savvas realize topic 5 assesment

(a) The value of sin 2θ is -3√(91)/50.

(b)  θ/2 is located in the third quadrant.

(c) The value of cos θ/2 is -√65/10.

We know that cos θ = 3/10, so we can find sin θ using the Pythagorean identity:

sin² θ + cos² θ = 1

sin²θ + (3/10)² = 1

sin²θ = 1 - (3/10)² = 91/100

sin θ = ±√(91)/10

Since 3π/2 < θ < 2π,

we know that sin θ < 0, so sin θ = -√(91)/10.

a) To find sin 2θ, we can use the double angle formula:

sin 2θ = 2 sin θ cos θ

sin 2θ = 2 (-√(91)/10) (3/10)

sin 2θ = -3√(91)/50

b) To find the quadrant in which θ/2 is located, we first need to find θ/2:

θ/2 = (3π/2 + 2π)/2 = 5π/4

5π/4 is in the third quadrant, so θ/2 is located in the third quadrant.

c) To find cos θ/2, we can use the half angle formula:

cos θ/2 = ±√((1 + cos θ)/2)

Since 3π/2 < θ < 2π, we know that cos θ < 0,

so we take the negative square root:

cos θ/2 = -√((1 + 3/10)/2)

cos θ/2 = -√(13/20)

Simplifying the radical by dividing both numerator and denominator by 4:

cos θ/2 = -√(13)/2√5

Multiplying numerator and denominator by √5 to rationalize the denominator:

cos θ/2 = -√65/10

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Tο determine which lοcatiοn typically has the cοοler temperature, we shοuld cοmpare the median temperature οf Sunny Tοwn tο the mean temperature οf Beach Tοwn. Thus, cοrrect οptiοns are:

Median is a measure οf central tendency that represents the middle value in a dataset when the values are arranged in οrder οf magnitude.

When cοmparing the data tο determine which lοcatiοn typically has the cοοler temperature, the measure οf center tο use depends οn the shape οf the distributiοn οf the data.

Fοr the Sunny Tοwn data, since the histοgram is symmetric, we can use the median as a measure οf center. The median οf the Sunny Tοwn data wοuld be the value that separates the data intο twο equal halves.

Fοr the Beach Tοwn data, since the histοgram is skewed, we can use the mean as a measure οf center. The mean οf the Beach Tοwn data wοuld be the average οf all the data pοints.

Therefοre, tο determine which lοcatiοn typically has the cοοler temperature, we shοuld cοmpare the median temperature οf Sunny Tοwn tο the mean temperature οf Beach Tοwn.

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The p-value, under the assumption that the null hypothesis is correct, is the likelihood of receiving a test statistic that is as extreme as the one that was observed.

A declaration or assumption that there is no significant difference or association between two or more variables or populations being studied is known as the null hypothesis in statistics. The default hypothesis that is evaluated against an alternative hypothesis is known as the null hypothesis and is frequently represented as "H0." Typically, the null hypothesis is formulated in light of prior research, theoretical predictions, or accepted wisdom. It posits that any observed differences or correlations between groups or variables are the result of chance or random variation and serves as the foundation for evaluating a hypothesis.

given

We can use a standard format that includes the following details to present the z test results:

This is a claim regarding the population parameter being tested, or the null hypothesis.

In this instance, the null hypothesis is that the population of police officers' mean physical fitness score is equal to 73.

This assertion defies the null hypothesis, according to the alternative. The alternative hypothesis in this situation is that the sample of police officers' mean physical fitness score is higher than 73.

The p-value, under the assumption that the null hypothesis is correct, is the likelihood of receiving a test statistic that is as extreme as the one that was observed.

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There are 200 thousands in 200,000.

Answer:

Step-by-step explanation:

answer 42

how to explain but the answe is 42

Answer:

Step-by-step explanation:

Fractions are parts or pieces of a complete thing.

In this problem, the fractions start out with varying denominators, and then end with the same denominator.  In order to go from different denominators to the same denominator, we must find a common denominator.

A common denominator is a number that denominators of fractions can equal when multiplied by something else.  Ex:  1/3 + 2/ 7  Multiples of 3:  3, 6, 9, 12, 15, 18, 21   Multiples of 7:  7, 14, 21.  Both 3 and 7 have 21 in common, so this would be the common denominator.

A common denominator is needed so fractions with different denominators can be combined.  To get from 6, 3, and 7 to 42, the multiples of 2, 3, and 7 must be listed.

Multiples of 6:  6, 12, 18, 24, 30, 36, 42

Multiples of 3:  3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 29, 42

Multiples of 7:  7, 14, 21, 28, 35, 42

All three numbers have 42 in common, so that is their common denominator.  Whatever was done to the denominator, must be done to the numerator.  Now, we must take the number that can be multiplied by the denominator to reach 42 and multiply it by the numerator.

[tex]\frac{1*7}{6*7}}[/tex]

[tex]=\frac{7}{42}[/tex]

[tex]\frac{2*14}{3*14}[/tex]

[tex]=\frac{28}{42}[/tex]

[tex]\frac{3*6}{7*6}[/tex]

[tex]=\frac{18}{42}[/tex]

So, to sum it up, when trying to add fractions with different denominators, find a common denominator by listing multiples of the denominators and multiply the numerator by the number the denominator was multiplied by to get the common denominator.

The logic in this fraction addition problem is to find a common denominator for all the fractions and then add the numerators. In this case, the common denominator is the least common multiple of the denominators, which is 42.

Now that all the fractions have a common denominator of 42, we can add the numerators to get the final answer:

[tex]\begin{aligned} \sf \dfrac{1}{6} + \dfrac{2}{3} + \dfrac{3}{7} = \dfrac{7}{42} + \dfrac{28}{42} + \dfrac{18}{42} = \bold{\dfrac{53}{42}} \end{aligned}[/tex]

Therefore, the sum of the fractions is 53/42.

I hope this helps!

In the given angles the required value of x is 17° respectively.

An angle is a figure in Euclidean geometry made up of two rays that share a common terminal and are referred to as the angle's sides and vertices, respectively.

Angles created by two rays are in the plane where the rays are located.

The meeting of two planes also creates angles. We refer to these as dihedral angles.

Based on the measurement, there are many kinds of angles in geometry.

The names of fundamental angles include acute, obtuse, right, straight, reflex, and full rotation.

So, we know that:

2x+5 + 8x+5 = Straight angle

2x+5 + 8x+5 = 180

Now, solve for x as follows:

2x+5 + 8x+5 = 180

10x+10 = 180

10x = 170

x = 17

Therefore, in the given angles the required value of x is 17° respectively.

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Therefore , the solution of the given problem of unitary method comes out to be  enable the current to travel, a connection, such as a wire or conductor, is required.

Utilizing previously well-known variables, this uniform convenience, or all crucial elements from a prior flexible study that followed a particular methodology event can all be used to achieve the goal. It will be possible to contact the entity again if the anticipated assertion outcome actually happens; if it doesn't, both important systems will surely miss the statement.

Here,

D. In order for a current to move, there must be an energy source, a recipient, and a connection.

Current is the movement of charged ions through a conductor, most frequently electrons.

The energy required to transport the charged particles is provided by an energy source, such as a battery or generator.

The current has a route through a recipient, which can be a machine or a circuit.

In order to finish the circuit and enable the current to travel, a connection, such as a wire or conductor, is required.

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The order of the matrix is 2 x 2.

What is a matrix and what is meant by its order?

In the field of mathematics, matrices are rectangular arrangements of numerical values, symbols, or algebraic expressions, organized in rows and columns. Matrices are used in many areas of mathematics and science, as well as in engineering, physics, and computer graphics.

The term 'order' in the context of matrices refers to the dimensions of the matrix, which are determined by the number of rows and columns it contains. It is denoted by m x n, where m is the number of rows and n is the number of columns. To find the order of a matrix, we simply count the number of rows and columns. The order is usually written as a pair of numbers in the form m x n.

Determine the order of the given matrix:

In the given problem, we are asked to find the order of the matrix:

[tex]\begin{pmatrix}9 &6 \\-4 & 8\end{pmatrix}[/tex]

we count the number of rows and columns.

This matrix has 2 rows and 2 columns, so its order is 2 x 2.

Therefore, the order of the matrix is 2 x 2.

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Answer: a )The answer is 196.126.

b )

Step-by-step explanation:

a) Multiply 33.1 and 18.2= 602.62

Divide the result by 5.8:

602.62 / 5.8 = 196.126

Therefore, 33.1x18.2 / 5.8 = 196.126.

Answer:

y = 5/6x + 1/6

Step-by-step explanation:

m = 5/6, x = 7, y = 6

y = mx + b

6 = 5/6(7) + b

b = 6 - 35/6

b = 36/6 - 35/6 = 1/6

y = 5/6x + 1/6

Answer:

Below

Step-by-step explanation:

Here is another way:

 Start with point ( 7,6)   slope ( m= 5/6)  form :

    (y-6) = 5/6 ( x-7)     expand

    y-6 = 5/6 x -  35/6     add 6 to both sides

       y = 5/6 x + 1/6     Done.

Answer:x=23

Step-by-step explanation:

x+1+7x-5=180

8x-4=180

8x=184

x=184/8

x=23

Answer:

For the first 40 hours that Mr. Ed works, he earns his regular rate of pay, which is $15.50 per hour. So, his regular pay for the week is:

40 hours x $15.50 per hour = $620

For the additional 5 hours he works, he earns overtime pay at a rate of time-and-a-half, which is 1.5 times his regular pay rate. So, his overtime pay for the week is:

5 hours x $15.50 per hour x 1.5 = $116.25

Therefore, Mr. Ed's total pay for the week in which he works 45 hours is:

$620 (regular pay) + $116.25 (overtime pay) = $736.25.

Answer:

volume = 290m

Step-by-step explanation:

box 1

[tex]vol= lwh\\=3*5*6\\=90m[/tex]

box 2

[tex]vol=lwh\\=8*5*5\\=200m[/tex]

total volume = 90+200 =290m

Answer:

Step-by-step explanation:

To calculate the amount of money Nathaniel needs to deposit today to fund this gift for his descendants, we can use the present value formula for a perpetuity:

Present Value = Annual Payment / Discount Rate

In this case, the annual payment is R380 000 and the discount rate (expected rate of return) is 6.45% in decimals, or 0.0645 as a fraction. So we can plug these values into the formula:

Present Value = R380 000 / 0.0645

This gives us a present value of approximately R5,899,225.68. Therefore, Nathaniel would need to deposit R5,899,225.68 today into the trust fund to provide R380 000 a year forever for his descendants, assuming a 6.45% expected rate of return.

If the graph of a polynomial function P(x) has x-intercepts at x = -4, x = 0, x = 5, the following must be true for P(x): D. The degree of the polynomial is greater than or equal to three.

Based on the graph of the polynomial f(x), the following is not a factor: B. (x - 1).

In Mathematics and Geometry, a polynomial graph simply refers to a type of graph that touches the x-axis at zeros, roots, solutions, x-intercepts, and factors with even multiplicities.

Generally speaking, the zero of a polynomial function simply refers to a point where it crosses or cuts the x-axis of a graph. Since the polynomial function with x-intercepts at x = -4, x = 0, and x = 5 cuts the x-axis at three (3) points. Therefore, the degree of this polynomial is either greater than or equal to three (3).

By critically observing the graph of the polynomial function shown above, we can logically deduce that its x-intercepts include the following;

x = -1 ⇒ x + 1 = 0.

x = -2 ⇒ x + 2 = 0.

x = 2 ⇒ x - 2 = 0.

x = 3 ⇒ x - 3 = 0.

Therefore, (x - 1) is not a factor.

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The angles 5 and 6 which are given in the diagram above are typical example of an adjacent angle.

An adjacent angle is defined as the type of angle that has a common sides which is shared by the two angles that are being compared.

A vertical angle is the type of angle that is usually opposite and they constructed by the intersection of two lines.

Therefore since a common sides exists between angle 5 and 6.

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Step-by-step explanation:

The equation 2x − 15 = 185 represents the relationship between Sam's and Martha's heights. We can solve for x, which represents Martha's height, by isolating x on one side of the equation:

2x − 15 = 185

Add 15 to both sides of the equation:

2x = 200

Divide both sides of the equation by 2:

x = 100

Therefore, Martha's height is 100 centimeters.

Answer:

18,739

Step-by-step explanation:

14 thousands = 14,000. 46 hundreds = 4,600. 13 tens = 130. 9 = 9. You add them all up and get 18,739.

Answer:

To calculate the expected return on the portfolio, we use the following formula:

Expected return = (weight of stock X * expected return of stock X) + (weight of stock Y * expected return of stock Y) + (weight of stock Z * expected return of stock Z)

Expected return = (0.35 * 0.09) + (0.2 * 0.15) + (0.45 * 0.12) = 0.0321 or 3.21%

To calculate the variance of the portfolio, we use the following formula:

Variance = (weight of stock X)^2 * variance of stock X + (weight of stock Y)^2 * variance of stock Y + (weight of stock Z)^2 * variance of stock Z + 2 * weight of stock X * weight of stock Y * covariance of stocks XY + 2 * weight of stock X * weight of stock Z * covariance of stocks XZ + 2 * weight of stock Y * weight of stock Z * covariance of stocks YZ

Assuming that the stocks are uncorrelated, the covariance terms will be zero. Also, we assume that the variances of the stocks are equal to the square of their standard deviations. Therefore, we can simplify the formula to:

Variance = (weight of stock X)^2 * standard deviation of stock X^2 + (weight of stock Y)^2 * standard deviation of stock Y^2 + (weight of stock Z)^2 * standard deviation of stock Z^2

Variance = (0.35)^2 * (0.09)^2 + (0.2)^2 * (0.15)^2 + (0.45)^2 * (0.12)^2 = 0.00060167 or 0.060167%

To calculate the standard deviation of the portfolio, we take the square root of the variance:

Standard deviation = sqrt(0.00060167) = 0.0245 or 2.45%

Therefore, the expected return of the portfolio is 3.21%, the variance is 0.060167% and the standard deviation is 2.45%.

Answer:

Step-by-step explanation:

To calculate the expected return on the portfolio, we need to take the weighted average of the individual stock returns based on their proportions in the portfolio. The expected return on the portfolio can be calculated as follows:

Expected return on portfolio = (weight of stock X × expected return on stock X) + (weight of stock Y × expected return on stock Y) + (weight of stock Z × expected return on stock Z)

Expected return on portfolio = (0.35 × 9%) + (0.20 × 15%) + (0.45 × 12%)

Expected return on portfolio = 3.15% + 3.00% + 5.40%

Expected return on portfolio = 11.55%

To calculate the variance and standard deviation of the portfolio, we need to use the formula that takes into account the individual stock variances, covariances, and the weights of the stocks in the portfolio. Assuming the covariances between the stocks are zero, we can use the following formulas:

Portfolio variance = (wX^2 * σX^2) + (wY^2 * σY^2) + (wZ^2 * σZ^2) + 2(wX * wY * σXY) + 2(wX * wZ * σXZ) + 2(wY * wZ * σYZ)

Portfolio standard deviation = sqrt(portfolio variance)

where wX, wY, and wZ are the weights of stocks X, Y, and Z in the portfolio respectively, σX, σY, and σZ are the standard deviations of returns on stocks X, Y, and Z respectively, and σXY, σXZ, and σYZ are the covariances between the returns on stocks X and Y, X and Z, and Y and Z respectively.

Since we assume the covariances between the stocks are zero, we can simplify the formulas as follows:

Portfolio variance = (wX^2 * σX^2) + (wY^2 * σY^2) + (wZ^2 * σZ^2)

Portfolio standard deviation = sqrt[(wX^2 * σX^2) + (wY^2 * σY^2) + (wZ^2 * σZ^2)]

Substituting the values, we get:

Portfolio variance = (0.35^2 * 0.09) + (0.20^2 * 0.15) + (0.45^2 * 0.12)

Portfolio variance = 0.0036475

Portfolio standard deviation = sqrt[(0.35^2 * 0.09) + (0.20^2 * 0.15) + (0.45^2 * 0.12)]

Portfolio standard deviation = sqrt(0.0036475)

Portfolio standard deviation = 0.06035

Therefore, the expected return on the portfolio is 11.55%, the portfolio variance is 0.0036475, and the portfolio standard deviation is 0.06035.

Step-by-step explanation:

2  2/3 =  8/3

8/3 * 1/4  = 8/12 =   2/3 pound in each portion

(x+2), (x+1), and (x-2) are factors of f(x) because they correspond to the roots of the polynomial.

What is Polynomial function ?

A polynomial function is a type of mathematical function that consists of a sum of terms, where each term is a product of a constant coefficient and one or more variables raised to non-negative integer powers. In other words, a polynomial function is an algebraic expression that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents.

Thank you for providing the graph of the polynomial f(x). Based on the graph, we can see that the polynomial has roots at x = -2, x = -1, x = 2, and x = 7. Therefore, (x+2), (x+1), and (x-2) are factors of f(x) because they correspond to the roots of the polynomial.

However, since there is no point on the graph where f(x) is equal to zero when x = -7, we can conclude that (-x+7) is NOT a factor of f(x).

Therefore, (x+2), (x+1), and (x-2) are factors of f(x) because they correspond to the roots of the polynomial.

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The product of the expressions are;

Step 1: (x + 2)(x + 3) and 3(x + 3)/4(x + 5)

Step 3: 4(x + 3) × 3(x + 3)/4(x + 5)

Step 4: 3/x + 5

It is important to note that algebraic expressions are described as expressions that are composed of variables, terms, coefficients, constants and factors.

From the information given, we have the fraction;

4x + 8/x² + 5x + 6 × 3x + 9/4x + 20

To determine the product, let us reduce the expressions to their lowest forms, we have;

4x + 8 = 4(x + 2)

x² + 5x + 6 = (x + 2)(x + 3)

3x + 9 = 3(x +3)

4x + 20 = 4(x + 5)

Substitute the expressions

4(x + 2)/(x + 2)(x + 3)  × 3(x +3)/4(x + 5)

divide the common terms

4(x + 3) × 3(x + 3)/4(x + 5)

Divide further, we have.

3/x + 5

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Answer: N

Step-by-step explanation: The set of natural numbers is represented by the letter “N”.  A natural number is a number that occurs commonly and obviously in nature. As such, it is a whole, non-negative number. The set of natural numbers, denoted N, can be defined in either of two ways: N = {0, 1, 2, 3, ...}

The average allowance for each student is $9.25.

To find the average allowance for each student, we need to add up all the allowances and divide by the total number of students.

Adding up the allowances:

$7 + $12 + $8.50 + $10 + $7 + $8 + $10.50 + $11 = $74

There are 8 students, so we divide the total allowance by 8 to get:

$74 / 8 = $9.25

Therefore, the average allowance for each student is $9.25.

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The maximum errοr οf estimate is 0.092 years οr apprοximateIy 1 mοnth.

The standard deviatiοn is a measure οf the amοunt οf variatiοn οr dispersiοn οf a set οf vaIues frοm their mean οr average. It is caIcuIated as the square rοοt οf the variance.

Tο find the maximum errοr οf estimate, we need tο use the fοrmuIa:

Maximum errοr οf estimate = z * (standard deviatiοn / √(sampIe size))

where z is the z-scοre assοciated with the given cοnfidence IeveI, standard deviatiοn is the pοpuIatiοn standard deviatiοn (0.8 years), and sampIe size is the number οf TVs sampIed (445).

At a cοnfidence IeveI οf 95%, the z-scοre is 1.96.

Substituting the vaIues in the fοrmuIa, we get:

Maximum errοr οf estimate = 1.96 * (0.8 / √(445))

= 0.092 years (rοunded tο 3 decimaI pIaces)

Therefοre, the maximum errοr οf estimate is 0.092 years οr apprοximateIy 1 mοnth. This means that we can be 95% cοnfident that the true mean rate at which the TVs need tο be serviced οr repIaced is within 1 mοnth οf the sampIe mean οf 3.5 years.

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The completed statement with regards to the area of the square and the rectangle are;

The estimated value of the length of the shaded square is 5·√5 inches. The estimated value of the area of the unshaded rectangle is 175 square inches.

The area of a square is the product of the side lengths which are congruent, therefore;

Area of a square = Side length, s × Side length, s = s²

The possible figure in the question includes;

A shaded square that is 125 square inches

An adjacent unshaded rectangle, that share a side with the square that has a side length of 7·√5 inches

Please find attached the possible drawing of the figure in the question, (not drawn to scale) obtained from a similar question posted online, created with MS Word.

Therefore;

The side length of the square = √(125) inches =  5·√5 inches

The estimated value of the side length of the square is; 5·√5 inches

The area of a rectangle = Length × Width

The length of the rectangle = 7·√5 inches

The width of the rectangle = 5·√5 inches

Therefore;

The area of the unshaded rectangle, therefore is; 5·√5 × 7·√5 = 175

The estimated area of the unshaded rectangle is 175 square inches

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Therefore, the coordinates of point B are approximately (12.67, 0.67).

In mathematics, an equation is a statement that asserts the equality of two expressions. Equations are written using an equal sign (=) between the two expressions that are being compared. The expressions on either side of the equal sign are called the left-hand side (LHS) and the right-hand side (RHS) of the equation. The equation is true if the LHS is equal to the RHS. Equations are used to represent various relationships and patterns in mathematics, and are an important tool in problem-solving and mathematical modeling.

Here,

(a) To calculate the length of OC, we need to find the coordinates of point O, which is the intersection of the diagonals of the parallelogram. The midpoint of AC is:

((12+5)/2, (0+5)/2) = (8.5, 2.5)

Therefore, the equation of the line passing through the midpoint of AC and perpendicular to AC is:

y - 2.5 = -(5-0)/(5-12) (x - 8.5)

Simplifying:

y - 2.5 = 5/7 (x - 8.5)

y = 5/7 x - 1

This is the equation of the line passing through O and parallel to AB. We can see that when x = 12, y = 5/7 * 12 - 1 = 7.4. Therefore, the coordinates of O are (12, 7.4).Using the distance formula, we can now calculate the length of OC:

OC = √((12-5)² + (7.4-0²) = √(109.96) ≈ 10.49

Therefore, the length of OC is approximately 10.49 units.

(b) The gradient of OC can be found using the coordinates of points O and C:

m = (y₂ - y₁)/(x₂ - x₁) = (7.4 - 5)/(12 - 5) = 0.4/1 ≈ 0.4

Therefore, the gradient of OC is approximately 0.4.

(c) The equation of line AB can be found using the coordinates of points A and B:

m = (y₂ - y₁)/(x₂ - x₁) = (y - 0)/(x - 12)

Simplifying and substituting the coordinates of point A:

0.4 = (y - 0)/(x - 12)

0.4(x - 12) = y

Therefore, the equation of line AB is y = 0.4x - 4.8.

(d) The coordinates of point B can be found by solving the system of equations formed by the equations of lines AB and OC. Equating y in both equations:

0.4x - 4.8 = 5/7 x - 1

Simplifying and solving for x:

0.3x = 3.8

x = 12.67

Substituting x in either equation:

y = 0.4(12.67) - 4.8 ≈ 0.67

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