using properties of logarithms and trigonometric identities in exercises 87, 88, 89, and 90, show that the two formulas are equivalent.

Explanation:

ASA = angle side angle

The marked side is between the marked angles.

The markings on the angles tell us which angles pair up to be congruent. The tickmark on the sides tell us the sides are congruent.

AAS is similar to ASA, but with AAS the marked side isn't between the marked angles.

We cannot use SAS since we don't know anything about another pair of sides.

The domain of the function f(x) = √(x - 5) is x ≥ 5, thus, x = 3 is not in the domain.

Remember that for a function y = f(x), we define the domain as the set of the possible inputs.

Here we have the square root function:

f(x) = √(x - 5)

Remember that the argument of a square root needs to be zero or larger, then the domain of the given function is:

x - 5 ≥ 0

Solving for x:

x ≥ 5

The domain is the set of values equal to or larger than 5, thus, the value 3 is not in the domain.

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The formula for calculating the number of cards each player receives in Bridge is N/4, where N = number of cards in the deck.

For a standard deck of 52 cards, each player would receive 52/4 = 13 cards. This can be further simplified to 13.33, however, since cards cannot be divided, the result is rounded down to 13 cards. Therefore, each player in a Bridge game with a standard deck of 52 cards will receive 13 cards.

The total number of cards in the deck is 52, so each player will get 13 cards. This can be expressed mathematically using the formula P = 52/4, where P is the number of cards per player. Therefore, P = 13. We can also use the formula to calculate the number of players if we know the total number of cards. For example, if we have 156 cards, the number of players can be calculated using the formula P = 156/4, which gives us P = 39. Therefore, there will be 39 players in the game.

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Answer:

  x = 30

Step-by-step explanation:

You are given an angle and the sum of that angle with another that is (3x-15)°, and you want the value of x.

The sum of the two angles is given as 100°.

  P +Q = 100°

  (3x -15)° +25° = 100° . . . . . . use the given values for angles and sum

  3x = 90 . . . . . . . . . . . . divide by ° and subtract 10

  x = 30 . . . . . . . . . divide by 3

The value of x is 30.

The theoretical probability is 25%, and the experimental probability is 27.5%.

Given is that There are 10 brown, 10 black, 10 green and 10 gold marbles in bag. A student pulled a marble, recorded the color, and placed the marble back in the bag. The table below lists the frequency of each color pulled during the experiment after 40 trials.

Thee given table is -

Outcome             Frequency

Brown                        13

Black                          9

Green                         7

Gold                           11

The probability of pulling out a gold ball will be -

P{gold} = 11/40 = 0.275 = 0.275 x 100 = 27.5%

P{gold} = 27.5%

Therefore, the theoretical probability is 25%, and the experimental probability is 27.5%.

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Answer:

50h(5h - 2)

Step-by-step explanation:

Open up all the parenthesis on each answer

only the first one is correct

Answer:

50h(5h-2)

Step-by-step explanation:

Solution given

250h² - 100h

50*5*h*h-50*2*h

taking common

50*h(5*h-2)

50h(5h-2)

here taking common and keeping remaining on bracket

50h(5h-2)

a. 45 m S and 35 m S: This vector combination resolves to 10 m S.

b. 15 m N and 60 m S: This vector combination resolves to 45 m S.

c. 5 m S, 10 m E, and 7m W: This vector combination resolves to 8 m E.

Vectors can be added, subtracted, and multiplied by scalars (numbers) to create new vectors. In physics, vectors are used to represent quantities such as velocity, force, and acceleration.

They can also be used in other fields such as engineering, computer graphics, and economics. Vectors can be represented in different ways, such as in Cartesian coordinates (x, y, z) or polar coordinates (magnitude, angle).

a. The vector combination of 45 m S and 35 m S is 10 m S. This is because the vectors are pointing in the same direction (south) and the magnitude of the resultant vector is the sum of the magnitudes of the individual vectors.

b. The vector combination of 15 m N and 60 m S is 45 m S. This is because the vectors are pointing in opposite directions (north and south) and the magnitude of the resultant vector is the difference of the magnitudes of the individual vectors.

c. The vector combination of 5 m S, 10 m E, and 7 m W is 8 m E. This is because the vectors are pointing in different directions (south, east, and west) and the magnitude of the resultant vector is found using the Pythagorean theorem.

The east and west vectors can be viewed as the legs of a right triangle, with the resultant vector being the hypotenuse. The magnitude of the resultant vector is the square root of the sum of the squares of the individual vectors.

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The correct statement about quantitative research is:

It uses formal questions and predetermined response options in questionnaires administered to large numbers of respondents.

We know that the quantitative research is nothing but the process of collecting and analyzing numerical data.

It is used to find the patterns, make predictions, test the relationships, and generalize results to given populations.

Whereas the qualitative research involves collecting and analyzing non-numerical data.

Quantitative research is mostly used in the natural sciences and social sciences.

It collects data using sampling methods, or by sending out online surveys/ polls, and questionnaires.

It uses formal questions and predetermined response options in questionnaires administered to large numbers of respondents.

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The complete question is:

Which of the following is true of quantitative research?

1)    It enables researchers and clients to get closer to their customers and potential customers than does qualitative research.

2)    It uses formal questions and predetermined response options in questionnaires administered to large numbers of respondents.

3)    It is most often used with exploratory research designs.

4)    It usually involves collecting detailed data from relatively small samples by asking questions or observing behavior.

5)    It is superior for studying topics that involve complex psychological motivations not easily reduced to survey formats and quantitative analyses.

The area of the ΔABC is 98 cm²

In a triangle, we can use the sine formula to find the area of the triangle when we know one angle and the two side lengths opposite and adjacent to that angle.

Area of ΔABC = (1/2) bc sinA

Substitute b = 28 cm, c = 14 cm, and m∠A = 30° into the formula:

Area of ΔABC = (1/2) × 28cm × 14cm × sin 30°

Area of ΔABC = 98 cm²

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The probability that at least one of the five female high school friends born in 2003 lives to be 100 years old is 0.054 or 5.4%.

Probability refers to the chance, odds, or likelihood that an expected outcome occurs when there are many possible outcomes.

Probability is a fractional value, lying between one and zero when the expected event is not 100% certain or uncertain.

Probability is computed as the quotient or ratio that results from dividing the expected value by the total weight.

The chance of British women born in 2003 living at least 100 years according to The Guardian website = 0.27

The probability that at least one of five female high school friends born in 2003 lives to be 100 years old = 0.27 x 1/5

= 0.054

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The value of X in the given system of matrix is,

[tex]X=\left[\begin{array}{ccc}\frac{-97}{145}&\frac{-45}{145} \\\\\frac{20}{29}&\frac{21}{29}\\\end{array}\right][/tex]

Matrices are one of the most powerful tools in mathematics. The evolution of the concept of matrices is the result of an attempt to obtain compact and simple methods of solving the system of linear equations.

We have 2 by 2 matrix in the form,

[tex]\left[\begin{array}{ccc}9&9\\-9&-9\\\end{array}\right] X+\left[\begin{array}{ccc}6&3\\7&9\\\end{array}\right] =\left[\begin{array}{ccc}-1&8\\-4&6\\\end{array}\right] X\\\\Assume, X=\left[\begin{array}{ccc}a&b\\c&d\\\end{array}\right] \\\\[/tex]

After multiplication,

[tex]9a+9c+6=-a+8c\\\\-9a-9c+7=-4a+6c\\\\9b+9d+3=-b+8d\\\\-9b-9d+9=-4b-6d\\\\[/tex]

After solving the given system of equations

[tex]a=\frac{-97}{145}\\\\b=-\frac{54}{145}\\\\c=\frac{20}{29}\\\\d={21}{29}\\\\Hence, \\X=\left[\begin{array}{ccc}\frac{-97}{145}&\frac{-45}{145} \\\\\frac{20}{29}&\frac{21}{29}\\\end{array}\right][/tex]

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Answer:

y = -5x + 7

Step-by-step explanation:

Parallel lines on a graph have the same slopes/gradients.

m1=m2

First you want to rewrite the equation of the graph given. So put the 5x to the other side and you will have:

y = -5x -2

This means that the equation that we need must have the gradient -5 (bc it is m in y=mx+c)

The equation will be y = -5x + c

However we must still find c aka the y intercept.

Sub the point we are given (1,2) into the equation

We will have:

2 = -5(1) + c

2 = -5 + c

7 = c

Now that we know c, we know what the equation of the line is. Just sub it in (without 1 and 2)

y = -5x + 7

The sample mean can be classified as a random variable because it is calculated from the sample which is built as random.

The mean of a data-set is obtained as the sum of all observations in the data-set divided by the number of observations.

Hence the sample mean is obtained as the sum of all observations of the sample divided by the sample size.

As the observations of the sample are taken at random, the sum of these observations will be a random variable, and so will the sample mean.

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The greatest number that evenly divides the sum of any three consecutive whole numbers is three.

This is referred to as a whole number that can be positive, negative, or zero and we should note that it doesn't a fractional number and examples include 0, 2, -3 etc.

The product of any three consecutive integers is always divisible by 3 and the addition of any 3 consecutive numbers together it will always equal a multiple of 3 which is therefore the reason why it was chosen as the correct choice.

An example is  n+2 is an integer, 3(n+2) is divisible by 3 in this scenario hence 3 is the correct choice.

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s3+5s2+8s+4

Step-by-step explanation:

Answer:

(1-e^(-2t))*u(t) - (1-e^(-t))*u(t)

Where u(t) is the step function

Step-by-step explanation:

use partial fraction decomposition. This involves expressing the rational function as the sum of simpler fractions, each with a pole at a different point in the complex plane. Once we have done this, we can use the formula for the inverse Laplace transform of a simple pole to find the inverse Laplace transform of the original function.

The number of vitamins (in grams) for a price of x dollars. Then the correct option is B. The inverse function is P⁻¹(x) = (x - 6.4) / 0.50. And at x = 9.2, the value P⁻¹(x) will be 5.4 grams.

Let the function f is given as

y = m(x + a) + c

Then the inverse function of the function f will be given by swapping x with y and y with x.

The function is given below.

P(n) = 0.50n + 6.4

The inverse of the function P(n) is given as,

x = 0.50P⁻¹(x) + 6.4

P⁻¹(x) = (x - 6.4) / 0.50

At x = 9.2, then we have

P⁻¹(9.2) = (9.2 - 6.4) / 0.50

P⁻¹(9.2) = 5.6 grams

The number of vitamins (in grams) for a price of x dollars.

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Answer: To find the surface area of the cylindrical vase, we can use the formula:

SA = B + Ph

Where B is the area of the base (the circular top of the vase), P is the perimeter of the circular top of the vase, and h is the height of the vase.

We know that the diameter of the vase is 4.3 inches and the height of the vase is 11 inches. To find the radius of the base, we can divide the diameter by 2:

r = 4.3 inches / 2 = 2.15 inches

The area of the base is given by πr², so:

B = πr² = π * 2.15² = 14.65 square inches

To find the perimeter of the circular top we can use 2πr:

P = 2πr = 2 * 3.14 * 2.15 = 13.64 inches

Now we can substitute these values into the formula:

SA = B + Ph = 14.65 + 13.64 * 11 = 260.04 square inches

Rounded to the nearest tenth, the surface area of the vase is 260.0 square inches.

Step-by-step explanation:

The probability that 8 or more of these students were classed at professional level is given as follows:

0%.

The mass probability formula, giving the probability of x successes, is of:

[tex]P(X = x) = C_{n,x}.p^{x}.(1-p)^{n-x}[/tex]

[tex]C_{n,x} = \frac{n!}{x!(n-x)!}[/tex]

The parameters are given by:

The values of the parameters for this problem are given as follows:

n = 10, p = 0.05.

Hence the probability of 8 or more of these students were classed at professional level is given as follows:

P(X >= 8) = P(X = 8) + P(X = 9) + P(X = 10)

In which:

Hence the probability is of 0%.

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The factorisation of  20xy - 54 can be expressed as 2( 10xy - 27 ).

The concept here is the factorisation . The breaking or decomposition of an entity (such as a number, a matrix, or a polynomial) into a product of another entity, or factors, whose multiplication results in the original number, matrix, etc., is known as factorization or factoring in mathematics.

A number or other mathematical object is factorized or factored when it is written as the product of numerous factors, typically smaller or simpler things of the same kind.

We were given, 20xy - 54, we can see that that factor of 2 is common for both 20xy and 54.

Then we can factorize as 2( 10xy - 27 ).

2( 10xy - 27 )

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The chi-square goodness-of-fit test requires a comparison of the observed values with their expected values in the f2 generation.

The chi-square goodness-of-fit test is used to determine how well observed values fit a model of independent assortment. The test requires a comparison of the observed values with their expected values in the f2 generation.

Option A is the best choice for this comparison, as it provides a direct comparison of the observed and expected values. This comparison allows for an assessment of the goodness-of-fit of the model, as any differences between the observed and expected values can be attributed to the model's performance.

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X and Y don't have a proportionate relationship.No, not all yx ratios are equal.

If the ratios of two or more numbers or variables are equal to one another, they are said to be in proportion.

The answer to the query is

8; 10, 12; 14, given x

y : 5, 7, 9, 11

Ratios between various x and y values are not comparable.

( 8 /5)

≠ (10 / 7)

≠ (12 /9)

≠(14 /11)

We get to the conclusion that x and y have no proportional relationship.

Hence, Option(1) is the correct answer.

X and Y don't have a proportionate relationship.No, not all yx ratios are equal.

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By applying the exponential properties, it can be concluded that the simplest rational exponent is [tex]x^{\frac{3}{4} }[/tex]

Exponential is a repeated multiplication method with the same number, which is notated in the form aⁿ, where a is called the base and n is the exponent or power.

Some properties of exponent:

[tex]a^{x}[/tex] x [tex]a^{y}[/tex] = [tex]a^{x+y}[/tex]

[tex]a^{x}[/tex] : [tex]a^{y}[/tex] = [tex]a^{x-y}[/tex]

[tex](a^{x}) ^{y}[/tex] = [tex]a^{xy}[/tex]

[tex](\frac{a}{b} )^{x}[/tex] = [tex]\frac{a^{x} }{b^{x} }[/tex]

a⁰ = 1

a⁻ˣ = 1 / aˣ

In mathematics, a radical is described as [tex]\sqrt[n]{x}[/tex], where n is a positive integer ≥ 1 and x is a real number. Then n is called the index of the radical, x is the radicand, and the symbol √ is the radical.

The radical can be written as a rational exponent as follows:

[tex]\sqrt[n]{x} = x^{\frac{1}{n} }[/tex]   or [tex]\sqrt[n]{x^{m} } = x^{\frac{m}{n} }[/tex]

From the problem, we can denote the expression as follows:

square root of x = √x  

the fourth root of x = ⁴√x

The product of the two expressions will be:

(√x)   x  (⁴√x)  = [tex]x^{\frac{1}{2}}[/tex]  x  [tex]x^{\frac{1}{4} }[/tex]

                       = [tex]x^{\frac{3}{4} }[/tex]

Thus the simplest rational exponent is [tex]x^{\frac{3}{4} }[/tex]

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Answer:

a)  Exponential growth function: 60%

b)  Exponential decay function: 20%

c)  Neither

d)  Exponential decay function: 45%

Step-by-step explanation:

[tex]\textsf{Exponential Growth}: \quad y=a(1+r)^x[/tex]

[tex]\textsf{Exponential Decay}: \quad y=a(1-r)^x[/tex]

where:

Given function:

[tex]f(x) = 0.3(1.6)^x[/tex]

The function is an exponential growth function and the growth rate is:

[tex]1+r=1.6 \implies r= 0.6 = 60\%[/tex]

Given function:

[tex]g(x) = 0.8^{0.5x} - 7[/tex]

The function is an exponential decay function and the decay rate is:

[tex]1-r=0.8 \implies r=0.2=20\%[/tex]

Given function:

[tex]h(x) = 3(-5)^x[/tex]

This is neither an exponential growth or decay function as exponential functions cannot have negative bases.

Natural base exponential function

[tex]y=ae^{rx}[/tex]

If a > 0 and r > 0, the function is an exponential growth function.

If a > 0 and r < 0, the function is an exponential decay function.

Given function:

[tex]f(x) = e^{-0.45x} + 4[/tex]

As a = 1 > 0 and r = -0.45 < 0, the function is exponential decay function and the decay rate is:

[tex]r=-0.45=45\%[/tex]

to find a given term, substitute the corresponding value of n into the ninth term formula. For example, if the ninth term is 3n + 2, the 10th term of the sequence can be found by substituting n = 10 into the ninth term. 3 x 10 + 2 = 32 and so, the tenth term of the sequence is 32.

To find the nth term of an arithmetic sequence, use the following formula:

[tex]an = a + (n - 1)d[/tex]

an(nth term) = a(first term) + (n(term you are looking for) - 1) * d(common difference)

This is a True statement that both the matrix equation and the augmented matrix translate into the same system of linear equations.

What are Linear equations?

An equation is said to be linear if the maximum power of the variable is consistently 1. Another name for it is a one-degree equation.

A rectangular matrix is a grouping of numbers into rows and columns. Systems of equations can be solved using matrices. But first, we need to understand how to use matrices to represent systems.

An augmented matrix can be used to express a set of equations. Each column in an augmented matrix represents a variable or the constant terms, and each row in an augmented matrix represents one equation in the system. As a result, it is clear that augmented matrices provide a convenient approach to express systems of equations.

Therefore it is true that the solution set of the equation's equation is the same as the solution set of the linear system whose augmented matrix is.

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Answer:

Linear equation with one variable

Step-by-step explanation:

Here, that one variable is x.

- 11x - 14x - 26 = - 25 - 25x - 1

- 25x - 26 = - 25x - 25 - 1

- 25x - 26 = - 25x - 26

Multiply - 1 on both sides,

25x + 26 = 25x + 26

This equation has infinite solutions.

Where value you substitute in place of x, you will the same answer on the both the sides.

The outcome that are there when a six sided dice is rolled four times is 2

A sample space is a collection or a set of possible outcomes of a random experiment. The p space is represented using the symbol, “S”. The subset of possible outcomes of an experiment is called events. A sample space may contain a number of outcomes that depends on the experiment.

The sample space of getting a 6 from a die is 1. A die has six sides and are labelled 1-6

The outcome of 4 times is is 4

Therefore for two rolls, it will be 4/2 = 2

Therefore the outcome to get 6 exactly two times is 2.

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