write the force equation for the following system b and b are the coefficient of viscious friction and force case by this friction is

The simplified form of the given expression as required to be determined in the task content is; 2x⁴√5.

It follows from the task content that the Simon form of the given expression √20x⁸ is required to be determined from the task content.

On this note, since the given expression is; √20x⁸.

We have that; = √ (4 × 5 × x⁸)

Therefore, since 4 and x⁸ are perfect squares; it follows that we have;

= 2x⁴ √5.

Ultimately, the simplified form of the given expression as required to be determined is; 2x⁴ √5.

Complete question; The correct expression is; √20x⁸.

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The formula for the real portion of a complex function can be used to write re(e(1/z)) in terms of x and y: f(z) + f(z*) = re(f(z)) / 2

where z* denotes z's complex conjugate.

By applying this formula to the provided function, we obtain:

re(e(1/z)) = (e(1/z) + e(1/z*)) / 2

re(e(1/z)) = (e(x-iy) + e(x+iy)) / 2

re(e(1/z)) = (ex (cos y + I sin y) + ex (cos y - I sin y)) /2

As a result, re(e(1/z)) can be represented as ex cos y in terms of x and y.

Because it fulfills Laplace's equation, which asserts that the total of a function's second-order partial derivatives is equal to zero, this function is harmonic in every domain excluding the origin.

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Yes, W_2 = {p_2 = a_2t_2 + a_1t + a_0 ∈ ℙ_2 | a_0 = 2} is a subspace of the vector space ℙ_2.

Yes, W_1 = {[a b c; d 0 0] ∈ M_{2x3} | b = a + c} is a subspace of the vector space M_{2x3}.

Vector spaces are closed under vector addition and scalar multiplication, and in this case, ℙ_2 is the set of all 2nd degree polynomials with the usual polynomial addition and scalar multiplication with reals.

Vector spaces are closed under vector addition and scalar multiplication, and in this case, M_{2x3} is the set of all 2x3 matrices with the standard matrix addition and scalar multiplication with reals.

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The primary purpose of statistical analysis is to transform information into data. Therefore, the correct option is a)transform information into data.

Statistical analysis is the method of using statistical techniques to collect and analyze data, evaluate its reliability and determine its statistical significance. The following are the primary purposes of statistical analysis:Provide summaries and descriptions of data: One of the most important applications of statistical analysis is to present data in a clear and concise manner. Summarizing the data can assist people in making sense of the data and drawing inferences from it.

For example, data can be summarized using graphs, charts, or tables. Identify patterns and relationships: The goal of statistical analysis is to identify any patterns or relationships that exist within the data. For example, statistical analysis can be used to determine if a product's sales are correlated with a specific time of year.

Test hypotheses and draw conclusions: Statistical analysis is used to test hypotheses and draw conclusions about a population or phenomenon. This is accomplished by using statistical techniques to determine whether or not the data supports a particular hypothesis. Statistical analysis can also be used to determine the probability of an event occurring in the future.

The primary purpose of statistical analysis is to transform information into data. Therefore, the correct option is a)transform information into data.

The complete question is as follows:

The primary purpose of statistical analysis is to:

a)transform information into data.
b)select samples and make inferences about populations
c)convert data into useful desicion-making information
d)perform statistical computations

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80 middle school students prefer the documentaries television genre.

None of the given options match exactly, but the closest one is 76.

A circle graph, also known as a pie chart, is a type of chart that displays data as a circular diagram, divided into slices to represent proportions of a whole. Each slice of the circle represents a percentage of the total data being represented, and the entire circle represents 100% of the data.

Based on the circle graph, the percentage of students who prefer documentaries is 16% (as the percentage for documentaries is missing from the given options, we need to calculate it).

To find out the actual number of students who prefer documentaries, we need to multiply this percentage by the total number of students in the sample:

16% of 500 = (16/100) x 500 = 80

Therefore, 80 middle school students prefer the documentaries television genre.

None of the given options match exactly, but the closest one is 76.

Circle graphs are commonly used to show proportions or percentages of different categories within a dataset. They are useful for displaying data in a way that is easy to understand, and they are often used in business and finance, as well as in education and research.

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The 1.5 liters of fruit juice is cheaper per liter compared to the 1 liter of fruit juice.

For the 1 liter of fruit juice, the cost per liter is R25/1 liter = R25/liter.

For the 1.5 liters of fruit juice, the cost per liter is R34/1.5 liters = R22.67/liter.

A supermarket is a large retail store that offers a wide range of food and household items to consumers. These stores are typically designed to be a one-stop shop for customers, allowing them to purchase everything they need in one convenient location. One of the primary advantages of shopping at a supermarket is the ability to choose from a wide variety of products at competitive prices.

Supermarkets usually have multiple departments, including a fresh produce section, a meat and seafood section, a bakery, and a deli. In addition to food items, supermarkets also offer a variety of household goods, such as cleaning supplies, personal care products, and pet food. Many supermarkets also offer loyalty programs, coupons, and other incentives to help customers save money.

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The estimated probability of a success is 40% since the numbers 1 and 2 represent a success out of the integers 1 to 5. Therefore, the success outcomes (1 and 2) make up 2 out of 5 possible outcomes, or 40%.

To find the estimated probability of a success, we need to determine the proportion of successes in the generated list.

Out of the numbers 1 to 5, two numbers represent success (1 and 2). Therefore, the probability of success for each individual number is 2/5 or 0.4.

Since we are considering a randomly generated list of integers, we can assume that each number is equally likely to be generated. So, the estimated probability of a success can be calculated by finding the proportion of 1's and 2's in the list.

Let's assume that the list has n elements. If we generate the list multiple times, we can expect that the proportion of successes will approach the true probability of success, which is 0.4.

For example, if we generate a list of 10 integers and get the following numbers: 2, 5, 1, 3, 1, 4, 2, 5, 3, 1, then we have 4 successes out of 10 numbers. So, the proportion of successes in this list is 4/10 or 0.4, which matches the true probability of success.

Therefore, the answer is A. 40%.

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Answer:

  (a) AC = 4√2 cm

  (b) AM = 2√2 cm

  (c) EM = √41 cm

  (d) EF = 3√5 cm

Step-by-step explanation:

You want to solve for various lengths in the right square pyramid shown with base edge 4 cm and lateral edge 7 cm.

Each right triangle can be solved for unknown lengths using the Pythagorean theorem: the square of the hypotenuse is the sum of the squares of the other two sides.

Right triangles of interest here are ...

  ADC . . . . for finding AC and AM (isosceles right triangle)

  CME . . . . for finding EM

  FME . . . . for finding EF

AC is the hypotenuse of ∆ADC, so ...

  AC² = AD² +DC²

  AC = √(4² +4²)

  AC = 4√2 . . . . cm

M is the midpoint of AC, so ...

  AM = AC/2 = (4√2)/2

  AM = 2√2 . . . . cm

FM is half the length of one side of the base, so is 2 cm. CM = AM = 2√2.

  CE² = CM² +EM²

  EM = √(CE² -CM²) = √(7² -(2√2)²)

  EM = √41 . . . . cm

EF is the hypotenuse of ∆EMF.

  EF² = EM² +FM²

  EF = √(EM² +FM²) = √(41 +2²) = √45

  EF = 3√5 . . . . cm

Answer:

X=8

Bcuz 8x=64

64÷8=8

So that the answer is 8

Answer:

[tex]\bf x=8[/tex]

Step-by-step explanation:

[tex]\bf 8x=64[/tex]

Divide both sides by 8:

[tex]\bf \cfrac{8x}{8}=\cfrac{64}{8}[/tex]

Simplify:-

[tex]\bf x=8[/tex]

______________________

Hope this helps!

The height of the rock formation is approximately 17.45 feet (rounded to two decimal places).

Trigonometry is a branch of mathematics that deals with the study of the relationships between the sides and angles of triangles. It is used to solve problems involving triangles, such as finding missing sides or angles, determining distances or heights, and more. Trigonometry has applications in various fields, such as engineering, physics, architecture, and astronomy, among others.

In the given question,

We can use trigonometry to solve this problem. Let h be the height of the rock formation in feet, and let x be the horizontal distance from Ramona to the base of the rock formation in feet. Then, we have:

tan(10) = h/x

Rearranging this equation, we get:

h = x * tan(10)

We need to find the value of h, so we need to find the value of x. We can use the angle of elevation and the distance that Ramona walked up the hill to find x. We have a right triangle with height h, base x, and hypotenuse 100 ft. The angle opposite the height h is 10 degrees. So, we have:

tan(10) = h/x

sin(10) = h/100

Rearranging the second equation, we get:

h = 100 * sin(10)

Substituting this into the first equation, we get:

x * tan(10) = 100 * sin(10)

Dividing both sides by tan(10), we get:

x = 100 * sin(10) / tan(10)

Plugging this value of x into the equation for h, we get:

h = x * tan(10) = (100 * sin(10) / tan(10)) * tan(10) = 100 * sin(10)

Therefore, the height of the rock formation is approximately 17.45 feet (rounded to two decimal places).

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The distance she must climb to reach the peak is greater than 2,663 feet. An inequality to show the remaining distance, d, in feet she must climb to reach the peak is d > 2,663.

Inequlity represents a relationship between two values that is not equal. That is inequlity means no equal and the symbols which used for not equal is ≠ and for comparison are < , > , ≤ , ≥. For example, ax > b etc. We have, Rina is climbing a mountain and mountain height present in above figure. See the figure carefully, the main two points in figure are the following: height of mountain peak

= 12,358 feet

height of base camp = 9,695 feet

we have to write an inequality for the remaining distance, d, in feet. Also, it is specified that she has not reached base camp yet. So, his climbed distance is less than 9,695 feet. Let the remaining distance be 'd feet '. From the above figures, d + base camp height > 12,358

Substract 9695 from both sides

=> d > 12,358 - 9,695

=> d > 2,663

Hence, required inequalty is d > 2,663.

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Complete question:

Rina is climbing a mountain. She has not yet reached base camp. Write an inequality to show the remaining distance, d, in feet she must climb to reach the peak. See tha above figure.

In the following question, among the various parts to solve- a) the probability that the individual waits more than 7 minutes is 0.3. b)the probability that the individual waits between 2 and 7 minutes is 0.5.

a) The probability that an individual will wait more than 7 minutes can be found as follows:

Given that the waiting time of an individual is a continuous uniform distribution and that a bus arrives at the bus stop every 10 minutes.Since the waiting time is a continuous uniform distribution, the probability density function (PDF) can be given as:fX(x) = 1/(b-a)where a = 0 and b = 10.

Hence the PDF of the waiting time can be given as:fX(x) = 1/10The probability that an individual waits more than 7 minutes can be obtained using the complementary probability. This is given by:P(X > 7) = 1 - P(X ≤ 7)The probability that X ≤ 7 can be obtained using the cumulative distribution function (CDF), which is given as:FX(x) = P(X ≤ x) = ∫fX(t) dtwhere x ∈ [a,b].In this case, the CDF of the waiting time is given as:FX(x) = ∫0x fX(t) dt= ∫07 1/10 dt + ∫710 1/10 dt= [t/10]7 + [t/10]10= 7/10Using this, the probability that an individual waits more than 7 minutes is:P(X > 7) = 1 - P(X ≤ 7)= 1 - 7/10= 3/10= 0.3So, the probability that the individual waits more than 7 minutes is 0.3.

b) The probability that the individual waits between 2 and 7 minutes can be calculated as follows:P(2 < X < 7) = P(X < 7) - P(X < 2)Since the waiting time is a continuous uniform distribution, the PDF can be given as:fX(x) = 1/10Using the CDF of X, we can obtain:P(X < 7) = FX(7) = (7 - 0)/10 = 0.7P(X < 2) = FX(2) = (2 - 0)/10 = 0.2Therefore, P(2 < X < 7) = 0.7 - 0.2 = 0.5So, the probability that the individual waits between 2 and 7 minutes is 0.5.

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The cat population in Catonsville in 2007 was 200. The answer is option B.

Population refers to the total number of people, animals, or objects in a particular group or area. It is the entire group or collection of individuals, things, or events that we are interested in studying or describing.

We can use the given equation to find the cat population at any given time in years since the beginning of 2010. However, to find the population in 2007, we need to make a conversion from years since the beginning of 2010 to years since the beginning of 2007.

From the beginning of 2007 to the beginning of 2010, there are 3 years, so if we subtract 3 from the number of years since the beginning of 2010, we will get the number of years since the beginning of 2007. Therefore, we need to substitute t = -3 into the given equation and solve for p:

[tex]p = 1600(2)^t\\p = 1600(2)^(-3)\\p = 1600(1/8)\\p = 200[/tex]

Therefore, the cat population in Catonsville in 2007 was 200. The answer is option B.

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Answer:

H = 3.5m + 90

m = 12 months

Step-by-step explanation:

We can use a linear equation (H = 3.5m + 90), where H is the height in centimeters and m is month.

We can let H = 132 to solve for m:

[tex]132=3.5m+90\\42=3.5m\\12=m[/tex]

Thus, it will take the boy 12 months to reach 132 cm.

The gross capital gain is option (D) $4330

Gross capital gain is the total amount of profit that an investor makes when they sell an asset, such as stocks, real estate, or other investments, for a higher price than the purchase price.

Zach's gross capital gain is the difference between the selling price and the purchase price of the shares.

Purchase price of 200 shares = 200 x $21.35 = $4,270

Selling price of 200 shares = 200 x $43 = $8,600

Therefore, the gross capital gain is

= $8,600 - $4,270

Subtract the numbers

= $4,330

Therefore, the correct option is (D) $4,330.

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Therefore, the expression Marie should have used in Step 3 is option B: 2/21 ÷ ¾, which represents the division of the fraction 21/2 by 3/4.

A fraction is a way of expressing a quantity that represents a part of a whole or a ratio between two quantities. It is usually written as one number (the numerator) over another number (the denominator), separated by a horizontal line or slash. For example, the fraction 2/3 represents two parts out of three equal parts of a whole, or a ratio of two to three. Fractions can be expressed in various forms, such as proper fractions (where the numerator is smaller than the denominator), improper fractions (where the numerator is greater than or equal to the denominator), and mixed numbers (where a whole number and a proper fraction are combined). Fractions are an important concept in many areas of mathematics, including arithmetic, algebra, geometry and calculus.

by the question.

To solve this, Marie can use the following steps:

Step 1: 10 ½ ÷ ¾

Step 2: 21/2 ÷ ¾

Step 3: (21/2 ÷ 3/4) (finding the quotient of two fractions)

Step 4: (21/2) × (4/3) (dividing by a fraction is the same as multiplying by its reciprocal)

Step 5: 14

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Answer: To convert the given equation, √y = (6/p)x - (2/q), into the linear form y = mx + c, we can use the following steps:

Square both sides of the equation to eliminate the square root:

√y = (6/p)x - (2/q)

√y^2 = (6/p)x - (2/q)^2

Simplifying the right-hand side, we get:

y = (36/p^2)x - (4/q) + 4/q^2

Rearrange the equation to the form y = mx + c:

y = (36/p^2)x + (4/q^2 - 4/q)

So the linear form of the given non-linear equation is y = (36/p^2)x + (4/q^2 - 4/q).

Step-by-step explanation:

Answer:

----------------------------------

Let the cost price of the camera be x.

When the selling price is 60000 there is a loss of 20%. Let's show this as equation and find the cost price:

The cost price is Rs 75000, and we need a profit of 12%, it gives us the selling price of:

As we have proved that the line created works exclusively with the three points by justifying how the x-value and y-value fit into the equation for the line.

Let's start by defining what a linear function is. A linear function is an equation of the form y = mx + b, where m is the slope of the line and b is the y-intercept. The slope represents the rate of change of the line, and the y-intercept represents the value of y when x is equal to zero. To create a linear function, we need two points on the line.

Now, let's create another linear function using points B and C:

slope (m) = (y₂ - y₁) / (x₂ - x₁) = (6 - 4) / (5 - 3) = 1

y-intercept (b) = y - mx = 4 - 1 * 3 = 1

Therefore, the linear function that passes through points B and C is also y = x + 1. We can check if point A lies on this line by substituting its x and y values into the equation:

2 = 1 + 1

This is true, so point A lies on the line created by points B and C. Therefore, we have also proved that the line created works exclusively with these three points.

In conclusion, we have created two linear functions using three points and proved that they work exclusively with those three points.

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Answer:

x = +8

x = -8

That's the thing that I only know. Im a seventh grader(turned 13) so I wouldn't know this fully. Ask someone else to get it fully.

Step-by-step explanation:

5x + 6x - 10 + 12 = 90

11x = -88

x = -8

or

x = 8

Answer:

5-8=-3

Step-by-step explanation:

-3-1=-4

So the transformed rectangle has vertices A'(40, 160), B'(40, 130), C'(80, 130), and D'(80, 160).

In mathematics, a transformation rule (also known as a transformation function, transformation formula, or simply a transformation) is a mathematical rule or formula that describes how to map or transform a set of points in one coordinate system to another set of points in a different coordinate system. Transformations can be applied to various mathematical objects, such as points, lines, curves, shapes, or functions, and can be used to achieve various purposes, such as to change the size, shape, position, or orientation of an object, to create a mirror image or a rotation of an object, or to change the coordinate system of an object.

Here,

There appears to be an error in the coordinates of points B and D that you provided, as they are both (-90,25). I will assume that the correct coordinates of point D are (-50,55) to form a rectangle ABCD.

To apply the given transformation rules to each of the four vertices of the rectangle, we add 130 to the x-coordinate and 105 to the y-coordinate. Therefore, the coordinates of the transformed rectangle are:

A' = (-90 + 130, 55 + 105) = (40, 160)

B' = (-90 + 130, 25 + 105) = (40, 130)

C' = (-50 + 130, 25 + 105) = (80, 130)

D' = (-50 + 130, 55 + 105) = (80, 160)

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Complete question:

Please help! The transformation rules says: (x, y)→(x + 130, y + 105 )  

Given rectangle ABCD with A(-90,55) B(-90,25) C(-50,25) D(-50,25).

Find the coordinates of transformed rectangle.

The greatest possible number of different types of this caterpillar that could exist is 120.

If we assume that the order of the parts does not matter and that all caterpillars with the same color arrangement are considered identical, we can use combinations to find the number of different types of caterpillars that could exist.

First, we need to choose 2 out of the 5 parts to be green, which can be done in 5 choose 2 ways:

5 choose 2 = (5!)/(2!(5-2)!) = 10

For each green-yellow arrangement there are;

3! ways to permute the yellow parts and

2! ways to permute the green parts.

Therefore, the total number of different types of caterpillars is:

10 × 3! × 2! = 10 × 6 × 2 = 120

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The point is where the vertex is situated [tex](2.5, 100)[/tex].

A vertex, which is a unique point on a mathematical object, is often where 2 or more lines or edges come together. The shapes that include vertices most frequently are graphs, polygons, polyhedra, and angles. Vertices in a graph are sometimes called nodes.

A parabola's vertex is the location where its symmetry line and parabola intersect. The vertices of a parabola which equation is provided in standard form will be the lowest (lowest point) and highest (highest point) points on the graph, respectively.

To graph the equation [tex]h(t) = -16t^2 + 80t[/tex] from [tex]t=0[/tex] to [tex]t=5[/tex] seconds, we can create a table of values by plugging in values of t and calculating the corresponding values of [tex]h(t)[/tex]:

t                   h(t)

0                      0

1                    64

2                   128

3                   144

4                   128

5                     80

To find the vertex, we can use the formula [tex]t = -b/(2a)[/tex], where [tex]a = -16[/tex] and [tex]b = 80[/tex].

[tex]t = -b/(2a) = -80/(2(-16)) = 2.5[/tex]

So the vertex occurs at [tex]t = 2.5[/tex] seconds.

Now we can plot the points from the table and graph the equation,

Graph of [tex]h(t)=-16t^2+80t[/tex] from [tex]t=0[/tex] to [tex]t=5[/tex] seconds

The vertex is located at the point [tex](2.5, 100)[/tex].

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We have that, finding the horizontal component of the reaction at A in pounds, it is 866.03 pounds, with zero horizontal component in the negative direction.

To determine the horizontal component of the reaction at A, we first need to draw a free-body diagram of the given structure. So, we can use the equation of equilibrium for the horizontal direction to find the horizontal component of the reaction at A. The equation of equilibrium for the horizontal direction is as follows:

∑Fx=0∑Fx=0

Equilibrium equation for the horizontal direction, where ∑Fx is the sum of all forces acting horizontally in the positive direction and in the negative direction. Let's draw the free body diagram of the given structure: Here, RAY is the horizontal component of the reaction at A. We can see that there is no force acting horizontally in the negative direction.

Therefore,∑Fx=RAx∑Fx=RAx

Now, we need to find the force that acts horizontally in the positive direction. In the free body diagram, we can see that:

∑Fx=1000cos30°=866.03∑Fx=1000cos30°=866.03

Therefore

∑Fx=RAx=866.03RAx=866.03

So the horizontal component of the reaction at A is 866.03 pounds. Therefore, the answer is 866.03 (positive).

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Answer: x < -46/17

Step-by-step explanation:

To solve the inequality:

1/3x - 1/4(x + 2) > 3x - 4/3

First, we simplify the left-hand side by finding a common denominator:

4(1/3x) - 3/4(x + 2) > 3x - 4/3

4/3x - 3/4x - 9/2 > 3x - 4/3

Next, we simplify the equation:

7/12x - 9/2 > 3x - 4/3

To isolate the variable x on one side of the inequality, we will move all the x terms to the left-hand side and all the constants to the right-hand side:

7/12x - 3x > 9/2 - 4/3

-17/12x > 23/6

Finally, we can solve for x by dividing both sides by -17/12, remembering to reverse the inequality because we are dividing by a negative number:

x < (23/6) ÷ (-17/12)

x < -46/17

Therefore, the solution to the inequality is:

x < -46/17

The solution to the system of equations 3x+2x=11 , y=x+3 is (x, y) = (1, 4) using the method of substitution.

To solve the system of equations using the method of substitution, we first need to solve for one variable in terms of the other in one of the equations. In this case, we can solve for y in terms of x in the second equation:

y = x + 3

We can then substitute this expression for y in the first equation:

3x + 2(x + 3) = 11

Simplifying the equation by distributing the 2:

3x + 2x + 6 = 11

Combining like terms:

5x + 6 = 11

Subtracting 6 from both sides:

5x = 5

Dividing both sides by 5:

x = 1

Now that we know the value of x, we can substitute it back into the second equation to find the value of y:

y = x + 3 = 1 + 3 = 4

In conclusion, the method of substitution involves solving one equation for one variable and substituting the resulting expression into the other equation.

By doing this, we can eliminate one variable and solve for the other. In this case, we solved for y in terms of x, substituted this expression for y in the first equation, and solved for x. Then, we substituted the value of x back into the second equation to solve for y.

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The probability that a randomly chosen student will spend between $300 and $400 on clothing in a year is 0.6827.

Here the given data are

mean [tex]\mu = $380[/tex],

standard deviation [tex](\sigma) = $50[/tex]

Let X be the random variable which denotes the amounts of money spent on clothing in a year by college students. The distribution of X is normal distribution.

We need to find the probability that a randomly chosen student will spend between $300 and $400 on clothing in a year. If we have the standard normal distribution, we can easily calculate the probability from the normal distribution table. Otherwise, we have to use the standard normal distribution and convert the values to standard units. This process is called standardization. We will use the z-score formula for standardization.

Let’s standardize the given values.

Lower value [tex](X_1) = $300[/tex]

Upper value [tex](X_2) = $400[/tex]

Population mean [tex](\mu) = $380[/tex]

Population standard deviation [tex](\sigma) = $50z_1 = (X_1 - \mu) / \sigma z_1 = ($300 - $380) / $50z_1 = -1.6z_2 = (X_2 - \mu) / \sigma z_2 = ($400-$380)/$50z_2 = 0.4[/tex]

Now we need to find the area between these two z-scores using the standard normal distribution table.The probability that a randomly chosen student will spend between $300 and $400 on clothing in a year is

[tex]P(-1.6 < Z < 0.4).P(-1.6 < Z < 0.4) = P(Z < 0.4) - P(Z < -1.6)\\P(Z < 0.4) = 0.6554\\P(Z < -1.6) = 0.0548\\P(-1.6 < Z < 0.4) = 0.6554 - 0.0548 = 0.6006[/tex]

Therefore, the probability that a randomly chosen student will spend between $300 and $400 on clothing in a year is 0.6006 (approx.) or 0.6827 (approx.).

Hence, the probability that a randomly chosen student will spend between $300 and $400 on clothing in a year is 0.6827.

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The value of angle x rounded to the nearest tenth of a degree is 53.6°.

How can trigonometric ratios be used to find the angle in a right triangle?

Trigonometric ratios are used to relate the sides of a right triangle to its angles. In a right triangle, one of the angles is always 90 degrees, which is the right angle. The three primary trigonometric ratios are sine, cosine, and tangent, which are defined as follows:

Sine (sin): The sine of an angle is the ratio of the length of the side opposite the angle to the length of the hypotenuse.

Cosine (cos): The cosine of an angle is the ratio of the length of the side adjacent to the angle to the length of the hypotenuse.

Tangent (tan): The tangent of an angle is the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

To find the value of an angle in a right triangle, we can use the inverse trigonometric functions, also known as arc functions or anti trigonometric functions. These functions are denoted as [tex]sin^{-1}[/tex], [tex]cos^{-1}[/tex] , and  [tex]tan^{-1}[/tex] , and they are used to find the angle whose trigonometric ratio is known.

Finding the value of x :

We can use the trigonometric ratio of sine to solve for x.

sin(x) = opposite/hypotenuse = MN/LN

[tex]sin(x) = 83/92[/tex]

Taking the inverse sine of both sides:

[tex]x = sin^{-1}(83/92)[/tex]

Using a calculator, we get:

[tex]x \approx 53.6^{o}[/tex]

Therefore, the value of x rounded to the nearest tenth of a degree is 53.6°.

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The dot plot has a bell shape. The dots are more heavily clustered around the central point, with fewer dots on the ends.

Heavily clustered refers to a situation when a large number of items, such as data points, are grouped together in a dense area. This can be seen in many forms of data visualisation, such as scatterplots, where items are represented in a graph and appear close together in a particular area. In some cases, the items can be so densely packed that they appear to form a single unit. Heavily clustered data can be used to identify areas of high concentration, or to identify trends or correlations.

This indicates that the data is normally distributed and that there is a higher probability of the data points falling around the central point. The dots are not evenly distributed, but rather grouped on one side.

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