prove that:
(P+1/P-1 + P+1/P+1)-(P-1/P+2 + P+1/P-2)=-4(2p² + 1)/(p²-4) (p² - 1)​

An appropriate measure of center for the data, and its value include the following: D. The median is the best measure of center, and it equals 19.

In Mathematics and Statistics, a box plot is sometimes referred to as box-and-whisker plot and it can be defined as a type of chart that can be used to graphically or visually represent the five-number summary of a data set with respect to locality, skewness, and spread.

Based on the information provided about the data set, the five-number summary for the given data set include the following:

Minimum (Min) = 12.

First quartile (Q₁) = 17.

Median (Med) = 19.

Third quartile (Q₃) = 20.

Maximum (Max) = 24.

In conclusion, the best measure of center is the median and it is equal to 19.

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A subscription to a live streaming program normally costs $29 but is on sale for 25% off. An 8% tax is added at checkout then the total cost is $23.49.

To calculate the total cost, first, we find the discounted price of the subscription. A 25% discount on $29 is calculated by multiplying $29 by 0.25 (25% as a decimal), which gives us a discount of $7.25. Subtracting the discount from the original price, we get the discounted price of $29 - $7.25 = $21.75.

Next, we need to add the 8% tax to the discounted price. To calculate the tax amount, we multiply $21.75 by 0.08 (8% as a decimal), which gives us $1.74. Adding this tax amount to the discounted price, we get the total cost as $21.75 + $1.74 = $23.49.

Step 1: Calculate the discount amount:

Discount = 25% of $29

Discount = 0.25 * $29

Discount = $7.25

Step 2: Find the discounted price:

Discounted Price = Original Price - Discount

Discounted Price = $29 - $7.25

Discounted Price = $21.75

Step 3: Calculate the tax amount:

Tax Amount = 8% of Discounted Price

Tax Amount = 0.08 * $21.75

Tax Amount = $1.74

Step 4: Calculate the total cost:

Total Cost = Discounted Price + Tax Amount

Total Cost = $21.75 + $1.74

Total Cost = $23.49

So, the total cost after applying the 25% discount and adding an 8% tax at checkout is $23.49.

- Discount = 0.25 * $29 = $7.25

- Discounted Price = $29 - $7.25 = $21.75

- Tax Amount = 0.08 * $21.75 = $1.74

- Total Cost = $21.75 + $1.74 = $23.49

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The surface area of the prism is 360in²

A prism is a solid shape that is bound on all its sides by plane faces.

The surface area of a prism is expressed as;

SA = 2B + ph

where h is the height of the prism and B is the base area , p is the perimeter of the base.

Base area = 1/2 × 5 × 12

= 5 × 6 = 30 in²

Perimeter of the base = 5+12+13

= 30 in²

height = 10 in

SA = 2 × 30+30× 10

= 60 + 300

= 360 in²

therefore the surface area of the prism is 360 in²

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Answer: ≅ $0.02

Step-by-step explanation:

First, lets find the area of the large triangle

0.5 x 14 x 11 = 77 mm^2

Next lets find the area of the middle rectangle:

4 x 14 = 56 mm^2

Last lets find the area of the small triange:

0.5 x 14 x 3 = 21 mm^2

So the total area is: 77 + 56 + 21 = 154 mm^2

the total cost is given to be: 1.52 + 0.42 + 0.92 = $2.86

dividing the cost by the area to find the cost/mm:

2.86/154 ≅ $0.02

Multiplying is necessary because each larger unit contains a certain number of the smaller units and we need to account for that conversion factor.

For example, when converting from meters to centimeters where we know that 1 meter is equal to 100 centimeters.

To convert 3 meters to centimeters, we need to multiply by the conversion factor of 100 which gives us:

= 3 meters * 100 centimeters/meter

= 300 centimeters

So, without the multiplication, we would not account for the fact that each meter contains 100 centimeters. This principle applies to all conversions from larger units to smaller units and it is important to remember to ensure accurate conversions.

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VZ is 92.5. WY is the midsegment of AVXZ, it means that WY is parallel to AX and its length is half the length of AX.

what is midsegment  ?

A midsegment of a triangle is a line segment connecting the midpoints of two sides of the triangle. It is also known as a midline. The midsegment of a triangle is parallel to the third side of the triangle, and its length is half the length of the third side.

In the given question,

Since WY is the midsegment of AVXZ, it means that WY is parallel to AX and its length is half the length of AX.

Let's call the length of AX as L. Then, we have:

WY = 11t = 1/2 L

Multiplying both sides by 2, we get:

22t = L

Now, we can use this information to find VZ. Since WY is also parallel to VZ, we have:

WV = VZ

Substituting WV = L/2 and VZ = 5t + 51, we get:

L/2 = 5t + 51

Substituting L = 22t, we get:

22t/2 = 5t + 51

11t = 5t + 51

6t = 51

t = 8.5

Substituting t = 8.5 in the equation for VZ, we get:

VZ = 5t + 51 = 5(8.5) + 51 = 92.5

Therefore, VZ is 92.5.

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Answer:

Put your calculator in degree mode.

sin(35°) = 7/r

r sin(35°) = 7

r = 7/sin(35°) = 12.2 cm

Answer:

Commutative property of addition: Changing the order of addends does not change the sum. For example, 4 + 2 = 2 + 4 4 + 2 = 2 + 4 4+2=2+44, plus, 2, equals, 2, plus, 4. Associative property of addition: Changing the grouping of addends does not change the sum

The matrix formed by performing the row operation 4R₁ + R₂ — R₂ on M will have R₁ = [1 0 3] and R₂ = [ -1 2 10]

Row operations are a set of operations that can be performed on the rows of a matrix in order to transform it into a row equivalent matrix, which has the same solution set as the original matrix.

performing the row operation 4R₁ + R₂ — R₂ on M, we have;

4(1) + (-5) = -1 {row 2 column 1}

4(0) + 2 = 2 {row 2 column 2}

4(3) + (-2) = 10 {row 2 column 3}

Therefore, the matrix formed by performing the row operation 4R₁ + R₂ — R₂ on M will have R₁ = [1 0 3] and R₂ = [ -1 2 10]

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The calculated value of the area of the real room is 10.67 cm²

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

In the drawing, 3cm represents 4cm

This means that

Scale factor = 4/3

Also, we have the dimension of the scale drawing to be

Dimension = 3cm by 2 cm

So, the area of the bedroom in the scale is

Area = 3cm * 2cm

For the actual room, we have

Area = 3cm * 2cm * (4/3)²

Evaluate the exponent

Area = 3cm * 2cm * (16/9)

Evaluate the products

Area = 10.67 cm²

This means that the real area is 10.67 cm²

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The area of the triangle is 128 square inches.

To find the area of a triangle, you can use the formula:

Area = (1/2) * base * height

In this case, the base of the triangle is 32 inches and the height is 8 inches.

Plugging these values into the formula, we get:

Area = (1/2) * 32 * 8

Area = 16 * 8

Area = 128 in²

So, the area of the triangle is 128 square inches.

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A triangle is shown in the image. A triangle with a height of 8 inches. The height is perpendicular to the base labeled 32 inches. The side from the top of the perpendicular side to the base is labeled 28 inches. What is the area of the triangle? 256 in2 224 in2 128 in2 112 in2

The graph of the linear equation y = (1/2)*x + 3 can be seen in the image at the end.

Here we want to graph the linear equation:

y = (1/2)*x + 3

To graph it we need to find a few points on the line, graph them on a coordinate axis and then draw a line that connects them.

The first point that we can use is the y-intercept, it is what we get when we evaluate in x = 0.

y = (1/2)*0 + 3

The point is (0, 3)

Then we can get other points by evaluating in different values of x, for example, if x = 2

y = (1/2)*2 + 3 = 1+ 3 =4

If x = 4

y = (1/2)*4 + 3 = 2 + 3 = 5

So we have the points (2, 4) and (4, 5)

Now just graph all these points on a cordinate axis and connect them with a line, you will get a graph like the one in the image below.

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Answer:

( -6, 6)

Step-by-step explanation:

This statement describes the relationship between symmetries and rotations

This statement describes a relation between two symmetry operations, denoted by [tex]s_i[/tex] and [tex]s_j[/tex], and a rotation operation denoted by [tex]r_{i-j}[/tex]. The notation [tex]"s_i \circ s_i"[/tex] and [tex]"s_j \circ s_i"[/tex] denotes the composition of these operations, which means applying one operation after the other.

The statement says that [tex]s_i[/tex] and [tex]s_j[/tex] are both symmetries, and that they have a specific relationship with each other: when [tex]s_i[/tex] is composed with itself, it gives the identity operation (denoted by 1), and when [tex]s_j[/tex] is composed with [tex]s_i[/tex], it also gives the identity operation.

Then the statement concludes that this relationship implies that [tex]s_j[/tex] can be obtained by first applying [tex]s_i[/tex] and then applying a rotation operation [tex]r_{i-j}[/tex], and that [tex]s_i[/tex] can be obtained by first applying a rotation operation [tex]r_{i-j}[/tex] and then applying [tex]s_j[/tex].

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Answer:

$176420

Step-by-step explanation:

Monthly payment = $992

Yearly payment = 992 × 12 = $11064

Total payment = 11064 × 30 = $331920

Interest = 331920 - 155500 = $176420. The interest seems too much, are you sure there's no mistake in the question? Maybe the monthly payment is more than what you've written here?

Answer:

[tex] {(x - 3)}^{2} + {(y - 4)}^{2} = 36[/tex]

Answer:

Step-by-step explanation:

We are given a figure, consisting of a right triangle with a circle cut out of it.

To Calculate : area of the shaded region below,

First Let's find the Area of the right triangle:

Area(triangle) = 1/2 × Base × Height

= 1/2 × 56 × 56

= 1/2 × 3136

= 3136/2

= 1568 m².

Now, We are diameter of the circle = 20 m

Radius of the circle = 20/2 = 10 m

Acc. to question we have to use 3.14 as an

approximation for pi.

Area (circle) = 3.14 × 10 × 10

Area (circle) = 3.14 × 100

Area (circle) = 314 m²

Area of shaded region = Area of triangle - Area of circle =

= 1568 m² - 314 m²

= 1.254 square meters

Therefore, The approximate area of the shaded region is 1.254 square meters.

Option (a) is the required answer.

An example of a function that models a linear relationship between two quantities, x and y is y = mx + b

We need to use the equation of a straight line, which is commonly expressed in slope-intercept form as:

y = mx + b

In this function, x represents the independent variable, m is the slope of the line, and b is the y-intercept. To use this function, we simply plug in the values of x, m, and b that correspond to the specific relationship we are modeling.

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The calculated R2 value is 0.621, indicating that 62.1% of the variability in tumor growth can be explained by the linear regression model.

In statistics, the coefficient of determination, denoted by R2, is a measure of how well the regression line fits the data. It represents the proportion of the variance in the dependent variable (y) that is predictable from the independent variable (x).

To calculate R2, we use the formula R2 = SSR/SST, where SSR is the sum of squares of regression and SST is the total sum of squares.

From the given data, SSR = 568.5 and SST = 914.8. Thus, R2 = 568.5/914.8 = 0.621, rounded to three decimal places.

This means that 62.1% of the variability in the tumor growth can be explained by the linear regression model. The remaining 37.9% of the variability is unexplained and may be due to other factors not included in the model.

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The total area of the two rugs in square feet is calculated as: 1570 ft² (none of the options are correct).

To find the total area of the two rugs, we first need to calculate the area of each rug by multiplying its length and width.

Area of the first rug = 15 ft x 10 ft = 150 ft²

Area of the second rug = 142 ft x 10 ft = 1420 ft²

Now, we can find the total area by adding the area of both rugs:

Total area = 150 ft² + 1420 ft² = 1570 ft²

Therefore, the answer is not one of the options provided.

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Answer:

A ribeye steak dinner costs $32.23 and a grilled salmon dinner costs $14.04.

Step-by-step explanation:

Let's use variables to represent the cost of each dinner. Let's say that the cost of a ribeye steak dinner is "r" and the cost of a grilled salmon dinner is "s".

From the first day's sales, we know:

From the second day's sales, we know:

We now have two equations with two variables, which we can solve using elimination or substitution.

Let's use elimination. If we multiply the first equation by 15 and the second equation by 11, we can eliminate the "r" variable:

Subtracting the second equation from the first, we get:

Dividing both sides by 133, we get:

So a grilled salmon dinner costs $14.04.

We can substitute this value back into one of the original equations to solve for "r". Let's use the first equation:

So a ribeye steak dinner costs $32.23.

Therefore, a ribeye steak dinner costs $32.23 and a grilled salmon dinner costs $14.04.

The woman's gross income, adjusted income and taxable income are:

The gross income to the woman is:

= Wages + interest

= 33, 900 + 2, 600

= $ 36, 500

The adjusted gross income :

= Gross income - Retirement plan contribution

= 36, 500 - 3, 600

= $ 32, 900

The taxable income ;

= AGI - Personal exemption - Deductions

= 32, 900 - 3, 500 - 5, 440

= $ 23, 960

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The slope of a line parallel to this line is 6/7.

The slope of a line perpendicular to this line is -7/6.

In Mathematics and Geometry, parallel lines can be defined as two (2) lines that are always the same (equal) distance apart and never meet.

In Mathematics and Geometry, two (2) lines are parallel under the following conditions:

Slope, m₁ = Slope, m₂

Based on the information provided about this line, we have the following equation in standard form;

6x - 7y = 4

By making y the subject of formula, we have:

7y = 6x - 4

y = 6x/7 - 4

In Mathematics and Geometry, a condition that is true for two lines to be perpendicular is given by:

m₁ × m₂ = -1

6/7 × m₂ = -1

6m₂ = -7

Slope, m₂ = -7/6

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Answer: The United States rule is a method for calculating interest and balances on loans that have partial payments during their term. The rule assumes that each partial payment is applied to the interest first, then to the principal. Here's how to use the United States rule to calculate the balance and interest for this loan:

Calculate the daily interest rate by dividing the annual interest rate by 360 (the number of days in a standard year for most loans).

Daily Interest Rate = 15.0% / 360 = 0.04167%

Calculate the interest due for the first partial payment of $5800 made on day 112. Since this payment is larger than the remaining principal, it will pay off the loan entirely and no interest will accrue beyond this date.

Calculate the interest due for the second partial payment of $4350 made on day 196.

Calculate the interest due on the remaining principal at maturity (day 280).

Add up the interest due from each period to get the total interest paid over the life of the loan.

Add up the principal amounts from each payment to get the total amount paid.

Subtract the total amount paid from the original principal to get the balance at maturity.

Balance at maturity = $1450.00 - $10,150.00 = -$8700.00 (which means the lender owes the borrower this amount)

Therefore, the balance at maturity is -$8700.00 (meaning the lender owes the borrower this amount) and the total interest paid over the life of the loan is $8.10.

Answer:

x^1/4 + 8x^1/4 + 12

Step-by-step explanation:

Using the FOIL method, we get x^1/4 + 8x^1/4 + 12

The probability that a student chosen randomly from the class has a brother is 57%.

Given is a data table summarizes how many students have a brother or a sister.

We need to find the probability that a student chosen randomly from the class has a brother,

So, we know, probability = favorable outcome / total outcome

Here, favorable outcome = having a brother = 12 + 4 = 16

Total outcomes = all the students = 12+4+10+2 = 28

So,

P(having a brother) = 16/28 = 0.57 = 57%

Hence the probability that a student chosen randomly from the class has a brother is 57%.

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The standard deviation for the given distribution is 8.739 using either the direct method or the shortcut method.

Standard deviation is defined as a measure of how spread out numbers are. It is also referred to as the square root of the Variance, and then the Variance is the average of the squared differences from the Mean.

To calculate the standard deviation for the given distribution, we will utilize either the direct method or the shortcut method.

Direct method:

Calculate the mean wage (F) for the distribution:

Mean = (20 + 22 + 27 + 30 + 31 + 32 + 35 + 45 + 40 + 48)/10

Mean = 34

Calculate the differences between each wage and the mean wage:

20 - 34 = -14

22 - 34 = -12

27 - 34 = -7

30 - 34 = -4

31 - 34 = -3

32 - 34 = -2

35 - 34 = 1

45 - 34 = 11

40 - 34 = 6

48 - 34 = 14

Square each of the differences:

(-14)² = 196

(-12)² = 144

(-7)² = 49

(-4)² = 16

(-3)² = 9

(-2)² = 4

1² = 1

11² = 121

6² = 36

14² = 196

Add up the squared differences: 196 + 144 + 49 + 16 + 9 + 4 + 1 + 121 + 36 + 196 = 756

Divide the sum of the squared differences by the number of workers: 756/10 = 75.6

Take the square root of the result to get the standard deviation:

√75.6 = 8.739

Shortcut method:

The mean wage (F) for the distribution:

(20 + 22 + 27 + 30 + 31 + 32 + 35 + 45 + 40 + 48)/10 = 34

The sum of the squares of the differences between each wage and the mean wage:

(20 - 34)² + (22 - 34)² + (27 - 34)² + (30 - 34)² + (31 - 34)² + (32 - 34)² + (35 - 34)² + (45 - 34)² + (40 - 34)² + (48 - 34)² = 756

Divide the sum by the number of workers: 756/10 = 75.6

Take the square root of the result to get the standard deviation:

Standard deviation = √75.6 = 8.739

Therefore,

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Complete question is:

Calculate standard deviation for the following

distribution by direct and shortcut method.

Assume mean as 32.

Serial No. of Workers

1

2

3

4

5

6

7

8

9

10

Wage (F)

20

22

27

30

31

32

35

45

40

48

The value of x in each expressions are:

1) 6

2) 1

3) -3

4) -21

5) -4

We have,

The expressions are:

1)

[tex]3^x = 9^3[/tex]

And,

9³ = (3²)³ = [tex]3^6[/tex]

So,

[tex]3^x = 3^6[/tex]

And,

x = 6

2)

[tex]4^{x + 1}[/tex] = 16

16 = 4²

So,

x + 1 = 2

x = 2 - 1

x = 1

3)

[tex](1/3)^x[/tex] = 27

27 = 3³

So,

[tex]3^{-1x}[/tex] = 3³

-x = 3

x = -3

4)

[tex]5^{3x}[/tex] = [tex]25^{x - 1}[/tex] =

[tex]5^{3x}[/tex] = [tex]5^{2x - 21}[/tex]

3x = 2x - 21

3x - 2x = -21

x = -21

5)

[tex]2^{-x}[/tex] = 16

16 = [tex]2^4[/tex]

So,

-x = 4

x = -4

Thus,

The value of x in each expression are:

1) 6

2) 1

3) -3

4) -21

5) -4

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The solution of the trigonometric equation on the interval (, 360) is

A trigonometric equation is an equation that contains trigonometric ratios.

Given the trigonometric equation

2sinx = 1 - cosx, we proceed to solve for x as follows.

Since 2sinx = 1 - cosx

Using sin²x = 1 - cos²x

⇒ sinx = √(1 - cos²x )

Substituting this into the equation, we have that

2sinx = 1 - cosx

2√(1 - cos²x ) = 1 - cosx

Squaring both sides, we have that

[2√(1 - cos²x )]² = (1 - cosx)²

4(1 - cos²x ) = (1 - cosx)²

4 - 4cos²x = 1 - 2cosx + cos²x

Collecting similar terms, we have that

1 - 4 - 2cosx + 4cos²x + cos²x = 0

- 3 - 2cosx + 5cos²x = 0

Re-arranging the equation, we have that

5cos²x - 2cosx - 3 = 0

Let cosx = y

So, we have that

5y² - 2y - 3 = 0

Factorizing, we have that

5y² - 2y - 3 = 0

5y² - 5y + 3y - 3 = 0

5y(y - 1) + 3(y - 1) = 0

(5y + 3)(y - 1) = 0

⇒ 5y + 3 = 0 or y - 1 = 0

⇒ 5y = -3 or y = 1

⇒ y = -3/5 or y = 1

Since y = cosx, we have that

cosx = -3/5 or cosx = 1

cos(180° - x) = 3/5 or cos(270° - x) = 3/5 or cosx = 1

Taking inverse cosine, we have that

180° - x = cos⁻¹(3/5) or 270° - x = cos⁻¹(3/5) or x = cos⁻¹(1)

180° - x = 53.13° or 270° - x = 53.13° or x = 90°

x = 180° - 53.13° or x = 270° - 53.13° or x = 90°

x = 126.87° or x = 216.87 or x = 90°

So, the solution is

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