Liam is standing on a cliff that is 2km tall, he looks out towards the sea from the top of a cliff and notices two cruise liners on is 5km away at a diagonal and the other is 6.8km away at a diagonal. what is the distance between the two cruise liners?

Step-by-step explanation:

suppose 2 more variable y and z

To evaluate the integral of (3x+6)/(2x²-3)dx, we can use partial fraction decomposition:

(3x+6)/(2x²-3) = A/(x-√(3)/2) + B/(x+√(3)/2)

Multiplying both sides by the denominator and simplifying, we get:

3x+6 = A(x+√(3)/2) + B(x-√(3)/2)

Setting x = √(3)/2, we get:

3√(3)/2 + 6 = B(√(3)/2-√(3)/2) = 0

So B = -2√(3). Setting x = -√(3)/2, we get:

-3√(3)/2 + 6 = A(-√(3)/2+√(3)/2) = 0

So A = 2√(3). Therefore, we have:

(3x+6)/(2x^2-3) = 2√(3)/(x-√(3)/2) - 2√(3)/(x+√(3)/2)

Integrating each term, we get:

∫(3x+6)/(2x²-3)dx = 2√(3)ln|x-√(3)/2| - 2√(3)ln|x+√(3)/2| + C

where C is the arbitrary constant.
To evaluate the integral of the function 3x + 6 with respect to x, we will use the integral symbol and find the antiderivative:

∫(3x + 6) dx

To find the antiderivative, we will apply the power rule, which states that the integral of x^n is (x^(n+1))/(n+1), and the constant rule, which states that the integral of a constant is the constant times the variable:

(3 * (x^(1+1))/(1+1)) + (6 * x) + C

Simplifying the expression:

(3x²)/2 + 6x + C

Here, C is the arbitrary constant. So, the evaluated integral of 3x + 6 is:

(3x²)/2 + 6x + C

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The given statement "A function f(x) = 3x² dominates g(x) = x²" is True as it grows faster than the other function.

To show that f(x) dominates g(x), we need to prove that there exists a constant c such that f(x) > c * g(x) for all x > 0.

Let's consider c = 3. Then, for all x > 0, we have:

[tex]f(x) = 3x^2 > 3x^2/1 = 3x^2 * 1 > x^2 * 3 = g(x) * 3[/tex]

A function dominates another function when it grows faster than the other function. In this case, f(x) = 3x² and g(x) = x². Since f(x) has a higher coefficient (3) than g(x) (1) for the x² term, it grows faster than g(x) as x increases.

Therefore, we have shown that f(x) > 3g(x) for all x > 0, which means that f(x) dominates g(x).

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Answer:

x=9

Step-by-step explanation:

These 2 angles are both on a straight line, meaning that the total angle sum is 180°.

We can write an equation:

180=(6x+16)+(12x+2)

combine like terms

180=18x+18

subtract 18 from both sides

162=18x

divide both sides by 18

9=x

Hope this helps! :)

The third term in the sequence has a value of 1.

Here, we have,

Sequence

Given:

f(3) = 1

Let f(x) b a function.

Here x is replaced by "3", f(3) represents the value of the 3rd term of the function, which is 1.  

In this case f(3) = 1 means the value of a  member of the sequence when x = 3 is 1.

For example, if the sequence is the values of x^2-8  from 1 to infinity, then f(1) would have a value 1-8 = -7 and f(3)  would be 9- 8 = 1

The third term in the sequence has a value of 1.

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Hence, 7 + 24 = 25 is a valid equation, and C is the correct response as the right triangle with sides of 7 inches, 25 inches, and 24 inches.

A right quadrilateral relationship between its sides is described by the Pythagorean Theorem, a fundamental theorem of geometry. According to this rule, the hypotenuse's square value, which is the side that forms the right angle, is the same as the total of the squared that compose the other two sides. In other words, the following is how the theorem can be expressed for a quadrilateral with leg of length a, b, and c and a hypotenuse of length c: [tex]a^2 + b^2 = c^2[/tex] . Although it's believed that the Greeks and romans and Indians knew about this theorem before the ancient Greek philosopher Plato, who is recognized with discovering it, gave it its name.

given

The Pythagorean Theorem's equation setup for a right triangle is as follows: [tex]a^2 + b^2 = c^2[/tex]

where c is the length of the hypotenuse and a, b, and c are the lengths of the right triangle's legs.

Right triangle with sides of 7 inches, 25 inches, and 24 inches is shown. Its legs are 7 inches and 24 inches, and its hypotenuse is 25 inches. Hence, we may construct the equation as follows:

[tex]7^2 + 24^2 = 25^2[/tex]

When we simplify this equation, we obtain:

49 + 576 = 625

625 = 625

Hence, 7 + 24 = 25 is a valid equation, and C is the correct response as the right triangle with sides of 7 inches, 25 inches, and 24 inches.

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Answer:

16

Step-by-step explanation:

4y = 2.6

y = 0.65

20y + 3

= 20 × 0.65 + 3

= 13 + 3

= 16

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Answer:

tan A: BC/AB

cos A: AB/AC

sin C: AB/AC

cos C: BC/AC

sin A: BC/AC

Step-by-step explanation:

sin of an angle: opposite/hypotenuse

cosine of an angle: adjacent/hypotenuse

tangent of an angle: opposite/adjacent

The algebraic expression for the weigh of both baskets where each one contains pumpkins and carrots in kilos is equals to the 7x + 2, in kilos.

We have two baskets where each one contains pumpkins and carrots. We have to determine the both baskets weigh together in kilos. Let's assume that

The number of pumpkins in second basket = x kilos

Now, according to first scenario, first basket contains the pumpkins twice as many kilos of pumpkin as in the second basket. That is the number of pumpkins in first basket = 2x kilos

In second case, the second basket there are three more kilos of carrots than there are kilos of pumpkin in the first. So, the number of carrots in second basket

= (3 + 2x ) kilos

In third case, the first basket has 4 kilos less carrot than the second, that is x

=( ( 3 + 2x) - 4 ) kg

Now, weigh of first basket = carrots + pumpkins = (2x + 2x - 1) kilos

= (4x - 1 ) kilos

Weigh of second basket = carrots + pumpkins = (3 + 2x) kilos + x kilos

= (3 + 3x) kilos

So, weigh of both baskets together

= (4x - 1 ) kilos + (3 + 3x) kilos

=( 7x + 2 ) kilos.

Hence, required expression is 7x + 2.

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Complete question:

You have two baskets. Each contains pumpkins and carrots. In the first basket there are twice as many kilos of pumpkin as in the second, and in the second there are three more kilos of carrots than there are kilos of pumpkin in the first. The first basket has 4 kilos less carrot than the second. How many kilos do both baskets weigh together? Represent it algebraically

The spherical coordinates for the given rectangular coordinates (7, 7, 2/7) are (ρ, θ, φ) = (10, 45°, 88.57°) or (10, π/4, 1.545) in radians.

We'll need to find ρ (rho), θ (theta), and φ (phi) using the given rectangular coordinates (x, y, z). Here's the step-by-step process:

1. Calculate ρ (rho): ρ is the distance from the origin to the point in 3D space. You can find it using the formula: ρ = √(x² + y² + z²)

In this case, x = 7, y = 7, and z = 2/7.

Plugging these values into the formula: ρ = √(7² + 7² + (2/7)²) ρ = √(49 + 49 + 4/49) ρ = √(98 + 4/49) ρ = √(4900/49) ρ = 10

2. Calculate θ (theta): θ is the angle in the xy-plane, measured from the positive x-axis. You can find it using the formula: θ = arctan(y/x)

In this case, x = 7 and y = 7.

Plugging these values into the formula: θ = arctan(7/7) θ = arctan(1) θ = 45° (in degrees) or π/4 (in radians)

3. Calculate φ (phi): φ is the angle between the positive z-axis and the line connecting the origin to the point. You can find it using the formula: φ = arccos(z/ρ)

In this case, z = 2/7 and ρ = 10.

Plugging these values into the formula: φ = arccos((2/7)/10) φ = arccos(1/35) φ ≈ 88.57° (in degrees) or ≈ 1.545 (in radians)

So, the spherical coordinates for the given rectangular coordinates (7, 7, 2/7) are (ρ, θ, φ) = (10, 45°, 88.57°) or (10, π/4, 1.545) in radians.

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The values of given conditions are:

1. median=70

2. Lower quartile=40

3. Upper quartile=87

4. Range=170

5. IQR=47

6. Lower outlier threshold=-20.5

7. Upper outlier threshold=160.5

In statistics, the median is the value separating the higher half from the lower half of a dataset. In other words, it is the middle value of a dataset when it is ordered in ascending or descending order.

Here,

To find the median time, we need to arrange the data in order from least to greatest and find the middle value.

10, 15, 30, 35, 40, 40, 45, 55, 60, 62, 64, 64, 66, 68, 70, 70, 72, 75, 75, 80, 82, 84, 90, 90, 105, 110, 120, 180

There are 28 values in the data set, so the median is the average of the 14th and 15th values:

Median = (70 + 70)/2

= 70

To find the lower quartile, we need to find the median of the lower half of the data set:

10, 15, 30, 35, 40, 40, 45, 55, 60, 62, 64, 64, 66, 68

There are 14 values in the lower half, so the lower quartile is the median of these values:

Lower quartile = (40 + 40)/2

= 40

To find the upper quartile, we need to find the median of the upper half of the data set:

72, 75, 75, 80, 82, 84, 90, 90, 105, 110, 120, 180

There are 14 values in the upper half, so the upper quartile is the median of these values:

Upper quartile = (84 + 90)/2

= 87

To find the range, we subtract the smallest value from the largest value:

Range = 180 - 10

= 170

To determine if there are any outliers in the data set, we need to calculate the interquartile range (IQR):

IQR = Upper quartile - Lower quartile

= 87 - 40

= 47

Any value that is more than 1.5 times the IQR below the lower quartile or above the upper quartile is considered an outlier.

Lower outlier threshold = Lower quartile - 1.5IQR

= 40 - 1.547

= -20.5

Upper outlier threshold = Upper quartile + 1.5IQR

= 87 + 1.547

= 160.5

To draw a box-and-whisker plot, we need to plot a box from the lower quartile to the upper quartile, with a line inside the box representing the median. We then draw whiskers extending from the box to the smallest and largest values that are not outliers. The box extends from 40 to 87, with a line at 70 representing the median. The whisker on the left extends to the smallest non-outlier value of 10, and the whisker on the right extends to the largest non-outlier value of 120.

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The product of the expression 5x(x²) is 5x³.

1. Write down the given expression: 5x(x²)
2. Apply the distributive property, which states that a(b + c) = ab + ac. In this case, we have a single term inside the parentheses, so the expression becomes: 5x * x²
3. Multiply the coefficients (numbers) together: 5 * 1 = 5
4. Multiply the variables together, which means adding the exponents since they have the same base (x): x¹* x² = x⁽¹⁺²⁾ = x³
5. Combine the result from steps 3 and 4: 5x³

The product of the expression 5x(x²) can be found by multiplying the coefficients (numbers) and adding the exponents of the variables (letters). In this case, we have 5 times x times x squared.

5 times x equals 5x, and x squared means x times x, so we can rewrite the expression as:

5x(x²) = 5x(x*x) = 5x³

So, the product of the expression 5x(x²) is 5x³.

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The volume of the block after cutting out the smaller cubes is 56/125 cubic centimeters.

The initial volume of the solid wooden cube is given by:

V_initial = (4/5 cm)³ = 64/125 cm³

To find the volume of each of the 8 smaller cubes cut out from the corners, we can use the formula:

V_small cube = (1/5 cm)³= 1/125 cm³

Since we cut out 8 smaller cubes, the total volume of the smaller cubes is:

V_small cubes = 8 x (1/125 cm³) = 8/125 cm³

To find the final volume of the block after cutting out the smaller cubes, we can subtract the volume of the smaller cubes from the initial volume of the block:

V_final = V_initial - V_small cubes

Substituting the values we obtained earlier, we get:

V_final = (64/125 cm³) - (8/125 cm³) = 56/125 cm³

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The correct option is (2) triangle, because the slice intersects the cube face at three non-collinear points.

When a cube is cut by a single slice passing through three non-collinear points on a face, the shape of the cross section will be the intersection of the slice with the cube face. In this case, the slice passes through points A, B, and C, which are non-collinear, and thus the shape of the cross section will be a triangle.

This is because a triangle is the only shape that can be formed by the intersection of a plane with three non-collinear points on a flat surface, such as the face of a cube. The other options of rectangle, square, and trapezoid are not possible since they cannot be formed by the intersection of a plane with three non-collinear points on a flat surface.

A rectangle can only be formed by the intersection of a plane with four points that form a right angle, a square can only be formed by the intersection of a plane with four points that form a right angle and are equidistant from each other, and a trapezoid can only be formed by the intersection of a plane with four points that are not collinear, but only two of which are parallel.

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Answer: Let's denote the number of cubic centimeters of the 20% salt solution as x and the number of cubic centimeters of the 60% salt solution as y.

We know that the total volume of the mixture is 100 cubic centimeters, so we have:

x + y = 100

We also know that the final solution should be a 30% salt solution. This means that the amount of salt in the final solution should be 0.3 times the total volume of the solution:

0.3(100) = 0.20x + 0.60y

where 0.20x represents the amount of salt in the 20% salt solution and 0.60y represents the amount of salt in the 60% salt solution.

We now have two equations with two unknowns:

x + y = 100

0.20x + 0.60y = 30

We can solve for x and y by using any method of linear equations, such as substitution or elimination.

Here, we will use substitution. Solving the first equation for x, we get:

x = 100 - y

Substituting this expression for x in the second equation, we get:

0.20(100 - y) + 0.60y = 30

Simplifying and solving for y, we get:

20 - 0.20y + 0.60y = 30

0.40y = 10

y = 25

So, we need 25 cubic centimeters of the 60% salt solution.

To find the amount of the 20% salt solution, we can substitute this value of y back into either equation:

x + y = 100

x + 25 = 100

x = 75

So, we need 75 cubic centimeters of the 20% salt solution.

Therefore, we need to mix 75 cubic centimeters of the 20% salt solution and 25 cubic centimeters of the 60% salt solution to obtain 100 cubic centimeters of a 30% salt solution.

The hire purchase price for the video recorder is 3,133.20.

To calculate the hire purchase price for the video recorder, follow these steps:

1. Calculate the initial deposit: 20% of the cash price (2,700) is (0.20 * 2,700) = 540.
2. Subtract the deposit from the cash price to get the outstanding balance: (2,700 - 540) = 2,160.
3. Calculate the interest for one year on the outstanding balance: 18% of 2,160 is (0.18 * 2,160) = 388.80.
4. Divide the interest by 12 to find the interest per month: (388.80 / 12) = 32.40.
5. Add the interest per month to the outstanding balance: (2,160 + 32.40 * 12) = 3,133.20.
6. The hire purchase price is 3,133.20, which is the total amount payable in 12 equal monthly instalments.

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Complete question:

A housewife purchased a videorecorder with a cash price of 8 2 700 under hire purchase terms She paid an initial deposit of 20% of the cash price and interest at 18% per annum on the outstanding balance is Charged. The Jamount payable is paid in 12 equal monththly instalments Calculate for the video recorder  The hire purchase price​

Faith needs a prosthetic hand because she lost her left hand at birth and requires assistance holding objects.

According to the article "7-Year-Old Girl Gets New Hand from 3-D Printer," Faith is a young girl who was born with a condition that caused her to lose her left hand. As a result, she has difficulty holding onto objects such as paper and bicycle handlebars. Faith had been waiting for an opportunity to design her own prosthetic hand, but the waiting list for other types of treatment was too long. Fortunately, a team of students and educators at a local university were able to create a prosthetic hand for her using a 3-D printer. The new hand will allow Faith to have greater independence and mobility, and she is excited to be able to participate in activities she was previously unable to do. This story is an example of how technology can be used to improve the lives of individuals with disabilities and provide them with greater opportunities to participate fully in everyday life.

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Answer: The mean amount of rainfall per day is 16 cm.

Step-by-step explanation: Finding the total of all the rainfall amounts and dividing it by the total number of days will estimate the mean amount of rain that falls each day. We will use the midpoint technique, which assumes that the rainfall values in each interval have equal distributions, to calculate the mean.

Here is how to calculate it:

Midpoint of 0 < r ≤ 10 = (0+10)/2 = 5

Midpoint of 10 < r ≤ 20 = (10+20)/2 = 15

Midpoint of 20 < r ≤ 30 = (20+30)/2 = 25

Midpoint of 30 < r ≤ 40 = (30+40)/2 = 35

Midpoint of 40 < r ≤ 50 = (40+50)/2 = 45

The formula for calculating average rainfall is (95 + 1315 + 525 + 235 + 1*45) / (9 + 13 + 5 + 2+1) = (45 + 195 + 125 + 70 + 45) / 30 = 480 / 30 = 16

Consequently, the estimated average daily rainfall is 16 cm.

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The mean number of points Kathleen made is 38.

To calculate the mean number of points Kathleen made, we will use the following terms: mean, sum, and total number of assignments.

The mean is the average value of a set of numbers. To find the mean, we need to sum all the given values and then divide the sum by the total number of values in the set.

Kathleen's assignment scores are 29, 38, 45, 42, and 36 points. To find the sum, we add these numbers together: 29 + 38 + 45 + 42 + 36 = 190 points.

Now, we need to determine the total number of assignments. Kathleen has completed five assignments. So, we will divide the sum of her points (190) by the total number of assignments (5) to find the mean.

Mean = Sum / Total number of assignments
Mean = 190 / 5
Mean = 38

The mean number of points Kathleen made on her assignments is 38 points. This indicates that on average, she scored 38 points per assignment. Calculating the mean gives us a general idea of her performance across all assignments, allowing us to gauge her overall progress.

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Therefore , the solution of the given problem of unitary method comes out to be  the carpenter divided the wood into 60 pieces.

The well-known minimalist approach, current variables, and any crucial elements from the initial Diocesan tailored query can all be used to accomplish the work. In response, you can be granted another chance to utilise the item. If not, important impacts on our understanding of algorithms will vanish.

Here,

Divide the overall length of the wood by the length of each piece to get how many pieces the carpenter cut.

Each piece measures 0.25 feet, or 1/4 of a foot.

By dividing the overall length of the wood by the length of each component, we can determine the number of pieces:

=> Quantity = Length overall / Length of each element

=> Piece count is 15 feet divided by 0.25 feet. or 15/0.25

=> There are 60 parts total.

So, the carpenter divided the wood into 60 pieces.

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A quadratic function should have an exponent of 2, a parabola ending its life by going down should have a leading coefficient sign of negative, and either y = x^2 or y = -x^2 can be a valid equation for a quadratic function, with the choice depending on the direction of the desired parabola.

1. The quadratic should have an exponent of 2, as a quadratic is a polynomial of degree 2.
2. A parabola ending its life by going down should have a leading coefficient sign of negative, as this indicates that the quadratic term has a negative coefficient and the parabola opens downwards.
3. Both equations, y = x^2 and y = -x^2, are valid equations for a quadratic function. The main difference between them is the direction in which the parabola opens. The equation y = x^2 represents a parabola that opens upwards, while y = -x^2 represents a parabola that opens downwards. The choice of which equation to use depends on the specific context and the direction of the desired parabola.

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The given expression x²y − 7xy + xyz is a 3rd degree polynomial.

To classify it by degree and term, let's first determine the degree of each term:

1. x²y: The degree is the sum of the exponents of the variables (x and y). Here, the degree is 2 (from x²) + 1 (from y) = 3.
2. 7xy: The degree is 1 (from x) + 1 (from y) = 2.
3. xyz: The degree is 1 (from x) + 1 (from y) + 1 (from z) = 3.

Now, we can classify the expression:

- Degree: Since the highest degree among the terms is 3, the expression is a 3rd-degree polynomial.
- Term: There are three terms in the expression, so it is a trinomial.

In summary, the given expression, x²y − 7xy + xyz, is a 3rd-degree trinomial.

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The probability of getting a sample mean greater than 85 thunderstorms per month for a sample of ten months is 0.0008.

The probability that the number of thunderstorms in a single month is greater than 85 can be found using the z-score formula.

z = (85 - 75) / 20 = 0.5

Using a standard normal distribution table, the probability of z being less than 0.5 is 0.6915. So the probability of having more than 85 thunderstorms in a single month is 1 - 0.6915 = 0.3085 or about 30.85%.

t = (85 - 75) / 2.00 = 5.00

Using a t-distribution table with 9 degrees of freedom, the probability of t being greater than 5.00 is very close to 0. Therefore, the probability of having a mean of more than 85 thunderstorms per month in a sample of ten months is extremely low.

Therefore, the probability of getting a sample mean greater than 85 thunderstorms per month for a sample of ten months is 0.0008.

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There are no other combinations to consider.

To determine the possible combinations, since Chris wants to list the thirteen states in chronological order as they continued to join the union, there is only one combination possible.

This is because the order is based on the dates the states joined, which is a fixed sequence. Here are the steps for listing the states:

1. Begin with Delaware, as it was the first state to join the union on December 17, 1787.
2. Follow with Pennsylvania, which joined on December 12.
3. Add New Jersey, which joined on December 18.
4. Continue listing the remaining ten states in the order they joined.

As the order is determined by the joining dates, there are no other combinations to consider.

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y=4x+3

Parallel lines have the same gradient - 4x

substitute the x and y values from the coordinates into y=mx+c

so

-5=(4×-2)+c

-5=-8+c

c=3

therefore, the answer is y=4x+3

Considering the function f(x) = x(x-4), if the point (2+c, y) is on the graph of f(x), then the following point will also be on the graph of f(x): (2-c, y). Explanation: Since f(x) is symmetric with respect to the vertical line x = 2 (due to the fact that f(x) = x(x-4) = (x-2+2)(x-2) = (x-2)^2 - 2^2), if the point (2+c, y) is on the graph, then its symmetric counterpart, (2-c, y), will also be on the graph.

The definition of a function in mathematics can also be interpreted as a relation that connects each member of x in a set called the domain with a single value f(x) from a second set called the codomain.

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The absolute value of x can be written in the form

x-(1/2)x=0 or |x| = (1/2)x

To write the absolute value in the form x-b=c where b is a number and c can be either a number or expression, you can use the following steps:

1. Start with the absolute value expression: |x|

2. Recall that the absolute value of a number is the distance of that number from zero on the number line. So, we can rewrite |x| as the distance between x and 0 on the number line.

3. To write this distance in the form x-b, we need to find a value for b that represents the midpoint between x and 0. That is, we need to find the number that is halfway between x and 0 on the number line.

4. The midpoint between x and 0 is given by the expression (x + 0)/2, which simplifies to x/2.

5. So, we can write the absolute value expression |x| as the distance between x and 0, which is the same as the distance between x and x/2 + x/2.

6. Simplifying this expression, we get:

|x| = |x - x/2 - x/2|

7. Rearranging terms, we get:

|x| = |(1/2)x - (1/2)x|

8. Finally, we can write the absolute value in the form x-b=c by setting b = (1/2)x and c = 0, which gives us:

|x| = |x - (1/2)x - 0| = |(1/2)x - 0|

So, the absolute value of x can be written in the form x-(1/2)x=0, or in other words:

|x| = (1/2)x

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The derivative of y = ln[tex](9x^2-7x+4)[/tex] is y' = (18x-7) / [tex](9x^2-7x+4)[/tex].

To differentiate f(x) = ln[tex]((x^8-2)/x[/tex]), we use the chain rule and the quotient rule:

f'(x) = [[tex](x^8[/tex]-2)/x]' / (x^8-2)/x + ln[tex]((x^8-2[/tex])/x)'

[tex]= [((x^8-2)'x - (x^8-2)x') / x^2] / (x^8-2)/x + [(1/x)'(x^8-2) - (1)'x(x^8-2)/x^2][/tex]

[tex]= [(8x^7)(x) - (x^8-2)] / x^2(x^8-2)/x + [(1/x)(x^8-2)/x^2] - (1)(x^8-2)/x^2[/tex]

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

[tex]= (7x^8-4) / (x^2(x^8-2)) - (x^8-2) / (x^3(x^8-2))[/tex]

Simplify to get:

[tex]f'(x) = (6x^8-4) / (x^3(x^8-2))[/tex]

Therefore, the derivative of f(x) = ln[tex]((x^8-2)/x) is f'(x) = (6x^8-4) / (x^3(x^8-2)).[/tex]

To differentiate y = ln[tex](9x^2-7x+4)[/tex], we use the chain rule:

y' =[tex][(9x^2-7x+4)' / (9x^2-7x+4)][/tex]

[tex]= [(18x-7) / (9x^2-7x+4)][/tex]

Simplify to get:

[tex]y' = (18x-7) / (9x^2-7x+4)[/tex]

Therefore, the derivative of y = [tex]ln(9x^2-7x+4) is y' = (18x-7) / (9x^2-7x+4).[/tex]

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The values of U and d for an arithmetic sequence with U20 = 100 and U25 = 115 is U =  43 and d = 3.

The formula for the nth term of an arithmetic sequence: Un = U1 + (n-1)d

We know that U20 = 100 and U25 = 115, so we can set up two equations using the formula above:

U20 = U1 + 19d = 100    

U25 = U1 + 24d = 115

We now have two equations with two variables (U1 and d) that we can solve for.

First, we'll isolate U1 in the first equation:

U1 = 100 - 19d

Then we'll substitute this expression for U1 into the second equation and solve for d:

100 - 19d + 24d = 115

5d = 15

d = 3

Substitute d = 3 in the equation, U1 = 100 - 19d

So, U1 = 100 - 19(3) = 43.

Therefore, the values of U and d for the arithmetic sequence are U= 43 and d = 3.

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The total mass of the wire is [tex]M = 9\rho * (\pi/2).[/tex]

Using the formula for finding the centroid of a two-dimensional object with uniform density:

cy = (1/Area) * ∫(y*dA)

The equation of the circle is [tex]x^2 + y^2 = 25[/tex]. Solving for y, we get:

[tex]y = \sqrt(25 - x^2)[/tex]

Since the wire is in the first quadrant, the limits of integration are 0 ≤ x ≤ 5 and 0 ≤ y ≤ [tex]\sqrt(25 - x^2).[/tex]

To find the area of the wire, we integrate:

[tex]Area = \int \int dA = \int 0^5 \int 0^{\sqrt(25-x^2)}dy dx[/tex]

[tex]= \int 0^{5 (sqrt(25-x^2))}dx[/tex]

[tex]= (1/2) * [25sin^{(-1)(x/5)} + x\sqrt(25-x^2)] from 0 to 5[/tex]

[tex]= (1/2) * [25\pi/2] = 25\pi/4[/tex]

To find the centroid (cy), we integrate:

[tex]cy = (1/Area) * \int(ydA) = (1/(25\pi/4)) * \int0^5 \int0^{\sqrt(25-x^2)} y dy dx[/tex]

[tex]= (4/25*\pi) * \int0^5 [(1/2)*y^2]_0^{\sqrt(25-x^2)} dx[/tex]

[tex]= (4/25\pi) * \int 0^5 [(1/2)(25-x^2)] dx[/tex]

[tex]= (4/25\pi) * [(25x - (1/3)*x^3)/2]_0^5[/tex]

[tex]= (4/25\pi) * [(255 - (1/3)*5^3)/2][/tex]

[tex]= 50/3[/tex]

Therefore, the centroid of the wire is cy = 50/3.

Now use the formula for the mass of a thin wire for total mass:

M = ∫ρ ds

Since the wire has uniform density, the linear density is constant and can be factored out of the integral:

M = ρ * ∫ds

The differential element of arc length is:

[tex]ds = \sqrt(dx^2 + dy^2) = \sqrt((-9sin t)^2 + (9cos t)^2) dt[/tex]

[tex]= 9\sqrt(sin^2 t + cos^2 t) dt = 9 dt[/tex]

Integrating from 0 to pi/2, we get:

[tex]M = \rho * \int ds = \rho * \int 0^{(\pi/2)} 9 dt[/tex]

[tex]= 9\rho * [t]_0^{(\pi/2)} = 9\rho * (\pi/2)[/tex]

Therefore, the total mass of the wire is [tex]M = 9\rho * (\pi/2).[/tex]

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