Consider the equation. −2(x−1)−3x=12(x+3) What is the value of x in the equation?

Since A1, A2, ..., Am are mutually exclusive and exhaustive, answers to both parts of the question is;

a) We can use the same argument to show that P(A2) = P(A3) = ... = P(Am) = 1/m.
b) We have proved that if A = A1 ∪ A2 ∪ ... ∪ Ah and (a) holds, then P(A) = h/m.

(a) Since A1, A2, ..., Am are mutually exclusive and exhaustive, we have:

P(S) = P(A1) + P(A2) + ... + P(Am)

Since P(A1) = P(A2) = ... = P(Am), we can rewrite the above equation as:

P(S) = m * P(A1)

Since S is the sample space and its probability is 1, we have:

P(S) = 1

Therefore, we can solve for P(A1) as:

P(A1) = 1/m

Similarly, we can use the same argument to show that P(A2) = P(A3) = ... = P(Am) = 1/m.

(b) Since A1, A2, ..., Am are mutually exclusive and exhaustive, we have:

P(S) = P(A1) + P(A2) + ... + P(Am)

Using (a), we know that P(Ai) = 1/m for i = 1, 2, ..., m. Therefore, we can rewrite the above equation as:

1 = m * (1/m) + P(Ah+1) + ... + P(Am)

Simplifying this equation, we get:

P(Ah+1) + ... + P(Am) = (m - h) * (1/m)

Since A = A1 ∪ A2 ∪ ... ∪ Ah, we can write:

P(A) = P(A1) + P(A2) + ... + P(Ah) = h * (1/m)

Therefore, we have proved that if A = A1 ∪ A2 ∪ ... ∪ Ah and (a) holds, then P(A) = h/m.

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To evaluate the integral using a linear change of variables, we need to:

(A) Find a transformation: Let u = x + y and v = x - y. The inverse transformation is x = (u + v)/2 and y = (u - v)/2.

(B) Transform the polygon region: The vertices (2, 0), (0, 2), (-2, 0), and (0, -2) transform to (2, 2), (2, -2), (-2, -2), and (-2, 2), forming a rectangular region with u = [-2, 2] and v = [-2, 2].

(C) Compute the Jacobian: J(u, v) = |∂(x, y)/∂(u, v)| = |(1/2, 1/2; 1/2, -1/2)| = 1/2.

(D) Convert the iterated double integral: ∬R e^(x² - y²) dA = ∬_{-2}^2 ∬_{-2}^2 e^((u+v)²/4 - (u-v)²/4) * (1/2) du dv.

(E) The final answer: The integral evaluates to 1/2 ∬_{-2}² ∬_{-2}² e^(uv) du dv = 2π(e⁴ - 1).

The integral evaluates to 2π(e⁴ - 1) using a linear change of variables with the given transformation, transformed region, Jacobian, and converted integral.

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Answer:

x = 13.8 ft

The triangle is obtuse

Step-by-step explanation:

Using the cosine rule to determine x:

[tex]x=\sqrt{(11.7)^{2}+(7.4)^{2} -2(11.7)(7.4) * cos90 } \\=13.8 ft\\[/tex]

Testing whether or not the Pythagoras theorem applies

[tex]r^{2} =x^{2} +y^{2} \\(13.8)^{2} = (7.4)^{2} +(11.7)^{2} \\190.44\neq 191.65[/tex]

Therefore the triangle is obtuse

Answer:

Step-by-step explanation:

The mean of a discrete uniform distribution is the average of the minimum and maximum values of the distribution.

In this case, X ranges from 12 to 24, so the minimum value is 12 and the maximum value is 24. Therefore, the mean is:

Mean = (12 + 24) / 2 = 18

So the answer is c. 18.0.

CD is the altitude to side AB of right [tex]\triangle[/tex]ABC, where m[tex]\angle[/tex]ACB = [tex]90^o[/tex] The value of BC is 7.28 units.

Value in math is a concept that describes the magnitude, or size, of a number. It can refer to absolute value, which is the actual number, or it can refer to relative value, which is the number compared to other numbers. Value is important in math because it is used to compare and measure different quantities. For example, in addition and subtraction, the value of the numbers being added or subtracted determines the answer. In multiplication, the value of the factors determines the product. Value is also important for performing calculations, such as finding averages, which requires knowledge of numbers and their relative values.
The given triangle is a right triangle, with ∠acb as the right angle. Using the Pythagorean Theorem, we can find the length of the side BC. The Pythagorean Theorem states that the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.
Therefore, BC² = AC² + BD²
Substituting the given values in the equation,
BC² = 52 + (5 1/3)²
Simplifying the equation,
BC² = 25 + 27.69
Therefore, BC² = 52.69
Taking the square root of both sides,
BC = √52.69
Therefore, BC = 7.28 units.
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The polar coordinates are (4√3, 5π/6). The correct answer is D. (4√3, 5π/6). The other given options are incorrect.

To convert Cartesian coordinates (-6, 2√3) to polar coordinates, we use the formulas:

r = √(x^2 + y^2)
θ = tan^-1 (y/x)

Plugging in the values, we get:

r = √((-6)^2 + (2√3)^2) = √(36 + 12) = 2√13
θ = tan^-1 (2√3/-6) = -π/3

However, since the point is in the second quadrant, we need to add π to the angle, giving us:

θ = -π/3 + π = 2π/3

Therefore, the polar coordinates of (-6, 2√3) can be expressed in two ways:

A. (4 √3, 3 π/4)
B. (3 √3, 3 π/4)

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When writing a rational number as a decimal, the decimal may either terminate or repeat indefinitely.

If the decimal repeats, there is a pattern of digits that repeat after a certain point. To indicate the repeating pattern, a bar is placed over the digits that repeat. This bar is typically placed over the smallest repeating pattern, which may be one or more digits.

For example, in the decimal representation of 1/3, the digit 3 repeats indefinitely, so the number is written as 0.333... with a bar over the 3. In the decimal representation of 2/7, the pattern 142857 repeats indefinitely, so the number is written as 0.285714285714... with a bar over the repeating pattern.

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Complete Question:

Writing Rational Numbers as Repeating Decimals. Highlight the number that repeats.

The price at which unitary elasticity occurs is $157.14 per unit.

To find the price at which unitary elasticity occurs, we need to first determine the elasticity of demand with respect to price. The elasticity of demand is defined as the percentage change in quantity demanded divided by the percentage change in price.

Let's rearrange the demand equation to solve for q

q = 86,000 - 550p

We can then take the derivative of q with respect to p:

dq/dp = -550

This tells us that the rate of change of quantity demanded with respect to price is a constant -550. To find the price at which unitary elasticity occurs, we need to find the price where the absolute value of the elasticity is equal to 1.

Using the formula for elasticity of demand

e = (dq/dp) × (p/q)

At unitary elasticity, e = -1, so:

-1 = (dq/dp) × (p/q)

Substituting in the expression for dq/dp and q, we get

-1 = (-550)  (p / (86,000 - 550p))

Simplifying this equation gives

p = $157.14

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The price at which unitary elasticity occurs is $157.14 per unit.

To find the price at which unitary elasticity occurs, we need to first determine the elasticity of demand with respect to price. The elasticity of demand is defined as the percentage change in quantity demanded divided by the percentage change in price.

Let's rearrange the demand equation to solve for q

q = 86,000 - 550p

We can then take the derivative of q with respect to p:

dq/dp = -550

This tells us that the rate of change of quantity demanded with respect to price is a constant -550. To find the price at which unitary elasticity occurs, we need to find the price where the absolute value of the elasticity is equal to 1.

Using the formula for elasticity of demand

e = (dq/dp) × (p/q)

At unitary elasticity, e = -1, so:

-1 = (dq/dp) × (p/q)

Substituting in the expression for dq/dp and q, we get

-1 = (-550)  (p / (86,000 - 550p))

Simplifying this equation gives

p = $157.14

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Answer:

Step-by-step explanation:

1008

The value of the car 20 years after it was purchased is approximately $4,100.

Depreciation refers to the decrease in the value of an asset over time due to wear and tear, obsolescence, or other factors. In the context of a car, depreciation means that its value decreases as it is used and ages.

To calculate the value of the car 20 years after it was purchased, we need to find out how many times the value is halved in 20 years. Since 3.5 years is the time it takes for the value to be halved, we can divide 20 by 3.5 to get the number of times the value is halved.

20 / 3.5 = 5.71 (rounded to two decimal places)

So, the value of the car after 20 years would be:

$29,000 / (2^5.71) = $4,090 (rounded to the nearest hundred dollars)

Therefore, the value of the car 20 years after it was purchased is approximately $4,100.

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The left null space is the null space of the transpose of the matrix, A^T. Since the original matrix is 3x4 and has rank 3, its nullity can be calculated as n - rank = 4 - 3 = 1. Thus, the dimension of the left null space is 1, and it is represented by R^1.

If a 3×4 matrix has rank 3, this means that there are 3 linearly independent columns. Therefore, the column space of the matrix is spanned by these 3 columns. In terms of the matrix itself, we can say that the column space is spanned by the columns corresponding to the pivot positions in the matrix after it has been reduced to row echelon form. So, in this case, we would be looking at the columns corresponding to the 3 pivot positions.

To find the left null space of the matrix, we need to find all vectors x such that A^T x = 0. Since A is a 3×4 matrix, its transpose is a 4×3 matrix. So we are looking for vector x that is in R^4 and satisfies the equation A^T x = 0. The left null space is the set of all such vectors.

To find the left null space, we can use the fact that the left null space is orthogonal to the row space of A. The row space of A is spanned by the rows corresponding to the pivot positions in the matrix after it has been reduced to row echelon form. Since the matrix has rank 3, there are only 3 pivot positions, so the row space has dimension 3.

Therefore, we can find a basis for the left null space by finding a basis for the orthogonal complement of the row space. We can use the Gram-Schmidt process to do this. Start with a basis for the row space, and then orthogonalize it by subtracting the projection onto each previous vector in the basis.

Once we have a basis for the left null space, we can determine its dimension. Since the matrix has 4 columns, the left null space has dimension 4 - rank(A) = 4 - 3 = 1. So the left null space is a one-dimensional subspace of R^4.

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Answer:

B. The length of the arc is 1.5 times longer than the radius.

C. The ratio of arc length to radius is 1.5.

The probability that Niamh chooses B after she had the hint is given as follows:

0.30 = 30%.

A probability is calculated as the division of the desired number of outcomes by the total number of outcomes in the context of a problem/experiment.

The sum of all the probabilities is given as follows:

1 = 100%.

Hence, initially, considering that there are four choices, the probability of each is given as follows:

1/4 = 0.25.

A decays by 0.15, while the increase in each of the other probabilities is given as follows:

0.15/3 = 0.05.

Hence the probability of B is given as follows:

p = 0.25 + 0.05

p = 0.3.

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Answer:

Total Surface Area is 20

Step-by-step explanation:

The formula for surface are with slant heigh is

SA = a^2 + 2×a×l

a = Base Edge (this case 2)

I = Slant Height (this case 4

2^2 + 2(2)(4) = 4+16=20

The probability of spinning a 5 first and then a 1 is:

(1/6) * (1/6) = 1/36

Expressed as a percent, this is:

(1/36) * 100% = 2.78%

So the probability of landing on a 5 and then a 1 is 2.78%

Answer:

32.2

Step-by-step explanation:

Answer:

Step-by-step explanation:

You want the measures of angles A and B in right triangle ABC with hypotenuse AB = 15, and side BC = 8.

The mnemonic SOH CAH TOA reminds you of the relationships between side lengths and trig functions in a right triangle:

  Sin = Opposite/Hypotenuse

  Cos = Adjacent/Hypotenuse

Here, the hypotenuse is given as AB=15. The side opposite angle A is given as BC=8, so we have ...

  sin(A) = 8/15   ⇒   A = arcsin(8/15) ≈ 32°

The side adjacent to angle B is given, so we have ...

  cos(B) = 8/15   ⇒   B = arccos(8/15) ≈ 58°

Of course, angles A and B are complementary, so we can find the other after we know one of them.

  B = 90° -A = 90° -32° = 58°

The measures of the angles are A = 32°, B = 58°.

__

Additional comment

The inverse trig functions can also be called arcsine, arccosine, arctangent, and so on. On a calculator these inverse functions are indicated by a "-1" exponent on the function name—the conventional way an inverse function is indicated when suitable fonts are available.

You will note the calculator is set to DEG mode so the angles are given in degrees.

This text presents information about two exponential functions f and g. Function f passes through the points (-1, 5) and (2, -1.5), and intercepts the x-axis at (1, 0) and the y-axis at (0, 2). Function g is a decreasing exponential function with a y-intercept of 5 and no x-intercept. The text asks to compare the end behavior of these two functions as x approaches negative and positive infinity. End behavior refers to the behavior of the function as x approaches either positive or negative infinity.

$15

$15 because they are asking you to do a subtraction the initial amount used to be $55then you have to substract $40 because they are saying $40 OFF

$15 because they are asking you to do a subtraction the initial amount used to be $55then you have to substract $40 because they are saying $40 OFF55 - 40= 15don't forget to add the $ sign

$15 because they are asking you to do a subtraction the initial amount used to be $55then you have to substract $40 because they are saying $40 OFF55 - 40= 15don't forget to add the $ sign !Hope I helped you

The decision variables for this problem are:

This problem has the following constraints:

Production time cannot exceed the available time on each machine:

5x1,1 + 8x2,1 ≤ 600

4x1,2 + 5x2,2 ≤ 600

Production cannot be negative:

x1,1 ≥ 0

x1,2 ≥ 0

x2,1 ≥ 0

x2,2 ≥ 0

Demand must be met for each product:

x1,1 + x1,2 ≥ 120

x2,1 + x2,2 ≥ 150

Demand cannot exceed the maximum demand for each product:

x1,1 + x1,2 ≤ 200

x2,1 + x2,2 ≤ 130

Therefore, this problem has 4 constraints.

The decision variables for this problem are x1,1 (the number of units of product 1 produced on machine 1), x1,2 (the number of units of product 1 produced on machine 2), x2,1 (the number of units of product 2 produced on machine 1), and x2,2 (the number of units of product 2 produced on machine 2).

Therefore, this problem has 4 decision variables.

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Let x be a random variable with pdf f(x) = 1/13, 21 P(X > 30) = 0.31.

We are given that X is a random variable with a probability density function (pdf) of f(x) = 1/13 for the interval 21  x  34.

We are asked to find P(X > 30), which means we need to find the probability of the random variable X being greater than 30. To do this, we will calculate the area under the PDF in the interval [30, 34].

Step 1: Determine the width of the interval [30, 34].
Width = 34 - 30 = 4

Step 2: Calculate the area under the PDF in the interval [30, 34].
Since the pdf is a constant value (1/13) within the given interval, we can calculate the area as follows:
Area = f(x) * width
Area = (1/13) * 4

Step 3: Round off the result to the second decimal place.
Area ≈ 0.31 (rounded to two decimal places)

So, P(X > 30) ≈ 0.31.

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Answer: The given statement "if every column of an augmented matrix contains a pivot then the corresponding system is consistent" is true. This is because when every column of an augmented matrix contains a pivot, it implies that there are no free variables in the system of equations represented by the matrix.

Step-by-step explanation: Since every variable has a pivot in the augmented matrix, there is a unique solution to the system of equations. This is the definition of a consistent system - one that has at least one solution.                                                                                                                  In summary, the statement is true because the presence of a pivot in every column of an augmented matrix guarantees a unique solution to the system of equations, which is the definition of a consistent system.

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None of the options A, B, C, or D are the necessary step to verify InB + 2 = -2t as a solution to dB/dt = -2B.

what is differential equations?

Differential equations are mathematical equations that describe the relationship between an unknown function and its derivatives (or differentials).

To verify that InB + 2 = -2t is a solution to dB/dt = -2B, we can substitute InB + 2 for B in the differential equation and check if it satisfies the equation.

So, let's first differentiate InB + 2 with respect to t:

d/dt (InB + 2) = 1/B * dB/dt

Using the given differential equation, we can substitute dB/dt with -2B:

d/dt (InB + 2) = 1/B * (-2B)

Simplifying this expression, we get:

d/dt (InB + 2) = -2

Now, substituting InB + 2 for B in the original differential equation, we get:

dB/dt = -2(InB + 2)

We can differentiate this expression with respect to B to get:

d/dB (dB/dt) = d/dB (-2(InB + 2))

d²B/dt² = -2/B

Since we have already established that d/dt (InB + 2) = -2, we can differentiate this expression with respect to t to get:

d²B/dt² = d/dt (-2) = 0

Therefore, d²B/dt² = -2/B if and only if d/dt (InB + 2) = -2.

Now, let's check if the given solution satisfies this condition. Substituting InB + 2 = -2t in d/dt (InB + 2), we get:

d/dt (InB + 2) = d/dt (In(-2t) + 2) = -2/t

Since -2/t is not equal to -2, the given solution does not satisfy the differential equation dB/dt = -2B, and hence, we cannot verify it as a solution.

Therefore, none of the options A, B, C, or D are the necessary step to verify InB + 2 = -2t as a solution to dB/dt = -2B.

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The current date using the TODAY function, you can enter the formula "=TODAY()" in cell H5. This will display the current date in the cell.

Switch to the Cost Estimates worksheet. In cell A9, create a formula using the AVERAGE function that calculates the average of the values in the range A5:A7, then copy your formula to cell 09.

cell A10, create a formula using the MAX function that identifies the maximum value in the range A5:A7 and then copy your formula to cell D10. In cell A11, create a formula using the MIN function that identifies the minimum value in the range A5:A7 and then copy your formula to cell 011.

In cell B13, create a formula using the VLOOKUP function that looks up the value from cell A11 in the range A5:B7, returns the value in column 2, and specifies an exact match. Copy the formula to cell E13. Switch to the Profit Projections worksheet. In cell H5, use the TODAY function to insert the current date.

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From the given choices, only P[{clubs}) = 0.7, P{{diamonds}) = 0.3 satisfies the requirements of the definition of probability.

The choice that fulfills the requirements of the definition of probability is:

P[{clubs}) = 0.7, P{{diamonds}) = 0.3.

For an event A in a sample space S, the probability of A, denoted by P(A), must satisfy the following conditions:

P(A) is a non-negative real number: This means that the probability of an event cannot be negative.

P(S) = 1: The probability of the sample space is always equal to 1. This implies that at least one of the events in the sample space must occur.

If A and B are two mutually exclusive events, then P(A or B) = P(A) + P(B): This means that the probability of either event occurring is equal to the sum of their individual probabilities.

In the given sample space S = {clubs, diamonds}, the probabilities of the two events must add up to 1, since there are only two possible outcomes.

Therefore, the probabilities of the events cannot be negative or greater than 1.

From the given choices, only P[{clubs}) = 0.7, P{{diamonds}) = 0.3 satisfies the requirements of the definition of probability.

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The exact values of P(X = k) for k = 0, 1, 2, ..., 16 are:

P(X = 0) = 1/65536

P(X = 1) = 1/4096

P(X = 2) = 15/8192

P(X = 3) = 455/65536

P(X = 4) = 3003/262144

P(X = 5) = 1001/65536

P(X = 6) = 2002/65536

P(X = 7) = 1716/65536

P(X = 8) = 6435/262144

P(X = 9) = 5005/262144

P(X = 10) = 3003/262144

P(X = 11) = 455/65536

P(X = 12) = 1001/65536

P(X = 13) = 15/8192

P(X = 14) = 1/4096

P(X = 15) = 1/65536

P(X = 16) = 1/65536

The number of possible outcomes when rolling a fair die once is 6, with 3 odd numbers (1, 3, and 5) and 3 even numbers (2, 4, and 6). Therefore, the probability of rolling an odd number is 3/6 = 1/2 and the probability of rolling an even number is also 1/2.

The number of odd numbers that appear in 16 tosses of a fair die is a binomial random variable with parameters n = 16 and p = 1/2. The probability mass function of X, the number of odd numbers, is given by:

P(X = k) = (16 choose k) [tex]*[/tex] (1/2)¹⁶, for k = 0, 1, 2, ..., 16.

To find the exact value of P(X = k), we need to substitute k into this formula and evaluate it. For example:

P(X = 0) = (16 choose 0) [tex]*[/tex] (1/2)¹⁶ = 1/65536

P(X = 1) = (16 choose 1) [tex]*[/tex] (1/2)¹⁶ = 16/65536 = 1/4096

P(X = 2) = (16 choose 2) [tex]*[/tex] (1/2)¹⁶ = 120/65536 = 15/8192

and so on, until

P(X = 16) = (16 choose 16) [tex]*[/tex] (1/2)¹⁶ = 1/65536

Therefore, the exact values of P(X = k) for k = 0, 1, 2, ..., 16 are:

P(X = 0) = 1/65536

P(X = 1) = 1/4096

P(X = 2) = 15/8192

P(X = 3) = 455/65536

P(X = 4) = 3003/262144

P(X = 5) = 1001/65536

P(X = 6) = 2002/65536

P(X = 7) = 1716/65536

P(X = 8) = 6435/262144

P(X = 9) = 5005/262144

P(X = 10) = 3003/262144

P(X = 11) = 455/65536

P(X = 12) = 1001/65536

P(X = 13) = 15/8192

P(X = 14) = 1/4096

P(X = 15) = 1/65536

P(X = 16) = 1/65536

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Answer:

X° = 72.6459

Step-by-step explanation:

To solve x we must use tan b/c it contain both side,

which is opposite and adjecent

tan ( x°) =16/5

tan ( x°) =16/5tan ( x°) = 3.2

tan ( x°) =16/5tan ( x°) = 3.2X °= tan^-1(3.2)

tan ( x°) =16/5tan ( x°) = 3.2X °= tan^-1(3.2)X° = 72.6459 round to 72.65°

This formula will give you the correct t statistic for comparing the means of two independent samples without assuming equal variances

To calculate the t statistic for the difference between means of independent samples without pooling, you can use the following formula:

t = (xbar1 - xbar2) / √[(s₁² / n₁) + (s₂² / n₂)]

Here,
- xbar1 and xbar2 are the sample means of the two groups,
- s₁² and s₂² are the sample variances for the two groups,
- n₁ and n₂ are the sample sizes for the two groups.

This formula will give you the correct t statistic for comparing the means of two independent samples without assuming equal variances (i.e., without pooling).

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The function g(t) = 3t⁵ + 40t⁴ + 150t³ + 120 is concave up on the interval (-∞, -2) and concave down on the interval (-2, ∞). There is an inflection point at t = -2.

1. Find the first derivative, g'(t) = 15t⁴ + 160t³ + 450t².
2. Find the second derivative, g''(t) = 60t³ + 480t² + 900t.
3. Factor out the common term, g''(t) = 60t(t² + 8t + 15).
4. Solve g''(t) = 0 to find critical points. In this case, t = 0 and t = -2.
5. Test the intervals to determine the concavity: For t < -2, g''(t) > 0, so it's concave up. For t > -2, g''(t) < 0, so it's concave down.
6. Since the concavity changes at t = -2, there is an inflection point at t = -2.

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Yes, the mysteries equal 0.06 of the total books.

Marcella said that the mysteries equal 0.06 of the total books.

To check the mysteries equal 0.06 of the total books is correct or not.

We can follow these steps:

1. Identify the total number of books and the number of mysteries: Marcella read 100 books, and 60 of them were mysteries.

2. Calculate the fraction of mysteries: Divide the number of mysteries (60) by the total number of books (100) to find the fraction of mysteries.

3. Compare the fraction with Marcella's claim: If the calculated fraction equals 0.06, then she is correct.

Now let's perform the calculations:

60 mysteries ÷ 100 total books = 0.6

Since 0.6 ≠ 0.06, Marcella's claim that the mysteries equal 0.06 of the total books is incorrect. In reality, mysteries make up 0.6 or 60% of the total books she read.

A model to support this answer could be a pie chart, where the circle represents the 100 books, and the mysteries portion is shaded in. By dividing the circle into 10 equal sections, the mysteries would fill 6 of those sections, which represents 60% of the total books.

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Answer:8

Step-by-step explanation:5 is 1st term 8 is second term 11 is third term…