06, 1/50, 2/25, 0.12, 16%

Median = (1/50 + 0.06) / 2

Median = 0.035

Therefore, the median of the portions of popped balloons is 0.035.

To find the MAD, we first need to find the absolute deviations of each portion from the mean. We can do this by subtracting the mean from each portion and taking the absolute value:

|0.16 - 0.0975| = 0.0625

|2/25 - 0.0975| = 0.0525

|0.06 - 0.0975| = 0.0375

|1/50 - 0.0975| = 0.0475

|0.12 - 0.0975| = 0.0225

|0.04 - 0.0975| = 0.0575

The MAD is the average of these absolute deviations:

MAD = (0.0625 + 0.0525 + 0.0375 + 0.0475 + 0.0225 + 0.0575) / 6

MAD = 0.046

Therefore, the MAD of the portions of popped balloons is 0.046.

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Answering the presented question, we may conclude that This greater expressions scale makes it easier to see the variations between the ratings of the three college students in a extra correct and informative way.

In mathematics, an expression is a collection of integers, variables, and complex mathematical (such as arithmetic, subtraction, multiplication, division, multiplications, and so on) that describes a quantity or value. Phrases can be simple, such as "3 + 4," or complicated, such as They may also contain functions like "sin(x)" or "log(y)". Expressions can be evaluated by swapping the variables with their values and performing the arithmetic operations in the order specified. If x = 2, for example, the formula "3x + 5" equals 3(2) + 5 = 11. Expressions are commonly used in mathematics to describe real-world situations, construct equations, and simplify complicated mathematical topics.

The difference in look between the two charts is due to the choice of the scales on the x and y-axes. In the left chart, the y-axis starts offevolved at 800 and has a range of only 200 points, whilst the x-axis starts offevolved at 1300 and has a vary of 200 points. This compressed scale makes the variations between the ratings of the three students appear larger than they absolutely are.

On the other hand, the proper chart has a y-axis that starts at zero and has a vary of 800 points, whilst the x-axis begins at 1200 and has a range of 800 points. This greater expanded scale makes it easier to see the variations between the ratings of the three college students in a extra correct and informative way.

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The difference in price between the two shops for 300 cm of ribbon is £3.60 - £2.40 = £1.20. Therefore, the answer is £1.20.

How to solve and what is Selling?

We can use proportions to find the cost of 300 cm of ribbon at each shop, and then subtract the cost at Shop B from the cost at Shop A to find the difference in price:

For Shop A:

140 cm of ribbon cost £1.68, so 1 cm of ribbon cost £1.68/140 = £0.012.

Therefore, 300 cm of ribbon would cost £0.012 x 300 = £3.60.

For Shop B:

215 cm of ribbon cost £1.72, so 1 cm of ribbon cost £1.72/215 = £0.008.

Therefore, 300 cm of ribbon would cost £0.008 x 300 = £2.40.

The difference in price between the two shops for 300 cm of ribbon is £3.60 - £2.40 = £1.20. Therefore, the answer is £1.20.

Selling is the process of exchanging goods or services for money or other valuable consideration. In business, selling is an essential part of the marketing and sales process, and involves identifying potential customers or clients, communicating with them about the features and benefits of the product or service being sold, and negotiating a price or other terms of the sale.

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The value of [tex](r_1+3)^{100}+(r_2+3)^{100}+...+(r_{100}+3)^{100}[/tex] is -1500.

What is function?

A mathematical phrase, rule, or law that establishes the link between an independent variable and a dependent variable (the dependent variable). In mathematics, functions exist everywhere, and they are crucial for constructing physical links in the sciences.

Here the given function is,

[tex]f(x)=(x^2+6x+9)^{50}-4x+3[/tex]

For any r , [tex](x^2+6x+9)^{50}-4x+3[/tex] is satisfy, Then take f(x)=0 then

=> [tex](x^2+6x+9)^{50}-4x+3=0[/tex]

Take x=r then

=> [tex](r^2+6r+9)^{50}-4r+3=0[/tex]

=> [tex](r^2+6r+9)^{50}=4r-3[/tex]

=> [tex]((r+3)^2)^{50}=4r-3[/tex]

=> [tex](r+3)^{100}=4r-3[/tex]

Then,

=> [tex]\sum_{i=1}^{100} (r_i+3)^{100}=\sum_{i=1}^{100} (4r_i-3)[/tex] = 4 × sum of roots - 300

Expanding [tex](x+3)^{100}-4x+3[/tex] using the binomial theorem, we get

=> f(x) = [tex]x^{100}+300x^{99}+....[/tex]

So sum of roots = -300 then

=> [tex](r_1+3)^{100}+(r_2+3)^{100}+...+(r_{100}+3)^{100} =[/tex] 4*(-300)-300=-1200-300

=> -1500.

Hence , The value of [tex](r_1+3)^{100}+(r_2+3)^{100}+...+(r_{100}+3)^{100}[/tex] is -1500.

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Therefore, the number of students who study only one subject is n(H-only) + n(G-only) = 14 + 28 = 42.

In mathematics, a set is a collection of distinct objects, which are called its elements. These objects can be anything, such as numbers, letters, or even other sets. Sets are usually denoted by capital letters and the elements of a set are listed within braces, separated by.

Given by the question.

i. n (H ∩ G) can be found using the formula:

n(H ∩ G) = n(H) + n(G) - n(H U G)

where n (H U G) represents the number of students who study either History or Government or both.

We know that n(H) = 2x and n(G) = 3x. Also, n(U) = 54, which means the total number of students is 54. Therefore, we can write:

n (H U G) = n(H) + n(G) - n (H ∩ G)

54 = 2x + 3x - x

54 = 4x

x = 13.5

Since x must be a whole number, we can round it up to 14. Therefore, n (H ∩ G) = x = 14.

ii. The number of students who study only one subject can be found by subtracting n (H ∩ G) from n(H) and n(G) respectively, and then adding the number of students who study neither History nor Government.

n(H-only) = n(H) - n (H ∩ G) = 2x - x = x = 14

n(G-only) = n(G) - n (H ∩ G) = 3x - x = 2x = 28

n(Neither) = n(U) - n (H U G) = 54 - 14 = 40

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Answer: 106 127 129 132 135 138 140 158

Step-by-step explanation: