Question 20 of 20
Two numbers have a sum of 2 and a difference of 8. Write a system and solve it to identify the two numbers.
5 and 3
O 11 and -9
-5 and 7
O5 and-3

1. Congruent triangles:
a) Triangle EAB and Triangle ECD
b) Triangle EAC and Triangle EBD

2. Triangle EAB ≅ Triangle ECD and Triangle EAC ≅ Triangle EBD by ASA Congruence Postulate.

In this scenario, we have two concentric circles and four tangents drawn from point E to these circles, creating points of tangency A, B, C, and D.

1. Pairs of congruent triangles:
a) Triangle EAB and Triangle ECD
b) Triangle EAC and Triangle EBD

2. Showing congruence for each pair:

a) To show that Triangle EAB and Triangle ECD are congruent, we can use the following information:
- EA and EC are both radii of the larger circle, so EA = EC (congruent radii).
- AB and CD are tangents to the smaller circle, so the segments are parallel and form corresponding angles at points A and C. Thus, Angle EAB and Angle ECD are congruent (alternate interior angles).
- EB and ED are both radii of the smaller circle, so EB = ED (congruent radii).
With this information, we can prove Triangle EAB ≅ Triangle ECD using the Angle-Side-Angle (ASA) Congruence Postulate.

b) To show that Triangle EAC and Triangle EBD are congruent, we can use the following information:
- EA and EB are both radii of the larger circle, so EA = EB (congruent radii).
- AC and BD are tangents to the larger circle, so the segments are parallel and form corresponding angles at points A and B. Thus, Angle EAC and Angle EBD are congruent (alternate interior angles).
- EC and ED are both radii of the smaller circle, so EC = ED (congruent radii).
With this information, we can prove Triangle EAC ≅ Triangle EBD using the Angle-Side-Angle (ASA) Congruence Postulate.

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The function f(x) = x² + 12x - 3 has two distinct real roots and no rational or complex roots. The possible rational roots are ±1 and ±3, but they are not actual roots of the equation.

To find the number of complex zeros and possible rational zeros of the function f(x) = x² + 12x - 3 = 0, we can use the same methods as in the previous question.

According to the rational root theorem, any rational root of the polynomial must be of the form p/q, where p is a factor of the constant term (-3) and q is a factor of the leading coefficient (1). The possible rational roots are therefore ±1, ±3.

To determine if there are any complex roots, we can use the quadratic formula. The solutions to the equation f(x) = 0 are given by

x = (-b ± √(b² - 4ac)) / 2a

Plugging in the coefficients of the polynomial, we get

x = (-12 ± √(12² - 4(1)(-3))) / 2(1)

x = (-12 ± √(156)) / 2

x = -6 ± 3√(13)

Since the discriminant (b² - 4ac) is positive, the square root is real, and the roots are two distinct real numbers. Therefore, there are no complex roots.

The possible rational roots we found earlier, ±1 and ±3, are not actual roots of the equation, since plugging them into the equation does not give us zero. Therefore, there are no rational roots either.

In conclusion, the equation f(x) = x² + 12x - 3 = 0 has two distinct real roots and no complex or rational roots.

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--The given question is incomplete, the complete question is given

" State the number of complex zeros and the possible rational zeros for the function f(x) = x² + 12x -3 = 0

show your work"--

The standard deviation for the data must be approximately $111.93.

Since we are given a range of values that encompasses about 95% of the data, we can assume that this range is within two standard deviations of the mean.

Thus, we can set up the following equation:

2σ = $444.47 - $220.61

Simplifying this equation, we get:

2σ = $223.86

σ = $111.93

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Rational expressions and equations can be used to describe relationships and solve problems that involve proportions or rates of change.

They involve expressions that are fractions of polynomials, where the numerator and denominator can be thought of as values that change according to some variable.

For example, rational expressions can be used to represent rates of change in physics or finance, such as the speed of an object or the interest rate on a loan. By setting up equations with rational expressions, we can solve for unknown values or relationships between variables.

Rational equations can also be used to describe relationships between quantities that are proportional, such as the volume of a container and the amount of liquid it can hold. By solving for an unknown variable in a rational equation, we can determine the relationship between the two quantities and use it to make predictions or solve practical problems.

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Answer is:

For the south region, the proportion of sales from current stores is approximately 58 %.

because this proportion is relatively close to the sales director's statistical model, the data is consistent with the model.

The given information states that the proportion of sales from current stores in the South region is approximately 58%. This value falls within the range of the sales director's statistical model, which could be an estimated value or a range of values based on past data or other sources. Since the proportion of sales falls within the expected range, the data is consistent with the model. This means that the actual value observed is not significantly different from the expected value, and there is no evidence to suggest that the model needs to be revised or updated.

Therefore, the answer to the second drop-down menu is "consistent". If the actual proportion of sales from current stores in the South region was significantly different from the expected value in the sales director's statistical model, then the answer would be "inconsistent". However, since the observed proportion is relatively close to the expected value, the data is consistent with the model.

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The inverse of the function f(x) = √(x+1) is [tex]f^-1[/tex](x) = x² - 1.

Given a function f(x), its inverse function, denoted as [tex]f^-1[/tex](x), is a function that takes the output of f(x) and returns the original input. In other words, if y = f(x), then x = [tex]f^-1[/tex](y).

To find the inverse of the function f(x), we follow these steps:

Step 1: Replace f(x) with y:

   y = √(x+1)

Step 2: Interchange x and y:

   x = √(y+1)

Step 3: Solve for y:

   x² = y+1

   y = x² - 1

Step 4: Replace y with f^-1(x):

   f^-1(x) = x² - 1

Therefore, the inverse of the function f(x) = √(x+1) is [tex]f^-1[/tex](x) = x² - 1.

To verify this, we can check that ([tex]f^-1[/tex] o f)(x) = x and (f o [tex]f^-1[/tex])(x) = x for all x in the domain of the functions:

([tex]f^-1[/tex] o f)(x) = [tex]f^-1[/tex](f(x)) = (√(x+1))² - 1 = x

(f o [tex]f^-1[/tex])(x) = f([tex]f^-1[/tex](x)) = √((x² - 1) + 1) = √(x²) = |x| + 1

Since the range of f(x) is [0, infinity), we restrict the domain of [tex]f^-1[/tex](x) to [1, infinity) to ensure that the inverse is a function.

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The surface area of the triangular prism is 720in²

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 p is the perimeter of the base and h is the height of the prism.

Base area = 1/2 bh

= 1/2 × 24 × 5

= 12 × 5

= 60 in²

Perimeter of the base = 24 +13 +13

= 50 in²

the height of the prism = 12 in

therefore SA = 2 × 60+ 12 × 50

= 120 + 600

= 720 in²

Therefore the surface area of the prism is 720 in²

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The critical point (180, 120) will yield a maximum revenue of 23,700. So, the correct answer is the (180, 120).

To determine which critical point will yield a maximum revenue, we need to evaluate the objective function at each point and compare the results.

Objective Function: Revenue = 75x + 85y

1. Point (0,0):
Revenue = 75(0) + 85(0) = 0

2. Point (180, 120):
Revenue = 75(180) + 85(120) = 13500 + 10200 = 23700

3. Point (300,0):
Revenue = 75(300) + 85(0) = 22500 + 0 = 22500

Comparing the revenues at each point:
(0,0) - Revenue = 0
(180, 120) - Revenue = 23700
(300,0) - Revenue = 22500

The critical point (180, 120) will yield a maximum revenue of 23,700. So, the correct answer is the (180, 120).

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Answer:

3.9

Step-by-step explanation:

pythagorean theorem states:

[tex]a^{2} +b^{2} =c^{2}[/tex]

We know that a is the height of the triangle, 2.

We also know that b is the base of the triangle, 3.

So, solve for c:

2²+3²=c²

simplify

4+9=c²

15=c²

take the square root of both sides

c=3.9

So, the spider and the fly are 3.9 ft apart from each other

Hope this helps :)

Answer:

3.6 feet

Step-by-step explanation:

The Pythagorean theorem states that a^2+b^2=c^2, where c is the hypotenuse, or longest side of the triangle opposite to the right angle, and a and b are the two legs of the triangle (the other two sides.)

In the question, sides a and b are given as 2ft and 3ft long. Plugging 2 and 3 into the equation, we will get that 2^2+3^2=c^2. 2 x 2 = 4 and 3 x 3 = 9, so 4 + 9 = c^2, or, c^2 = 13.

However, this is still c^2, not c, so we must take the square root of both sides. This is because the equation is now essentially saying that c x c = 13, so if we find a number that when multiplied by itself equals 13, that number will be c. to do this, we must take the square root of 13. This will give us an irrational number- approximately 3.605551275. The question asks to round it to the nearest tenth though, so this will simply become 3.6.

In the context of this question, it means that the spider and the fly are 3.6ft apart.

This is a statistical question because it involves collecting data. Most employees work around 40 hours each week.

a. This is a statistical question because it involves collecting data on the number of hours worked by employees each week and analyzing the distribution or pattern of the data.

b. Here's a dot plot of the data provided:

   37: •
   38: ••
   39: ••
   40: ••••••••
   41: •••
   42: ••
   43: •
   47: •

The distribution of the data in the dot plot shows that the majority of employees worked around 40 hours each week, as this is where the data is most heavily clustered. Most of the data are clustered around 40 hours, with a peak at 40 hours. There is a gap between 43 and 47 hours.

c. Based on the distribution of the data, most employees work about 40 hours each week.

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Step-by-step explanation:

if this factorisation because I think you alr simplified, but here's the answer for factorisation anyways.

you can text me below in the comments section if that's not you want, I will try to answer if I can!!!

Answer:

3y(y - 2)(y - 1)

Step-by-step explanation:

Simplify by factoring

3y(y² - 3y + 2)

= 3y(y - 2)(y - 1)

The equation that represents the total number of miles, m, Victoria should expect to be able to drive if she uses g gallons of gasoline is m = (19g).

We can determine the proportionality constant by dividing the total number of miles driven (76) by the number of gallons of gasoline burned (4), which gives us 19 miles per gallon (76/4 = 19).

We can then use this proportionality constant to create the equation m = (19g), where m represents the total number of miles Victoria should expect to be able to drive if she uses g gallons of gasoline. This equation tells us that for every additional gallon of gasoline burned, Victoria should expect to be able to drive an additional 19 miles.

For example, if Victoria were to burn 6 gallons of gasoline, she should expect to be able to drive 114 miles (6 x 19 = 114). This equation assumes that Victoria's car has a constant fuel efficiency of 19 miles per gallon.

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A 20-ounce smoothie from Blender Splendor contains 8 ounces of fruit juice.

To find out how many ounces of fruit juice are in a 20-ounce smoothie from Blender Splendor, given that 40% of the smoothie is fruit juice, follow these steps:

1. Convert the percentage to a decimal: 40% = 0.40
2. Multiply the total ounces of the smoothie by the decimal: 20 ounces * 0.40 = 8 ounces

Your answer: A 20-ounce smoothie from Blender Splendor contains 8 ounces of fruit juice.

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Herberto travelled through Amusement Park (AP), City Park (P), Landmark (L), High School (HS), City Hall (CH).

Herberto can take the following path to travel from the amusement park (AP) to the city hall (CH):

Amusement Park (AP) -> City Park (P) -> Landmark (L) -> High School (HS) -> City Hall (CH)

This path takes him through the city park, then to the landmark, followed by the high school, and finally to the city hall.

Please note that without a specific map or layout of Knappville, there could be multiple valid paths from the amusement park to the city hall. The path provided above is just one example.

When Herberto travels from the amusement park to the city hall, he can choose various paths depending on the specific layout of Knappville. The path mentioned earlier is just one example, but the actual paths could differ based on the town's infrastructure and the locations of the landmarks.

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The values of distance and orbital periods of planets Mercury is 0.39AU and 0.242yrs,Venus 0.72AU distance and 0.616yrs,Earth 1AU distance and 1yrs,Mars 1.52AUdistance and 1.88,Jupiter 5.2AUdistance and 1.88 ,Saturn N/Adistance and NA years as infromation is not provided,Uranus 19.18AUdistance and 84yrs,Neptune 30.06AUdistance and165yrs.

Based on the given information, here are some key facts about the planets in our solar system:

- Mercury is the closest planet to the Sun, with a distance from the Sun of 0.39 astronomical units (AU). It has an orbital period of 0.242 years.
- Venus is the second planet from the Sun, with a distance of 0.72 AU. Its orbital period is 0.616 years.
- Earth is the third planet from the Sun, with a distance of 1 AU. Its orbital period is 1 year.
- Mars is the fourth planet from the Sun, with a distance of 1.52 AU. Its orbital period is 1.88 years.
- Jupiter is the fifth planet from the Sun, with a distance of 5.2 AU. Its orbital period is 11.86 years.
- Saturn is the sixth planet from the Sun, with a distance of N/A AU. Its orbital period is N/A.
- Uranus is the seventh planet from the Sun, with a distance of 19.18 AU. Its orbital period is 84 years.
- Neptune is the eighth planet from the Sun, with a distance of 30.06 AU. Its orbital period is 165 years.

Therefore the the distance and orbital periods are in increasing arrangement mercury, venus, earth, mars, jupiter, saturn ,uranus, neptune distance from the sun.

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In the given question, if y, p and q vary jointly and p is 14 when y and q are equal to 2 and  p and y are equal to 7, we get q is equal to 14 using the  joint variation formula.

To solve this problem, we need to use the formula for joint variation, which states that y, p, and q vary jointly if there exists a constant k such that ypk = kq.

In this case, we know that when y=2 and q=2, p=14. So we can set up the equation: 2*14*k = 2kq

Simplifying this, we get: 28k = 2kq
Dividing both sides by 2k, we get: 14 = q
So when p=7 and y=7, we can use the same equation: 7*14*k = 7kq

Simplifying this, we get: 98k = 7kq
Dividing both sides by 7k, we get: q = 14

Therefore, when p and y are equal to 7, q is equal to 14.

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Each car on Mabel's train is a little less than half a foot long i.e., 3/8 feet long, or 0.375 feet long. This is solved by using simple mathematical operations.

To find the length of each car on Mabel's train, we can use the formula:

Total length of train = length of one car × number of cars

We are given that Mabel's train has 8 cars and is 3 feet long, so we can substitute those values into the formula and solve for the length of one car:

3 = x × 8

where x is the length of one car.

To solve for x, we can divide both sides by 8:

3/8 = x

Therefore, each car on Mabel's train is 3/8 feet long, or 0.375 feet long.

In other words, each car is a little less than half a foot long.

To help visualize this, imagine dividing the 3-foot length of the train into 8 equal parts. Each part would be 3/8 feet long, which is the length of one car. So when Mabel puts all 8 cars together, they form a train that is 3 feet long.

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The area and cost of each of the lots is given below

The area and cost of each of the lots is calculated below

To calculate the rectangle trapezoid, it will be:

B = 140m

h = 50m

b = 70m

Note that Area of Lot 1 : A =(B+b)h/2

Hence Height will be:

A= (140m+ 70m) 50m/2

A = 5250 m²

Cost: C = 5250m² *800 pesos = $4,200,000

Lot 2 is another rectangle trapezoid:

B = 150m

b = 80m

h = 50m

Area: A= (150m+ 80m)50m/2

A = 5750 m²

Cost: C = 5750m²  x 800 pesos = $4,600,000

So, for Lot 3 which is a rectangle:

b = 70m

to = 50m

Area: A =a x b

W=50m x 70m

A  = 3500 m²

Cost: C = 3500m² x 800 pesos = $2,800,000

Lot 4 is a right triangle:

b = 70m

h = 50m

Area: A =b x h/2

A= (50m x 70m)/2

A = 1750 m²

Cost: C = 1750 m² x 800 pesos = $1,400,000

So for Lot 5 that is a rhomboid:

b = 150 base meters

h=50m

Area: A=b x h

A= (150m  x  50m

H = 7500 m²

Cost: C = 7500m² x  800 pesos = $6,000,000

So, for Lot 6 that is an isosceles triangle:

b = 140m

h= 50m

Area: A =b x h/2

A= (140m x  50m)/2

A = 3500 m²

Lastly, the Cost: C = 3500m²  x 800 pesos = $2,800,000

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See transcribed text below

The family of one of your colleagues, who had a piece of land with 7 adjoining lots, and they found themselves in need of selling them. They established a price per square meter of 800 pesos and they have to calculate the cost, based on the square meters that each lot has. Before continuing, review the terrain sketch and answer the questions

What is the area and cost of each of the lots?

Answer:

Shape 1 and Shape 3

Step-by-step explanation:

An obtuse angle is an angle that is greater than 90° but less than 180°.

We can see that the first shape has 2 angles that are greater than 90°, making this a correct choice.
The second shape has 4 boxes, meaning the angles are exactly 90°, making this incorrect.

The third shape has 6 angles that are greater than 90°, making this another correct choice.

The last shape has all 3 angles under 90°, making this also incorrect.

So, the 1st and 3rd shapes are correct.

Hope this helps! :)

The total cost to buy t movie tickets for $8.50 each is o only one, the number of ounces of tomato juice, j, in 2 quarts o only one, the number of pizzas to order, p, if each person eats 2 slices more than one, the greatest number of $6 notebooks, n, ed can buy with $36 only one, and the area, a, of a wall with a length of 5 meters and a height of 3 meters o only one.

1. The total cost to buy t movie tickets for $8.50 each:
- Only one value for the variable

The cost to buy a specific number of movie tickets at a fixed price will have only one total cost value.

2. The number of ounces of tomato juice, j, in 2 quarts:
- Only one value for the variable

There are 32 ounces in a quart, so 2 quarts will always have 64 ounces.

3. The number of pizzas to order, p, if each person eats 2 slices:
- More than one value for the variable

The number of pizzas needed depends on the number of people attending, which can vary, leading to different values for p.

4. The greatest number of $6 notebooks, n, Ed can buy with $36:
- Only one value for the variable

Since each notebook costs $6, Ed can buy exactly 6 notebooks with $36 (36/6 = 6).

5. The area, a, of a wall with a length of 5 meters and a height of 3 meters:
- Only one value for the variable

The area of a rectangle is calculated by multiplying its length and height. In this case, the area will always be 15 square meters (5m x 3m = 15m²).

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The type of variable being studied in this case is a categorical variable, specifically a nominal variable.

Categorical variables are variables that can be classified or grouped. The categories in this example are the various modes of transportation utilized by off-campus students (drive alone, carpool, public transportation, walk, or a paid driving service).

Nominal variables are categorical variables with no intrinsic order or numerical value. Because the various modes of transportation in this study have no numerical value or inherent order, they are classified as nominal variables.

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The unit rate for chapters per hour is calculated as 16/21

In order to find the unit rate for the number of chapters Gregory reads in an hour, we must firstly divide the total amount of chapters read by the amount of hours devoted to the activity:

= 2/3 chapters ÷ 7/8 hours

= 2/3 chapters × 8/7 hours

= 16/21 chapters per hour

Consequently, the unit rate for chapters per hour is calculated as 16/21, which already exists in its most simplified form.

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The coordinates of the resulting point, R′ are R' = (-9, -2)

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

The point R(-2, 9) is rotated 90° clockwise around the origin.

This means thar

R = (-2, 9)

Rotation rule = 90° clockwise around the origin.

The rotation rule of 90° clockwise around the origin is

(x,y) becomes (y,-x)

So, we have

R' = (y, -x)

Substitute the known values in the above equation, so, we have the following representation

R' = (-9, -2)

Hence, the coordinates of the resulting point, R′ are R' = (-9, -2)

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Sylvia can fit 9 large cubic boxes and 240 small rectangular boxes into her car trunk.

We first need to convert all measurements to the same unit. Since the large cubic boxes are given in inches, we need to convert the dimensions of the trunk to inches as well.

The trunk measures 51 inches wide (4.25 ft x 12 in/ft), 48 inches deep (4 ft x 12 in/ft), and 53.33 inches high (4*1/3 ft x 12 in/ft).

Next, we need to determine how many large boxes can fit in the trunk. Since each large box is 16 inches wide,

we can fit 3 boxes across the width of the trunk:

(51 inches ÷ 16 inches = 3.19)

Similarly, we can fit 3 boxes deep and 3 boxes high.

Therefore, the maximum number of large boxes Sylvia can fit in her trunk is:

27 (3 x 3 x 3 = 27)

After fitting as many large boxes as possible, we can calculate the remaining volume of the trunk and use this to determine how many small boxes can fit.

The remaining volume is approximately 59,712 cubic inches (51 in x 48 in x 20 in).

Dividing this by the volume of each small box:

(3 in x 6 in x 4 in = 72 cubic inches)

We can fit 829 small boxes in the remaining space.

Therefore, Sylvia can fit a total of 27 large boxes and 829 small boxes in her trunk.

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The difference between the median number of turkey sandwiches sold and the median number of ham sandwiches sold is 3.

First, let's find the median number of turkey and ham sandwiches sold.
To find the median:
1. Arrange the data in ascending order.
2. If there's an odd number of data points, the median is the middle value. If there's an even number of data points, the median is the average of the two middle values.

Turkey sandwiches sold: 7, 8, 12, 13, 15
Ham sandwiches sold: 6, 8, 9, 11, 16
Now let's find the medians:

Turkey median: There are 5 data points, which is odd, so the median is the middle value, 12.
Ham median: There are 5 data points, which is odd, so the median is the middle value, 9.
Finally, let's find the difference between the medians:
Difference = Turkey median - Ham median
Difference = 12 - 9
Difference = 3
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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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The survey results show that the estimates for the number of students in the school who prefer rap music range from 110 to 154. The center of the estimates is around 121, with a variation of about 66.

To estimate the number of students who prefer rap music, multiply the proportion who prefer rap in each sample by the total number of students in the school (440)

Sample 1: 0.25 x 440 = 110

Sample 2: 0.20 x 440 = 88

Sample 3: 0.30 x 440 = 132

Sample 4: 0.35 x 440 = 154

The center of the estimates is the average of the four estimates: (110 + 88 + 132 + 154) / 4 = 121. Variability can be described using the range (154 - 88 = 66) or the standard deviation of the estimates, which requires additional calculations.

To know more about standard deviation:

https://brainly.com/question/23907081

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