Solve. Write the solution set in interval notation.
2 − 5(x + 1) ≥ 3(x − 1) − 24

Answer:

We are not given the ratio of cordial to water used in the mixture, so we can assume that the entire bottle of cordial is mixed with water to make the drink.

Since the bottle of cordial holds 800 ml of cordial, the total volume of the mixture would be 800 ml + volume of water added. Let's call the volume of water added x.

Therefore, the total volume of the drink would be 800 ml + x.

We are asked to find the minimum capacity of the container in liters, so we need to convert the total volume of the drink from milliliters to liters:

800 ml + x = (800 + x)/1000 liters

Now we can set up an inequality to find the minimum value of x that would make the total volume of the drink at least 1 liter:

800 ml + x ≥ 1000 ml

Simplifying this inequality, we get:

x ≥ 200 ml

Therefore, the minimum volume of water that needs to be added to the cordial to make a drink with a total volume of at least 1 liter is 200 ml.

So the minimum capacity of the container would be:

800 ml + 200 ml = 1000 ml = 1 liter

Therefore, the minimum capacity of the container in liters would be 1 liter.

Step-by-step explanation:

The answer of the given question based on the Equation is given the solution to the given equation is n = 21/4.

A quadratic equation is type of equation in an algebra that involves variable (usually represented by x) that  raised to the second power (i.e., squared). The general form of the quadratic equation are given,

[tex]ax^2+bx+c = 0[/tex]

The highest power of  variable is 2, hence , name "quadratic".

To solve this equation, we can start by simplifying each of the terms on both sides of the equation:

3n + √(n² + 8) + 3√(n² + 8) - √(n² + 8) = 8

Combining like terms, we get:

3n + 3√(n² + 8) = 8

Subtracting 3√(n² + 8) from both sides, we get:

3n = 8 - 3√(n² + 8)

Dividing both sides by 3, we get:

n = (8/3) - (√(n² + 8)/3)

Now, we can rearrange this equation to isolate the radical term on one side of the equation:

√(n² + 8)/3 = (8/3) - n

by Squaring both the sides of equation, we get:

n² + 8 = 64/9 - (16n/3) + n²

Simplifying this equation, we get:

16n/3 = 56/9

Dividing both sides by 16/3, we get:

n = 21/4

Therefore, the solution to the given equation is n = 21/4.

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The values of the length and angle are;

AD = 22 units

m < ADC = 106 degrees

The properties of a kite are given as;

From the information given, we have that;

AB = 8x - 2

AD = 6x + 4

Equate the sides

8x - 2 = 6x + 4

collect like terms

2x = 6

x = 3

AD = 22 units

Also,

m < ABC = m < ADC

Substitute the values

12x + 10 = 15x - 14

collect like terms

-3x = -24x

x = 8

m < ADC = 106 degrees

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Answer: AD = 22 units

Step-by-step explanation:

m < ADC = 106 degrees

AB = 8x - 2

AD = 6x + 4

Equate any sides

8x - 2 = 6x + 4

Collect terms

2x = 6

x = 3

AD = 22 units

m < ABC = m < ADC

Substitute the values

12x + 10 = 15x - 14

Collect terms

-3x = -24x

x = 8

m < ADC = 106 degrees

X = 7 & -7/2

First, use the distributive property to simplify the equation:

2 - 5/(x+7) = -7/(x-3)(x+7)

2(x-3)(x+7) - 5(x-3) = -7(x+7)

2x2 + 7x - 6x - 15 = -7x - 49

2x2 -13x - 49 = 0

This equation can be solved by factoring the binomial:

2x2 -13x - 49 = 0

2x(x-7) - 7(x-7) = 0

(2x-7)(x-7) = 0

Therefore, the solution for x is x = 7 and x = -7/2.

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a. The scale factor is 2.

b. The lengths of GH, HJ, and JF are 13 m, 4 m, and 8 m respectively.

A scale factor is a number that is used to resize, a geometric figure. When a figure is scaled, all of its dimensions are multiplied by the scale factor.

The resulting figure is similar to the original figure, but it may be larger or smaller, depending on the value of the scale factor.

Here we have

ABCD ∼ FGHJ

Since both figures are similar

The ratio of the corresponding sides will be equal

a. Scale factor = AB/FG = 8m/ 4m = 2

b. Calculating JF, HJ, and GH  

As we know the ratio of the corresponding sides is equal

=> AB/FG = BC/GH = CD/HJ = AD/JF

From the figure,

=> 8 m /4 m = 26 m/GH = 8 m/HJ = 16/JF  

=>  26 m/GH = 8 m/HJ = 16/JF = 2

=> 26 m/GH = 2

=> GH = 13 m

=> 8 m/HJ = 2  

=> HJ = 4

=>  16/JF = 2  

=> JF = 8 m

Therefore,

a. The scale factor is 2.

b. The lengths of GH, HJ, and JF are 13 m, 4 m, and 8 m respectively.

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There are 120 different ways to arrange a mother, a father, and their 3 kids in a row.

There are several ways to arrange a mother, a father, and their 3 kids in a row. One way is to use the permutation formula, which is given by:

P = n! / (n - r)!

Where n is the total number of items and r is the number of items to choose from. In this case, n = 5 (mother, father, and 3 kids) and r = 5 (since we are choosing all 5 items).

Plugging in the values into the formula, we get:

P = 5! / (5 - 5)!
P = 5! / 0!
P = 120 / 1
P = 120

Therefore, there are 120 different ways to arrange a mother, a father, and their 3 kids in a row.

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a) The monthly interest rate for the $40,000 loan repaid monthly over 20 years with a monthly payment of $544.53 is 0.00153. b) The monthly interest rate for the $40,000 loan repaid monthly over 20 years with a monthly payment of $544.53 comes out as 0.00153 percent

To find the monthly payment needed to repay a $40,000 loan over 20 years with a 0.5% interest rate per month, we can use the following formula: Monthly payment = (Loan amount * Monthly interest rate) / (1 - (1 + Monthly interest rate)^-Number of payments)

Plugging in the given values: Monthly payment = ($40,000 * 0.005) / (1 - (1 + 0.005)^-240) Monthly payment = $200 / (1 - 0.334) Monthly payment = $200 / 0.666, Monthly payment = $300.30

Therefore, the monthly payment needed to repay the $40,000 loan over 20 years with a 0.5% interest rate per month is $300.30.To find the monthly interest rate for a $40,000 loan repaid monthly over 20 years with a monthly payment of $544.53, we can use the same formula and rearrange it to solve for the monthly interest rate:

Monthly interest rate = (Monthly payment * (1 - (1 + Monthly interest rate)^-Number of payments)) / Loan amount. Plugging in the given values:

0.005 = ($544.53 * (1 - (1 + 0.005)^-240)) / $40,000

0.005 * $40,000 = $544.53 * (1 - (1 + 0.005)^-240)

$200 = $544.53 * (1 - (1 + 0.005)^-240)

$200 / $544.53 = 1 - (1 + 0.005)^-240

0.367 = 1 - (1 + 0.005)^-240

(1 + 0.005)^-240 = 0.633

1 + 0.005 = 0.633^-240

0.005 = 0.633^-240 - 1

0.005 = -0.367

Monthly interest rate = -0.367 / -240

Monthly interest rate = 0.00153

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Answer:

Let's assume that Mia's present age is x.

According to the problem, Rosie is 6 years older than Mia, so Rosie's present age is x + 6.

In 4 years, Mia's age will be x + 4, and Rosie's age will be (x + 6) + 4 = x + 10.

The problem also states that Rosie's age in 4 years will be three times Mia's present age, so we can write the equation:

x + 10 = 3x

Simplifying this equation, we can subtract x from both sides to get:

10 = 2x

Dividing both sides by 2, we get:

x = 5

Therefore, Mia is currently 5 years old. To check, we can verify that in 4 years, Rosie will be three times Mia's age:

Rosie's age in 4 years = (5 + 6) + 4 = 15

Mia's age in 4 years = 5 + 4 = 9

Rosie's age in 4 years is indeed three times Mia's age, so our solution is correct.

The response to the given question would be that Therefore, after equation growing to 174 in 2003, the wolf population in Yellowstone National Park dropped from 150 in 2002 to 118.32 in 2005.

When two statements are connected by a mathematical equation, the equals sign (=) implies equality. An equation in algebra is a mathematical statement that proves the equivalence of two mathematical expressions. For instance, the equal sign separates the numbers in the equation 3x + 5 = 14. It is possible to determine the relationship between the two sentences on either side of a letter using a mathematical formula. The logo for the particular piece of software is frequently the same. as 2x - 4 = 2, for instance.  

beginning with a wolf population of 150 in 2002:

The number of wolves grew by 16% in 2003. We may multiply the initial population by 1.16 (100% + 16% = 116%) to determine the increase:

150 times 1.16 is 174 wolves.

There was a 32% decline in the wolf population between 2003 and 2005. We may multiply the population in 2003 by 0.68 (100% - 32% = 68%) to determine the decline:

174 times 0.68 equals 118.32 wolves (rounded to two decimal places)

Therefore, after growing to 174 in 2003, the wolf population in Yellowstone National Park dropped from 150 in 2002 to 118.32 in 2005.

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By answering the above question, we may infer that So the perimeter of the regular decagon is approximately 38.2 units (rounded to the nearest tenth).

In geometry, a decagon is either a decagon or not. There are 144° of inner angles total in a simple decagon. A regular decagon that self-intersects is known as a decagram. A polygon with 10 sides, ten internal angles, and ten vertices is called a decagon. Geometry may contain the form known as a decagon. It also has ten horns and ten horns. A dodecagon is a polygon with twelve sides. Some unusual types of dodecagons are shown in the photographs above. Particularly, a regular dodecagon has angles that are equally placed around a circle and sides that are of the same length.

Each interior angle of a regular decagon measures:

[tex]$$(n-2)\times180^\circ/n = (10-2)\times180^\circ/10 = 144^\circ$$\\$$\cos(72^\circ) = \frac{x}{2y}$$[/tex]

Solving for x, we get:

[tex]$$x = 2y\cos(72^\circ)$$[/tex]

We can use the fact that[tex]$\cos(72^\circ) = \frac{1+\sqrt{5}}{4}$[/tex](which can be derived using the golden ratio) to get:

[tex]$$x = 2y\cos(72^\circ) = 2y\cdot\frac{1+\sqrt{5}}{4} = \frac{y}{2}(1+\sqrt{5})$$\\$$R = \frac{x}{2\sin(180^\circ/10)} = \frac{x}{2\sin(36^\circ)}$$\\[/tex]

We can use this formula to find[tex]$y$:[/tex]

[tex]$$y = R = \frac{x}{2\sin(36^\circ)} = \frac{x}{2\sin(\frac{1}{2}\times72^\circ)} = \frac{x}{2\cos(72^\circ/2)}$$[/tex]

We can use the half-angle identity [tex]$\cos(\theta/2) = \sqrt{\frac{1+\cos(\theta)}{2}}$ to simplify this expression:[/tex]

[tex]$$y = \frac{x}{2\cos(72^\circ/2)} = \frac{x}{2\sqrt{\frac{1+\cos(72^\circ)}{2}}} = \frac{x}{2\sqrt{\frac{1+\frac{1+\sqrt{5}}{4}}{2}}} = \frac{x}{2\sqrt{\frac{3+\sqrt{5}}{4}}} = \frac{x}{\sqrt{3+\sqrt{5}}}$$[/tex]

Putting it all together, we have:

[tex]$$\text{Perimeter} = 10x = 10\cdot\frac{y}{2}(1+\sqrt{5}) = 5\sqrt{10+2\sqrt{5}}\approx 38.2$$[/tex]

So the perimeter of the regular decagon is approximately 38.2 units (rounded to the nearest tenth).

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Answer:

1. around 4.86 or 4.9 for the nearest tenth

2. 17 months

Step-by-step explanation:

1. (1.7×10^6)/(3.5×10^5) = (1.7/3.5)×(10^6/10^5) = 0.4857×10 = 4.857

Therefore, 1.7×10^6 is about 4.857 times as great as 3.5×10^5.

2.  start by finding out how much Erica still owes after the down payment:

Total cost - Down payment = $1,867 - $320 = $1,547

divide the amount still owed by the monthly payment to find out how many months Erica will be paying:

$1,547 ÷ $91 per month = 17 months (rounded up)

Therefore, Erica will be paying for the bike for 17 months

Plugging in \( x = 1 \) gives us \[ -144 = (1+1)(1-4)(1^2 - 6(1) + 13) = (2)(-3)(8) = -48a \] Solving for \( a \) gives us \[ a = \frac{-144}{-48} = 3 \] So, the expression for \( f(x) \) is \[ f(x) = 3(x+1)(x-4)(x^2 - 6x + 13) \]

To find the expression for \( f(x) \), we need to use the fact that the zeros of a polynomial are the values of \( x \) that make the polynomial equal to zero. This means that if \( -1, 4, \) and \( 3+2i \) are zeros of \( f(x) \), then the factors of \( f(x) \) are \[ (x+1)(x-4)(x-(3+2i))(x-(3-2i)) \] Now, we can simplify the last two factors by multiplying them together: \[ (x-3-2i)(x-3+2i) = (x-3)^2 - (2i)^2 = x^2 - 6x + 9 - 4i^2 = x^2 - 6x + 9 + 4 = x^2 - 6x + 13 \] So, the expression for \( f(x) \) is \[ f(x) = (x+1)(x-4)(x^2 - 6x + 13) \] Now, we can use the fact that \( f(1) = -144 \) to find the value of the leading coefficient of \( f(x) \). Plugging in \( x = 1 \) gives us \[ -144 = (1+1)(1-4)(1^2 - 6(1) + 13) = (2)(-3)(8) = -48a \] Solving for \( a \) gives us \[ a = \frac{-144}{-48} = 3 \] So, the expression for \( f(x) \) is \[ f(x) = 3(x+1)(x-4)(x^2 - 6x + 13) \] This is the final answer.

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The probabilities are:

(a) P(diamond or heart) = 0.5

(b) P(diamond or heart or spade) = 0.75

(c) P(seven or heart) ≈ 0.308.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

(a) To compute the probability of randomly selecting a diamond or heart from a standard deck of 52 cards, we need to count the number of diamond cards and heart cards in the deck. There are 13 diamond cards and 13 heart cards, so the total number of cards that are either diamonds or hearts is 13 + 13 = 26. Therefore, the probability of randomly selecting a diamond or heart is:

P(diamond or heart) = number of diamond cards + number of heart cards / total number of cards

P(diamond or heart) = 26/52

P(diamond or heart) = 1/2

P(diamond or heart) = 0.5

(b) To compute the probability of randomly selecting a diamond or heart or spade from a standard deck of 52 cards, we need to count the number of cards that belong to any of these three suits. There are 13 diamond cards, 13 heart cards, and 13 spade cards in the deck, so the total number of cards that are either diamonds, hearts, or spades is 13 + 13 + 13 = 39. Therefore, the probability of randomly selecting a diamond or heart or spade is:

P(diamond or heart or spade) = number of diamond cards + number of heart cards + number of spade cards / total number of cards

P(diamond or heart or spade) = 39/52

P(diamond or heart or spade) = 3/4

P(diamond or heart or spade) = 0.75

(c) To compute the probability of randomly selecting a seven or a heart from a standard deck of 52 cards, we need to count the number of seven cards and the number of heart cards in the deck. There are 4 seven cards and 13 heart cards, but we need to be careful not to double-count the seven of hearts. Therefore, the total number of cards that are either sevens or hearts is 4 + 12 = 16. Therefore, the probability of randomly selecting a seven or a heart is:

P(seven or heart) = number of seven cards + number of heart cards - number of seven of hearts / total number of cards

P(seven or heart) = 16/52

P(seven or heart) = 4/13

P(seven or heart) ≈ 0.308

Hence, The probabilities are:

(a) P(diamond or heart) = 0.5

(b) P(diamond or heart or spade) = 0.75

(c) P(seven or heart) ≈ 0.308.

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The equation of a circle with center at (4, 8) and passing through (-10, -5) is [tex](x-4)^2 + (y-8)^2 = (6)^2.[/tex]

To find this equation, we need to use the distance formula: [tex]d = sqrt((x2-x1)^2 + (y2-y1)^2).[/tex]

We can substitute in the coordinates of the center of the circle and the point it passes through:

[tex]d = sqrt((-10-4)^2 + (-5-8)^2)[/tex]
[tex]d = sqrt( (6)^2 + (13)^2 )[/tex]

d = sqrt(36 + 169)

d = sqrt(205)

d = 14.2

Now, we have the radius of the circle. We can use this to form the equation of the circle:

[tex](x-4)^2 + (y-8)^2 = (14.2)^2[/tex]

[tex](x-4)^2 + (y-8)^2 = (6)^2[/tex]

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The probability that none of the 3 detectors would work if there was smoke in the factory is 0.25³, or 0.015625 (1.56%).

Probability is a branch of mathematics that deals with the likelihood of certain outcomes or events occurring. Probability is a measure of how likely an event is to occur, expressed as a number between 0 and 1. An event that is certain to occur has a probability of 1, while an event that is impossible to occur has a probability of 0. The probability of an event occurring can be calculated by dividing the number of successful outcomes by the total number of possible outcomes.

This probability indicates that the factory does have a safety problem, as there is a 1.56% chance that none of the smoke detectors would sound an alarm if there was smoke present. This presents a significant risk to the factory, as it could lead to an undetected fire that may cause damage to the facility or put employees in danger.

The company should look into ways to reduce the risk of the smoke detectors not working. One way to do this would be to increase the number of smoke detectors in the factory room, as the more detectors there are, the lower the probability that none of them will work. The company could also consider using more reliable smoke detectors that have a higher probability of working correctly when needed. Ultimately, the company should seek to reduce the risk of none of the detectors working by taking steps to increase the probability that at least one of them will sound an alarm in the event of smoke or a fire.

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Jean-Luc's account will be worth approximately $740.55 after 5 years of continuous  at a compounding annual interest rate of 1.6%.

Compound interest is the addition of interest to the principal amount of a loan or investment. In other words, it is the interest earned on both the principal amount and any accumulated interest from previous periods.

Interest rate is the amount charged by a lender to a borrower for the use of money or the amount earned by an investor for lending money or investing in an asset. It is usually expressed as a percentage of the principal amount and is typically calculated on an annual basis.

In the given question,

The formula for continuous compounding is:

A = Pe^(rt)

where A is the amount of money in the account after t years, P is the principal amount (the initial deposit), r is the annual interest rate (as a decimal), and e is the mathematical constant approximately equal to 2.71828.

Plugging in the given values, we get:

A = 625 * e^(0.016*5)

Simplifying this expression, we get:

A = 625 * e^(0.08)

Using a calculator, we can evaluate this expression to get:

A ≈ $740.55

Therefore, Jean-Luc's account will be worth approximately $740.55 after 5 years of continuous compounding at an annual interest rate of 1.6%.

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The values of k that make the system inconsistent are k = 0 and k = 2.

The system obtained from the given augmented matrix is:
```
1 2 -1 | 4
0 -1 1 | 3
0 0 k^2-2k | -3k-3
```
For the system to be inconsistent, the last equation must be a contradiction, meaning that the coefficients of the variables are all zero but the constant term is nonzero. In other words, we need `k^2-2k = 0` and `-3k-3 ≠ 0`.

Solving for k in the first equation, we get:
```
k^2-2k = 0
k(k-2) = 0
```
So k = 0 or k = 2.

However, we also need `-3k-3 ≠ 0`, which means that k ≠ -1. Therefore, the values of k that make the system inconsistent are k = 0 and k = 2.

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Answer: To determine the effective rate of Zoe's loan, we can use the formula:

Effective rate = (1 + (nominal rate/number of compounding periods))^number of compounding periods - 1

Since the nominal rate is compounded monthly, the number of compounding periods is 12.

Plugging in the values, we get:

Effective rate = (1 + (0.039/12))^12 - 1

Effective rate = 0.0407 or 4.07%

Therefore, the effective rate of Zoe's loan is 4.07%, rounded to the nearest hundredth. The correct response is 3.97%.

Step-by-step explanation:

Answer:

Let's say that the class has a total of 7x students, where 2x wear glasses and 5x don't.

After 5 of the students who didn't wear glasses now do, the number of students who don't wear glasses becomes 5x - 5.

So, the new ratio of students who wear glasses to those who don't wear glasses is:

2x : (5x - 5)

We can simplify this ratio by dividing both terms by the greatest common factor, which is x:

2 : (5 - 5/x)

We can't simplify this any further, but we can see that the ratio is not fixed and depends on the value of x.

The sequence of transformations that maps △RST to △R′S′T′ is a translation of 6 units to the right and 2 units down, followed by a reflection across the y-axis, and finally a rotation of 90 degrees counterclockwise about the origin.

What is the sequence of transformation?

In mathematics, a sequence of transformations refers to a series of geometric transformations performed on a shape to create a new shape. These transformations can include translations, rotations, reflections, and dilations.

The sequence of transformations is the order in which the transformations are performed. The order matters because different sequences of transformations can lead to different final shapes.

For example, to transform a triangle into a new position, we might first perform a translation to move the triangle to a new location, then perform a rotation to change its orientation, and finally perform a reflection to flip the triangle across a line. The sequence of transformations in this case is translation-rotation-reflection.

Sequences of transformations are used in geometry to analyze and describe shapes and to solve problems related to symmetry, congruence, and similarity. They are also used in computer graphics and animation to create 2D and 3D shapes that can be moved and transformed on a screen.

To map △RST to △R′S′T′, we need to perform a sequence of transformations that includes translations, rotations, reflections, and/or dilations. Here's one possible sequence of transformations:

Translation: We can translate △RST by 6 units to the right and 2 units down to get a new triangle that has R at (3, -3), S at (5, -3), and T at (2, -7).

Reflection: We can reflect the translated triangle across the y-axis to get a new triangle that has R′ at (-3, -3), S′ at (-5, -3), and T′ at (-2, -7).

Rotation: We can rotate the reflected triangle 90 degrees counterclockwise about the origin to get a new triangle that has R′ at (3, -3), S′ at (3, -1), and T′ at (-1, -4).

Therefore, the sequence of transformations that maps △RST to △R′S′T′ is a translation of 6 units to the right and 2 units down, followed by a reflection across the y-axis, and finally a rotation of 90 degrees counterclockwise about the origin.

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The area of rectangle S is 84.15 square unit.

Area is the region enclosed by the shape of an object. The space covered by a figure or any two-dimensional geometric shape in the plane is the area of ​​the shape, or

Area is a measure of the surface area of ​​a shape. To find the area of ​​a rectangle or square, you need to multiply the length and width of the rectangle or square.

Solution according to the given information in the question:

Area of rectangle R = 187

Ratio of rectangle R = 17

Ratio of rectangle S = 7.65

Area of rectangle S = (7.65/17)×187

                                 = 84.15 square unit

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the solution is: population in 2015 will be 40,115,896

The exponential function is an illustration of a mathematical function that may be used to identify if something is increasing or decreasing exponentially. An exponential function employs exponents, as suggested by its name.

But, you should be aware that an exponential function does not have a constant base and a variable exponent (if a function has a variable as the base and a constant as the exponent then it is a power function but not an exponential function).

the median population change in California from 2000 to 2009 in percentage terms.

Given that there are 9 years, add up all the percentages, and then divide by 9.

1.97+1.71+1.65+1.42+1.22+1.02+1.07+1.22+0.93/9 = 1.3567%

From 2009 through 2015, there were six different time periods.

In the future, 37,000,000 x (1 + 0.013567) x 6 = 40,115,896.06

Hence, the solution is: population in 2015 will be 40,115,896

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The amount we would have after 40 years will be $8183.27

Exponential growth is a pattern of data that shows greater increases with passing time, creating the curve of an exponential function.

Given that, an amount increasing exponentially every two years and with a rate of 15% and the amount is $500, we need to find the amount we would have after 40 years.

Since, the amount is increasing exponentially every two years, therefore,

T = 40 / 2 = 20 years

A = P(1+0.15)²⁰

A = 500(1+0.15)²⁰

A = 500(1.15)²⁰

A = 8183.27

Hence, the amount we would have after 40 years will be $8183.27

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Answer:

62

Step-by-step explanation:

x2 a 4 - 91

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1. The probability of getting a z-score of 0.189 or less is 0.5753.  

2. The probability of getting a z-score of 1.13 or less is 0.8708.

3. The probability of getting a z-score of 3.02 or less is 0.9982.

Normally distributed data refers to data that follows a normal distribution, also known as a Gaussian distribution or bell curve.

In a normal distribution, the data is symmetric around the mean and the majority of the data is clustered around the mean with decreasing density as the data moves further away from the mean.

Here we have

The mean price for liquid crystal display (LCD) computer monitors is $ 170  and the standard deviation is $ 53.  

Find the z-score associated with $ 180 using the formula:

z = (x - μ) / σ

where x is the value we need to find the probability, μ is the mean, and σ is the standard deviation.

z = (180 - 170) / 53

z = 0.189

Hence, using a standard normal distribution table or calculator, we can find that the probability of getting a z-score of 0.189 or less is 0.5753.  

To find the probability that the mean cost of 9 randomly selected LCD computer monitors is less than $ 180, find the standard deviation σ/√n.

The standard deviation of the sample means is σ/√n = $53/√9 = $17.67.

Now we need to find the z-score associated with $ 180 using the formula:

z = (x - μ) / σ/√n

where x is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

z = (180 - 170) / (53 / √9)

z = 1.13

Hence, using a standard normal distribution table or calculator, we can find that the probability of getting a z-score of 1.13 or less is 0.8708.  

To find the probability that the mean cost of 36 randomly selected LCD computer monitors is less than $ 180, we can use the same formula as in part 2, with n = 36:

z = (x - μ) / σ/√n

z = (180 - 170) / (53 / √36)

z = 3.02

Hence, using a standard normal distribution table or calculator, we can find that the probability of getting a z-score of 3.02 or less is 0.9982.

Therefore,

1. The probability of getting a z-score of 0.189 or less is 0.5753.  

2. The probability of getting a z-score of 1.13 or less is 0.8708.

3. The probability of getting a z-score of 3.02 or less is 0.9982.

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The simplest form of the expression (x^(2)+8x+16)/(x^(2)-4x-32) is (x+4)/(x-8) with restrictions x≠8 and x≠-4.

The expression (x^(2)+8x+16)/(x^(2)-4x-32) can be simplified by factoring the numerator and denominator.

First, we can factor the numerator:
(x^(2)+8x+16) = (x+4)(x+4)

Next, we can factor the denominator:
(x^(2)-4x-32) = (x-8)(x+4)

Now, we can simplify the expression by canceling out the common factor of (x+4):
(x+4)(x+4)/(x-8)(x+4) = (x+4)/(x-8)

Therefore, the simplest form of the expression is (x+4)/(x-8).

However, there are restrictions on the variable x. The denominator of the expression cannot equal zero, so we must find the values of x that make the denominator zero and exclude them from the domain of the expression.

To find the restrictions, we set the denominator equal to zero and solve for x:
(x-8)(x+4) = 0

This gives us two solutions:
x = 8
x = -4

Therefore, the restrictions on the variable are x≠8 and x≠-4.

In conclusion, the simplest form of the expression (x^(2)+8x+16)/(x^(2)-4x-32) is (x+4)/(x-8) with restrictions x≠8 and x≠-4.

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The correct slope and y-intercept for the linear equation y=2x+7 are 2 and 7, respectively.

The y-intercept of a graph is the point where the graph crosses the y-axis. It is written as (0, b), where b is the y-intercept. The y-intercept is the value of y when x is equal to zero. It can be used to determine the equation of a line when two points on the line are known.

The slope of a linear equation is the coefficient of the x variable, which is 2 in this case. The y-intercept is the constant term, which is 7 in this case.

So, the slope is 2 and the y-intercept is 7.

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The result of blending 50 percent of each animation of the pairs that follow will be a new animation that combines the features of both animations. This is known as "root motion blending" and is used to create smooth transitions between different animations.

The new animation will have a duration that is the average of the two animations being blended (for example, blending IDLE and WALK_1 will result in a 3s duration animation) and will include features from both animations. For example, blending WALK_1 and WALK_2 will result in a walking animation that is at a pace between relaxed and fast. Similarly, blending SPRINT_1 and SPRINT_2 will result in a sprinting animation that includes both a straight sprint and a slight right turn. The root motion of the new animation will be the average of the root motions of the two animations being blended.

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The explicit formula for the nth term of the sequence 7, 35, 175 is:

[tex]a_n = 7 * 5^{n-1}[/tex]

What is the geometric sequence?

A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed number called the common ratio. The general formula for a geometric sequence is:

a, ar, ar², ar³, ...

where a is the first term and r is the common ratio.

To find the explicit formula for the sequence 7, 35, 175, we need to identify the pattern in the sequence.

Notice that each term is obtained by multiplying the previous term by 5. Specifically, the first term (7) is multiplied by 5 to get the second term (35), and the second term is multiplied by 5 to get the third term (175).

So, the sequence is a geometric sequence with a common ratio of 5.

To find the explicit formula for a_n, we can use the formula:

[tex]a_n = a_1 * r^{n-1}[/tex]

where a1 is the first term, r is the common ratio, and n is the term we want to find.

In this case, a1 = 7 and r = 5.

Substituting these values into the formula, we get:

[tex]a_n = 7 * 5^{n-1}[/tex]

Therefore, the explicit formula for the nth term of the sequence 7, 35, 175 is:

[tex]a_n = 7 * 5^{n-1}[/tex]

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