Omar Cuts A Piece Of Wrapping Paper with the shape and dimensions as shown .Find the area of the wrapping paper. Round your answer to the nearest tenth if needed

The probability that a student gets a lunch with chips and apple juice is 1/12, and the probability that a student gets a lunch without chips is 1/2.

There are 3 choices of sandwiches, 2 choices of sides, and 2 choices of drinks, so there are a total of 3x2x2 = 12 possible lunch combinations. To find the probability that a student gets a lunch that includes chips and apple juice, we need to count the number of lunch combinations that include chips and apple juice, and then divide by the total number of possible lunch combinations.

Number of lunch combinations that include chips and apple juice = 1 (chips and apple juice is only one combination)

Total number of possible lunch combinations = 12

Probability of getting a lunch that includes chips and apple juice = 1/12

To find the probability that a student gets a lunch that does not include chips, we need to count the number of lunch combinations that do not include chips, and then divide by the total number of possible lunch combinations,

Number of lunch combinations that do not include chips = 6 (3 choices of sandwiches x 2 choices of drinks) Total number of possible lunch combinations = 12, probability of getting a lunch that does not include chips = 6/12 = 1/2

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The annual rate of inflation between the two time periods is approximately 3.05%.

To calculate the annual rate of inflation, we can use the formula:

Annual Inflation Rate = ((Current Price - Past Price) / Past Price) * 100 / Number of Years

Plugging in the values, we have:

((3.25 - 1.75) / 1.75) * 100 / 20 ≈ 0.153 * 100 / 20 ≈ 3.05%

Therefore, the annual rate of inflation between the two time periods is approximately 3.05%.

Inflation refers to the general increase in prices over time, which leads to a decrease in the purchasing power of money. In this case, the cost of a gallon of milk has increased from $1.75 to $3.25 over 20 years. By calculating the annual rate of inflation, we find that prices have been rising at an average rate of 3.05% per year during this period.

This means that the cost of goods and services, including milk, has increased by an average of 3.05% each year due to inflation. It highlights the importance of considering inflation when comparing prices and understanding the impact it has on the value of money over time.

In conclusion, based on the given information, the annual rate of inflation between the two time periods is approximately 3.05%, indicating the increase in the cost of a gallon of milk over 20 years.

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The lowest price she can charge per gift box to earn $2,000 in profit in her first month is approximately $3.2.

Jill is starting her own business called Fuzzy Socks Box, where she'll knit and sell gift boxes of fuzzy socks online.

Jill's profit per gift box is x - 2 dollars. To earn $2,000 in profit in her first month, she needs to sell

2000 / (x - 2) gift boxes.

According to her survey, the number of gift boxes she'll sell is

-10x + 350

Setting these two expressions equal to each other, we can find the lowest price Jill can charge per gift box to earn $2,000 in profit

-10x + 350 = 2000 / (x - 2)

Multiplying both sides by (x - 2), we get

-10[tex]x^{2}[/tex] + 12x - 470 = 0

Solving this quadratic equation, we find that

x = (6 +[tex]\sqrt{196[/tex] )/5 ≈ 3.2

or

x = (6 -[tex]\sqrt{196[/tex] )/5 ≈ 0.7

Since Jill cannot charge a negative price for her gift boxes, the lowest price she can charge per gift box to earn $2,000 in profit in her first month is approximately $3.2.

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Lana would pick a non-striped marble about 294 times.

The probability of picking a marble that is not striped is 100% - 8% = 92% = 0.92. This means that for each pick, the probability of getting a non-striped marble is 0.92.

If Lana picks a marble and replaces it 320 times, the number of times she would pick a non-striped marble can be predicted by multiplying the probability of getting a non-striped marble by the number of picks.

So, the number of times she would pick a non-striped marble is:

0.92 x 320 = 294.4

Rounding to the nearest whole number = 294.

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Answer:

89.6°

Step-by-step explanation:

180-90.4=89.6°

Answer: 89.6

Step-by-step explanation:

Supplementary means angles adding up to 180

180-90.4  = x

x=89.6   this is your supplemental angle

From the given data 11 is the mean of the data set.

What is Mean ?

In statistics, the mean (also called the average) is a measure of central tendency that represents the typical value in a data set. It is calculated by adding up all the values in the data set and dividing by the number of values.

To find the value of x in the data set, we can use the formula for the mean (also called the average). The mean is calculated by adding up all the values in the data set and dividing by the number of values. In other words:

mean = (sum of all values) : (number of values)

We are given that the mean of the data set is 11, so we can write:

11 = (12 + 9 + 12.5 + 13 + x + 10 + 11 + 11 + 7 + 9) : 10

Here, we have 10 values in the data set (including the unknown value x), so we divide the sum of all the values by 10 to find the mean.

To solve for x, we can start by simplifying the right-hand side of the equation:

110 = 75.5 + x

Next, we can isolate x by subtracting 75.5 from both sides:

x = 34.5

Therefore, the value of x that makes the mean of the data set equal to 11 is x = 34.5.

In other words, if we replace the unknown value x with 34.5, the resulting data set will have a mean of 11. This means that the sum of all the values in the data set will be 110, since:

12 + 9 + 12.5 + 13 + 34.5 + 10 + 11 + 11 + 7 + 9 = 110

And when we divide this sum by 10, we get:

110 : 10 = 11

Therefore, From the given data 11 is the mean of the data set.

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If the vertex of a parabola is at (5,0) and it opens UPWARD, it will have one solution.

Since the parabola opens upward, its vertex is the minimum point, and therefore, it will have only one solution.

A parabola is a U-shaped curve in mathematics that can be formed by the graph of a quadratic function. It is a type of conic section, along with circles, ellipses, and hyperbolas.

Mathematically, a parabola can be defined as the set of all points in a plane that are equidistant from a fixed point called the focus and a fixed line called the directrix. The focus lies on the axis of symmetry of the parabola, and the directrix is perpendicular to the axis of symmetry.

If the vertex of a parabola is at (5,0) and it opens UPWARD, it will have one solution.

Your answer: one

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For the function y = f(x) = 3x² the Δx is  4, Δy is 72, and Δy/Δx is 18  between x1 = 1 and x2 = 5.

To find the values of Δx, Δy, and Δy/Δx for the function y = f(x) = 3x² between x1 = 1 and x2 = 5.

Δx represents the change in x between x1 and x2,

It can be calculated as Δx = x2 - x1 = 5 - 1 = 4.

Δy represents the change in y (or the output of the function f(x)) between x1 and x2, and can be calculated as Δy = f(x2) - f(x1).

We can find the value of f(x) by substituting x = 1 and x = 5 into the equation f(x) = 3x²:

f(1) = 3(1)² = 3

f(5) = 3(5)² = 75

Therefore, Δy = f(x2) - f(x1) = 75 - 3 = 72.

Δy/Δx represents the average rate of change of y with respect to x between x1 and x2,

It can be calculated as Δy/Δx' = [f(x2) - f(x1)] / [x2 - x1].

We can substitute the values of Δy and Δx into this equation to get:

Δy/Δx' = [f(x2) - f(x1)] / [x2 - x1] = [75 - 3] / (5 - 1) = 72 / 4 = 18.

Therefore, the values of Δx, Δy, and Δy/Δx are 4, 72, and 18, respectively.

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The total surface area of the figure shown as a rectangular prism is 208. 6 cm

The formula for calculating the total surface area of a rectangular prism is expressed as;

TSA =  2 (lh +wh + lw )

Such that the parameters are;

Now, substitute the values, we have;

TSA = 2(2(9) + 7.9(9) + 2(7.6)

expand the bracket, we have;

TSA = 2(18 + 71.1 + 15.2)

add the values

TSA = 2(104. 3)

TSA = 208. 6

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To select all the tables that show quadratic functions, look for tables with a second-degree polynomial equation in the form of "y = ax² + bx + c".

Tables that show quadratic functions:

(a).

x y

-2 8

-1 3

0 0

1 1

2 4

This table shows a quadratic function in the form of y = x² - 2x.

(b)

x y

-3 0

-2 1

-1 4

0 9

1 16

2 25

3 36

This table shows a quadratic function in the form of y = x².

A quadratic function is a second-degree polynomial function that can be expressed in the general form of "y = ax² + bx + c", where a, b, and c are constants.

In this form, the variable "x" is squared, and the coefficient "a" determines whether the parabola opens upward or downward.

To identify tables that show quadratic functions, we need to look for tables that display data points that follow a quadratic pattern.

That is, the dependent variable (y) changes in a way that corresponds to a quadratic equation.

In the first table, the values of y correspond to the quadratic equation y = x² - 2x. The second table shows a set of data points that corresponds to the quadratic function y = x².

Therefore, these two tables show quadratic functions.

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Answer:

This relation is a function.

Algebraically, this function is written as

y = 5x.

A z-score of 0 means that the sample proportion is equal to the population proportion. Therefore, after using sampling distribution we can conclude that we agree with Aiden's expectation that the percentage for this sample will be 45%.

Based on the given information, we can assume that the population proportion of students who brought a can of food to contribute to the canned food drive is 0.45. Aiden picks a representative sample of 25 students, and he expects the percentage for this sample will be 45%.

We can use the sampling distribution formula to calculate the expected sample proportion:

SE = sqrt[p(1-p) / n]

where p is the population proportion, n is the sample size, and SE is the standard error.

Plugging in the values, we get:

SE = sqrt[0.45(1-0.45) / 25] = 0.0984

Next, we can use the normal distribution to find the z-score corresponding to a sample proportion of 0.45:

z = (0.45 - 0.45) / 0.0984 = 0

A z-score of 0 means that the sample proportion is equal to the population proportion. Therefore, we can conclude that we agree with Aiden's expectation that the percentage for this sample will be 45%.

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The equation would be: y = 3 + x

To include the terms you mentioned, we can set up an equation to represent the relationship between the number of weeks Stacy volunteers (x) and the number of cards she has (y).

Since Stacy starts with 3 collector's cards and receives 1 more card each week she volunteers, the equation would be:

y = 3 + x

In this equation, x represents the number of weeks Stacy volunteers, and y represents the total number of collector's cards she has.

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The inequality that is true is Negative 2 and three-fourths less-than negative 1 and one-half.

The first inequality from the number line can be shown to be :

( 1 / 4 ) < - 1 1 / 2

This is not possible as a negative cannot be larger than a positive.

The second inequality is:

- 2. 75 < - 1. 5

This is true as larger negative numbers are lower than smaller negative numbers.

The third inequality is:

- 2. 25 > - 1. 25

This is not possible for the reason explained.

In conclusion, option B is correct.

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Answer: 0 g * m/s

Step-by-step explanation:

You first want to multiply 100g, by 12 m/s. This gives you the momentum of the first bird, the answer being 1200 g * m/s

Second, You was to multiply 30g, by 40 m/s. This gives you the momentum of the second bird, The answer also being 1200 g * m/s.

Then, subtract The final answers, and you get 0 g * m/s.

=(100 g)(12 m/s​)+(30 g)(−40 m/s​)

=1200 g⋅ m/s​+(−1200 g⋅ m/s​)

=0 g⋅ m/s

I am not an expert, so I may have gotten some things incorrect.

If this doesn't make sense please consult an expert :,)

The relative speed of the birds is 52 m/s.

We are given that;

Number of blackbirds= 100g

Speed= 12m/s

Now,

The relative speed of the blackbird as seen by the bluebird is:

vblackbird relative to bluebird​=vblackbird​−vbluebird​

vblackbird relative to bluebird​=−12−40

vblackbird relative to bluebird​=−52 m/s

This means that the blackbird is moving to the left at 52 m/s as seen by the bluebird. The relative speed of the bluebird as seen by the blackbird is:

vbluebird relative to blackbird​=vbluebird​−vblackbird​

vbluebird relative to blackbird​=40−(−12)

vbluebird relative to blackbird​=52 m/s

This means that the bluebird is moving to the right at 52 m/s as seen by the blackbird. The relative speed of either bird is equal to the magnitude (absolute value) of their relative velocity.

Therefore, by the speed the answer will be 52 m/s.

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If Holly cuts 6 ribbons into fifths for her craft project, after cutting, she has a total of 30 one-fifth-size ribbons (6 ribbons x 5 cuts each).

Holly has cut 6 ribbons into fifths for her craft project. This means that she has a total of 30 ribbons, each with a size of one-fifth. To understand this better, we can break it down into fractions. Each ribbon is one-fifth of a whole ribbon, and since Holly has cut 6 ribbons, she has 6 times one-fifth, which equals 30. So, to answer the question, Holly has 30 ribbons, each with a size of one-fifth. These ribbons can be used for various crafts, such as creating bows, wrapping presents, or decorating cards. The possibilities are endless, and with 30 ribbons, Holly can get creative and make a lot of beautiful crafts.

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The center is located at (2, −4), and the radius is 4.

To find the center and radius of this equation, we need to complete the square for both the x and y terms.

Starting with the x terms:

x^2 - 4x

To complete the square, we need to add and subtract (4/2)^2 = 4:

x^2 - 4x + 4 - 4

Now we can simplify:

(x - 2)^2 - 4

And for the y terms:

y^2 + 8y

We need to add and subtract (8/2)^2 = 16:

y^2 + 8y + 16 - 16

Simplifying:

(y + 4)^2 - 16

Now we can rewrite the original equation:

(x - 2)^2 - 4 + (y + 4)^2 - 16 = -4

Combining like terms:

(x - 2)^2 + (y + 4)^2 = 4

This is in the form of a circle:

(x - h)^2 + (y - k)^2 = r^2

Where the center is (h, k) and the radius is r.

So the center is located at (2, -4) and the radius is 4.

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The value of m by the given data is m=2

We are given that;

Number of elephants= 125000gram

The mass of african elephant= 3 x 10

Now,

To find the values of b and c, we need to write the given masses in scientific notation, which means that the coefficient must be between 1 and 10. For example, 6 x 10 grams is not in scientific notation, because 6 is not between 1 and 10. To fix this, we need to move the decimal point one place to the left and increase the exponent by one. We get:

6x106 grams=6.0x106 grams=6.0x106−1×101 grams=6.0x105×101 grams=6.0x105+1 grams=6.0x107 grams

Similarly, for the silverback gorilla, we get:

2x105 grams=2.0x105 grams=2.0x105−1×101 grams=2.0x104×101 grams=2.0x104+1 grams=2.0x105 grams

Therefore, the values of b and c are b=7,c=5

To find the value of m, we need to divide the mass of the African elephant by the mass of the silverback gorilla and write the result in scientific notation. We get:

Mass of silverback gorillaMass of African elephant​=2.0x105 grams6.0x107 grams​=2.06.0​×105107​=3.0×107−5=3.0×102

Therefore, by unitary method the answer will be m=2.

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if one side of triangle is approximately 10 feet, the height of the attic is approximately 8.66 feet.

To find the height h of an equilateral triangle, we need to know the length of one of its sides. However, this information is not given in the question.

We can use the Pythagorean theorem to find an approximate value of h. Let's assume that the length of one side of the equilateral triangle is 10 feet (this is just an arbitrary value for illustration purposes).

Then, the height h can be found using the formula:

h = √(10² - (10/2)²)

h = √(100 - 25)

h = √75

h ≈ 8.66 feet

So, if one side of the equilateral triangle is approximately 10 feet, the height of the attic is approximately 8.66 feet. However, if the length of the side is different, the height will also be different.

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Answer:

x = 4 and y = 9

Step-by-step explanation:

Let x and y be the two numbers. Also,

Let the first number be x and the second number be y.

Converting the word expressions to an equation we get:

                   y = x + 5

               3x + 2y = 30

Only thing left is to substitute y = x + 5 into the second equation

     3x + 2y = 30

     3x + 2(x + 5) = 30   (substituting y as x + 5)

     3x + 2x + 10 = 30   (multiplying by both x and 5)

     5x = 30 - 10     (collecting like terms to one side)

     5x = 20

     [tex]\frac{5x}{5} = \frac{20}{5}[/tex]     (divide both sides by 5 to get the value of x)

     x = 4

 Now that we know the value of x, we can find y.

      y = x + 5

      y = 4 + 5

      y = 9

If you want to make sure your answer is correct, substitute x and y into the equation.

  3x + 2y = 30

  3(4) + 2(9) = 30

  12 + 18 = 30

  30 = 30

The solution for the differential equation is y = (23/48) x + 4.

(A) Re-writing the differential equation in terms of differentials, we get:

dy = (23/48) dx

Here, dy and dx represent infinitesimal changes in the variables y and x, respectively.

(B) Integrating both sides of the equation with respect to their respective variables, we get:

∫dy = ∫(23/48)dx

On the left-hand side, the integral of dy is simply y (plus a constant of integration), while on the right-hand side, we can pull the constant factor (23/48) outside the integral:

y + C1 = (23/48) ∫dx

Integrating the right-hand side with respect to x, we get:

y + C1 = (23/48) x + C2

where C1 and C2 are constants of integration.

(C) To solve for y, we can isolate it on one side of the equation by subtracting C1 from both sides:

y = (23/48) x + (C2 - C1)

Next, we can use the initial condition y(0) = 4 to solve for the constant C2 - C1:

y(0) = (23/48) (0) + (C2 - C1) = C2 - C1

Since y(0) = 4, we have:

4 = C2 - C1

Therefore, C2 - C1 = 4, and we can substitute this back into the expression for y to get the final solution:

y = (23/48) x + 4

So the solution for the differential equation with initial condition y(0) = 4 is y = (23/48) x + 4.

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2 1/7 x 4.3 (repeating the three) as a mixed number in simplest form is 9 2/7.

First, we can simplify the mixed number 4.3 (repeating the three) as follows:

Let x = 4.3 (repeating the three)

Then 10x = 43.33333...

Subtracting x from 10x, we get:

10x - x = 43.33333... - 4.33333...

9x = 39

x = 4.33333... / 9

x = 4 1/3

Now, we can multiply 2 1/7 by 4 1/3:

2 1/7 x 4 1/3 = (15/7) x (13/3)

= (15 x 13) / (7 x 3)

= 195 / 21

To write this as a mixed number in simplest form, we can divide 195 by 21 and write the quotient as a mixed number:

195 ÷ 21 = 9 with a remainder of 6

So, 195 / 21 = 9 6/21, which can be simplified to 9 2/7.

Therefore, 2 1/7 x 4.3 (repeating the three) as a mixed number in simplest form is 9 2/7.

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The image of point (-3, 8) after a rotation of 180 counterclockwise about the origin is (3, -8)

Transformation is the movement of a point in the coordinate plane from one location to another. Transformation can either be reflection, rotation, translation and dilation.

Rotation is the flipping of a figure about a point in the coordinate plane; this point of rotation is usually origin.

(x, y) → (-x, -y) represents a rotation 180° counterclockwise.

The image of point (-3, 8) after a rotation of 180 counterclockwise about the origin is (3, -8)

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a The purpose of this report is to investigate the advantages and limitations of the annual budgeting system to determine whether the elimination of the system is a good idea.

b Report on the Advantages and Limitations of an Annual Budgeting System This report investigates the advantages and limitations of an annual budgeting system. The purpose is to assess the feasibility of eliminating the annual budgeting system at Brooklane Ltd. By analyzing academic sources, we can gain insights into the potential benefits and drawbacks of such a decision.

The annual budgeting system has several advantages. Firstly, it allows organizations to set financial goals and targets for the coming year. This helps organizations to allocate resources and prioritize their spending to achieve their goals.

Limitations of an Annual Budgeting System is that despite its advantages, the annual budgeting system also has limitations. Firstly, it can be time-consuming and costly to prepare, especially for larger organizations. This can divert resources from other important areas of the organization.

In conclusion, the annual budgeting system has both advantages and limitations.

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In the scenario you mentioned, the part of the Federal Reserve's congressional mandate that is triggered is both price stability and maximum sustainable employment.

The Federal Reserve has a dual mandate from Congress which consists of:

1. Price stability: This objective aims to maintain a stable inflation rate, ensuring that the purchasing power of money remains relatively constant over time. Price stability helps households and businesses make informed decisions regarding investments and consumption.

2. Maximum sustainable employment: This objective focuses on maintaining a high level of employment while avoiding unsustainable economic growth that could lead to high inflation. Achieving maximum sustainable employment involves minimizing unemployment while not causing inflation to rise significantly.

In the scenario you mentioned, the Federal Reserve would need to carefully balance its monetary policy to ensure that both objectives, price stability and maximum sustainable employment, are achieved. This might involve adjusting interest rates or using other monetary policy tools to influence economic activity, employment levels, and inflation.

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The solution of the system of equations is

A system of equations is a set of two or three equations.

Given the system of equations

x + y = 5 (1)

y + z = 6 (2)

z + x = 9 (3)

Givent he guide (1) - (2) x - z = - 1 (4), we proceed to solve the system of equations

Now, taking equations (3) and (4), we have that

z + x = 9 (3)

x - z = - 1 (4)

Adding them we have that

z + x = 9 (3)

+

x - z = - 1 (4)

2x = 9 - 1

2x = 8

x = 8/2

x = 4

From equation (3)

z = 9 - x

So, substituting the value of x into the equation, we have that

z = 9 - x

z = 9 - 4

z = 5

From equation (2)

y = 6 - z

So, substituting z into the equation, we have that

y = 6 - z

= 6 - 5

= 1

So,

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The mass of the lamina that occupies the region D is 31/2 units and the center of mass is located at (1/2, 2/e).

We can find the mass of the lamina by integrating the density function over the region D:

m = ∫∫D p(x,y) dA

where dA is the area element in polar coordinates, which is equal to r dr dθ. The region D can be described as 0 ≤ x ≤ 1 and 0 ≤ y ≤ e, so the integral becomes:

m = ∫0^1 ∫0^e 31y dy dx

Solving the integral, we get:

m = 31/2

To find the center of mass, we need to find the x-coordinate and y-coordinate separately:

x = (1/m) ∫∫D x p(x,y) dA

y = (1/m) ∫∫D y p(x,y) dA

For the x-coordinate, we have:

x = (1/m) ∫0^1 ∫0^e x(31y) dy dx

Simplifying, we get:

x = (1/m) ∫0^1 31/2 x dx

x = 1/2

For the y-coordinate, we have:

y = (1/m) ∫0^1 ∫0^e y(31y) dy dx

Simplifying, we get:

y = (1/m) ∫0^1 31/3 e^3 dx

y = (2/e)

Therefore, the center of mass is located at (1/2, 2/e).

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The graphs of the sinusoidal function for the question 1 to 4 created with MS Excel are attached

5. The function is; y = cos(θ) + 2

6. The function is; y = sin(2·θ) - 4

A sinusoidal function is a smooth periodic or repetitive function based on the sine or cosine of an angle.

The equations for the graphs in the sinusoidal function indicates that we get;

The period = 2·π/B

The horizontal shift = -C/B

The vertical shift = D

Where;

B = The coefficient of the angle, θ

C = The constant within the parenthesis

D = The constant term outside the sine or cosine function parenthesis

1. y = 2·sin(2·θ + 1)

The function indicates that the amplitude is 2, the period is 2·π/2 = π, the vertical shift is 0, and the horizontal shift is -1/2

Please find attached the graph of the function, created with MS Excel

2. y = -cos(θ) - 2

The amplitude is 1

The period is 2·π

The horizontal shift is 0

The vertical shift is -2

Please find attached the graph of the function y = -cos(θ) - 2, created with ms Excel

3. y = 4·cos(3·θ -2)

The amplitude is 4

The period is 2·π/3

The horizontal shift is 0

The vertical shift is -2

Please find attached the graph of the function y = 4·cos(3·θ -2), created with MS Excel

4. y = 3·sin(6·θ) - 1

The amplitude is 3

The period is π/3

The horizontal shift is 0

The vertical shift is -1

Please find attached the graph of the function y = 3·sin(6·θ) - 1, created with MS Excel

5. The points on the graph are;

Peak; (0, 3)

The next adjacent trough; (180, 1)

The adjacent peak; (360, 3)

Therefore;

The amplitude is 1

The period is 360°

The horizontal shift is 2

The vertical shift is 0

The peak point at θ = 0, indicates;

The function is; y = cos(θ) + 2

6. The points on the graph are;

Peak; (45, -3)

The next adjacent trough; (135, -5)

The adjacent peak; (225, -3)

Therefore;

The amplitude is 1

The period is 180 = π radians

The horizontal shift is 0

The vertical shift is -4

The function is; y = sin(2·θ) - 4

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If the base of the roof is 32 feet long, then the area of the triangular roof is 256 square feet.

In the given scenario, we are provided with information about a triangular roof. It is mentioned that the base of the roof is 32 feet long, and we need to determine the area of the triangular roof.

To calculate the area of a triangle, we need to know the length of the base and the height. In this case, we are given that the height is half the length of the base, which can be expressed as h = (1/2) * b.

Since we are given that the base length, b, is 32 feet, we can substitute this value into the height equation to find the height of the triangle: h = (1/2) * 32 feet = 16 feet.

Now that we have the base length (b = 32 feet) and the height (h = 16 feet), we can use the formula for the area of a triangle: A = (1/2) * b * h.

Substituting the values, we have A = (1/2) * 32 feet * 16 feet = 256 square feet.

Therefore, the area of the triangular roof is determined to be 256 square feet. This represents the amount of surface area covered by the triangular section of the roof.

Let the base of the triangular roof be denoted by b, and its height be denoted by h. We are given that

h = (1/2) * b, and that

b = 32 feet.

We can use this information to find the area A of the roof as follows:

A = (1/2) * b * h

 = (1/2) * 32 feet * (1/2) * 32 feet

 = 256 square feet

Therefore, the area of the triangular roof is 256 square feet.

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The opposite angles in an inscribed quadrilateral are supplementary, meaning their sum is 180 degrees.

An inscribed quadrilateral is a quadrilateral whose vertices all lie on a circle. Let's label the vertices of the quadrilateral as A, B, C, and D, in clockwise order.

Draw the circle that contains all four vertices, and label the center of the circle as O.

Now, draw chords AC and BD that cross at point P. Each chord divides the quadrilateral into two triangles. Notice that angle AOC and angle BOD are both central angles that subtend the same arc, CD.

Therefore, these angles have the same measure, and we can write:

angle AOC = angle BOD = x

Similarly, we can show that angle AOB = angle COD = y.

Now, consider the two triangles APC and BPD. These triangles share the side P D and have the same angle APD, which is equal to angle AOC + angle BOD, or 2x.

Therefore, by the angle-sum property of triangles, the other angles in each triangle must also be equal. We can write:

angle APC = angle BPD = (180 - 2x)/2 = 90 - x

Similarly, consider the two triangles APB and CPD. These triangles share the side P C and have the same angle APC, which we just found to be 90 - x.

Therefore, by the angle-sum property of triangles, the other angles in each triangle must also be equal. We can write:

angle APB = angle CPD = (180 - (90 - x))/2 = 90 + x/2

Finally, notice that angle APB + angle CPD = (90 + x/2) + (90 - x/2) = 180, so the opposite angles in the quadrilateral are indeed supplementary.

Therefore, the main answer is: The opposite angles in an inscribed quadrilateral are supplementary, meaning their sum is 180 degrees.

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