Nicole needed $1,000 for an unexpected car repair. Her only option was to take a payday loan from Staten Island Loan Stars. She must pay back the loan in 3 months. The fee is $400. What is the APR for this loan, to the nearest percent?

To solve the question, we need to consider the transformation described and determine the function that represents the stretched graph.

The absolute value parent function is [tex]\displaystyle\sf f(x) = |x|[/tex]. According to the question, this parent function is horizontally stretched by a factor of 5 to create the graph of [tex]\displaystyle\sf g(x)[/tex].

A horizontal stretch by a factor of 5 means that the x-values on the graph are multiplied by 5. In other words, the transformation can be represented as [tex]\displaystyle\sf g(x) = |5x|[/tex].

Therefore, the correct function that represents the graph of [tex]\displaystyle\sf g(x)[/tex], the stretched version of [tex]\displaystyle\sf f(x) = |x|[/tex], is [tex]\displaystyle\sf g(x) = |5x|[/tex].

The correct answer is option D: [tex]\displaystyle\sf g(x) = |5x|[/tex].

Answer:

Factorizing each expression:

x^2 + 8x + 15 = (x+3)(x+5)

x^2 - 2x - 15 = (x-5)(x+3)

x^2 + 9x - 36 = (x+12)(x-3)

x^2 - 12x - 45 = (x-15)(x+3)

So the factorized expressions are:

- x^2 + 8x + 15 = (x+3)(x+5)

- x^2 - 2x - 15 = (x-5)(x+3)

- x^2 + 9x - 36 = (x+12)(x-3)

- x^2 - 12x - 45 = (x-15)(x+3)

The number of sodas sold at the baseball game was 119, while the number of hot dogs sold was 72.

Let's assume the number of sodas sold as 'x' and the number of hot dogs sold as 'y'.

According to the problem, the total number of sodas and hot dogs sold is 191, so we can write the equation:

x + y = 191   ...(1)

The problem also states that the number of hot dogs sold was 47 less than the number of sodas sold. Mathematically, we can express this as:

y = x - 47   ...(2)

To find the values of x and y, we can solve the system of equations (1) and (2). Substituting equation (2) into equation (1), we have:

x + (x - 47) = 191

Simplifying the equation:

2x - 47 = 191

2x = 191 + 47

2x = 238

Dividing both sides by 2:

x = 238/2

x = 119

Substituting the value of x back into equation (2):

y = 119 - 47

y = 72

As a result, the total amount of sodas sold is 119, and the total amount of hot dogs sold is 72.

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Answer: 6 feet

Step-by-step explanation:

The height of the man and the length of his shadow are proportional.

This means that we can use proportions to find the height of the man if we know the length of his shadow.

We can set up a proportion as follows: height of man/length of shadow = (which includes the triangle formed by the man, his shadow, and the man/length constant.

Let's call the constant "k". We don't know the value of "k" yet, but we do know that it will be the same for all similar triangles ground).

Using the measurements given in the problem, we get 6 / 12 = simplifying the left side of the equation gives 1 / 2 = kNow we know that k = 1/2.

We can use this to find the height of the man. Let's call the height "h". We can set up another proportion using the value we just found:h / 12 = 1 / 2

Cross-multiplying gives us: 2h = 12

Simplifying further: h = 6 Therefore, the height of the man is 6 feet.

Answer:

The opportunity cost of attending summer school in this scenario is $4,000. Opportunity cost is the value of the next best alternative that must be given up in order to pursue a certain action. In this case, by choosing to attend summer school, you are giving up the opportunity to earn $3,000 from your usual summer job. Additionally, you have to pay $1,000 for tuition, bringing the total opportunity cost to $3,000 + $1,000 = $4,000.

Step-by-step explanation:

The opportunity cost of attending summer school in this scenario is $4,000, which includes both the foregone earnings from the summer job and the cost of tuition.

The potential earnings from a job that you choose not to do are considered an opportunity cost in economics. In this situation, the opportunity cost of attending summer school is both the $3,000 you lose from not working your summer job and the tuition fee of $1,000 you pay for the school. Economic theory suggests that we consider both of these costs because both are expenses that are a direct result of your decision to attend summer school. Therefore, the total opportunity cost in this scenario is $4,000 ($3,000 + $1,000).

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To achieve a comparable rate of return, Hank would need to earn an interest rate of approximately 0.86% per month, compounded monthly on his ordinary annuity.

To find the interest rate, compounded monthly, that Hank would need to earn on an ordinary annuity for a comparable rate of return, we can use the present value formula for an ordinary annuity.

First, let's calculate the present value of Hank's payments. He made payments of $219 per month for 30 years, so the total payments amount to $219 * 12 * 30 = $78840.

Now, we need to find the interest rate that would make this present value equal to the selling price of the property, which is $195,258.

Using the formula for the present value of an ordinary annuity, we have:

PV = P * (1 - (1+r)[tex]^{(-n)})[/tex]/r,
where PV is the present value, P is the payment per period, r is the interest rate per period, and n is the number of periods.

Plugging in the values we have, we get:
$78840 = $219 * (1 - (1+r)[tex]{(-360)}[/tex])/r.

Solving this equation for r, we find that Hank would need to earn an interest rate of approximately 0.86% per month, compounded monthly, in order to have a comparable rate of return.

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The value of x for measure of the angle m∠CED subtended by the arc CD at the circumference is equal to 0.4 to the nearest tenth.

The angle subtended by an arc of a circle at it's center is twice the angle it substends anywhere on the circles circumference. Also the arc measure and the angle it subtends at the center of the circle are directly proportional.

arc CD = 2(m∠CED)

Also arc CD = 84°

84° = 2(12x + 37)°

12x + 37 = 84/2 {divide through by 2}

12x + 37 = 42

12x = 42 - 37 {collect like terms}

12x = 5

x = 5/12 {divide through by 12}

x = 0.4167

Therefore, the value of x for measure of the angle m∠CED subtended by the arc CD at the circumference is equal to 0.4 to the nearest tenth.

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In the expression 7x - 12, the terms, coefficients, and constants can be identified as follows:

Terms:

7x: This is a term consisting of the variable x multiplied by the coefficient 7.

-12: This is a term without a variable, and it is a constant term.

Coefficients:

The coefficient of the term 7x is 7.

It represents the factor by which the variable x is multiplied.

Constants:

The constant in the expression is -12.

It is a term without a variable, representing a fixed value.

or the term 7x.

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Alvarado can create 20 different arrangements of plants on the new display stand.

To determine the number of different arrangements of plants that Alvarado can create, we can use the concept of combinations.

When selecting 3 plants out of 6, the order of the plants doesn't matter, since all that matters is which plants are chosen. Therefore, we need to calculate the number of combinations.

The formula for calculating combinations is:

C(n, r) = n! / (r!(n - r)!)

Where:

n is the total number of items (in this case, the total number of plants)

r is the number of items chosen (in this case, the number of plants to be placed on the display stand)

! represents factorial, which is the product of all positive integers up to a given number

Using the values from the problem:

n = 6 (total number of plants)

r = 3 (number of plants to be placed on the display stand)

C(6, 3) = 6! / (3!(6 - 3)!)

C(6, 3) = 6! / (3! × 3!)

C(6, 3) = (6 × 5 × 4) / (3 × 2 × 1)

C(6, 3) = 20

Therefore, Alvarado can create 20 different arrangements of plants on the new display stand.

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The measure that changes the most when the value of 82° is changed to 94° is the Median, which becomes 84°.

To determine which measure changes the most when the value of 82° is changed to 94°, we need to calculate the measures for both the original and new data.

Given the average high temperatures: 58, 61, 71, 77, 91, 100, 105, 102, 95, 82, 66, 57.

First, let's calculate the original measures:

Mean: The mean is the average of all the values.

Mean = (58 + 61 + 71 + 77 + 91 + 100 + 105 + 102 + 95 + 82 + 66 + 57) / 12 = 81.4°.

Median: The median is the middle value when the data is arranged in ascending order.

Arranging the data in ascending order: 57, 58, 61, 66, 71, 77, 82, 91, 95, 100, 102, 105.

The median is the 6th value, which is 82°.

Range: The range is the difference between the highest and lowest values.

Range = Highest value - Lowest value

= 105 - 57

= 48°.

IQR (Interquartile Range): The IQR is the difference between the third quartile (Q3) and the first quartile (Q1).

Calculating Q1 and Q3:

Q1 = 66, Q3 = 95.

IQR = Q3 - Q1

= 95 - 66

= 29°.

Now, let's calculate the new measures after changing the value of 82° to 94°:

Mean: The mean is unaffected by changing a single value.

Median: Since we only changed one value, the new median will still be the same as the original median, which is 82°.

Range: The range is unaffected by changing a single value.

IQR (Interquartile Range): The IQR is unaffected by changing a single value.

Therefore, the measure that changes the most when the value of 82° is changed to 94° is the Median, which becomes 84°.

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The missing length of the right triangle is 8√3

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

The right triangle

The missing length of the right triangle can be calculated using the following Pythagoras theorem

x² = difference of squares of the other sides

Using the above as a guide, we have the following:

x² = (16)² - (8)²

Evaluate

x² = 192

Take the square roots

x = 8√3

Hence, the missing length of the right triangle is 8√3

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Answer:

[tex]\displaystyle y=\frac{x^2}{4}-1[/tex]

Step-by-step explanation:

[tex]\displaystyle r=\frac{2}{1-\sin\theta}\\\\r-r\sin\theta=2\\\\\sqrt{x^2+y^2}-y=2\\\\\sqrt{x^2+y^2}=y+2\\\\x^2+y^2=y^2+4y+4\\\\x^2=4y+4\\\\\frac{x^2}{4}=y+1\\\\\frac{x^2}{4}-1=y[/tex]

Note that [tex]r=\sqrt{x^2+y^2}[/tex] and [tex]y=r\sin\theta[/tex]

The predicted number of employed graphic designers in 2023 can be calculated by applying the annual growth rate of 9% to the number of designers in 2012. Based on this calculation, it is estimated that there will be approximately 35,889 graphic designers employed in 2023.

1. Determine the initial number of graphic designers in 2012: The question states that there were 23,900 designs in the industry in 2012.

2. Calculate the annual growth rate: The question mentions that the graphic design industry has an annual growth rate of 9%. This growth rate represents an increase of 9% each year.

3. Calculate the growth in the number of graphic designers from 2012 to 2023: To find the growth in the number of graphic designers, multiply the initial number of designers (23,900) by the growth rate (9%) for each year. Repeat this process for each year from 2012 to 2023.

4. Add the growth in the number of designers to the initial number of designers: Sum up the growth in the number of designers for each year and add it to the initial number of designers in 2012.

5. Calculate the predicted number of employed graphic designers in 2023: The sum obtained in step 4 represents the estimated number of graphic designers employed in 2023.

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9.  42 chocolate lava cakes and 15 caramel apple tarts were sold.

10.Brian needs to sell approximately 14,286 meals to break even.

9. Let's denote the number of chocolate lava cakes as x and the number of caramel apple tarts as y. We can set up a system of equations based on the given information:

x + y = 57 (sum of the two desserts is equal to 57)

x - y = 27 (difference of the two desserts is 27)

We can solve this system of equations using the method of substitution or elimination. Let's use the method of elimination:

Add the two equations:

(x + y) + (x - y) = 57 + 27

2x = 84

x = 42

Substitute the value of x into one of the equations to find y:

42 - y = 27

y = 42 - 27

y = 15

Therefore, 42 chocolate lava cakes and 15 caramel apple tarts were sold.

10. To determine the number of meals Brian needs to sell to break even, we can calculate the break-even point by comparing the total cost and the total revenue.

Total cost = $250,000 (start-up cost) + ($17.50 per meal) * N (number of meals)

Total revenue = ($35.00 per meal) * N (number of meals)

To break even, the total cost should be equal to the total revenue:

$250,000 + ($17.50 per meal) * N = ($35.00 per meal) * N

Simplifying the equation:

$250,000 = $35.00N - $17.50N

$250,000 = $17.50N

N = $250,000 / $17.50

N ≈ 14,285.71

Since we cannot sell a fraction of a meal, we round the number of meals up to the nearest whole number.

Therefore, Brian needs to sell approximately 14,286 meals to break even.

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Step-by-step explanation:

You didn't include a diagram, but a dodecagon is a 12 sided polygon.

The SUM of the interior angles =   (n-2) * 180    where n =12  

(n-2)* 180

(12-2) * 180 =   1800    <====this is the SUM of the 12 interior angles

         so EACH angle is   1800 / 12 = 150 degrees

The calculated value of y when x = -10 is -30

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

y varies directly with x

This means that

y = kx

Where

k = constant of variation

So, we have

24 = 8k

k = 3

This also means that

y = 3x

When x = -10, we have

y = 3 * -10

Evaluate

y = -30

Hence, the value is -30

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Question

Suppose y varies directly with x and y=24 when x=8.

What is the value of y when x = -10?

The total cost of 21 bottles of cola at £1.59 each is £33.39.

1. Determine the cost per bottle: Each bottle of cola costs £1.59.

2. Multiply the cost per bottle by the number of bottles: £1.59 x 21 = £33.39.

3. Therefore, the total cost of 21 bottles of cola is £33.39.

In the given scenario, each bottle of cola is priced at £1.59. To calculate the total cost of 21 bottles, we need to multiply the cost per bottle by the number of bottles. By multiplying £1.59 by 21, we get £33.39 as the final answer.

Certainly! Here's an expanded explanation in 100 additional words:

To calculate the total cost of 21 bottles of cola, we need to multiply the cost per bottle by the number of bottles. In this case, each bottle costs £1.59. By multiplying £1.59 by 21, we can determine the total cost.

To perform the calculation, we can use the distributive property of multiplication. We multiply the cost per bottle (£1.59) by the number of bottles (21) individually, and then sum up the results.

First, we multiply the ones place: 9 multiplied by 21, which equals 189. Since 9 multiplied by 1 equals 9, we write down the 9 in the ones place and carry over the 18 to the tens place.

Next, we multiply the tens place: 5 multiplied by 21, which equals 105. Adding the carried-over 18, we get 123. We write down the 3 in the tens place and carry over the 12 to the hundreds place.

Finally, we multiply the hundreds place: 1 multiplied by 21, which equals 21. Adding the carried-over 12, we get 33.

Combining the results, we have £33.39 as the total cost of 21 bottles of cola.

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Part A: The mean and median for Bay Side School are 3.8 and 5, respectively, while for Seaside School they are 4.93 and 5.

Part B: The range and IQR for Bay Side School are 8 and 4, respectively, while for Seaside School they are 8 and 2.

Part C: Bay Side School is a better choice for smaller class sizes due to its smaller mean, smaller range, and larger IQR.

Part A: Measures of Center

Bay Side School:

Mean = 3.8 (The sum of all class sizes divided by the number of classes)

Median = 5 (The middle value when class sizes are arranged in ascending order)

Seaside School:

Mean = 4.93 (The sum of all class sizes divided by the number of classes)

Median = 5 (The middle value when class sizes are arranged in ascending order)

Part B: Measures of Variability

Bay Side School:

Range = 8 (The difference between the maximum and minimum class sizes)

Interquartile Range (IQR) = 4 (The difference between the first quartile and third quartile)

Seaside School:

Range = 8 (The difference between the maximum and minimum class sizes)

Interquartile Range (IQR) = 2 (The difference between the first quartile and third quartile)

Part C: Conclusion and Reasoning

If you are interested in a smaller class size, Bay Side School is a better choice. Bay Side School has a smaller mean (3.8) compared to Seaside School (4.93), indicating a generally smaller class size. Additionally, the median class size is the same for both schools, but the range and IQR are smaller for Bay Side School.

This suggests that class sizes at Bay Side School have less variation and are more consistently smaller. Therefore, based on the provided data, Bay Side School is the better choice for smaller class sizes.

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Answer: Changing the radius of an object by a given amount has a greater effect on the volume than changing the height by the same amount. The volume of a cylinder is given by the formula V = πr²h, where V is the volume, r is the radius, and h is the height. If we change the radius by a given amount, say x, the new radius would be r+x. Hence, the new volume would be V' = π(r+x)²h = π(r²+2rx+x²)h = V + 2πrxh + πx²h. We can see that the volume change equals 2πrxh + πx²h. The first term is proportional to both the radius and the height, whereas the second term is proportional to the square of the radius and the height. Assuming that the height change is also x, the new volume would be V'' = πr²(h+x) = V + πr²x. We can see that the volume change is proportional to the radius squared and the change in height. Therefore, changing the radius by a given amount has a greater effect on the volume than changing the height by the same amount.

According to the information we can infer that the correct match is screwdriver: 250mm, height of a kitchen worktop: 0.9m, the overall length of a JCB 3cx Backhoe loader digger: 5620mm, and height of a door: 2m

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The mean score is  = 53.16.the average of the two middle values:  53.5. there is no repeated value, so there is no mode.

To describe Michael's scores on the math tests, we can consider different measures of central tendency, namely the mean, median, and mode. Each measure provides a different perspective on the typical or representative value of the data.

The mean is calculated by summing up all the scores and dividing by the number of scores. It is affected by extreme values and can be skewed if there are outliers. In this case, Michael's scores are 24, 28, 21, 79, 84, and 93. The mean score is (24 + 28 + 21 + 79 + 84 + 93) / 6 = 53.16.

The median is the middle value when the scores are arranged in ascending or descending order. It is less affected by extreme values or outliers compared to the mean. To find the median, we sort the scores: 21, 24, 28, 79, 84, 93. Since there is an even number of scores, we take the average of the two middle values: (28 + 79) / 2 = 53.5.

The mode represents the most frequently occurring value in the data set. In this case, there is no repeated value, so there is no mode.

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Answer:

The correct answer is A. triangles.

Step-by-step explanation:

Answer:

A)triangles

Step-by-step explanation:

CPCTC is a theorem that states that if two triangles are congruent, then their corresponding parts are also congruent. This means that if you have two triangles that are identical in shape and size, you can conclude that their corresponding angles and sides are equal.

The value of [tex]x_5[/tex], rounded to the nearest decimal place using Newton's method of approximation is 1.466.

The correct answer is option B.

To find the value of [tex]x_5[/tex]using Newton's method of approximation, we need to iterate the following formula:

[tex]x_(_n_+_1_) = x_n - f(x_n) / f'(x_n)[/tex]

where f'(x) represents the derivative of the function f(x). Let's calculate the values step by step.

Given function: f(x) = [tex]x^3 - x^2 - 1[/tex]

Step 1: Find the derivative of f(x)

f'(x) = [tex]3x^2 - 2x[/tex]

Step 2: Initialize the starting value

[tex]x_0[/tex]= 1

Step 3: Calculate [tex]x_1[/tex]

[tex]f(x_0) = f(1) = (1^3) - (1^2) - 1 = 1 - 1 - 1 = -1[/tex]

[tex]f'(x_0) = f'(1) = (3(1^2)) - (2(1)) = 3 - 2 = 1[/tex]

[tex]x_1 = x_0 - f(x_0) / f'(x_0)[/tex]

   = 1 - (-1) / 1

   = 2

Step 4: Calculate [tex]x_2[/tex]

[tex]f(x_1) = f(2) = (2^3) - (2^2) - 1 = 8 - 4 - 1 = 3[/tex]

[tex]f'(x_1) = f'(2) = (3(2^2)) - (2(2)) = 12 - 4 = 8[/tex]

[tex]x_2 = x_1 - f(x_1) / f'(x_1)[/tex]

   = 2 - 3 / 8

   = 1.625

Step 5: Repeat the process until we reach [tex]x_5[/tex]

Performing the calculations for [tex]x_3, x_4, andx_5[/tex], we find:

[tex]x_3[/tex] ≈ 1.465

[tex]x_4[/tex] ≈ 1.466

[tex]x_5[/tex]≈ 1.466

After applying Newton's method of approximation to the function f(x) = [tex]x^3 - x^2 - 1,[/tex]starting with,[tex]x_0 = 1,[/tex] we iteratively calculated the values of [tex]x_1, x_2, x_3, x_4, and x_5[/tex]. The final approximation, [tex]x_5[/tex], is approximately 1.466 when rounded to the nearest decimal place. This aligns with option B as the correct answer.

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Answer:

FALSE

Step-by-step explanation:

Types of Polygons

Triangle (3sides)

Quadrilateral  (4sides)

Pentagon  (5sides)

Hexagon  (6 sides)

Heptagon (7sides)

Octagon  (8 sides)

Nonagon  (9sides)

Decagon  (10 sides)

'All polygons are quadrilaterals' is a false statement.

A closed shape with four sides, four vertices, and four angles is known as a quadrilateral. It is formed by adjoining four non-collinear points. The sum of all the inner angles of a quadrilateral is always 360°.

A polygon is a two-dimensional object made up of straight lines and can have greater than four sides. Similarly, polygons with different numbers of sides are given distinct names, such as hexagons, heptagons, and octagons for 6, 7, and 8 sides, respectively.

All polygons with greater than or less than four angles and sides are not quadrilaterals. A triangle, for example, has three sides, whereas a quadrilateral has four sides.

Therefore, not all polygons are quadrilaterals, but all quadrilaterals are polygons.

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Answer:

:) ok let's see
i'll do my best.
pls dont sue me.

the graph is going down from what I can see, so I guess it would have a negative correlation.

If we wish to label the strength of the association, for absolute values of r, 0-0.19 is regarded as very weak, 0.2-0.39 as weak, 0.40-0.59 as moderate, 0.6-0.79 as strong and 0.8-1 as very strong correlation, but these are rather arbitrary limits, and the context of the results should be considered.

But, judging from the image, I would say it would have a weak correlation, because the coordinated are not that very well together.

So ultimately, we'll go for answer D.
let me know it u think it's something else.
and don't forget to mark me brainliest!

Answer:

Step-by-step explanation:

It sounds like you are describing a line graph that shows the amount of rainfall over a period of six weeks. The Y axis is labeled in inches, and the X axis is labeled in weeks. Each week is represented by a dot on the graph, with the amount of rainfall for that week indicated by the height of the dot on the Y axis. The dots are connected by a line to show the change in rainfall over time. Based on the information you provided, it looks like the correct answer is B, with week 2 having a dot at 3 inches of rainfall.

The equation y = 10x represents a proportional relationship. The constant of proportionality is 10.

A proportional relationship means for every change in x, y also changes proportionately. Let us consider an example

If y = 20

then proportionality constant say k = 10

then x will be 2.

if y = 30

then proportionality constant remains the same ie 10

x will be 3

Here it is evident that as you increase x, y also increase. As you decrease x, y also decreases in a proportionate manner.

k here links x and y variables in a proportionate relationship. Therefore k= 10 is the ratio between y and x.

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1) 2;5;5;2;3:

Mean = 3.4

Median = 3

Mode = 2 and 5

Range = 3

2) 5;5;5;5;6:

Mean = 5.2

Median = 5

Mode = 5

Range = 6 - 5 = 1

3) 4;4;5;5;9;3:

Mean = 5

Median = 4.5

Mode = 4 and 5

Range = 9 - 3 = 6

4) 9;8;7;4;10;2:

Mean = 6.7

Median = 7.5

Mode = No mode

Range = 8

5) 2;5;3;6;2;8;3:

Mean = 4.1

Median = 3

Mode = 2 and 3

Range = 6

6) 4;8;2;3;7;10;5;6:

Mean = 5.6

Median = 5.5

Mode = No mode

Range = 8

Mean is the average of a data set. To calculate the mean, you add up all the values in the data set and then divide by the number of values.

Median is the middle value in a data set when the values are ordered from least to greatest.

Mode is the value that appears most often in a data set.

Range is the difference between the biggest number and smallest number in a data set.

1) 2;5;5;2;3:

Mean = (2 + 5 + 5 + 2 + 3)/5

Mean = 17/5

Mean = 3.4

Arrange in order

2, 2, 3, 5, 5

Median = 3

Mode = 2 and 5 (they both appear most i.e. two times)

Range = 5 - 2 = 3

2) 5;5;5;5;6:

Mean = (5 + 5 + 5 + 5 + 6)/5

Mean = 26/5

Mean = 5.2

5, 5, 5, 5, 6

Median = 5

Mode = 5

Range = 6 - 5 = 1

3) 4;4;5;5;9;3:

Mean = (4 + 4 + 5 + 5 + 9 + 3)/6

Mean = 30/6

Mean = 5

Arrange in order

3, 4, 4, 5, 5, 9

Median = (4+5)/2

Median = 9/2

Median = 4.5

Mode = 4 and 5 (they both appear most i.e. two times)

Range = 9 - 3 = 6

4) 9;8;7;4;10;2:

Mean = (9 + 8 + 7 + 4 + 10 + 2)/6

Mean = 40/6

Mean = 6.7

Arrange in order

2, 4, 7, 8, 9, 10

Median = (7+8)/2

Median = 15/2

Median = 7.5

Mode = No mode

Range = 10 - 2 = 8

5) 2;5;3;6;2;8;3:

Mean = (2 + 5 + 3 + 6 + 2 + 8 + 3)/7

Mean = 29/7

Mean = 4.1

Arrange in order

2, 2, 3, 3, 5, 6, 8

Median = 3

Mode = 2 and 3

Range = 8 - 2 = 6

6) 4;8;2;3;7;10;5;6:

Mean = (4 + 8 + 2 + 3 + 7 + 10 + 5 + 6)/8

Mean = 45/8

Mean = 5.6

Arrange in order

2,3, 4, 5, 6, 7, 8, 10

Median = (5 + 6)/2

Median = 11/2

Median = 5.5

Mode = No mode

Range = 10 - 2 = 8

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The polar form of (2cis60°)(5cis30°) is √(75 - 15√3) cis(π/4). The rectangular form of (2cis60°)(5cis30°) is (-5 + 5√3)/2 + (15√3 + 5)i/2.

Polar form:

(2cis60°) and (5cis30°) can be expressed in polar form as follows:

2cis60° = 2(cos60° + isin60°) = 2(1/2 + i√3/2) = 1 + i√3

5cis30° = 5(cos30° + isin30°) = 5(√3/2 + i/2) = (5√3)/2 + (5i)/2

Thus, (2cis60°)(5cis30°) = (1 + i√3)((5√3)/2 + (5i)/2)

To multiply these complex numbers, we can use FOIL method as we do with algebraic expressions to find:

= [(1)(5√3)/2 + (1)(5i)/2 + (i√3)(5√3)/2 + (i√3)(5i)/2]

= [(5√3)/2 + (5i)/2 + (5i√3)/2 − (15/2)]

= [(5√3)/2 − (15/2) + (5i)/2 + (5i√3)/2]

Now, let's determine the magnitude and argument of this complex number:

Magnitude:

|z| = √[(5√3)/2 − (15/2)]² + [(5/2)√3]²

= √[75/4 - 15√3 + 75/4]

= √(75 - 15√3)

Argument:

θ = tan^⁻1((5/2)√3 / [(5√3)/2 − (15/2)])

= tan^⁻1(1)

= π/4

Rectangular form:

To find the rectangular form, we can use the magnitude and argument we just found, and the following formula:

z = r(cosθ + isinθ)

So, we have:

(2cis60°)(5cis30°) = √(75 - 15√3) cis(π/4)

= √(75 - 15√3)(cos(π/4) + isin(π/4))

= (√(75 - 15√3)/2)(1 + i) + (√(75 - 15√3)/2)(√3 - i)

= ([(5√3)/2 − (15/2)] + [(5/2)√3]) + ([(5√3)/2 − (15/2)]√3 + (5/2))

= (-5 + 5√3) + (15√3 + 5)i/2

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Step-by-step explanation:

The rectangular form of a cis equation is

[tex]a + bi[/tex]

Given that we have

[tex]2cis60 \times 5cis30[/tex]

We can just multioly the magnitudes

[tex]2 \times 5 = 10[/tex]

And add the angles

[tex]60 + 30 = 90[/tex]

And that gives us our new equestion

[tex]10cis(90)[/tex]

Remember that

cis is basically

[tex] \cos( \alpha ) + i \sin( \alpha ) [/tex]

So our answer in polar form is

[tex]10( \cos(90) + i \sin(90) )[/tex]

To convert to rectangular form, distribute the 10

[tex]10 \cos(90) = a[/tex]

[tex]10i \sin(90) = bi[/tex]

So

[tex]a = 0[/tex]

[tex]b = 10[/tex]

So, our equation in rectangular form is

[tex]10i[/tex]