How to get the perimeter of something

The combined weight of the 3/4 lb bags is 1 1/2 lb.

To get combined weight of the 3/4 lb bags, we must know number of bags that weigh 3/4 lb to calculate their total weight.

Given the list of bags and weights:

[tex]1/4 lb, 1/4 lb, 3/4 lb, 1/2 lb, 1/4 lb, 3/4 lb[/tex]

There are two bags that weigh 3/4 lb.

We wlll calculate their total weight as follows:

Total weight of 3/4 lb bags:

= 2 * 3/4 lb

= 6/4 lb

= 1 1/2 lb.

Full question:

A Student makes bags of candy for Spring. The amount of candy in each bag is listed below. 1/4 lb, 1/4 lb, 3/4 lb, 1/2 lb, 1/4 lb, 3/4 lb. What is the combined weight of the 1/4 bags?

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The quadratic function for the path of the basketball as it is thrown indicates;

The path of the basketball will not make the shot

The height reached is about 5.24 meters

A quadratic function is a function of the form; f(x) = a·x² + b·x + c, where a ≠ 0, and a, b, c, are numbers.

The function for the path of the basketball is; b(x) = (-1/7)·x² + 1.36·x + 2

The function for the location of the basketball net is; n(x) = 3 and (8 < x < 8.5), where;

n(x) = The vertical height of the basketball

Plugging in the value of the n(x) = b(x), to check if equations have a common solution, we get;

b(x) = n(x) = 3 = (-1/7)·x² + 1.36·x + 2

(-1/7)·x² + 1.36·x + 2 - 3 = 0

(-1/7)·x² + 1.36·x - 1 = 0

(1/7)·x² - 1.36·x + 1 = 0

Solving the above equation, we get;

x = (119 - √(9786))/(25) ≈ 0.803, and x = (119 + √(9786))/(25) ≈ 8.717

Therefore, the x-coordinates of the height of the path of the basketball when the height is 3 meters are 0.803 and 8.717, neither of which are within the range (8 < x < 8.5), therefore, the baseketball will not go through the net and the path will not make the shot.

Bonus; The x-coordinates of the highest point of a quadratic function, f(x) = a·x² + b·x + c is; -b/(2·a)

Therefore, the x-value at the highest point of the equation, b(x) = (-1/7)·x² + 1.36·x + 2 is; x = -1.36/(2 × (-1/7)) = 1.36 × 7/2 = 9.52/2 = 4.76

The height of the highest point is; b(9.52) = (-1/7)·(4.76)² + 1.36·(4.76) + 2 ≈ 5.24 meters

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Answer:

Hi

Please mark brainliest ❣️

Thanks

Step-by-step explanation:

True

Reason

Use the last person time to minus the first person time which is

53.3 - 51.3

Answer:

[tex]\frac{1}{4}[/tex]

Step-by-step explanation:

using the rule of logarithms

[tex]log_{b}[/tex] x = n ⇒ x = [tex]b^{n}[/tex]

also the rule of exponents/ radicals

[tex]\sqrt[n]{a^{m} }[/tex] = [tex]a^{\frac{m}{n} }[/tex]

let

[tex]log_{5}[/tex][tex]\sqrt[4]{5}[/tex] = n , then

[tex]\sqrt[4]{5}[/tex] = [tex]5^{n}[/tex] , that is

[tex]5^{\frac{1}{4} }[/tex] = [tex]5^{n}[/tex]

since bases on both sides are the same, both 5 , then equate the exponents

n = [tex]\frac{1}{4}[/tex] ( = 0.25 )

Then , if KL is 45 units long, then KM will be the angle bisector of LKN.

To find the value of X that makes KM the angle bisector of LKN, we need to use the angle bisector theorem. According to the theorem, if a line bisects an angle of a triangle, then it divides the opposite side into segments that are proportional to the other two sides of the triangle.

Let's apply this theorem to triangle LKN. Suppose that KM bisects angle LKN, and let X be the length of KL. Then, according to the angle bisector theorem:

LK / KN = KL / KN
LK = KL * (KN / KN)
LK = KL

This tells us that LK and KL have the same length, which means that triangle LKN is isosceles. Therefore, we can write:

LK = KN

Now, we can use the fact that the sum of the angles of a triangle is 180 degrees to find the measure of angle LKN. We know that angle LKN is bisected by KM, so we can write:

angle LKM = angle MKN

Let's call this angle x. Then, we have:

angle LKN = 2x

And since triangle LKN is isosceles, we also have:

angle LNK = angle KLN = (180 - angle LKN) / 2
angle LNK = (180 - 2x) / 2
angle LNK = 90 - x

Now, we can use the fact that the sum of the angles of a triangle is 180 degrees to write:

angle LKM + angle LKN + angle LNK = 180

Substituting the values we found earlier, we get:

x + 2x + (90 - x) = 180

Simplifying, we get:

x = 45
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The smallest possible number of candies Dzani bought is 11

From the question, we have the following parameters that can be used in our computation:

Represent the number of jars with x

So, we have

4x + 3 = 5x + 1

Collect the like terms

So, we have

5x - 4x = 3 - 1

Evaluate

x = 2

So, we have

Candles = 4 * 2 + 3

or

Candles = 5 * 2 + 1

Evaluate

Candles = 11

Hence, the smallest possible number of candies Dzani bought is 11

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Answer:

85 feet

Step-by-step explanation:

To find the distance traveled by the laser pointer's beam, we can use the Pythagorean theorem. The laser beam forms a right triangle with the tower as one side and the distance traveled as the hypotenuse.

Let's denote the distance traveled by the laser beam as x. We can set up the following equation using the Pythagorean theorem:

x^2 = 40^2 + 75^2

Simplifying this equation:

x^2 = 1600 + 5625

x^2 = 7225

Taking the square root of both sides:

x = √7225

x = 85

Therefore, the laser pointer's beam has traveled approximately 85 feet.

The value of the angle S in the parallelogram is 45°.

Given is a parallelogram QRST where angle T = 135° we need to find the value of angle S,

Since the angle S and T are the adjacent angles and we know that the adjacent angles in a parallelogram are supplementary,

So,

∠T + ∠S = 180°

135° + ∠S = 180°

∠S = 45°

Hence the value of the angle S in the parallelogram is 45°.

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5. Area = 63.75 square units

Perimeter = 34 units

Sum of interior angles = 30 degrees

6. Area = 61. 40 square units

7. Length = 58. 2 units

We need to know that formula for calculating the area of a trapezoid is expressed as;

5. Area = a + b/2(h)

Substitute the values, we have;

Area = 6.5 + 10.5/2(7.5)

Add the values

Area = 63.75 square units

Perimeter = a + b+ c + d

Perimeter = 8.5 +6.5 + 10. 5 + 8.5

Add the values

= 34 units

6. Area of a sector is  = (θ/360º) × πr²

= 110/360 × 3.14 × 8²

Find the square and multiply

= 61. 40 square units

7. Length of an Arc = θ × (π/180) × r

Length = 210 × 3.14/180 × 16

Multiply the values

Length = 58. 2 units

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Step-by-step explanation:

At t = 4   bird is at  6 m

at t= 14 bird is at 29 meters

   bird taveled  29 -6 = 23 meters   in 10 seconds

      velocity = 23 m / 10 s = 2.3 m/ s

The 5 × 10⁻³ cm radius of one spherical cancerous cell indicates that the number of cells in the initial volume of 0.3 cm³ of the tumor is; 572,958 cells

The volume, V, of a sphere is the product of the cube of the radius of the sphere, pi, and (4/3); V =  (4/3) × Pi × (The radius of the sphere, r)³

The volume of each cancerous cell can be calculated as follows;

The shape of each cancerous cell = A sphere

Volume of a sphere = (4/3) × π × r³

Where;

r = The radius of the sphere

The radius of the cancerous sphere, r = 5 × 10⁻³ cm

The volume of each cancerous cell = (4/3) × π × (5 × 10⁻³)³

The number of cancerous cells = The initial volume of the tumor ÷ The volume of one cancerous cell

The number of cancerous cells = 0.3/((4/3) × π × (5 × 10⁻³)³) ≈ 572958

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The solutions are x = 29° and y = 27°.

In the given quadrilateral, the angles are mentioned.

So,

Q = 79°

R = 109°

S = (6y - 61)°

T = (4x - 45)°

A quadrilateral is a polygon with four vertices, four sides, and four angles with a total angle of 360 degrees.

The quadrilateral makes two triangles when its diagonals are drawn. These two triangles have an angle total of 180 degrees. The quadrilateral's total angle sum is hence 360°.

So,

T + R = 180°

(4x - 45) + 109 = 180

4x + 64 = 180

4x = 180 - 64

4x = 116

Therefore, x = 116/4

x = 29°

S + Q = 180°

(6y - 61) + 79 = 180

6y + 18 = 180

6y = 180 - 18

6y = 162

Therefore,

y = 162/6

y = 27°

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The probability of picking a white marble is 3/10

From the question, we have the following parameters that can be used in our computation:

four green marbles, six white marbles, and ten red marbles

This means that

Total = 4 + 6 + 10

Evaluate the sum

Total = 20

The probability of picking a white marble is

P = White/Total

substitute the known values in the above equation, so, we have the following representation

P = 6/20

Evaluate

P = 3/10

Hence, the probability of picking a white marble is 3/10

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The simplified expression is 6x. Simonne simplifies the expression 12 minus 3 times negative 2x plus 4 to 6x by following the order of operations, performing the necessary multiplication and subtraction.

To simplify the expression 12 minus 3 times negative 2x plus 4, Simonne follows the order of operations (also known as PEMDAS) to ensure the correct simplification.

PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Let's break down the steps:

Step 1: Simplify within parentheses

The expression within the parentheses is -2x + 4. Since there is no exponent or multiplication/division to perform within the parentheses, we move to the next step.

Step 2: Apply the multiplication

The multiplication operation is between 3 and the expression within parentheses (-2x + 4). We distribute the 3 to each term within the parentheses by multiplying each term by 3:

3*(-2x) + 3*4

This simplifies to:

-6x + 12

Step 3: Apply subtraction

The original expression is 12 minus the result from Step 2 (-6x + 12):

12 - (-6x + 12)

To subtract a negative, we can rewrite the expression as:

12 + (-1)*(-6x + 12)

Next, we distribute the -1 to each term within the parentheses:

12 + (-1)*(-6x) + (-1)*12

Simplifying, we have:

12 + 6x - 12

The 12 and -12 cancel each other out, leaving us with:

6x.

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To find the area of the front of the podium add the area of two triangle and one rectangle.

To find the area of the front of the podium by decomposing it into two triangles and a rectangle, you can follow these steps:

Measure the dimensions: Measure the necessary dimensions of the triangles and rectangle. Typically, you would need to measure the base and height of each triangle and the length and width of the rectangle.

Calculate the area of each shape: Use the appropriate formulas to calculate the area of each shape.

The formula for the area of a triangle is (1/2) x base x height,

and for a rectangle, it is length x width.

Sum up the areas: Add up the areas of the two triangles and the rectangle to find the total area of the front of the podium.

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The coordinates of the vertices of trapezoid J'K'L'M' with its graph include the following:

J' (-3, 0).

K' (-3, -3).

L' (-8, 6).

M' (-8, 1).

In Mathematics and Geometry, the rotation of a point 90° about the center (origin) in a clockwise direction would produce a point that has these coordinates (y, -x).

By applying a rotation of 90° clockwise to the vertices of trapezoid JKLM, the coordinates of the vertices of the image are as follows:

(x, y)               →                ((y - b) + a, -(x - a) + b)

J (2, 1)           →     ((1 - 3) - 1, -(2 + 1) + 3) = (-2 - 1, -3 + 3) = J' (-3, 0).

K (5, 1)           →     ((1 - 3) - 1, -(5 + 1) + 3) = (-2 - 1, -6 + 3) = K' (-3, -3).

L (8, -4)           →     ((-4 - 3) - 1, -(8 + 1) + 3) = (-7 - 1, -9 + 3) = L' (-8, 6).

M (1, -4)           →     ((-4 - 3) - 1, -(1 + 1) + 3) = (-7 - 1, -2 + 3) = M' (-8, 1).

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Evaluating the function f(x) = -2x + 5 when x = -1 is 7

Evaluating a function involves calculating the value of the function for a given input or argument. When you evaluate a function, you substitute the input value into the function's expression and perform the necessary computations to determine the corresponding output value.

A function is typically defined by an equation or rule that relates the input value (often represented as x) to the output value (often represented as f(x)). By evaluating the function, you can determine the specific output value corresponding to a particular input

In this problem given, we can evaluate the function by substituting the value for x into it.

f(x) = -2x + 5

f(-1) = -2(-1) + 5

f(-1) = 2 + 5

f(-1) = 7

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There are 5 chances out of 21 of picking an A, E, I, O, or U.

There are 5 vowels in the English alphabet (A, E, I, O, U) out of a total of 26 letters. So, the probability of picking a vowel is 5/26.

To find the odds in favor of picking a vowel, we divide the probability of picking a vowel by the probability of not picking a vowel:

odds in favor = probability of picking a vowel / probability of not picking a vowel

The probability of not picking a vowel is 1 - 5/26 = 21/26.

So, the odds in favor of picking an A, E, I, O, or U are:

5/26 / 21/26 = 5/21

This means that there are 5 chances out of 21 of picking an A, E, I, O, or U.

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The equation of the ellipse that has the center at (0, 0) the vertex at (-8, 0), and the covertex at (0, 4) is; [tex]\frac{x^2}{64} + \frac{y^2}{16} = 1[/tex]

The standard form of the equation of an ellipse is as follows;

[tex]\frac{x^2}{a^2} + \frac{y^2}{b^2}=1[/tex]

Where;

a = The semi major axis

b = The semi minor axis

The semi major axis is the distance from the center of the ellipse to the vertex, and the semi minor axis is the distance from the center to the co-vertex.

The coordinate of the center of the ellipse = (0, 0)

The coordinates of the vertex = (-8, 0)

The coordinates of the co-vertex = (0, -4)

The ellipse has an horizontal major axis, therefore;

a = |0 - (-8)| = 8, and b = |0 - 4| = 4

The equation of the ellipse is therefore;

[tex]\frac{x^2}{8^2} + \frac{y^2}{4^2}=\frac{x^2}{64} + \frac{y^2}{16} = 1[/tex]

[tex]\frac{x^2}{64} + \frac{y^2}{16} = 1[/tex]

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Answer:

-5+√2 and -5-√2

Step-by-step explanation:

With the quadratic formula:

[tex]\displaystyle x^2 + 10x + 25 = 2\\\\x^2+10x+23=0\\\\x=\frac{-10\pm\sqrt{10^2-4(1)(23)}}{2(1)}=\frac{-10\pm\sqrt{100-92}}{2}=\frac{-10\pm\sqrt{8}}{2}=\frac{-10\pm2\sqrt{2}}{2}=-5\pm\sqrt{2}[/tex]

We can also complete the square (which is faster):

[tex]x^2+10x+25=2\\(x+5)^2=2\\x+5=\pm\sqrt{2}\\x=-5\pm\sqrt{2}[/tex]

The square root of 9 by long division method will give an answer of 3.

1: Group the digits of 9 into pairs from right to left. Since 9 is a single-digit number, we can consider it as a pair by itself.

2: Find the largest number whose square is less than or equal to the leftmost pair. In this case, the largest number whose square is less than or equal to 9 is 3.

3: Write 3 as the divisor on the left, and the quotient on the top right.

3 √ 9 |  0

4: Multiply the divisor (3) by the quotient (3), and write the result under the 9.

3 √ 9 |  0

      - 9

5: Subtract the result (9) from the leftmost pair (9), and write the difference (0) below the line.

3 √ 9 |  0

      - 9

       0

6: Bring down the next pair (0) to the right of the difference.

3 √ 9 |  0

      - 9

       0

     -----

7: Double the quotient (3) and write a blank space to the right.

3 √ 9 |  0

      - 9

       0

     -----

       0

      ?

8: Find a digit to fill in the blank space, such that when the new divisor (37) is multiplied by the digit, the result is less than or equal to the current dividend (0). In this case, the largest digit we can use is 0.

9: Write the digit (0) as the new quotient, and write the product of the new divisor (37) and the new quotient (0) below the line.

3 √ 9 |  0

      - 9

       0

     -----

       0

      0

10: Subtract the product (0) from the current dividend (0), and write the difference (0) below the line.

3 √ 9 |  0

      - 9

       0

     -----

       0

      0

      - 0

11: Since there are no more pairs to bring down, we have reached the end of the division. The square root of 9 is 3.

√9 = 3.

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In mathematics, the term "difference" is commonly associated with subtraction. Subtraction is an arithmetic operation that involves finding the difference between two numbers or quantities.

When we subtract one number from another, we are essentially finding the difference between them. The result of a subtraction operation is called the difference. For example, in the expression 10 - 5, the difference is 5, indicating that the number 5 is the result of subtracting 5 from 10.

Subtraction is denoted by the minus sign "-" or by using the word "minus" or "subtract" in mathematical expressions. For instance, in the equation 8 - 3, we are subtracting 3 from 8 to find the difference.

The concept of difference is fundamental in various mathematical applications. It allows us to compare quantities, measure distances, determine changes, and solve various problems. For instance, when analyzing data sets, we often calculate the differences between values to understand patterns, trends, or changes over time.

The difference between two numbers can be positive, negative, or zero. If the minuend (the number from which subtraction is performed) is greater than the subtrahend (the number being subtracted), the difference is positive. If the subtrahend is greater, the difference is negative. When both numbers are equal, the difference is zero.

For example, in the subtraction 7 - 3, the difference is 4. In the subtraction 5 - 9, the difference is -4. And in the subtraction 6 - 6, the difference is 0.

Overall, the concept of difference is a crucial aspect of subtraction, providing a measure of the numerical distance between two quantities. It allows us to express the result of a subtraction operation and plays a significant role in various mathematical calculations and analyses.

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Answer:

Hi

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Thanks

Step-by-step explanation:

The answer is NO

Reason

x= 2 y= 1

Now input in the first inequality

y≤ -x + 4

1 ≤ -2 +4

1≤ 2 i.e 1 is less than two

Next inequality

y≤ x +1

1 ≤ 2 + 1

1≤ 3 i.e 1 is less than 3

But 1 is not equal to 3 and also not equal to 2

Hence our answer is NO

The solution for y is y = log(N - (4^y))/log(8y).

The given expression is (8y + 4) ° y = N. To solve for y, we can simplify the expression first. We will use the exponent rules to solve this. According to the exponent rules, if we have a power raised to another power, we can multiply the exponents.

Therefore, we can write (8y + 4) ° y as (8y + 4)^(y).Now we can distribute the exponent to both terms inside the parentheses, giving us (8^y)(y^y) + (4^y) = N. We can simplify this expression further by multiplying the coefficients, which gives us (8^y)(y^y) + (4^y) = N.

We can then solve for y by rearranging the terms. We can subtract (4^y) from both sides, giving us (8^y)(y^y) = N - (4^y).

We can then take the logarithm of both sides, giving us y log(8y) = log(N - (4^y)).

Finally, we can solve for y by dividing both sides by log(8y), giving us y = log(N - (4^y))/log(8y).

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The simplified form of the expression -2(-8x+7)-(-3x - 7) is 19x - 7.

To simplify the expression -2(-8x+7)-(-3x - 7), we can start by applying the distributive property and simplifying the terms.

First, we distribute -2 to (-8x+7):

-2*(-8x) + (-2)*7 - (-3x) - (-7)

This simplifies to:

16x - 14 + 3x + 7

Next, we combine like terms:

(16x + 3x) + (-14 + 7)

This simplifies to:

19x - 7

Therefore, the simplified form of the expression -2(-8x+7)-(-3x - 7) is 19x - 7.

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The number with a z-score of -2.2 is given as follows:

X = 1.2.

The z-score formula is defined as follows:

[tex]Z = \frac{X - \mu}{\sigma}[/tex]

In which:

The parameters for this problem are given as follows:

[tex]\mu = 10, \sigma = 4[/tex]

Hence the score X is given as follows:

-2.2 = (X - 10)/4

X - 10 = 8.8

X = 1.2.

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56

One measure of the center is the mean.

Mean

The mathematical mean of data is also called the average. There are 3 main measures of center: the mean, median, and mode. The most common of these 3 is mean. However, each measurement has its own advantages and disadvantages. Since there are no outliers in this data set, the mean is a good measure of the center.

Finding Mean

In order to find the mean of a data set, we need to add all the terms and divide by the number of terms. In this case, there are 7 terms in the set. The first step is to add all the terms.

Then, divide the sum by 7.

So, the mean of the data set is 56.

The value of the green box is 0.

To solve the equation[tex]{3x-6} +{ x-2}= \frac{3x-6 + x - 2}{1} = 180$[/tex], we first simplify the left side of the equation:

[tex]{3x-6}{x-2} = \frac{(3x-6) + (x-2)}{x-2} = \frac{4x-8}{x-2}$$[/tex]

Now we can rewrite the original equation as:

[tex]\frac{4x-8}{x-2} = 180$$[/tex]

Next, we can multiply both sides of the equation by x-2 to eliminate the denominator:

[tex]$$(4x-8) = 180(x-2)$$[/tex]

Expanding the right side gives:

[tex]$$(4x-8) = 180x - 360$$[/tex]

Simplifying and moving all the terms to the left side gives:

[tex]$$176x - 352 = 0$$[/tex]

Adding 352 to both sides gives:

176x = 352

Finally, dividing both sides by 176 gives:

[tex]x = \frac{352}{176} = 2[/tex]

Therefore,  since there is no constant term on the left side of the equation after combining like terms.

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Answer:

[tex]\frac{2}{3}\ln(x+1)-\frac{1}{x-2} -\frac{2}{3}\ln(x-2)+C[/tex]

Step-by-step explanation:

Integrate the following expression.

[tex]\int \frac{-x+5}{(x^2-4x+4)(x+1)}[/tex]

[tex]\hrulefill[/tex]

In order to integrate this function we are gonna have to use partial fraction decomposition. Start by factoring the the denominator completely.

[tex]\int \frac{-x+5}{(x^2-4x+4)(x+1)}dx\\\\\Longrightarrow \int \frac{-x+5}{(x-2)^2(x+1)}dx[/tex]

Now we can apply partial fractions. Partial fractions allows us to split up complex fractions, in doing so this will make them easier to integrate.

[tex]\frac{-x+5}{(x-2)^2(x+1)}=\frac{A}{x+1}+\frac{B}{(x-2)^2}+\frac{C}{x-2}\\\\\Longrightarrow \frac{-x+5}{(x-2)^2(x+1)}=\frac{A}{x+1}+\frac{B}{(x-2)^2}+\frac{C}{x-2} \Big](x-2)^2(x+1)\\\\\Longrightarrow \boxed{-x+5=A(x-2)^2+B(x+1)+C(x-2)(x+1)}[/tex]

Expand the right-hand-side and use the comparison method to find the values of the undetermined coefficients, A, B, and C.

[tex]-x+5=A(x-2)^2+B(x+1)+C(x-2)(x+1)\\\\\Longrightarrow -x+5=Ax^2-4Ax+4A+Bx+B+Cx^2-Cx-2C\\\\\Longrightarrow \boxed{0x^2-x+5(1)=(A+C)x^2+(-4A+B-C)x+(4A+B-2C)(1)}[/tex]

We can now form a system of equations.

For x^2 terms:

[tex]A+C=0[/tex]

For x terms:

[tex]-4A+B-C=-1[/tex]

For #'s:

[tex]4A+B-2C=5[/tex]

[tex]\Longrightarrow \left\{\begin{array}{ccc}A+C=0\\-4A+B-C=-1\\4A+B-2C=5\end{array}\right[/tex]

You can use any method of choice to solve the system of equations. I am going to put the system in a matrix and use my calculator to row reduce.

[tex]\Longrightarrow \left[\begin{array}{ccc}1&0&1\\-4&1&-1\\4&1&-2\end{array}\right] =\left[\begin{array}{c}0\\-1\\5\end{array}\right]\\\\ \\ \ \ \ \Longrightarrow \left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right] =\left[\begin{array}{c}\frac{2}{3} \\1\\-\frac{2}{3}\end{array}\right]\\\\\\\therefore \boxed{A=\frac{2}{3}, \ B=1, \ \text{and} \ C=-\frac{2}{3}}[/tex]

Now we can split up the fraction.

[tex]\frac{-x+5}{(x-2)^2(x+1)}=\frac{A}{x+1}+\frac{B}{(x-2)^2}+\frac{C}{x-2}\\\\\Longrightarrow \boxed{\frac{-x+5}{(x-2)^2(x+1)}=\frac{\frac{2}{3} }{x+1}+\frac{1}{(x-2)^2}+\frac{-\frac{2}{3}}{x-2}}[/tex]

We can integrate the three fractions separately.

[tex]\Longrightarrow \int \frac{\frac{2}{3} }{x+1}dx+ \int \frac{1}{(x-2)^2}dx+\int \frac{-\frac{2}{3}}{x-2}dx\\\\\Longrightarrow \boxed{\frac{2}{3}\int \frac{1 }{x+1}dx+ \int \frac{1}{(x-2)^2}dx-\frac{2}{3}\int \frac{1}{x-2}dx}\\\\[/tex]

For the first integral let u=x+1 => du=dx, for the second let v=x-2 => dv=dx, and for the third integral let w=x-2 => dw=dx

[tex]\Longrightarrow \frac{2}{3} \int \frac{1 }{u}du+ \int \frac{1}{v^2}dv-\frac{2}{3}\int \frac{1}{w}dw\\\\\Longrightarrow \boxed{\frac{2}{3} \int \frac{1 }{u}du+ \int v^{-2}dv-\frac{2}{3}\int \frac{1}{w}dw}[/tex]

Use the rules of integration to integrate.

[tex]\boxed{\left\begin{array}{ccc}\text{\underline{Natural Log Rule:}}\\\\ \int\frac{1}{x}dx=\ln(x) \end{array}\right} \ \ \boxed{\left\begin{array}{ccc}\text{\underline{Power Rule:}}\\\\ \int x^ndx=\frac{x^{n+1}}{n+1} \end{array}\right}[/tex]

[tex]\frac{2}{3} \int \frac{1 }{u}du+ \int v^{-2}dv-\frac{2}{3}\int \frac{1}{w}dw\\\\\Longrightarrow \frac{2}{3}\ln(u)-v^{-1}-\frac{2}{3}\ln(w)+C\\\\\Longrightarrow \frac{2}{3}\ln(x+1)-(x-2)^{-1}-\frac{2}{3}\ln(x-2)+C\\\\\Longrightarrow \frac{2}{3}\ln(x+1)-\frac{1}{x-2} -\frac{2}{3}\ln(x-2)+C\\\\\therefore \boxed{\boxed{\int \frac{-x+5}{(x^2-4x+4)(x+1)}=\frac{2}{3}\ln(x+1)-\frac{1}{x-2} -\frac{2}{3}\ln(x-2)+C}}[/tex]

Thus, the given integral is solved where "C" is some arbitrary constant that can be found given an initial condition.

Boone and Greenville are 36 inches apart from each other.

We have the scale

2 inches = 25 miles

So, for 1 inch the scale = 25/2 = 12.5 miles

Boone and Greenville are 450 miles apart.

Then, the measurement of distance between Boone and Greenville on the map

= 450 / 12.5

= 36 inches

Thus, Boone and Greenville are 36 inches apart from each other.

Learn more about Scale Factor here:

https://brainly.com/question/29464385

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