if $m$ workers can complete a job in $d$ days, how many days will it take $n$ workers, working at the same rate, to complete one-third of the job? express your answer as a common fraction in terms of $d$, $m$ and $n$, in alphabetical order where applicable.

The function rule for a dilation centered at the origin with a scale factor of -k can be written as: f(x) = -kx

A dilation refers to the transformation in which the size of an object changes but the shape remains the same.

In other words, a dilation is a transformation that changes the size of an object.

The scale factor of a dilation is the factor by which the size of the object is changed.

If the scale factor is negative, then the object is not only scaled but also flipped about the origin.

The function rule for a dilation centered at the origin with a scale factor of -k is given by the formula, f(x) = -kx.

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The probability of having 4 successes in 5 trials, where the probability of success on each trial is 0.2, can be calculated using the binomial probability formula. The main answer is that the probability is approximately 0.0262.

To explain further, let's break down the calculation. The binomial probability formula is P(x) = C(n, x) * p^x * (1-p)^(n-x), where P(x) represents the probability of having x successes in n trials, C(n, x) is the number of combinations of n items taken x at a time, p is the probability of success on each trial, and (1-p) is the probability of failure on each trial.

In this case, x = 4, n = 5, and p = 0.2. Plugging these values into the formula, we get P(4) = C(5, 4) * 0.2^4 * (1-0.2)^(5-4). Calculating further, C(5, 4) = 5 (since there are 5 ways to choose 4 items out of 5), 0.2^4 = 0.0016, and (1-0.2)^(5-4) = 0.8^1 = 0.8. Multiplying these values, we find P(4) = 5 * 0.0016 * 0.8 = 0.0064.

Therefore, the probability of having 4 successes in 5 trials with a success probability of 0.2 is approximately 0.0064 or 0.64%.

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Answer:

see attachment

Step-by-step explanation:

The expressions would be as follows:

1.5 x regular pay rate = 1.5R (where R represents the regular pay rate)

2 x regular pay rate = 2R (where R represents the regular pay rate)

To calculate the values, we can multiply the regular pay rate by the given multipliers:

1.5 x regular pay rate = 1.5 * regular pay rate

2 x regular pay rate = 2 * regular pay rate

Since the regular pay rate is not specified, we can represent it as "R" for simplicity.

1.5 x regular pay rate = 1.5R

2 x regular pay rate = 2R

So, the expressions would be as follows:

1.5 x regular pay rate = 1.5R (where R represents the regular pay rate)

2 x regular pay rate = 2R (where R represents the regular pay rate)

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There are 37 ways in which the seats on the carousel can be randomly filled by 9 people, considering both scenarios of people sitting on the bench seat or not.

To determine the number of ways the seats on the carousel can be randomly filled by 9 people, we need to consider the different combinations of people sitting on the horses and the bench seat.

There are two scenarios to consider:

Scenario 1: Two people sitting on the bench seat:

In this case, we need to select 2 people out of the 9 to occupy the bench seat, which can be done in C(9, 2) = 36 ways. The remaining 7 people will occupy the horse seats.

Scenario 2: No one sitting on the bench seat:

Here, all 9 people will occupy the horse seats, and the bench seat remains empty.

To get the total number of ways, we sum up the possibilities from both scenarios:

Total number of ways = Number of ways in Scenario 1 + Number of ways in Scenario 2

                   = 36 + 1

                   = 37

Therefore, there are 37 ways in which the seats on the carousel can be randomly filled by 9 people.

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There are 120 possible outcomes for the Junior Student Council elections.

To find the number of possible outcomes for the given situation, we need to multiply the number of options for each position.

Number of options for secretary = 3

Number of options for treasurer = 4

Number of options for vice president = 5

Number of options for class president = 2

To find the total number of possible outcomes, we multiply the number of options for each position:

Total number of possible outcomes = (Number of options for secretary) x (Number of options for treasurer) x (Number of options for vice president) x (Number of options for class president)

Total number of possible outcomes = 3 x 4 x 5 x 2

                                    = 120

Therefore, there are 120 possible outcomes for the Junior Student Council elections.

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The exact value of sin(θ/2) is ±(3/√10).

To find the exact value of sin(θ/2), we can use the half-angle formula for sine:

sin(θ/2) = ±√[(1 - cosθ) / 2]

Given that cosθ = -4/5 and 90° < θ < 180°, we can determine the value of sin(θ/2) using the half-angle formula.

First, let's find sin(θ) using the Pythagorean identity:

sinθ = ±√(1 - cos²θ)

sinθ = ±√(1 - (-4/5)²)

= ±√(1 - 16/25)

= ±√(9/25)

= ±3/5

Since 90° < θ < 180°, we know that sinθ < 0. Therefore, sinθ = -3/5.

Now we can substitute this value into the half-angle formula:

sin(θ/2) = ±√[(1 - cosθ) / 2]

= ±√[(1 - (-4/5)) / 2]

= ±√[(1 + 4/5) / 2]

= ±√[(9/5) / 2]

= ±√(9/10)

= ±(3/√10)

Thus, the exact value of sin(θ/2) is ±(3/√10).

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The valid conclusion that can be drawn from the given statements is "∠2 ≅ ∠1". This conclusion was drawn using the Law of Detachment.

The first statement is a universal conditional statement, which means that it is true for all vertical angles. The second statement is a particular statement, which means that it is true for a specific pair of angles, ∠1 and ∠2.

The Law of Detachment states that if a universal conditional statement is true and the hypothesis of that statement is also true, then the conclusion of that statement must also be true. In this case, the universal conditional statement is "Vertical angles are congruent" and the hypothesis is "∠1 ≅ ∠2". Since the universal conditional statement is true and the hypothesis is true, the conclusion "∠2 ≅ ∠1" must also be true.

Therefore, the valid conclusion that can be drawn from the given statements is "∠2 ≅ ∠1". This conclusion was drawn using the Law of Detachment.

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Perimeter:

2 units -> 12 units

4 units -> 24 units

8 units -> 48 units

Area:

2 units -> 10.39 square units

4 units -> 41.57 square units

8 units -> 166.28 square units

Analyzing the patterns in the table, we can observe the following:

Perimeter: When the side length of the regular hexagon is doubled, the perimeter also doubles. For example, when the side length is 2 units, the perimeter is 12 units. When it is doubled to 4 units, the perimeter becomes 24 units. This doubling pattern continues when the side length is doubled to 8 units, resulting in a perimeter of 48 units. Therefore, we can conjecture that doubling the side length of a regular hexagon doubles its perimeter.

Area: When the side length of the regular hexagon is doubled, the area is quadrupled. For instance, when the side length is 2 units, the area is approximately 10.39 square units. When the side length is doubled to 4 units, the area becomes approximately 41.57 square units, which is four times the initial area. Similarly, when the side length is doubled again to 8 units, the area becomes approximately 166.28 square units, which is again four times the previous area. Hence, we can conjecture that doubling the side length of a regular hexagon results in its area being multiplied by four.

These patterns can be explained by considering the properties of regular polygons. In a regular hexagon, all sides are congruent, and the perimeter is the sum of all the side lengths. Therefore, when each side length is doubled, the perimeter doubles as well. Regarding the area, a regular hexagon can be divided into six congruent equilateral triangles. The area of an equilateral triangle is proportional to the square of its side length. When the side length is doubled, the area of each equilateral triangle is quadrupled, resulting in the overall area of the hexagon being multiplied by four.

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c. One of the two input terminals of a single-ended amplifier is connected to ground.

e. Neither input terminal of a differential amplifier is connected to ground, and the differential amplifier responds to the difference between the voltages applied to its input terminals.

In a single-ended amplifier, one of the input terminals is typically connected to a reference point, such as ground. This means that the input signal is referenced to the ground potential, and the amplifier amplifies the signal relative to this reference.

In a differential amplifier, neither of the input terminals is connected to ground. Instead, the differential amplifier measures the voltage difference between the two input terminals.

The amplifier amplifies this voltage difference, while rejecting any common-mode signals that are present on both input terminals.

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How is a single-ended amplifier different from a differential amplifier? Select two correct definitions. Check all that apply.

a. Two of the three input terminals of a differential amplifier is connected to ground.

b. Neither input terminal of a single-ended amplifier is connected to ground, and the single-ended amplifier responds to the difference between the voltages applied to its input terminals.

c. One of the two input terminals of a single-ended amplifier is connected to ground.

d. One of the two input terminals of a differential amplifier is connected to ground.

e. Neither input terminal of a differential amplifier is connected to ground, and the differential amplifier responds to the difference between the voltages applied to its input terminals. Submit Request Answer

To find an equation of the line containing the point (6, -3) and parallel to the line 3x - 5y = 8, the equation of the line is y = ____.

To determine the equation of the line parallel to 3x - 5y = 8, we first need to find the slope of the given line. The given line is in standard form, so we can rewrite it in slope-intercept form (y = mx + b) by solving for y.

Starting with 3x - 5y = 8:

-5y = -3x + 8

Dividing both sides by -5 gives us:

y = (3/5)x - 8/5

The slope of the given line is 3/5. Since parallel lines have the same slope, the parallel line we seek will also have a slope of 3/5.

Now that we have the slope, we can use the point-slope form of a line to write the equation of the parallel line. The point-slope form is:

y - y1 = m(x - x1)

Using the given point (6, -3), we substitute the coordinates and the slope into the point-slope form:

y - (-3) = (3/5)(x - 6)

Simplifying, we have:

y + 3 = (3/5)(x - 6)

Finally, we can convert the equation to slope-intercept form by expanding and simplifying:

y + 3 = (3/5)x - 18/5

y = (3/5)x - 18/5 - 3

y = (3/5)x - 33/5

Therefore, the equation of the line containing the point (6, -3) and parallel to the line 3x - 5y = 8 is y = (3/5)x - 33/5.

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After evaluate the expression for the given values.

2y + 3x, if y = 3 and x = -1, we get the final answer is 3.

To evaluate the expression for the given values.

2y + 3x, if y = 3 and x = -1.

Plugging these values in given equation:

2 * (3) + 3 * (-1)

6 - 3

3.

Therefore,  the expression for the given values. 2y + 3x, if y = 3 and x = -1 is 3.

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Answer:

x = 5(5/15) = 5.333

Step-by-step explanation:

13x +2x +10 = 90

15x 10 = 90

15x = 90 -10

15x = 80

x = 80/15

x = 5(5/15)

Chris's research costs can be calculated using the percentage increase in his return and the desired hourly wage. The cost of his research will be $375.45. As a result of the research, Chris's return will increase by $257.96. On a strict economic basis, Chris should perform the proposed research as the increase in return outweighs the cost.

To calculate the cost of Chris's research, we need to determine the amount of time he devotes to it. Let's assume he spends x hours on research. The cost of his research can be calculated by multiplying his desired hourly wage ($31.00) by the number of hours spent:

Cost of research = $31.00 × x

Now, to find x, we need to consider the percentage increase in Chris's return. The percentage increase is given as a range from 6.1% to 8.7%. Let's take the average of these percentages, which is (6.1% + 8.7%) / 2 = 7.4%. This means Chris's return will increase by 7.4% as a result of the research.

To find x, we can set up the following equation:

1.074 × initial return = final return

Simplifying the equation, we have:

initial return = final return / 1.074

Since the initial return is given as a percentage, we can express it as 100% (or 1 in decimal form). The final return can be expressed as 100% + 7.4% = 107.4% (or 1.074 in decimal form).

So, the equation becomes:

1 = 1.074 / initial return

Solving for initial return, we find:

initial return = 1.074

Now, we can substitute the initial return into the equation for cost of research:

Cost of research = $31.00 × x = $31.00 × (initial return - 1) = $31.00 × (1.074 - 1) = $31.00 × 0.074 = $2.294

Rounding this to the nearest cent, the cost of Chris's research is approximately $2.29.

Next, to find the increase in Chris's return as a result of the research, we can calculate:

Increase in return = final return - initial return = 1.074 - 1 = 0.074

Finally, we can calculate the increase in dollars:

Increase in dollars = Increase in return × initial return = $31.00 × 0.074 ≈ $2.30

Rounding this to the nearest cent, Chris's return will increase by approximately $2.30.

On a strict economic basis, Chris should perform the proposed research. The cost of research is $2.29, while the increase in return is $2.30. Therefore, the increase in return outweighs the cost, resulting in a positive net benefit.

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The number of solutions for the system of equations is 2.

Given data:

To determine the number of solutions for the system of equations:

y = -(1/4)x² - 2x

y = x² + 3/4

The points of intersection between the two equations on the coordinate plane is determined as:

Setting the right sides of the equations equal to each other,

-(1/4)x² - 2x = x² + 3/4

To simplify the equation, multiply both sides by 4 to eliminate the fractions:

-1x² - 8x = 4x² + 3

Rearranging terms,

5x² + 8x + 3 = 0

This is a quadratic equation in standard form. To solve it, factor it or use the quadratic formula. Let's use factoring:

(5x + 3)(x + 1) = 0

Setting each factor equal to zero,

5x + 3 = 0 or x + 1 = 0

Solving for x,

x = -3/5 or x = -1

Now , substitute these values of x back into either of the original equations to find the corresponding y-values.

For x = -3/5:

y = -(1/4)(-3/5)² - 2(-3/5)

= -(1/4)(9/25) + 6/5

= -9/100 + 120/100

= 111/100

For x = -1:

y = (-1)² + 3/4

= 1 + 3/4

= 7/4

Hence, the system has two solutions: (-3/5, 111/100) and (-1, 7/4).

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Answer:

B = -331

4 x -331 = -1324

For the value θ = -105°, the rounded values of cos θ, sin θ, and tan θ are approximately 0.26, -0.97, and 3.73, respectively.

To find the values of cos θ, sin θ, and tan θ for θ = -105°, we use a calculator and round the answers to the nearest hundredth.

cos (-105°) ≈ 0.26

The cosine function gives the ratio of the adjacent side to the hypotenuse in a right triangle with angle θ. For θ = -105°, we find the corresponding angle in the unit circle and determine the cosine value, which is approximately 0.26.

sin (-105°) ≈ -0.97

The sine function gives the ratio of the opposite side to the hypotenuse in a right triangle with angle θ. For θ = -105°, we find the corresponding angle in the unit circle and determine the sine value, which is approximately -0.97.

tan (-105°) ≈ 3.73

The tangent function gives the ratio of the opposite side to the adjacent side in a right triangle with angle θ. For θ = -105°, we find the corresponding angle in the unit circle and determine the tangent value, which is approximately 3.73.

Therefore, for θ = -105°, the rounded values of cos θ, sin θ, and tan θ are approximately 0.26, -0.97, and 3.73, respectively.

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Equation: 3x² - x + 3 = 0. The discriminant is -35, indicating no real solutions, only two imaginary solutions.

Let's calculate the discriminant of the equation 3x² - x + 3 = 0.

The discriminant (Δ) is given by the formula Δ = b² - 4ac, where a, b, and c are the coefficients of the quadratic equation.

For our equation 3x² - x + 3 = 0, we have:
a = 3
b = -1
c = 3

Substituting these values into the discriminant formula, we get:
Δ = (-1)² - 4(3)(3)
  = 1 - 36
  = -35

The discriminant of the equation is -35. Since the discriminant is negative, it indicates that the equation has no real solutions.

Instead, it has two complex solutions because a negative discriminant implies that the roots will be imaginary.

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The expression 21/√3, after rationalizing the denominator, simplifies to (21 x √3) / 3.

Given that a fraction 21 / √3 we need to rationalize,

To rationalize the denominator of the expression 21/√3, we need to eliminate the square root in the denominator.

We can do this by multiplying both the numerator and denominator by the conjugate of √3, which is also √3.

Let's perform the multiplication:

(21/√3) x (√3/√3)

Multiplying the numerators and the denominators separately, we get:

(21 x √3) / (√3 x √3)

Simplifying further, we have:

(21 x √3) / 3

So, the expression 21/√3, after rationalizing the denominator, simplifies to (21 x √3) / 3.

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The values of x, y, and z that make the equation true are : x = 1/3

y = 23/3  z =5

To determine the values of x, y, and z that make the given equation true, we can set the corresponding elements on both sides of the equation equal to each other.

From the given equation:

z - 3 = 2

3x = 1

0 = 8

-10 - 4x + 2y = 4

-4 = y + 12

From the first equation, we find:

z = 2 + 3

z = 5

From the second equation, we have:

3x = 1

x = 1/3

From the third equation, we see that:

0 = 8

This equation is not true for any value of x, y, or z. Therefore, there is no solution.

From the fourth equation, we find:

-10 - 4x + 2y = 4

-4x + 2y = 14

Substituting the value of x we found earlier:

-4(1/3) + 2y = 14

-4/3 + 2y = 14

2y = 14 + 4/3

2y = 42/3 + 4/3

2y = 46/3

y = (46/3) / 2

y = 23/3

Therefore, the values of x, y, and z that make the equation true are:

x = 1/3

y = 23/3

z = 5

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To complete the task, you need to collect the diameter and circumference measurements of ten round objects using a millimeter measuring tape.

Then, you can calculate the value of C/d (circumference divided by diameter) for each object and record the result. Here's a step-by-step guide:

Gather ten round objects of different sizes for measurement.

Use a millimeter measuring tape to measure the diameter of each object. Place the measuring tape across the widest point of the object and record the measurement in millimeters (mm).

Next, measure the circumference of each object using the millimeter measuring tape. Wrap the tape around the outer edge of the object, making sure it forms a complete circle, and record the measurement in millimeters (mm).

For each object, divide the circumference (C) by the diameter (d) to calculate the value of C/d.

C/d = Circumference / Diameter

Round the result of C/d to the nearest hundredth (two decimal places) for each object and record the value.

Repeat steps 2-5 for the remaining nine objects.

Once you have measured and calculated C/d for all ten objects, record the results for each object.

Remember to use consistent units (millimeters) throughout the measurements to ensure accurate calculations.

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The capital asset pricing model provides a risk-return trade-off in which risk is measured in terms of beta.

The capital asset pricing model (CAPM) is a financial model that establishes a relationship between the expected return of an investment and its systematic risk. According to CAPM, the expected return of an asset is determined by the risk-free rate of return, the market risk premium, and the asset's beta. Beta is a measure of systematic risk and represents the asset's sensitivity to market volatility.

The main idea behind CAPM is that investors should be compensated for taking on additional risk. The model suggests that the expected return of an asset increases as its beta, or systematic risk, increases. This means that assets with higher betas are expected to provide higher returns to compensate for the additional risk they carry. On the other hand, assets with lower betas are expected to have lower returns as they are less sensitive to market volatility.

By incorporating beta as a measure of risk, CAPM provides a risk-return trade-off where investors can evaluate the expected return of an investment based on its level of systematic risk. This allows investors to make informed decisions by considering the balance between risk and potential reward.

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The expression [tex]\(\left(\frac{1}{32}\right)^x \cdot \left(\frac{1}{2}\right)^{9x-5}\)[/tex]can be rewritten as[tex]\(\left(\frac{1}{2}\right)^{f(x)}\) where \(f(x) = 14x\).[/tex]

To rewrite the expression \(\left(\frac{1}{32}\right)^x \cdot \left(\frac{1}{2}\right)^{9x-5}\) as \(\left(\frac{1}{2}\right)^{f(x)}\), we need to determine the value of \(f(x)\) in terms of \(x\) that corresponds to the given expression.

Let's break down the given expression and find the relationship between \(f(x)\) and \(x\):

1. \(\left(\frac{1}{32}\right)^x\)

  This term can be rewritten as \(\left(\frac{1}{2^5}\right)^x\) since 32 is equal to \(2^5\).

  Using the property of exponents, we have \(\left(\frac{1}{2}\right)^{5x}\).

2. \(\left(\frac{1}{2}\right)^{9x-5}\)

  This term can be rewritten as \(\left(\frac{1}{2}\right)^{9x} \cdot \left(\frac{1}{2}\right)^{-5}\).

  Simplifying \(\left(\frac{1}{2}\right)^{-5}\), we get \(\left(\frac{1}{2^5}\right)^{-1}\), which is equal to \(2^5\).

  Therefore, \(\left(\frac{1}{2}\right)^{-5} = 2^5\).

  Substituting this back into the expression, we have \(\left(\frac{1}{2}\right)^{9x} \cdot 2^5\).

Now, let's combine the simplified terms:

\(\left(\frac{1}{2}\right)^{5x} \cdot \left(\frac{1}{2}\right)^{9x} \cdot 2^5\)

Using the laws of exponents, we can add the exponents when multiplying powers with the same base:

\(\left(\frac{1}{2}\right)^{5x + 9x} \cdot 2^5\)

Simplifying the exponent, we get:

\(\left(\frac{1}{2}\right)^{14x} \cdot 2^5\)

Finally, we can rewrite this expression as:

\(\left(\frac{1}{2}\right)^{f(x)}\)

where \(f(x) = 14x\) and the overall expression becomes \(\left(\frac{1}{2}\right)^{f(x)} \cdot 2^5\).

In summary, the expression \(\left(\frac{1}{32}\right)^x \cdot \left(\frac{1}{2}\right)^{9x-5}\) can be rewritten as \(\left(\frac{1}{2}\right)^{f(x)}\) where \(f(x) = 14x\).

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The answer is:

[tex]\sf{-6x(10x^3+9)}[/tex]

Work/explanation:

To factor an expression completely, we find its GCF, and factor it out.

Let's do it with the expression we have here: [tex]\sf{-60x^4+54x}[/tex].

I begin by finding the GCF. In this case, the GCF is 6x.

Next, I divide each term by -6x:

[tex]\sf{-60x^4\div-6x=\bf{10x^3}[/tex]

[tex]\sf{54x\div-6x=9}[/tex]

I end up with:

[tex]\sf{-6x(10x^3+9)}[/tex]

Hence, the factored expression is [tex]\sf{-6x(10x^3+9)}[/tex].

The smallest angle in degrees will be B 22.62 degrees

The ratio of the three sides of the triangle is 5:12:13.

It is clear that it is a right-angled triangle, since

[tex]5^{2}+12^{2}=25+144=169=13^{2}[/tex]

Hence one of the angles is 90°.

∴The sum of the other two angles=180°-90°=90°

∴ The smallest angle of the triangle ≤ 90°

Now, the angles can be given by [tex]tan^{-1}\frac{5}{12}[/tex] and [tex]tan^{-1}\frac{12}{5}[/tex].

Now, we know that in its domain the inverse of tan(x) is an increasing function. Hence the smallest angle will be given by [tex]tan^{-1}\frac{5}{12}[/tex].

Calculating the value we get [tex]tan^{-1}\frac{5}{12}[/tex]=22.62° approximately.

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Here is a sequence that is not an arithmetic sequence:

1, 4, 5, 8, 10

The explicit formula for this sequence is 2^n - 1, where n is the term number. The recursive formula is a_n = 2a_{n-1} - a_{n-2}.

Here is an explanation of the explicit formula:

The first term of the sequence is 1, which is just 2^0 - 1. The second term is 4, which is 2^1 - 1. The third term is 5, which is 2^2 - 1. The fourth term is 8, which is 2^3 - 1. The fifth term is 10, which is 2^4 - 1.

Here is an explanation of the recursive formula:

The first two terms of the sequence are 1 and 4. The third term is 5, which is equal to 2 * 4 - 1. The fourth term is 8, which is equal to 2 * 5 - 4. The fifth term is 10, which is equal to 2 * 8 - 5.

As you can see, the recursive formula generates the terms of the sequence by multiplying the previous term by 2 and then subtracting the previous-previous term. This produces a sequence that is not an arithmetic sequence, because the difference between consecutive terms is not constant.

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Answer:

Step-by-step explanation:

The strategy of assigning each goalie a number and using a spinner to choose the goalie for each game can result in a fair decision if the spinner is unbiased and all goalies have an equal chance of being selected for each game. Let's analyze the factors involved:

1. Equal talent: If the three goalies are indeed equally talented, then assigning each of them a number and using a spinner gives them an equal opportunity to play in each game. This aspect ensures fairness in terms of distributing playing time among the goalies.

2. Uninjured and eligible: Assuming all goalies are uninjured and eligible to play the entire season, there are no external factors that could impact the fairness of the decision-making process. As long as the goalies remain healthy and meet the eligibility criteria, the strategy remains fair.

3. Spinner bias: The fairness of the decision depends on the spinner being unbiased. If the spinner is properly constructed and evenly balanced, each goalie has an equal chance of being selected for each game. It's crucial to ensure that the spinner is not rigged or biased towards any particular goalie.

Overall, if all three goalies are equally talented, the spinner is unbiased, and there are no external factors influencing the decision, the strategy of using a spinner to choose the goalie for each game can result in a fair decision. However, it is important to regularly check and maintain the fairness of the spinner to ensure the integrity of the decision-making process.

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She needs 7 mice for 5 snakes.

To solve this, we can use the following equation:

number of mice = number of snakes + 2

If there are 5 snakes, then she needs 5 + 2 = 7 mice.

The explanation is as follows:

* She needs one mouse for each snake.

* She also needs two extra mice.

* Therefore, she needs a total of 1 + 2 = 3 mice for each snake.

* If there are 5 snakes, then she needs 5 * 3 = 15 mice.

* However, we need to round up to the nearest tenth, so she needs 7 mice.

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The solutions to the system of equation 9x² + 24x + 16 = 36 are x = 2/3 and x = -10/3.

To solve the equation 9x² + 24x + 16 = 36, we need to simplify the equation and find the values of x that satisfy it. After simplification, the equation can be rewritten as 9x² + 24x - 20 = 0.

To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. Factoring may not be straightforward in this case, so we can resort to the quadratic formula:

The quadratic formula states that for an equation in the form ax² + bx + c = 0, the solutions for x can be found using the formula:

x = (-b ± √(b² - 4ac)) / (2a)

Applying this formula to our equation, where a = 9, b = 24, and c = -20, we get:

x = (-24 ± √(24² - 4 * 9 * -20)) / (2 * 9)

Simplifying further:

x = (-24 ± √(576 + 720)) / 18

x = (-24 ± √1296) / 18

Since the square root of 1296 is 36, we have:

x = (-24 ± 36) / 18

This gives us two possible solutions:

x₁ = (-24 + 36) / 18 = 12 / 18 = 2 / 3

x₂ = (-24 - 36) / 18 = -60 / 18 = -10 / 3

Therefore, the solutions to the system of equation 9x² + 24x + 16 = 36 are x = 2/3 and x = -10/3.

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The equation of the plane that passes through the points (1, 1, 1), (2, 0, 3), and (-1, 4, 2) is -x + 2y + z - 2 = 0.

To find the equation of the plane that passes through the given points, we can use the formula for the equation of a plane:

Ax + By + Cz + D = 0

We can substitute the coordinates of the points into this equation to form a system of equations. Let's label the points as follows:

Point 1: (x1, y1, z1) = (1, 1, 1)

Point 2: (x2, y2, z2) = (2, 0, 3)

Point 3: (x3, y3, z3) = (-1, 4, 2)

Substituting these values into the equation, we get:

A(1) + B(1) + C(1) + D = 0 ...(1)

A(2) + B(0) + C(3) + D = 0 ...(2)

A(-1) + B(4) + C(2) + D = 0 ...(3)

Simplifying these equations, we have:

A + B + C + D = 0 ...(1)

2A + 3C + D = 0 ...(2)

-A + 4B + 2C + D = 0 ...(3)

Now, we can solve this system of equations to find the values of A, B, C, and D.

One possible solution is A = -1, B = 2, C = 1, and D = -2.

Therefore, the equation of the plane that passes through the points (1, 1, 1), (2, 0, 3), and (-1, 4, 2) is -x + 2y + z - 2 = 0.

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