Use substitution to find the indefinite integral. szve ? 2 zV9z+ - 5 dz [21922-5 Zy9z 9z- 5 dz = =]

The area of the sandbox whose base is 3.5 meter and height is 2.4 meter is 4.2 m².

Given:

Base = 3.5 m

Height = 2.4 m

First, the area of the sandbox formula:

Area = 1/2 x base x height.

Next,  substitute b = 3.5 meters and h = 2.4 meters.

Area = 1/2 * (3.5) * (2.4).

Now, simplify by multiplying 1/2, 2.4, and 3.5.

Area = 1/2 x 2.4 x 3.5

Area = 4.2 square meters.

Thus, The area of the sandbox is 4.2 m².

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The question attached here seems to be incomplete, the complete question is:

Lokota wants to build a sandbox for his little brother. Determine the amount of sand he needs by finding the area of the sandbox. Use the drop-down menus to complete the statements.

First, write the formula: A = 1/2 bh
Next, use parentheses when you substitute __ for b and __ for h.

Now, simplify by ___ 1/2, 2.4, and 3.5.

The area of the sandbox is ___ m²

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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The shape of the distribution  in histogram is skewed left and mean=43.071429 and standard deviation= 15.886445 are used together to measure the center and spread of the data.

What is histogram?

A grouped frequency distribution with continuous classes is graphically represented by a histogram. It is an area diagram, and its size is proportional to the frequencies in the associated classes. Due to the base's coverage of the spaces between class boundaries, all of the rectangles in such representations are contiguous.

We can make a sideways histogram. (makes it easier if typing text, but if you are writing on paper then you don't have to make it sideways) Each row will represent a range of 10 (1st row is 10 to 19, 2nd row is 20 to 29, etc)

=> 13 23 28 33 34 38 44 45 49 50 55 59 65 67

The data looks like it is slightly skewed left (hump towards the right [down]). The mean and the standard deviation are used together to measure the center and spread of the data.

Mean = [tex]\frac{ 13+23+28+33+34+38+44+45+49+50+55+59+65+67}{14}=\frac{603}{14}[/tex] = 43.071429

Standard deviation = 15.886445

Hence the shape of the distribution  in histogram is skewed left and mean=43.071429 and standard deviation= 15.886445 are used together to measure the center and spread of the data.

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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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The cost of each item, obtained from the equation for the sum of the costs of the item are;

A brush costs $3.5

A sketchbook costs $10.5

A paint set costs $21

An equation is a mathematical statement that expresses equivalence between two expression joined by an '=' sign.

The amount Camille brought to the art supply = $39.50

The cost of the brush = (1/3) × The cost of the sketchbook

Cost of the sketchbook = (1/2) × Cost of the paint set

Amunt Camille had left over = $4.50

The cost of the items Camille bought = $39.50 - $4.5 = $35

Let x represent the cost of the brush, let y represent the cost of the sketchbook and let z represent the cost of the paint set

Therefore, we get the following equation; x + y + z = 35

x = (1/3)·y

y = (1/2)·z

Which indicates;

x = (1/3) × (1/2)·z = (1/6)·z

From which we get; (1/6)·z + (1/2)·z + z = 35

(5/3)·z = 3

z = 35 × 3/5 = 21

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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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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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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 testimonies of Nazi officials at the Nuremberg Trials were incredibly important because they provided valuable insight into the atrocities committed by the Nazi regime.

Their testimonies helped to dispel any claims of denial that the events of the Holocaust had occurred. The confessions given by the Nazi officials gave detailed accounts of the crimes committed by the regime, and helped to hold those responsible accountable for their actions. It is important to note that while some Nazis did receive lighter sentences because of their confessions, the vast majority of those involved were held fully responsible for their crimes. Overall, the testimonies of Nazi officials played a crucial role in bringing justice to the victims of the Holocaust and shedding light on the horrific actions of the Nazi regime.

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Point B is not the midpoint of line AC, because angle AOB is not half of angle  AOC.

If point B is the midpoint of line AC, then angle AOB must be equal to angle  BOC.

The value of angle AOC is calculated as follows;

let angle AOC = θ

cos θ = 100 yds / 500 yds

cos θ = 0.2

θ = cos⁻¹ (0.2)

θ = 78.5⁰

The value of length AC is calculated as follows;

AC = √ (500² - 100²)

AC = 489.9

If point B is the midpoint, then AB = BC = 489.9/2 = 244.95

The value of angle AOB is calculated as follows;

tan β = AB/AO

tan β = 244.95/100
tan β = 2.4495

β = arc tan (2.4495)

β = 67.8⁰

Half of angle AOC = 78.5⁰/2 = 39.25⁰

β  ≠ 39.25⁰

So point B is not midpoint of line AC, since angle AOB is not half of angle  AOC.

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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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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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Based on the equation,  the expected average number of leaves on a tree 8 weeks after spring has started would be approximately 1966.64.

To write a linear equation that can be used to approximate the data distribution, we need to use the two given points on the scatter plot. Let's assume the first point is (0, 1700) and the second point is (6, 1900).

The slope of the line passing through these points can be calculated as:

slope = (1900 - 1700) / (6 - 0) = 200 / 6 = 33.33 (approx)

Using the point-slope form of a linear equation, we can write:

y - 1700 = 33.33(x - 0)

Simplifying, we get:

y = 33.33x + 1700

Therefore, the linear equation that can be used to approximate the data distribution is: Y = 33.33x + 1700 (Option C)

To find the expected average number of leaves on a tree 8 weeks after spring has started, we need to substitute x = 8 in the above equation and solve for Y:

Y = 33.33(8) + 1700 = 1966.64 (approx)

Therefore, the expected average number of leaves on a tree 8 weeks after spring has started would be approximately 1966.64.

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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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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.

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 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 image coordinates of N′M′O′ if the preimage is translated 7 units to the left is D- N′(−11, −2), M′(−8, −1), O′ (−11, −5)

A triangle is seen as a closed, two-dimensional geometric figure that has three straight sides and three angles.

To get the image coordinates of the preimage translated 7 units to the left, we simply subtract 7 from the x-coordinates of each vertex:

N' = (Nx - 7, Ny) = (−4 - 7, −2) = (−11, −2)

M' = (Mx - 7, My) = (−1 - 7, −1) = (−8, −1)

O' = (Ox - 7, Oy) = (−4 - 7, −5) = (−11, −5)

Therefore, the image coordinates of NMO after the translation 7 are: N′(−11, −2), M′(−8, −1), O′ (−11, −5)

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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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The first derivative is dy/dx = [tex]-20x^3[/tex], and the second derivative is d^2y/dx^2 = -60x^2.

The function you provided is: y =[tex]-5x^4 + 1[/tex]

To find the first derivative (dy/dx), we'll use the power rule which states that if y = x^n, then dy/dx =[tex]n * x^(n-1)[/tex].

Applying this rule to each term, we get: dy/dx = [tex]d(-5x^4)/dx + d(1)/dx[/tex]dy/dx =[tex]-5(4x^(4-1)) + 0[/tex] (since the derivative of a constant is 0) dy/dx = [tex]-20x^3[/tex]

Now, to find the second derivative [tex](d^2y/dx^2)[/tex], we'll differentiate the first derivative again using the power rule: [tex]d^2y/dx^2 = d(-20x^3)/dx d^2y/dx^2 = -20(3x^(3-1)) d^2y/dx^2 = -60x^2[/tex]

So, the first derivative is dy/dx = [tex]-20x^3[/tex], and the second derivative is [tex]d^2y/dx^2 = -60x^2.[/tex]

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The approximate area of the triangle is 21. 5 square units (option c).

Triangles are three-sided polygons that can have different shapes and sizes. Now, let's focus on your question about finding the area of a triangle.

To begin with, the area of a triangle is given by the formula:

Area = (base × height) ÷ 2

where the base is the length of the side that is perpendicular to the height. The height, on the other hand, is the distance between the base and the opposite vertex.

In your problem, the base of the triangle is given as 7 units, and the height is given as 6.1 units. So, we can substitute these values into the formula to get:

Area = (7 × 6.1) ÷ 2

Area = 21.35 square units

Therefore, the approximate area of the triangle is 21.35 square units. In the answer choices provided, the closest option is C, which is 21.5 square units.

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Complete Question:

What is the approximate area of the triangle when base = 7 and height = 6.1?

A. 12. 5 square units

B. 18 square units

C. 21. 5 square units

D. 31 square units

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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1) The discount percentage is 16.07%.

2) The discounted price of F₁, F₂ is $470,000

3) The amount that Adam had after purchasing F₁, F₂ is $530,000.

The discount percentage is a product of the discount amount divided by the original cost, multiplied by 100.

Price of item F = $280,000

Discounted price = $235,000

Discount amount = $45,000 ($280,000 - $235,000)

Discount rate = 16.07% ($45,000/$280,000 x 100)

Discounted price of F₁, F₂ = $470,000 ($235,000 x 2)

The amount that Adam had before the purchase = $1,000,000

The amount he had after purchasing F₁, F₂ = $530,000 ($1,000,000 - $470,000).

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F(x) and G(x) are polynomial functions in which the expression f(x) + g(x) is always a polynomial and f(x)/g(x) is sometimes a polynomial.

Let F(x) be a polynomial = 2x + 4

g(x) be a polynomial = 6x² + 12x

putting the value in the expression

f(x) + g(x) = 2x + 4 + 6x² + 12x

f(x) + g(x) = 6x² + 14x + 4

6x² + 14x + 4 is a polynomial

Now, putting the value in the equation

f(x)/g(x) = 2x + 4/6x² + 12x

Taking 3x common from 6x² + 12

We get 3x(3x+4)

f(x)/g(x) = 2x+4/3x(2x+4)

f(x)/g(x) = 1/3x

1/3x is not a polynomial

Hence, it sometimes a polynomial.

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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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The equation of the tangent line is y = x - 1.

To find the equation of the tangent line of y=xlog(x) at the point (1,0), we will first need to find the derivative of the function y=xlog(x) with respect to x.

Step 1: Find the derivative of y=xlog(x) with respect to x.
Using the product rule, (uv)' = u'v + uv', where u=x and v=log(x).

u' = derivative of x with respect to x = 1
v' = derivative of log(x) with respect to x = 1/x

Now, apply the product rule:
y' = u'v + uv' = 1*log(x) + x*(1/x) = log(x) + 1

Step 2: Find the slope of the tangent line at the point (1,0).
Evaluate y' at x=1:
y'(1) = log(1) + 1 = 0 + 1 = 1

The slope of the tangent line at (1,0) is 1.

Step 3: Find the equation of the tangent line.
We will use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) is the point (1,0) and m is the slope (1).

y - 0 = 1(x - 1)
y = x - 1

The equation of the tangent line of y=xlog(x) at the point (1,0) is y = x - 1.

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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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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

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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