Fill in the P (X=x) values to give a legitimate probability distribution for the discrete random variable X, whose possible values are -3, 1, 3, 5, and 6.

P(X=x)
-3: 0.14
1:
3: 0.13
5: 0.30
6:

The scale factor of the image after the dilation is k = 2

Given data ,

Polygon ABCD with vertices at A(1,-1) B(3,-1), C(3,-2) and D(1-2)

Now , the dilated polygon is represented as A'B'C'D'

where the coordinates are A’(2,-2), B(6,-2), C’(6,-4) and D’(2,-4).

Now , we can compare the corresponding side lengths of the two polygons to determine the scale factor:

Scale factor = (corresponding side length of dilated polygon) / (corresponding side length of original polygon)

So, A'/A = 2 / 1

A' = 2A

So, the scale factor is k = 2

Hence , the dilation is solved.

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The graph should visually represent the shaded regions A and B, as well as their overlap in region AB. It's important to note that without a specific scale or values, the graph may not accurately depict the exact shape and position of the shaded regions.

To graph the system of inequalities 2x - y < 4 and x + y > -1, we can start by graphing each inequality separately and then determining the overlapping region.

For the inequality 2x - y < 4:

Start by graphing the line 2x - y = 4. Choose two points on the line, such as (0, -4) and (2, 0), and connect them to draw the line.

Since the inequality is "less than" (<), shade the region below the line. Label this shaded region as A.

For the inequality x + y > -1:

Graph the line x + y = -1 using points such as (-2, 1) and (0, -1). Connect the points to draw the line.

Since the inequality is "greater than" (>), shade the region above the line. Label this shaded region as B.

Finally, identify the overlapping region of the shaded regions A and B. This region represents the solution to the system of inequalities and is labeled as AB. This region indicates the values of x and y that satisfy both inequalities simultaneously.

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Answer:

Step-by-step explanation:

2x − y < 4

x + y > −1

In each graph, the area for f(x) is shaded and labeled A, the area for g(x) is shaded and labeled B, and the area where they have shading in common is labeled AB.

The 36th derivative of f(x) = cos(2x) is [tex]2^{17}[/tex]* cos(2x).

To find the 36th derivative of the function f(x) = cos(2x), we can apply the chain rule repeatedly. The chain rule states that if we have a composite function y = f(g(x)), then its derivative is given by dy/dx = f'(g(x)) * g'(x).

Let's start by finding the first few derivatives of f(x) = cos(2x):

f'(x) = -2sin(2x)

f''(x) = -4cos(2x)

f'''(x) = 8sin(2x)

f''''(x) = 16cos(2x)

We observe a pattern where the derivatives of cos(2x) alternate between sin(2x) and cos(2x), with the signs changing accordingly.

Based on this pattern, we can see that the 36th derivative will be:

f^(36)(x) =[tex](-1)^{17} * 2^{17} *[/tex] cos(2x)

Simplifying this expression, we have:

f^(36)(x) = [tex]2^{17} * cos(2x)[/tex]

Therefore, the 36th derivative of f(x) = cos(2x) is[tex]2^{17[/tex] * cos(2x).

It's important to note that in this case, the number 36 is even, and since the derivatives of cos(2x) follow a repeating pattern every 4 derivatives, the sign (-1) raised to the power of 17 accounts for the change in sign in the 36th derivative.

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The circle should be a circle centered at point (4,-1.5) and have a radius of 2.The equation (x - 4)^2 + (y + 1.5)^2 = 4 represents a circle with a center at (4, -1.5) and a radius of 2.

To identify the graph that shows the position and radius of the wheels, we look for a graph that depicts a circle with a center at (4, -1.5) and a radius of 2.

Among the options provided, the graph that corresponds to the given equation is likely the one that shows a circle with its center at (4, -1.5) and a radius of 2.

The equation provided is a standard form of a circle equation which states that the center of the circle is (4,-1.5) and the radius is 2 (square root of 4).  Any graph with a circle centered at (4,-1.5) and a radius of 2 will represent the position and radius of the wheels of the test model.

Of the options given in the answer section, option (D) is the graph that shows the position and radius of the wheels since it represents a circle with center (4,-1.5) and a radius of 2.

It is also important to note that a circle equation in standard form is (x-h)^2+(y-k)^2=r^2, where (h,k) is the center of the circle and r is the radius.

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Based on the calculations using unit pricing, Toilet paper B has a lower unit price and is the better deal between the two. Customers will save more money if they purchase toilet paper B rather than toilet paper A because the price per mega roll is lower. (option b)

Saving money while shopping is one of the best ways to reduce your expenses and have more disposable income for other things. Unit pricing is a pricing system that displays prices in standard units, allowing customers to compare prices across different brands and package sizes and save money. To determine which toilet paper deal is better, we can use the unit price method, which divides the price by the number of units in the package. The toilet paper with the lower unit price is the better deal.
The formula for unit pricing is as follows:
Unit price = total price ÷ number of units
Using the above formula, we can calculate the unit price of toilet paper A and toilet paper B as follows:
For Toilet paper A:
Unit price = $4.59 ÷ 6 mega rolls
Unit price = $0.765 per mega roll
For Toilet paper B:
Unit price = $9.02 ÷ 12 mega rolls
Unit price = $0.751 per mega roll
Therefore, based on the calculations, Toilet paper B has a lower unit price and is the better deal between the two. Customers will save more money if they purchase toilet paper B rather than toilet paper A because the price per mega roll is lower. (option b)
Unit pricing is a great way to compare prices, and consumers should always use it to determine the better deal between two products, as this will help them save money and stick to their budget.

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The maximum profit is achieved when the manufacturer spends $4,000 on development and $4,000 on promotion.

The profit is given by

(150 - 50) * (320y/(y+2) + 160x/(x+4)) - x - y

= 100 * (320y/(y+2) + 160x/(x+4)) - x - y

= 32000y/(y+2) + 16000x/(x+4) - x - y

We want to maximize this function subject to the constraint that x + y = 8.

We can do this by using Lagrange multipliers. The Lagrangian is given by

L = 32000y/(y+2) + 16000x/(x+4) - x - y + lambda(x + y - 8)

Setting the partial derivatives of L with respect to x, y, and lambda equal to zero, we get the following equations:

32000/(y+2) - 1 - lambda = 0

16000/(x+4) - 1 - lambda = 0

x + y - 8 = 0

Solving these equations, we get x = 4, y = 4, and lambda = 0.

Therefore, the maximum profit is achieved when the manufacturer spends $4,000 on development and $4,000 on promotion.

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Answer:

centre = (- 5, - 2 ) , radius r = 18

Step-by-step explanation:

the equation of a circle in standard form is

(x - h)² + (y - k)² = r²

where (h, k ) are the coordinates of the centre and r the radius

(x + 5)² + (y + 2)² = 324 ← is in standard form , that is

(x - (- 5))² + (y - (- 2))² = 18²

with centre = (- 5, - 2 ) and r = 18

Answer:

Center = (-5, -2)

Radius = 18

Step-by-step explanation:

The equation of a circle in standard form is:

[tex]\boxed{\begin{minipage}{4 cm}\underline{Equation of a circle}\\\\$(x-h)^2+(y-k)^2=r^2$\\\\where:\\ \phantom{ww}$\bullet$ $(h, k)$ is the center. \\ \phantom{ww}$\bullet$ $r$ is the radius.\\\end{minipage}}[/tex]

If we compare the given equation of (x + 5)² + (y + 2)² = 324 to the standard form, we get:

Therefore, the center of the circle is (-5, -2).

To find the radius, square root 324:

[tex]r^2=324 \implies r=\sqrt{324}=18[/tex]

Therefore, the radius of the circle is 18.

When using the measures of centers, the first step you do with the data is to find the average value of the dataset. This is done by calculating the mean or arithmetic average of the data.

The mean is obtained by adding up all the values in the dataset and then dividing by the total number of values present in the dataset.

The formula for the mean is: Mean = Sum of all values in the dataset / Total number of values in the dataset.

The mean is a good indicator of the central tendency of the data, as it is sensitive to all the values present in the dataset.

However, it can be affected by outliers, which are values that are significantly different from the other values in the dataset.

In such cases, the median, which is the middle value when all the values in the dataset are arranged in ascending or descending order, can be a more appropriate measure of center.

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X is an inductive set, then {X [tex]\in[/tex] x is transitive} is also an inductive set. Consequently, every n [tex]\in[/tex] N is transitive.

To prove the statement, we need to demonstrate that if X is an inductive set, then the set {[tex]X \in x[/tex]is transitive} is also an inductive set.

Let's break down the proof into two parts:

If X is an inductive set, then {[tex]X \in x[/tex] is transitive} is a subset of X:

To show that {[tex]X \in x[/tex]is transitive} is a subset of X, we need to prove that every element in {[tex]X \in x[/tex] is transitive} is also an element of X.

If X is an inductive set, it means that X contains the empty set (∅) and for every element x in X, the successor of x (denoted as S(x)) is also in X. Now, consider an arbitrary element y in {[tex]X \in x[/tex] is transitive}. By definition, y is a transitive set.

Since X is inductive, it contains the empty set and for every element in X, its successor is also in X. Thus, y must also be in X, and {[tex]X \in x[/tex] is transitive} is a subset of X.

{[tex]X \in x[/tex] is transitive} is an inductive set:

To show that {[tex]X \in x[/tex] is transitive} is an inductive set, we need to demonstrate that it satisfies the properties of an inductive set.

First, we prove that the empty set (∅) is an element of {EX: x is transitive}. Since the empty set is transitive (it vacuously satisfies the definition of transitivity), it belongs to {[tex]X \in x[/tex] is transitive}.

Second, we prove that for every element y in {[tex]X \in x[/tex] is transitive}, its successor S(y) is also in {[tex]X \in x[/tex]  is transitive}. Let y be an arbitrary element in {[tex]X \in x[/tex] is transitive}.

By definition, y is a transitive set. We need to show that S(y) is also a transitive set. Since X is inductive, it means that for every element x in X, its successor S(x) is also in X. Applying this property to y, we conclude that S(y) is in X. Since S(y) is in X, it is also in [tex]X \in x[/tex] is transitive}. Hence, {[tex]X \in x[/tex] is transitive} satisfies the property of an inductive set.

By proving both parts, we have shown that if X is an inductive set, then {[tex]X \in x[/tex] is transitive} is also an inductive set. Consequently, every [tex]n \in N[/tex] is transitive.

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The central angle measure, x, for the Baseball slice in the circle graph is 50.4 degrees.

Given that, a survey was conducted in which 2400 people were asked to name their favorite sport to watch and the table below shows their answers: Sport percentage of people are:

Football 29%

Soccer 9%

Baseball 14%

Basketball 8%

Hockey 8%

Other 32%

We need to find the central angle measure, x, for the Baseball slice in the circle graph.

For this, we need to use the formula that gives the central angle measure in degrees for a sector of a circle:

Central angle measure = (Percentage/100) × 360°

Now, using the above formula, we can calculate the central angle measure for each sport as shown below:

Football: (29/100) × 360° = 104.4°

Soccer: (9/100) × 360° = 32.4°

Baseball: (14/100) × 360° = 50.4°

Basketball: (8/100) × 360° = 28.8°

Hockey: (8/100) × 360° = 28.8°

Other: (32/100) × 360° = 115.2°

The sum of all central angle measures should be 360°, which is the measure of a full circle.

So we can check if the calculations are correct: 104.4° + 32.4° + 50.4° + 28.8° + 28.8° + 115.2° = 360°

We see that the sum is indeed 360°, so the calculations are correct. The central angle measure for the Baseball slice is 50.4°.

Therefore, the central angle measure, x, for the Baseball slice in the circle graph is 50.4 degrees.

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The point in the solution set from the graph is (0, 0)

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

y < -5/2x - 2

y ≥ -1/2x + 2

Next, we plot the graph of the system of the inequalities

The graph is given

From the graph, we have solution to the system to be the shaded region

The point in the solution set from the graph is (0, 0)

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The midpoint of a line segment is the point that is exactly halfway between the two endpoints. To find the midpoint of the line segment joining points A(2,-7) and B(2,3), we can use the midpoint formula:

Midpoint = [(x1 + x2) / 2, (y1 + y2) / 2]

where (x1, y1) and (x2, y2) are the coordinates of the two endpoints.

Substituting the values for A and B, we get:

Midpoint = [(2 + 2) / 2, (-7 + 3) / 2]

Midpoint = [4 / 2, -4 / 2]

Midpoint = [2, -2]

Therefore, the midpoint of the line segment joining points A and B is (2, -2).

a. The graph represents a proportional relationship.

b. The graph does not represent a proportional relationship.

c. The graph does not represent a proportional relationship.

d. The graph represents a proportional relationship.

A proportional relationship is a relationship in which a constant ratio between the output variable and the input variable exists.

The equation that defines the proportional relationship is a linear function with slope k and intercept zero presented as follows:

y = kx.

Hence graphs a and d represent proportional relationship, as they have intercepts at the origin.

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How would you describe the difference between the graphs of f(x) = 2x² and g(x) = -2x²: D. g(x) is a reflection of f(x) over the x-axis.

In Mathematics and Geometry, a reflection over or across the x-axis is represented by this transformation rule (x, y) → (x, -y).

This ultimately implies that, a reflection over or across the x-axis would maintain the same x-coordinate while the sign of the y-coordinate changes from positive to negative or negative to positive.

In this context, we can reasonably infer and logically deduce that a graph of transformed function g(x) = -2x² would be created by applying a reflection over or across the x-axis to the graph of the parent function f(x) = 2x² as shown in the image attached below.

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The total animals = 139, and we can substitute this value into equation (3) to obtain:x = 2(139) - 279 => x = -1This doesn't make sense since we can't have negative chickens.

Let the number of chickens be x and the number of cows be y in the backyard. We can create two equations based on the given information:The total number of animals is x + y = total animals ...

(1)The total number of eyes and legs can also be expressed as 2x + 2y = 366 (since each animal has two eyes) and 2x + 4y = 558 (since each chicken has 2 legs and each cow has 4 legs).

Simplifying the equations, we get:x + y = total animals ... (1)x + 2y = 279 ... (2)From equation (1),

we can substitute y = total animals - x in equation (2) to obtain:x + 2(total animals - x) = 279 => 2total animals - x = 279 => x = 2total animals - 279 ... (3)We also know that x + y = total animals, so y = total animals - x.

Substituting equation (3) into this expression, we get:y = total animals - (2total animals - 279) => y = 279 - total animals ... (4)

To find the value of x, we need to know the value of total animals. We can obtain this value by adding equations (1) and (4):x + y + (279 - total animals) = total animals => x + y = 279/2 => x + y = 139.5

Since x and y are both whole numbers, they must add up to an even number. The only possible value is 70 + 69 = 139.  Therefore, there must be a mistake in the problem statement or calculations.

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The correlation coefficient (r) is -0.96.

Given the data:

x: 0, 1, 2, 2, 3, 4, 5, 5, 6

y: 12, 9.5, 9, 8.5, 8.5, 6, 5, 5, 3.5

Step 1: Calculate the mean (average) of x and y.

Mean of x = (0 + 1 + 2 + 2 + 3 + 4 + 5 + 5 + 6) / 9 = 3

Mean of y  = (12 + 9.5 + 9 + 8.5 + 8.5 + 6 + 5 + 5 + 3.5) / 9 = 7

Step 2: Calculate the deviations from the mean for x and y.

Deviation of x (dx) = x - X

Deviation of y (dy) = y - Y

x: -3, -2, -1, -1, 0, 1, 2, 2, 3

y: 5, 2.5, 2, 1.5, 1.5, -1, -2, -2, -3.5

Step 3: Calculate the product of the deviations of x and y.

(dx * dy): -15, -5, -2, -1.5, 0, -1, -4, -4, -10.5

Step 4: Calculate the sum of the products of the deviations (Σ(dx * dy)).

Σ(dx * dy) = -15 - 5 - 2 - 1.5 + 0 - 1 - 4 - 4 - 10.5 = -43

Step 5: Calculate the standard deviations of x (σx) and y (σy).

Standard deviation of x (σx) = √(Σ(dx²) / (n-1))

Standard deviation of y (σy) = √(Σ(dy²) / (n-1))

Calculating dx²:

dx^2: 9, 4, 1, 1, 0, 1, 4, 4, 9

Σ(dx²) = 33

Calculating dy²:

dy²: 25, 6.25, 4, 2.25, 2.25, 1, 4, 4, 12.25

Σ(dy²) = 61

Calculating the standard deviations:

σx = √(33 / (9-1)) = √(33 / 8) ≈ 1.84

σy = √(61 / (9-1)) = √(61 / 8) ≈ 2.70

Step 6: Calculate the correlation coefficient (r).

r = Σ(dx dy) / (√(Σ(dx²)  Σ(dy²)))

r = -43 / (√(33  61))

r ≈ -43 / (√(2013))

r ≈ -43 / 44.87

r ≈ -0.96

Therefore, the correlation coefficient (r) is approximately -0.96.

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first off, let's take a peek at the picture above

hmmm the hyperbola is opening sideways, that means it has a horizontal traverse axis, it also means that the positive fraction will be the one with the "x" variable in it.

now, the length of the horizontal traverse axis is 4 units, from vertex to vertex, that means the "a" component of the hyperbola is half that or 2 units, and 2² = 4, with a center at the origin.

[tex]\textit{hyperbolas, horizontal traverse axis } \\\\ \cfrac{(x- h)^2}{ a^2}-\cfrac{(y- k)^2}{ b^2}=1 \qquad \begin{cases} center\ ( h, k)\\ vertices\ ( h\pm a, k)\\ c=\textit{distance from}\\ \qquad \textit{center to foci}\\ \qquad \sqrt{ a ^2 + b ^2} \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \cfrac{(x- 0)^2}{ 2^2}-\cfrac{(y- 0)^2}{ (\sqrt{3})^2}=1\implies {\Large \begin{array}{llll} \cfrac{x^2}{4}-\cfrac{y^2}{3}=1 \end{array}}[/tex]

A statement that correctly compares the graph of function g with the graph of function f include the following: C. The graph of function g is a vertical compression of the graph of function f.

In Mathematics and Geometry, a dilation is a type of transformation which typically transforms the dimension (size) or side lengths of a geometric object, without affecting its shape.

Therefore, the dimension or side lengths of the dilated geometric object would be stretched or compressed (shrunk) depending on the scale factor that is applied.

When the parent function [tex]f(x) = e^x -4[/tex] is vertically compressed by a scale factor of 1/2, the transformed function g(x) is given by the following equation;

g(x) = kf(x)

g(x) = 1/2f(x)

[tex]g(x) = \frac{1}{2} e^x -4[/tex]

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Answer:
The factors correlate with the x-intercepts. For example, with the factor (x - 3), there is an x-intercept at (3, 0). Likewise, with the factor (x + 2), there is an x-intercept at (-2, 0). This suggests that the opposite sign of the factors are the x-intercepts in the graph.

Answer: y=-3x+b

Step-by-step explanation:

You didn't give the answers, but it would look something like this:

y=-3x+b

3 1/2 cups can be expressed as the multiplication expression: 3 + 1/2.

To express 3 1/2 cups as a multiplication expression using the unit "1 cup" as a factor, you can write it as:

3 1/2 cups = (3 + 1/2) cups = 3 cups + 1/2 cup

Since there are 1 cup in each term, we can rewrite it as:

3 cups + (1/2) cup

Now, we can express each term as a multiplication expression:

3 cups = 3 * 1 cup = 3

(1/2) cup = (1/2) * 1 cup = 1/2

Putting it all together, the multiplication expression is:

3 * 1 cup + (1/2) * 1 cup = 3 + 1/2

Therefore, 3 1/2 cups can be expressed as the multiplication expression: 3 + 1/2.

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The general rule for the given Sequence is a_n = 1 – 2n, where n = 1, 2, 3, ... This equation allows us to calculate any term in the sequence by substituting the corresponding value of n.

The given sequence –1, –3, –5, –7, –9,... follows a specific pattern or rule. To determine the general rule for n = 1, 2, 3,..., we can observe the sequence and identify the relationship between the terms.

Looking at the sequence, we notice that each term is decreasing by 2. Starting with –1, we subtract 2 to get the next term –3. Then, we subtract 2 again to get –5, and so on. Therefore, the rule for generating the terms in this sequence is to start with –1 and subtract 2 from each subsequent term.

We can express this rule algebraically as follows:

a_n = a_1 + (n – 1) * d

Where:

a_n represents the nth term of the sequence,

a_1 represents the first term of the sequence,

n represents the position of the term in the sequence,

d represents the common difference between the terms.

In this case, a_1 = –1 and d = –2 (since each term is decreasing by 2). Plugging these values into the equation, we get:

a_n = –1 + (n – 1) * (–2)

Simplifying further, we have:

a_n = –1 – 2n + 2

a_n = 1 – 2n

Therefore, the general rule for the given sequence is a_n = 1 – 2n, where n = 1, 2, 3, ... This equation allows us to calculate any term in the sequence by substituting the corresponding value of n.

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Note the full question may be :

From the following sequence: – 1, – 3, – 5, – 7, – 9,…, What is the general rule for n=1, 2, 3,…?

Answer:

The answer is 11

Step-by-step explanation:

brother only=12-8=4

sister only=13-8=5

brother and sister =8

neither of them=x

28=4+8+5+x

28=17+x

x=28-17

x=11

Answer:

  11

Step-by-step explanation:

You want to know the number of students with no brother or sister, given that 12 of 28 have at least one brother, 13 have at least one sister, and 8 have both.

The attachment shows the two-way table that can be formed from the given information. The four given numbers are shown in green at lower right. The other numbers in the table are filled in to make the totals accurate.

The number of students not having either a brother or sister is 11.

__

Additional comment

In terms of probability, which is the fraction that each event is of the total, you have ...

  P(B∪S) = P(B) +P(S) -P(B∩S) . . . . . . . useful relation to memorize

  P(B'∩S') = 1 -P(B∪S) = 1 -(12/28 +13/28 -8/28) = 11/28

This tells you 11 of the 28 students do not have a brother or sister.

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The magnitude of angle Z include the following:

cosZ = 0.471

sinZ = 0.88

tanZ = 1.875

In order to determine the magnitude of angle Z, we would apply the basic trigonometric ratio because the given side lengths represent the adjacent side, opposite side, and hypotenuse of a right-angled triangle.

cos(θ) = Adj/Hyp

Where:

For cosZ, we would apply the cosine trigonometric ratio as follows;

cosZ = 13.6/28.9

cosZ = 0.4706

Z = cos⁻¹(0.4706)

Z = 61.93°.

For sinZ, we would apply the sine trigonometric ratio as follows;

sinZ = 25.5/28.9

sinZ = 0.8824

Z = sin⁻¹(0.8824)

Z = 118.07.

For tanZ, we would apply the sine trigonometric ratio as follows;

tanZ = 25.5/13.6

tanZ = 1.875

Z = tan⁻¹(1.875)

Z = 61.93.

Answer:

a. How would you categorize this situation as a function? Circle one:

Linear function

Quadratic function

Exponential function

None of the above

The given situation can be categorized as a linear function.

Step-by-step explanation:

ham

Brooke earned $9.00 in interest over one year on her account with $90 at a 10% annual interest rate, compounded annually.

To solve this problem, we can use the formula for compound interest:

A =[tex]P(1 + r/n)^(^n^t^)[/tex]

where A is the amount of money in the account after t years, P is the principal (the initial amount of money), r is the annual interest rate as a decimal, n is the number of times the interest is compounded per year, and t is the time period in years.

In this case, we want to find the interest earned in one year, so we can set t = 1. We know that Brooke has $90 in her account, so P = 90. The annual interest rate is 10%, or 0.1 as a decimal. The interest is compounded annually, so n = 1.

We can plug these values into the formula and solve for the amount of money in the account after one year:

A = [tex]90(1 + 0.1/1)^(^1^*^1^)[/tex]

A = 90(1.1)

A ≈ 99

The amount of money in the account after one year is approximately $99. So, Brooke earned $99 - $90 = $9 in interest. To the nearest cent, this is $9.00.

In conclusion, using the formula for compound interest, we can calculate the interest earned on an account given the initial balance, annual interest rate, number of compounding periods per year, and time period.

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The statements that are true about triangle PQR are QR = (sqrt(3))/2 * PQ and PR = (sqrt(3))/2 * PQ.The correct answer is option B and D.

Let's analyze the statements one by one:

A. PQ = 2PR:

This statement is not true. In a 30-60-90 triangle, the ratio of the lengths of the sides is as follows: opposite the 30-degree angle is x, opposite the 60-degree angle is x√3, and the hypotenuse (opposite the 90-degree angle) is 2x.

Therefore, PQ = x, and PR = x√3. Since √3 is not equal to 2, this statement is false.

B. QR = (sqrt(3))/2 * PQ:

This statement is true. In a 30-60-90 triangle, the ratio of the lengths of the sides is as follows: opposite the 30-degree angle is x, opposite the 60-degree angle is x√3, and the hypotenuse (opposite the 90-degree angle) is 2x.

Therefore, QR = x√3/2 = (sqrt(3))/2 * x = (sqrt(3))/2 * PQ. This statement holds true.

C. OR = 2PR:

We don't have any information regarding the length of OR, so we cannot determine if this statement is true or false based on the given information.

D. PR = (sqrt(3))/2 * PQ:

This statement is true. In a 30-60-90 triangle, the ratio of the lengths of the sides is as follows: opposite the 30-degree angle is x, opposite the 60-degree angle is x√3, and the hypotenuse (opposite the 90-degree angle) is 2x. Therefore, PR = x√3 = (sqrt(3))/2 * 2x = (sqrt(3))/2 * PQ. This statement is correct.

E. QR = sqrt(3) * PR:

This statement is not true. In a 30-60-90 triangle, the ratio of the lengths of the sides is as follows: opposite the 30-degree angle is x, opposite the 60-degree angle is x√3, and the hypotenuse (opposite the 90-degree angle) is 2x. Therefore, QR = x√3, and PR = x√3. So, QR = PR, but not QR = sqrt(3) * PR.

F. PQ = sqrt(3) * PR:

This statement is not true. In a 30-60-90 triangle, the ratio of the lengths of the sides is as follows: opposite the 30-degree angle is x, opposite the 60-degree angle is x√3, and the hypotenuse (opposite the 90-degree angle) is 2x. Therefore, PQ = x, and PR = x√3. So, PQ = PR/√3, but not PQ = sqrt(3) * PR.

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The probable question may be:

In triangle PQR , m angle P = 60 deg , m angle Q = 30 deg and m angle R = 90 deg Which of the following statements about triangle PQR are true?

Check all that apply

A. PQ = 2PR

B.QR = (sqrt(3))/2 * PQ

C. OR = 2PR

D.PR = (sqrt(3))/2 * PQ

E. QR = sqrt(3) * PR

F. PQ = sqrt(3) * PR

The angle measure of an arc π meters is 45°.

Given that, the diameter of a circle is 8 meters.

Radius of a circle = 8/2 = 4 meters

The formula to find the arc length of a circle is θ/360° ×2πr.

Here, θ/360° ×2πr.

π=θ/360° ×2π×4

1=θ/360° ×8

360°=8θ

θ=360°/8

θ=45°

Therefore, the angle measure of an arc π meters is 45°.

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