(07.01, 07.02 MC)

An expression is shown below:

6x2y − 3xy − 24xy2 + 12y2

Part A: Rewrite the expression by factoring out the greatest common factor. (4 points)

Part B: Factor the entire expression completely. Show the steps of your work. (6 points)

Answer:

Step-by-step explanation:

Given rectangle RSTU with diagonals SU = 4x+2 and RT= 5x-7 that meet at point V with angle VTS = 39°, you want the measures of RV and angle VSR.

The diagonals of a rectangle bisect each other and are congruent:

  SU = RT

  4x +2 = 5x -7

  9 = x

And RV = RT/2:

  RV = (5x -7)/2 = (5·9 -7)/2

  RV = 19

The base angles in each of the isosceles triangles are congruent. That means ∠VST = ∠VTS = 39°. The angle of interest, ∠VSR is the complement of angle VST, so is ...

  ∠VSR = 90° -39°

  ∠VSR = 51°

The distance from the bottom of the slide to the bottom of the ladder is approximately 160.9 feet.

We can use the Pythagorean theorem to find the distance from the bottom of the slide to the bottom of the ladder. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is the distance we want to find, and the other two sides are the length of the slide (150 feet) and the height of the ladder (58 feet). Therefore, we have:

Distance² = 150² + 58²

Distance²= 22,500 + 3,364

Distance² = 25,864

Taking the square root of both sides, we get:

Distance = 160.9 feet (rounded to the nearest tenth)

Therefore, the distance from the bottom of the slide to the bottom of the ladder is approximately 160.9 feet.

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The area of the region outside r=8sin(θ) and inside r=24sin(θ) is 160π/3 square units.

To find the area, we first need to find the limits of integration for θ. The two curves intersect at θ=π/6 and θ=11π/6. Thus, we integrate from θ=π/6 to θ=11π/6. Next, we use the formula for the area of a polar region, which is given by: A = ∫[a,b] 1/2[r(θ)]^2 dθ

where r(θ) is the distance from the origin to the curve as a function of θ. In this case, we have:

A = ∫[π/6,11π/6] 1/2[(24sin(θ))^2 - (8sin(θ))^2] dθ

Simplifying the expression and evaluating the integral, we get:

A = (160π)/3

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Answer:116

angles in a triangle is 180.

Answer:

1=116

2=32

Thank You

The probability that a student learns more than 11 years is approximately 0.0548 or 5.48%

To find the probability that a student learns more than 11 years, we need to calculate the area under the normal distribution curve to the right of 11.

Given that the mean of the distribution is 7 years and the standard deviation is 2.5 years, we can standardize the value of 11 using the formula:

z = (x - μ)/σ

= (11 - 7)/2.5

= 1.6

We can then use a standard normal distribution table or calculator to find the probability that a standard normal random variable is greater than 1.6. Using a calculator, we get:

P(Z>1.6) = 0.0548

Therefore, the probability that a student learns more than 11 years is approximately 0.0548 or 5.48%

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The sample space of this experiment is {1,1}, {1,2}, {1,3}, {1,4}, {2,1}, {2,2}, {2,3}, {2,4}, {3,1}, {3,2}, {3,3}, {3,4}, {4,1}, {4,2}, {4,3}, {4,4}.

To explain further, the experiment involves selecting two parts from a box containing four identical parts numbered 1, 2, 3, and 4. The selection is done with replacement, meaning that after each part is selected, it is put back into the box before the next selection. Also, the order of the parts matters, so selecting part 1 first and part 2 second is different from selecting part 2 first and part 1 second.

The sample space of an experiment refers to the set of all possible outcomes. In this case, there are 16 possible outcomes, as shown above. Each outcome is equally likely to occur, assuming that the parts are truly identical and the selection process is random.

Knowing the sample space is important in probability theory because it allows us to calculate the probability of each possible outcome and make predictions about the likelihood of certain events occurring. For example, we can calculate the probability of selecting two parts with a sum greater than 6 or the probability of selecting two identical parts.

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The solution is:  the total surface area of the cylinder is: 208.92 unit^2.

Here, we have,

we know that,

Area of ends:

Area of Circle = πr²

given,  radius is 3.5 and the height is 6

so, we get,

Area of end = π3.5²=49/4π

There are two ends so we multiply that by 2 to get 49/2π

Area of Rest:

First, we need to find the circumference using the equation: πd

πx7=7π

Then to find the area we just need to multiply 7π by the height

7π x 6 = 42π

Total surface area

we now just need to add them together

49/2π + 42π = 133/2π

                    = 208.92

Hence, The solution is:  the total surface area of the cylinder is: 208.92 unit^2.

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The surface area of the given cylinder is 208.81 square units.

Given that, the radius of a cylinder is 3.5 units and the height is 6 units.

We know that, the total surface area of a cylinder is 2πr(r + h).

Here, surface area = 2×3.14×3.5×(3.5+6)

= 2×3.14×3.5×9.5

= 208.81 square units

Therefore, the surface area of the given cylinder is 208.81 square units.

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The required scientific notation is written as the weight of cans = 2.07837 x 10⁹ g.

The weight of one empty can is 0.03417 grams. The number of cans recycled in 2011 was 6.1 x 10¹⁰. To find the total weight of the cans recycled in 2011, we need to multiply these two numbers together.

Weight of cans = number of cans * weight of one can

Weight of cans = 6.1 x 10¹⁰* 0.03417

Weight of cans = 2.07837 x 10⁹ grams

The weight of the cans recycled in 2011 is 2.07837 x 10⁹ grams. In scientific notation, this is written as the weight of cans = 2.07837 x 10⁹ g.

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The Laplace transform of f(t) is a function of s, given by 5 / (s² + 100).

To find ℒ{f(t)} for the function f(t) = sin(5t) cos(5t), we can first use the trigonometric identity:
sin(2θ) = 2 sin θ cos θ
We can apply this identity to the product of sin(5t) and cos(5t) in f(t):
sin(5t) cos(5t) = 1/2 sin(2(5t))
Using the Laplace transform property for a scaled and shifted function:
ℒ{sin(at)} = a / (s² + a²)
We can find ℒ{1/2 sin(2(5t))} as:
1/2 ℒ{sin(10t)} = 1/2 × 10 / (s² + 10²) = 5 / (s² + 100)
Therefore, we can write ℒ{f(t)} as:
ℒ{sin(5t) cos(5t)} = ℒ{1/2 sin(2(5t))} = 5 / (s² + 100)
So the Laplace transform of f(t) is a function of s, given by 5 / (s² + 100).

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Answer: B.

Step-by-step explanation:

D. C. B.

100%

Therefore, the integral (0,5) 3/2 x-6 can be interpreted as the area of the upper triangle minus the area of the lower triangle, which is (-3.75) - (-15) = 11.25.

The integral (0,5) 3/2 x-6 represents the area under the curve of the function 3/2 x-6 from x=0 to x=5. This area can be split into two triangles, one above the x-axis and one below it. The area of the lower triangle is given by 1/2 base x height, where the base is 5-0=5 and the height is the value of the function at x=0, which is -6. So the area of the lower triangle is 1/2 (5)(-6) = -15.

The area of the upper triangle is given by the same formula, where the base is still 5 but the height is now the value of the function at x=5, which is -3/2. So the area of the upper triangle is 1/2 (5)(-3/2) = -3.75.

Therefore, the integral (0,5) 3/2 x-6 can be interpreted as the area of the upper triangle minus the area of the lower triangle, which is (-3.75) - (-15) = 11.25.

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To find the eigenvalues of matrix a, we can start by finding the characteristic polynomial det(a - λI), where I is the identity matrix and λ is an unknown constant.

Using the cofactor expansion method along the first row, we get:

det(a - λI) = (0.3 - λ)(-1)^(1+1) det(0.1 0.4 0 0.4) + (-1)^(1+2) (0 - λ) det(0 0.4 0.1 0.4) + (0.2)(-1)^(1+3) det(0 0.1 0.4 0.1; 0.4 0 0.4 0; 0 0.4 0.1 0.4; 0.4 0 0 0.1)

Simplifying this expression, we get:

det(a - λI) = (0.3 - λ)[(0.1)(0.4)(0.4) + (0.4)(0.4)(0.1) + (0.4)(0.1)(0.4)] - (0.2)(0.4)(0.1)(0.4) - (0.4)(0.4)(0.1)(0.1)

det(a - λI) = -λ^3 + 1.2λ^2 - 0.4λ

Next, we can solve for the roots of this polynomial by setting it equal to zero:

-λ^3 + 1.2λ^2 - 0.4λ = 0

Factorizing out a λ term, we get:

λ(-λ^2 + 1.2λ - 0.4) = 0

Using the quadratic formula to solve for the roots of -λ^2 + 1.2λ - 0.4, we get:

λ = 0.2, 0.4, 0.6

Therefore, the eigenvalues of matrix a are λ = 0.2, 0.4, and 0.6.

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Mr. Adams divides 183 markers equally among the 24 students in his class. To find the least number of extra markers in the box, divide the total markers (183) by the number of students (24). The result is 7 with a remainder of 15. So, the least number of extra markers in the box is 15.

Mr. Adams divides 183 markers equally among the 24 students in his class, which means each student gets 7 markers. However, since 7 does not divide evenly into 183, there will be some markers left over. To determine the least number of extra markers in a box, we need to find the remainder when 183 is divided by 24.
Using long division, we get:

   24 | 183
       -----
         7  6
         -----          
This means that there are 6 markers left over that Mr. Adams puts in a box. Therefore, the least number of extra markers in a box is 6.
Mr. Adams divides 183 markers equally among the 24 students in his class. To find the least number of extra markers in the box, divide the total markers (183) by the number of students (24). The result is 7 with a remainder of 15. So, the least number of extra markers in the box is 15.

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a. The distribution of X is normal with mean 9.7 ppm and standard deviation 1.5 ppm: X ~ N(9.7, 1.5)

b. The distribution of the sample mean is also normal with mean μ = 9.7 ppm and standard deviation σ/√n = 1.5/√36 = 0.25 ppm: ~ N(9.7, 0.25)

c. We need to find P(X > 9.4).Looking up the standard normal distribution table, we find P(Z > -0.2) = 0.5793. Therefore, the probability that one randomly selected city's waterway will have more than 9.4 ppm pollutants is 0.5793.

d.  Therefore, the probability that the average amount of pollutants is more than 9.4 ppm is 0.8849.

e. Yes, the assumption that the distribution is normal is necessary because we are using the central limit theorem to approximate the distribution of the sample mean. The central limit theorem applies only when the sample size is sufficiently large (n ≥ 30) and the population distribution is approximately normal.

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The stencil patterns per square feet she have when completed are:

[tex]\[ \text{Number of stencil patterns per square foot} = \frac{{13n}}{{160}} \text{ patterns/ft}^2 \][/tex]

To find the number of stencil patterns per square foot, we first need to calculate the total area of the floor in square feet. The floor measures [tex]8[/tex] feet by [tex]20[/tex] feet, so its total area is given by:

[tex]\[ \text{Area of the floor} = \text{Length} \times \text{Width} = 8 \text{ ft} \times 20 \text{ ft} = 160 \text{ ft}^2 \][/tex]

Next, we need to determine the total number of stencil patterns that will be used. The homeowner has [tex]13[/tex] different stencils. However, we don't know how many times each stencil will be repeated, so we'll assume that each stencil is used an equal number of times.

Let's denote the number of times each stencil is used as [tex]n[/tex]. Then the total number of stencil patterns used is given by [tex]\( 13 \times n \)[/tex].

To find the number of stencil patterns per square foot, we divide the total number of stencil patterns by the total area of the floor:

[tex]\[ \text{Number of stencil patterns per square foot} = \frac{{13 \times n}}{{\text{Area of the floor}}} = \frac{{13 \times n}}{{160 \text{ ft}^2}} \][/tex]

Since we don't have a specific value for [tex]n[/tex], we can express the answer in terms of [tex]n[/tex]:

[tex]\[ \text{Number of stencil patterns per square foot} = \frac{{13n}}{{160}} \text{ patterns/ft}^2 \][/tex]

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This means that Jason is in the 92nd percentile. The answer is d. 92nd.

What is mean?

In statistics, the mean (also known as the arithmetic mean or average) is a measure of central tendency that represents the sum of a set of numbers divided by the total number of numbers in the set.

To calculate the mean of a set of numbers, you add up all the values in the set, and then divide the sum by the total number of values.

Assuming a normal distribution, we know that approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations.

Since Jason's exam score is 1.41 standard deviations above the mean, we can say that approximately 92% of the data falls below his score (since 1.41 standard deviations above the mean is approximately the same as the mean plus 1.41 standard deviations). This means that Jason is in the 92nd percentile.

Therefore, the answer is d. 92nd.

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Therefore, the exact length of the polar curve r = 3 sin(θ), where 0 ≤ θ ≤ π/3, is π units.

To find the exact length of the polar curve r = 3 sin(θ), where 0 ≤ θ ≤ π/3, we can use the arc length formula for polar curves:

Length = ∫[θ1 to θ2] √(r^2 + (dr/dθ)^2) dθ

In this case, we have:

r = 3 sin(θ)

dr/dθ = 3 cos(θ)

Substituting these values into the arc length formula, we get:

Length = ∫[0 to π/3] √((3 sin(θ))^2 + (3 cos(θ))^2) dθ

Simplifying, we have:

Length = ∫[0 to π/3] √(9 sin^2(θ) + 9 cos^2(θ)) dθ

Length = ∫[0 to π/3] √(9 (sin^2(θ) + cos^2(θ))) dθ

Length = ∫[0 to π/3] √(9) dθ

Length = ∫[0 to π/3] 3 dθ

Length = 3θ |[0 to π/3]

Length = 3(π/3 - 0)

Length = π

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The component that is the reason why AE (Aggregate Expenditure) may be different from GDP (Gross Domestic Product) is unplanned inventory investment.

Unplanned inventory investment occurs when actual sales differ from expected sales, leading to unplanned changes in inventory levels. When firms produce more output than what consumers are willing to buy, the unsold goods accumulate as inventory. On the other hand, when the demand for goods exceeds the production levels, firms may run out of inventory.

The difference between actual inventory levels and planned inventory levels can lead to unplanned changes in inventory investment, which affects GDP. If actual inventory levels are greater than planned inventory levels, this indicates that firms have produced more than what consumers are willing to buy. Therefore, firms will reduce production in the future, leading to a decrease in GDP. Conversely, if actual inventory levels are lower than planned inventory levels, this indicates that firms have produced less than what consumers are willing to buy. Therefore, firms will increase production in the future, leading to an increase in GDP. Thus, unplanned inventory investment plays a significant role in the difference between AE and GDP

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So, the first partial derivatives of the function f(x,y)=x^4+6xy^5 is ∂f/∂y = 30xy^4.

To find the first partial derivatives of the function f(x,y)=x^4+6xy^5, we need to take the partial derivative with respect to x and y separately.

Starting with the partial derivative with respect to x, we treat y as a constant and differentiate x^4 to get:
∂f/∂x = 4x^3 + 6y^5

Next, we take the partial derivative with respect to y, treating x as a constant and differentiating 6xy^5 to get:
∂f/∂y = 30xy^4

So the first partial derivatives of the function f(x,y)=x^4+6xy^5 are:
∂f/∂x = 4x^3 + 6y^5
∂f/∂y = 30xy^4

Thus, the  first partial derivatives of the function f(x,y)=x^4+6xy^5 is ∂f/∂y = 30xy^4.

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By translating the graph of y = √x, a function that represents it is [tex]y=\sqrt{x+4}[/tex].

In Mathematics and Geometry, the translation of a graph to the left simply means subtracting a digit from the numerical value on the x-coordinate of the pre-image;

g(x) = f(x + N)

On the other hand, the translation a geometric figure or graph upward simply means adding a digit to the numerical value on the positive y-coordinate (y-axis) of the pre-image; g(x) = f(x) + N.

Based on the information provided, we have the following transformation:

(x, y)                               →                  (x - 4, y)

y = √x

[tex]y=\sqrt{x+4}[/tex].

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The other 28% represents the energy loss or inefficiency of the blender.

This means that only 72% of the input energy is effectively converted into useful work, while the remaining 28% is dissipated in the form of heat or other forms of energy loss. This energy loss is typically attributed to factors such as mechanical friction, heat generation, and

It could be due to factors such as friction, heat generation, or mechanical losses within the blender's components. This energy is not effectively converted into the desired blending action and is instead lost as waste.

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Step-by-step explanation:

so, from year 0 to year 1 after arrival the population grows by 5% (= factor 1.05).

as growth means the number of people before plus the additional 5%.

so, we multiply 15,400 by (1 + 0.05) or simply by 1.05.

and in year 2 after arrival we multiply that result again by 1.05.

which is the year 0 number multiplied by 1.05 × 1.05 or simply 1.05².

in year 3 after arrival that result gets multiplied by 1.05. or year 0 multiplied by 1.05×1.05×1.05 or simply 1.05³.

and so on, and so on.

so, we get an exponential function with starting value of 15,400 :

y = 15,400 × (1.05)^x

to calculate the local population x years after the arrival of the large company.

for x = 0 we get the starting value of 15,400.

The ratio of the critical load using the arrangement (a) to the critical load using the arrangement (b) is 16:9.

The critical load of a column depends on its effective length and its moment of inertia. The moment of inertia of the column in arrangement (a) is 2I, where I is the moment of inertia of each individual plank. The moment of inertia of the column in arrangement (b) is I. Therefore, the critical load of the column in arrangement (a) is 16 times that of the critical load of the column in arrangement (b), because the critical load varies as the moment of inertia. Hence, the ratio of the critical load using the arrangement (a) to the critical load using the arrangement (b) is 16:9.

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Therefore, we can write the expression g(x) as g(x) = f(h(x)), where h(x) is the inside function. In this expression, h(x) represents the input to f(x), and f(x) represents the outer function that is being applied to the input.

In order to identify the "inside function" h(x) in the expression g(x) = f h(x), we need to understand what a composition functions means.

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To calculate the overall probability, we multiply the individual probabilities together:

(3/11) * (1/10) * (7/9) = 21/990 ≈ 0.0212

Therefore, the closest option is 0.02.

The value of x in the quadratic equation using quadratic formula to the nearest hundredths place is x = 1.29 and x = 2.71.

The correct answer choice is option B.

2x² - 8x + 7 = 0

[tex]x = \frac{ -b \pm \sqrt{b^2 - 4ac}}{ 2a }[/tex]

[tex]x = \frac{ -(-8) \pm \sqrt{(-8)^2 - 4(2)(7)}}{ 2(2) }[/tex]

[tex]x = \frac{ 8 \pm \sqrt{64 - 56}}{ 4 }[/tex]

[tex]x = \frac{ 8 \pm \sqrt{8}}{ 4 }[/tex]

[tex]x = \frac{ 8 \pm 2\sqrt{2}\, }{ 4 }[/tex]

[tex]x = \frac{ 8 }{ 4 } \pm \frac{2\sqrt{2}\, }{ 4 }[/tex]

[tex]x = 2 \pm \frac{ \sqrt{2}\, }{ 2 }[/tex]

[tex]x = 2.70711[/tex]

or

[tex]x = 1.29289[/tex]

Hence,

Approximately, x = 1.29 or x = 2.71

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To find the average value of a function on an interval, we need to integrate the function over that interval and divide the result by the length of the interval. So, to find the average value of the function y=x^2(x^3+1)^(1/2) on the interval [0,2], we need to evaluate the definite integral:

(1/2) * ∫[0,2] x^2(x^3+1)^(1/2) dx

We can use a substitution u = x^3+1 and du = 3x^2 dx to simplify the integral:

(1/6) * ∫[1,9] (u-1)^(1/2) du

Now, we can use the power rule to integrate:

(1/6) * (2/3)*(u-1)^(3/2) |_1^9

= (1/9) * [(9-1)^(3/2) - (1-1)^(3/2)]

= (1/9) * [8^(3/2) - 0]

= 8/9 * sqrt(2)

So, the average value of the function on the interval [0,2] is:

(1/2) * [8/9 * sqrt(2)] / (2-0)

= 4/9 * sqrt(2)

Therefore, the average value of the function y=x^2(x^3+1)^(1/2) on the interval [0,2] is 4/9 * sqrt(2).

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The three numbers that can form a triangle are as follows:

The triangle inequality theorem states that in a triangle the sum of lengths of any two sides is greater than the length of the third side.

Therefore, the triangle inequality theorem can be used to check if the length of the three number can form a triangle.

Hence, if the lengths of a triangle are a, b and c, the triangle inequality theorem states that:

b + c > a

a + c > b

a + b > c

Therefore, the measure that forms a triangle are as follows:

9, 17, 6 can't form a triangle because 9 + 6 < 17.

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Answer:

-7+7=2x

Step-by-step explanation:

0=2x

0/2=2x/2

0/2=x

The flux of the vector field F = xy i + yz j + zx k out of a sphere of radius 5 centered at the origin is [?]. To find the flux of a vector field through a surface, we need to evaluate the surface integral of the dot product between the vector field and the outward normal vector of the surface.

In this case, the vector field F = xy i + yz j + zx k and the surface is a sphere of radius 5 centered at the origin.

To calculate the flux, we need to compute the dot product of the vector field F and the outward normal vector of the sphere at each point on the surface. The outward normal vector is given by the unit radial vector pointing away from the origin.

The flux integral can be expressed as:

Flux = ∬ F · dA

where dA is the vector differential area on the surface of the sphere.

To evaluate this integral over the entire surface of the sphere, we can use spherical coordinates. However, since the surface of the sphere is symmetric, the flux through one hemisphere will be equal to the flux through the other hemisphere. Therefore, we can calculate the flux through one hemisphere and multiply it by 2 to get the total flux.

The detailed calculation involves setting up the integral in spherical coordinates, determining the limits of integration, and evaluating the dot product. The resulting flux will depend on the specific limits and calculations performed.

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