Here are descriptions of data sets. Select all descriptions of data sets that could be graphed as dot plots.

The values of θ  for different conditions were calculated with the help of trigonometric functions.

The solution of trigonometric functions is the values of angles where the function value becomes zero.

Given function is:

A) f (θ) = 2 sin θ + √3

f (θ) = 2 sin θ + √3 = 0

2 sin θ = -  √3

sin θ = - √3 / 2

θ = nπ + π/3

where n=1,3,5,7......

So, at θ = nπ + π/3,  pogo stick's spring will be equal to its non-compressed length.

B)If the angle was doubled, the function will look like this,

f (θ) = 2 sin 2θ + √3

2 sin 2θ + √3 = 0

2 sin 2θ = -  √3

sin 2θ = - √3 / 2

θ = nπ/2 + π/6

where  n=1,3,5,7....

if n=1,  θ = 2π/3

n=2,  θ = 7π/6

n=3,  θ = 5π/6

So, These are solution in [0,2π}.

C) y = 2 cos θ + 1

And, y = sin2 θ - 1

Hence, Equate both equation,

2 cos θ + 1 = sin2θ - 1

2 cos θ = 2 sinθ cos θ - 2

cos θ = sinθ cosθ - 1

cosθ (1 - sin θ) = - 1

cosθ = - 1

θ = nπ

where n=1,2,3.....

So, θ = nπ , the lengths of springs from the original pogo stick and the toddler's pogo stick are equal.

Hence, the values of θ  for different conditions were calculated with the help of trigonometric functions.

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Starting with the given equation: 2 sin x cos x = cos x

By dividing both sides by cos x, we may simplify: 2 sin x = 1

Dividing both sides by 2:

sin x = 1/2

The values of x that fulfil sin x = 1/2 must now be found, and they are:

x = π/6 + 2πn or x = 5π/6 + 2πn, where n is any integer.

Therefore, the general solution is:

x = π/6 + 2πn or

x = 5π/6 + 2πn, where n is any integer.

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Answer:

x = 155° , y = 25° , z = 155°

Step-by-step explanation:

in a parallelogram

• opposite angles are congruent

• consecutive angles sum to 180°

then

x + 25° = 180° ( subtract 25° from both sides )

x = 155°

y = 25°

z = 155°

The probability that a student is on the sports team, given that the student plays an instrument, is approximately 35.3% when rounded to one decimal place.

To find the probability that a student is on the sports team given that the student plays an instrument, we can use conditional probability.

Let's denote:

A = Event that a student plays an instrument

B = Event that a student is on the sports team

We are given the following information:

P(A) = 68% = 0.68 (probability of a student playing an instrument)

P(A ∩ B) = 24% = 0.24 (probability of a student playing an instrument and being on a sports team)

The conditional probability P(B|A) represents the probability of event B occurring given that event A has already occurred. It is calculated as:

P(B|A) = P(A ∩ B) / P(A)

Substituting the given values:

P(B|A) = 0.24 / 0.68 ≈ 0.3529

To express the probability as a percentage, we multiply by 100:

P(B|A) ≈ 0.3529 * 100 ≈ 35.3%

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Plotting the polar coordinates: (-5, 5pi/4) matched with the image at

C (image plotted is attached)

Polar coordinates are a coordinate system used to describe the position of a point in a two-dimensional plane.

Unlike rectangular coordinates (x, y), which use Cartesian coordinates to represent a point's position as distances along the x and y axes, polar coordinates represent a point's position using a distance from the origin (r) and an angle from the positive x-axis (θ).

The graph calculator showed the graph matching with the option C

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Answer:8π

Step-by-step explanation:

To find the area of the circle circumscribed about an isosceles right triangle, we first need to determine the length of the hypotenuse of the triangle.

Given that the area of the isosceles right triangle is 8, we can use the formula for the area of a triangle:

Area = (1/2) * base * height

Since the triangle is isosceles and right-angled, the base and height are equal. Let's denote the base and height as 'x'.

8 = (1/2) * x * x

16 = x^2

x = √16

x = 4

Therefore, the length of the hypotenuse of the triangle is 4√2 (using the Pythagorean theorem).

Now, the diameter of the circle is equal to the hypotenuse of the triangle. So, the diameter of the circle is 4√2.

The radius of the circle is half the diameter, so the radius is (4√2)/2 = 2√2.

The area of a circle is calculated using the formula:

Area = π * radius^2

Plugging in the value of the radius, we get:

Area = π * (2√2)^2

Area = π * 4 * 2

Area = 8π

Therefore, the area of the circle circumscribed about the isosceles right triangle with area 8 is 8π.

By algebra properties, the simplified form of the rational equation 3 / (x + 1) - 1 / (x - 1) - 2 · x / (x² - 1) is equal to - 4 / (x² - 1). (Correct choice: C)

In this question we find a rational equation, whose simplified form must be found by algebra properties. First, write the entire expression:

3 / (x + 1) - 1 / (x - 1) - 2 · x / (x² - 1)

Second, factor all denominators:

3 / (x + 1) - 1 / (x - 1) - 2 · x / [(x + 1) · (x - 1)]

Third, use sum of fractions with different denominators:

[3 · (x - 1) - (x + 1) - 2 · x] / [(x + 1) · (x - 1)]

Fourth, simplify the resulting expression:

(3 · x - 3 - x - 1 - 2 · x) / [(x + 1) · (x - 1)]

- 4 / [(x + 1) · (x - 1)]

- 4 / (x² - 1)

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Answer:

Step-by-step explanation:

If the diameter is 4 meters, then the radius has to be 2 because the radius is half of the diameter.

The value of x and y is x ≈ 7.87 and y ≈ 25.38, the correct option is B.

We are given that;

Hypotenuse is 27 base is y and height is x the angle between base and hypotenuse is 17 degree

Now,

The Pythagoras theorem states that the square of the longest side must be equal to the sum of the square of the other two sides in a right-angle triangle.

|AC|^2 = |AB|^2 + |BC|^2  

To find x and y, we can use the Pythagorean theorem and the trigonometric ratios for a right triangle. The Pythagorean theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. The trigonometric ratios relate the angles and sides of a right triangle.

Using the Pythagorean theorem, we can write:

y^2 + x^2 = 27^2

y^2 + x^2 = 729

Using the trigonometric ratios, we can write:

sin(17°) = x/27

cos(17°) = y/27

Solving for x and y, we get:

x = 27 sin(17°) ≈ 7.87

y = 27 cos(17°) ≈ 25.38

Therefore, by pythagoras theorem answer will be x ≈ 7.87 and y ≈ 25.38.

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The equation of the line on the graph passing through the points (-3, 3) and (3, -1) is  [tex]y = -\frac{2}{3}x + 1[/tex].

The formula for equation of line is expressed as;

y = mx + b

Where m is slope and b is y-intercept.

From the image, the line runs through points (-3, 3) and (3, -1).

First, we determine the slope (m) using the given points:

[tex]m = \frac{y_2 -y_1}{x_2 - x_1} \\\\m = \frac{-1 - 3}{3-(-3)} \\\\m = \frac{-4}{3+3} \\\\m = \frac{-4}{6} \\\\m =-\frac{2}{3}[/tex]

Now, plug a point (-3,3) and the slope m = -2/3 into the point-slope form and solve for y:

y - y₁ = m( x - x₁ )

[tex]y - 3 = -\frac{2}{3}( x - (-3)) \\\\y - 3 = -\frac{2}{3}( x + 3) \\\\y - 3 = -\frac{2}{3}x - 2\\\\y = -\frac{2}{3}x - 2+3\\\\y = -\frac{2}{3}x + 1\\[/tex]

Therefore, the equation of the line is [tex]y = -\frac{2}{3}x + 1[/tex].

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The cost to manufacture a unit of energy drink, the selling price and the fixed monthly cost, indicates;

a. C(x) = 1.20·x + 8,000

b. R(x) = 2·x

c. Please find attached the graph of the Revenue and Cost functions showing the break-even point, created with MS Excel.

d. 10,000 units

e. $20,000

The cost function is an function that describes the cost of producing a specified number if units of a services or goods.

a. The cost of production = The variable cost + The fixed cost

The variable cost = The cost per unit × The number of units produced

The cost per unit = $1.20

The fixed cost per month = $8,000

Let x represent the number of units produced per month, we get;

Cost of production, C(x) = 1.20·x + 8,000

b. The revenue function, R(x) = The selling price per item sold × The number of items sold

The selling price per energy drink = $2

Let x represent the number of units sold, we get;

Therefore, the revenue function is; R(x) = 2·x

c. Please find attached the graph for the Revenue and Cost functions, showing the break-even point, created with MS Excel

d. The coordinates of the break-even point on the graph is; (10,000, 20,000), therefore, the number of energy drinks the manufacturer has to sell to break even is; x = 2,000 units of the energy drinks.

e. The revenue the manufacturer has to take to break-even is equivalent to the cost at the same number of units sold, which is $20,000

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Part 1: Oliver must sell more than 33 pineapples to make a profit.

Part 2: Oliver must sell at least 33 pineapples to cover the cost of producing 50 pineapples and make a profit.

Part 1:

The inequality is 6p > 200.

To solve for p, we divide both sides of the inequality by 6. This gives us p > 33.33.

Therefore, Oliver must sell more than 33 pineapples to make a profit.

Part 2:

To cover the cost of producing 50 pineapples, Oliver must sell at least 50 * $6 = $300 worth of fruit.

Since he must sell more than $200 worth of fruit to make a profit, he must sell at least 200 / 6 = 33.33 pineapples.

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The solution to the equation 3x + 10x = 8x is x = 2

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

3x + 10x = 8x

Collect the like terms

So, we have

8x - 3x = 10

When the like terms are evaluated, we have

5x = 10

Divide both sides by 5

x = 2

Hence, the solution to the equation is x = 2

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(a) The algebraic expression, -5x - 2 + 5x + 2 is simplified as 0.

(b) The algebraic expression, 5x -2  - (5x + 2) is simplified as -4.

The given algebraic expression is simplified by adding similar terms together, as it will make the expression to be in simplest form.

The given algebraic expressions are;

-5x - 2 + 5x + 2

5x -2  - (5x + 2)

The first algebraic expression is simplified as follows;

-5x - 2 + 5x + 2

collect similar terms;

(-5x + 5x) + (-2 + 2)

= 0 + 0

= 0

The second algebraic expression is simplified as follows;

5x - 2 - (5x + 2)

= 5x - 2 - 5x - 2

collect similar terms;

= (5x - 5x) + (-2 - 2)

= 0 - 4

= - 4

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Answer: It is around 3000 dolallars for you

Step-by-step explanation:

I am a Tax Preitor

The probability that Ms. Browning will select a blue marker both times is 1/9.

To calculate the probability that Ms. Browning will select a blue marker both times, we need to consider the total number of possible outcomes and the number of favorable outcomes.

Ms. Browning has 3 markers in the box: 1 red, 1 blue, and 1 yellow. Since she selects a marker, uses it, and puts it back, the total number of possible outcomes for the first selection is 3.

For each outcome of the first selection, Ms. Browning again has 3 markers to choose from for the second selection.

Therefore, the total number of possible outcomes for the second selection is also 3.

To calculate the probability of selecting a blue marker both times, we need to determine the number of favorable outcomes. There is only 1 blue marker in the box, and Ms. Browning needs to select it twice.

Therefore, the number of favorable outcomes is 1.

The probability of an event is given by the ratio of favorable outcomes to total outcomes.

In this case, the probability of selecting a blue marker both times is:

Probability = (Number of favorable outcomes) / (Number of total outcomes)

= 1 / (3 × 3)

= 1 / 9

Therefore, the probability that Ms. Browning will select a blue marker both times is 1/9.

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Paul will spend $1382.4 constructing the patio.

To calculate how much Paul will spend constructing the patio, we need to determine the area of the patio and then multiply it by the cost per square foot.

Looking at the scale provided, we can see that 1 cm represents 3 ft. We can use this scale to find the dimensions of the patio in feet.

The length of the patio is given as 8 cm, so the actual length in feet would be:

Length = 8 cm × 3 ft/cm = 24 ft.

Similarly, the width of the patio is given as 6.4 cm, so the actual width in feet would be:

Width = 6.4 cm × 3 ft/cm = 19.2 ft.

Now, we can calculate the area of the patio in square feet by multiplying the length and width:

Area = Length × Width = 24 ft × 19.2 ft = 460.8 sq ft.

The cost to construct the patio is $3 per square foot, so we can calculate the total cost by multiplying the area by the cost per square foot:

Total Cost = Area × Cost per square foot = 460.8 sq ft × $3/sq ft.

Total Cost = $1382.4.

Therefore, Paul will spend $1382.4 constructing the patio.

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The angle m∠J  in the right angle triangle is 58.1 degrees.

One of the angle of a right tangle triangle is 90 degrees. The sum of

angles in a triangle is 180 degrees.

Therefore, the side length LK can be found using Pythagoras's theorem and the angle can be found using trigonometric ratios.

Hence,

tan ∠J = opposite / adjacent

tan ∠J = 15.1 / 9.4

∠J = tan⁻¹ 1.60638297872

∠J = 58.0909229196

Therefore,

m∠J = 58.1 degrees

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The excess amount in the form of interest is K58,784.92.

To calculate the monthly installment amount and the excess amount in the form of interest, we can use the loan amount, interest rate, and tenure.

Loan amount: Total expenditure - Medical insurance coverage

Loan amount = K85,800 - K20,000

Loan amount = K65,800

Interest rate per month: 17% / 12 months = 1.42%

Tenure: 12 years × 12 months

= 144 months

Monthly interest rate = 1.42% = 0.0142

Number of months = 144

Monthly installment = (K65,800 × 0.0142) / (1 - (1 + 0.0142)⁻¹⁴⁴)

Monthly installment = K763.43

So, the monthly installment amount is approximately K763.43.

To calculate the excess amount in the form of interest, we can subtract the total interest paid from the loan amount.

Total interest paid = (Monthly installment × Number of months) - Loan amount

Total interest paid = (K763.43 × 144) - K65,800

Total interest paid= K58,784.92

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Answer:11

Step-by-step explanation:

The resulting matrix is shown below:

3 0   1   1

4 20 2 20

3 -2  0   6

A matrix is described as a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, which is used to represent a mathematical object or a property of such an object.

We can write  resulting matrix after replacing R2 with R2 + 3R3 as:

3 0 1 1

4 2 2 2

3 -2 0 6

We Perform the row operation:

R2 = R2 + 3R3

3 0 1 1

4 2 2 2 + 3(6) = 20

3 -2 0 6

In conclusion, the resulting matrix is:

3 0   1    1

4 20 2 20

3 -2  0   6

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Answer: Hope this helps :)

Step-by-step explanation:

The slope of the given line y = 5x - 3 is 5. The slope of a line perpendicular to it would be the negative reciprocal of 5, which is -1/5. Using the point-slope form of a line, we can find the equation of the line perpendicular to y = 5x - 3 that passes through the point (-3,4) as follows: y - y1 = m(x - x1), where (x1,y1) is the point (-3,4) and m is the slope -1/5. Substituting these values into the equation gives us: y - 4 = (-1/5)(x + 3). Simplifying this equation gives us: y = (-1/5)x - 3/5 + 4, or y = (-1/5)x + 17/5.

Answer:

To find the equation of a line perpendicular to y=5x-3 and which passes through (-3,4), we first need to find the slope of the line y=5x-3. The slope of this line is 5.

Since we want to find the equation of a line that is perpendicular to y=5x-3, we know that the slope of this new line will be the negative reciprocal of 5. The negative reciprocal of 5 is -1/5.

Now that we know the slope of the new line, we can use the point-slope form of a line to find its equation. The point-slope form of a line is y-y1=m(x-x1), where m is the slope of the line and (x1,y1) is a point on the line.

Using the point (-3,4) and the slope -1/5, we get:

y-4=-1/5(x-(-3)

y-4=-1/5(x+3)

y-4=-1/5x-3/5

y=-1/5x+23/5

Therefore, the equation of a line perpendicular to y=5x-3 and which passes through (-3,4) is y=-1/5x+23/5.

The cube root of -2i,

⇒ ∛2[ cos(π/6) + i sin(π/6) ]

We can represent -2i in polar form as,

r(cosθ + i sinθ)

by computing its magnitude and angle.

The magnitude of -2i is 2

Since the absolute value of any imaginary number is equal to its magnitude.

To find the angle θ,

We can use the fact that the tangent of an angle is equal to the ratio of the opposite side to the adjacent side.

In this case,

The opposite side is -2 and the adjacent side is 0.

Therefore, we have tanθ = -2/0, which is undefined.

However,

We can use the fact that the cube root of a complex number is equal to the cube root of its magnitude times exp(iθ/3) to find the cube root of -2i.

So, the cube root of -2i is equal to the cube root of 2 times exp(iθ/3).

Now, we need to find the value of θ/3. Since θ is undefined,

we will represent it as π/2 + 2πn,

where n is any integer.

So, θ/3 = (π/2 + 2πn)/3 = π/6 + 2πn/3.

Therefore, the cube root of -2i is equal to

⇒  ∛2 [ cos(π/6 + 2πn/3) + i sin(π/6 + 2πn/3) ]

Now put n = 0

⇒ ∛2[ cos(π/6) + i sin(π/6) ]

This is the final form of the cube root of -2i in the required form.

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Answer:

2/3 * 660

Step-by-step explanation:

If we convert 65% to a decimal, we get 0.65.

First, it will helpful to find exactly what is 65% of 668:  

0.65 * 668 = 434.2 and thus 65% of 668 is exactly 434.2.

Similarly, if you convert 2/3 to a decimal, you get 0.66 repeating.

2/3 * 660 = 440 which is only 10.2 units larger than 434.2.  Thus, 2/3 * 660 is the most accurate estimate of 65% of 668.

The function has a maximum value of 1

Option D is the correct answer.

We have,

The y-coordinate on the graph is the function value.

Now,

From the graph of the function,

The maximum is at the point = (-2, 1)

This means,

1 is the maximum value.

And,

From the graph of the function,

The minimum value can be calculated because the graph is infinitely down the graph.

Thus,

The function has a maximum value of 1

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Answer:

input 1 : output 3

Step-by-step explanation:

For the graph y = x+2/x where x = 1, y = 3. So find which equation when input 1, outputs 3.

The transformation of f(x) to g(x) is f(x) is shifted down by 4 units to g(x) and shifted right by 5 units

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

The functions f(x) and g(x)

Where, we can see that

f(x) = |x|

g(x) = |x + 5| - 3

So, we have

vertical difference = 0 - 3 = -3

horizontal difference = 5 - 0 = 5

This means that the transformation of f(x) to g(x) is f(x) is shifted down by 4 units to g(x) and shifted right by 5 units

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Answer:

To solve the inequality 8x + 2 < 10x - 4, we can start by subtracting 8x from both sides to get 2 < 2x - 4. Then, we can add 4 to both sides to get 6 < 2x. Finally, we can divide both sides by 2 to get 3 < x. This means that the solution set for the inequality is all values of x that are greater than 3. In interval notation, the solution set is (3, ∞).

Step-by-step explanation:

The value of the median of the trapezoid is, 14 units

In the given diagram a trapezoid has been given with two bases,

AB = 7 units and CD = 21 units

We have to find, the median of the trapezoid.

Since, We know that median of a trapezoid is half of the sum of bases.

So, We get;

median of ABCD = 1/2 (AB + CD)

                                = 1/2 ( 7 + 21 )

                                = 1/2 ( 28 )

                                = 14 units.

Thus,. The value of the median of the trapezoid is, 14 units

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The compound inequality representing the range for the skull size of cats is: 39.32 ≤ x ≤ 43.01.

To find the compound inequality that represents the range for the skull size of cats, we need to solve the given compound inequality:

8 ≤ 1.03x - 32.4 ≤ 12

Let's solve it step by step:

Add 32.4 to all parts of the compound inequality:

8 + 32.4 ≤ 1.03x - 32.4 + 32.4 ≤ 12 + 32.4

40.4 ≤ 1.03x ≤ 44.4

Divide all parts of the compound inequality by 1.03 (the coefficient of x):

40.4/1.03 ≤ (1.03x)/1.03 ≤ 44.4/1.03

39.32 ≤ x ≤ 43.01

Therefore, the range for the skull size of cats can be represented by the compound inequality:

39.32 ≤ x ≤ 43.01

Rounded to the nearest hundredth, the compound inequality representing the range for the skull size of cats is:

39.32 ≤ x ≤ 43.01

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