Explain why 1/2 is not equivalent to 2/5.

The simplest form of (4y^2)^3(3y^2) is 192 y⁸.

What is an expression?

Mathematical expressions consist of at least two numbers or variables, at least one arithmetic operation, and a statement. It's possible to multiply, divide, add, or subtract with this mathematical operation. An expression's structure is as follows: Expression: (Math Operator, Number/Variable, Math Operator)

Given expression is (4y²)³(3y²).

The law of exponent formula are:

(a^m)^n = a^(mn)

(a^m)(a^n) = a^(m+n).

Now applying the formula (a^m)^n = a^(mn) in (4y²)³

(4y²)³ = 4³× y²ˣ³ = 64y⁶

Putting (4y²)³ = 64y⁶ in (4y²)³(3y²):

(4y²)³(3y²) = 64y⁶ × (3y²)

Now applying the formula (a^m)(a^n) = a^(m+n):

(4y²)³(3y²) = (64× 3) y⁶⁺²

(4y²)³(3y²) = 192 y⁸

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The volume of the rectangular pyramid is 900 cubic yards.

The standard formula of the volume of the rectangular pyramid is written as

V = 1/3s²h

where s refers the base edge and h refers the height.

Here we have given that the square base has side lengths 15 yards and the height is 12 yards.

Now, we have to find the volume of the rectangular pyramid

As we know the the value of base edge is 15 yard and the height is 12 yards.

Now, we have to apply the values on the formula then we get,

=> V = 1/3 x 15² x 12

It can be simplified as,

=> V = 4 x 225

When we do the appropriate arithmetic operation, then we get,

=> V = 900 cubic yard.

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divide the volume by the total number of points to normalize it. The resulting value is the estimated volume of the bounded object.

def top(x, y):

return x+y

def bottom(x, y):

return x*2 + y*2

import numpy as np

xvals = np.random.uniform(-1, 1, size=10000)

yvals = np.random.uniform(-1, 1, size=10000)

# Initialize the volume

volume = 0

# Calculate the volume using Monte Carlo

for x, y in zip(xvals, yvals):

if top(x, y) > bottom(x, y):

volume += 1

# Normalize the volume by the total number of points

volume /= 10000

print(volume)

The Monte Carlo method is used to estimate the volume of a bounded object by randomly generating points within a known box. First, we generate a list of random (x, y) coordinates using numpy's uniform function. We then initialize the volume to 0 and loop through the list of coordinates and increment the volume by 1 if the point lies above the surface. Finally, we divide the volume by the total number of points to normalize it. The resulting value is the estimated volume of the bounded object.

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Answer:

(14, -3) (0, -5) (15, -25)

Step-by-step explanation:

for the 4th you just need a positive X and negative Y

(x, -y)

have a great day!

By using the alternating series estimation theorem we found that the number of terms we need to add in order to find the sum with an error less than 0.0001 is 7.

We must employ one of the convergence tests to determine whether the provided series is convergent or divergent. The alternating series test, which states that a series converges if it takes the form ∑ [tex](-1)^{n}[/tex] x an, where 'an' is a decreasing sequence of positive terms, is the most applicable test for this series.

∑ [tex](-1)^{n}[/tex] x an

a n =  [tex]\frac{(n + 1)}{(n^2 + 2n + 3)}[/tex]

by taking the derivative of the expression for a n, this sequence is decreasing and showing that it is always negative. given by a'n =  [tex]\frac{(n + 1)}{(n^2 + 2n + 3)}[/tex] [tex]- \frac{(n + 1) X 2)}{(n^2 + 2n + 3)^2}[/tex]

=  [tex]\frac{(-2n - 2)}{(n^2 + 2n + 3)^2}[/tex]

for all values of n it is negative, therefore the sequence a n is decreasing. Thus, given series satisfies the conditions of the alternating series test. Therefore, the series is convergent.

The alternating series estimation theorem can be used to determine how many terms are required to be added in order to find the total with an error of less than 0.0001. This asserts that the first disregarded term in an alternating series is smaller than the error in the series' sum. The error in the sum of the series with N + 1 terms is therefore smaller than the value of the (N + 1)th term in the series if we add N terms to the series and the error is less than E.

Find the first term in the series that is less than 0.0001 in order to calculate the number of terms we must add to the series in order to find the total with an error smaller than 0.0001.

given series that a 1 = [tex]\frac{2}{4}[/tex] = 0.5, a 2 = [tex]\frac{3}{9}[/tex] = 0.333, a 3 = [tex]\frac{4}{16}[/tex] = 0.25, and a 4 = [tex]\frac{5}{25}[/tex] = 0.2. Since the first term that is less than 0.0001 is a 7 = [tex]\frac{8}{64}[/tex] = 0.125, Adding, 7 terms to the series in order to find the sum with an error less than 0.0001.

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Answer:

44

Step-by-step explanation:

The correct statement regarding how can a translation and a rotation be used to map triangle HJK to triangle LMN is given as follows:

The translation just changes the position of the figure, hence it has to be among equivalent vertices of the triangles.

The equivalent vertices are given as follows:

The equivalent sides are given as follows:

HK and LN.

Thus the rotation is about K until HK lies on the line containing LN, and the third option is correct.

The problem is given by the image shown at the end of the answer.

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The simplified form of the given expression as required in the task content is; 8.

It follows from the task content that the simplified form of the given expression; -4 ( 2 - (³√8 × 6) ) / 5 is to be determined.

Therefore, By solving the innermost parentheses; we have;

-4 ( 2 - (2 × 6) ) / 5

= -4 ( 2 - 12 ) / 5

= -4 ( -10 ) / 5

= 40 / 5

= 8.

Therefore, the simplified form of the expression is; 8.

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To avoid bias - increase the sample size and take reviews randomly

The dining hall manager decided to do a survey to estimate the proportion of students living in the dormitories who think that the quality of the food should be improved. one evening, the manager asked the first 100 students entering the dining hall

There is a bias if taken data contain only the first 100 students, primary reasons students come in a friend circle to the dining hall, possibly that whole friend circle did/ didn't like food. Chances are the first hundred students like the food and the remaining students don't like, vice - versa.

To avoid bias - increase the sample size and take reviews randomly

Therefore, the manager should increase the sample size and take reviews randomly to avoid sampling bias.

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Answer: y=-3.46,3.46

Step-by-step explanation:

isolate y

5y^2=60

y^2=12

y=-3.46,3.46

x=-[tex]\sqrt{3}[/tex] ,[tex]\sqrt{3}[/tex],-2,2

What is a quadratic equation ?

Quadratics can be defined as a polynomial equation of a second degree, which implies that it comprises a minimum of one term that is squared. It is also called quadratic equations. The general form of the quadratic equation is:

ax² + bx + c = 0

Given:

y=[tex]x^{4} -7x^{2} +12[/tex]

Let [tex]x^{2}[/tex] = k

So , y = [tex]k^{2}[/tex] -7k+12

On simplifying/factorising the given equation  ,we get

y =  [tex]k^{2}[/tex] - 4k - 3k +12

On taking common ,we get

y = k(k-4) -3(k-4)

y =(k-3)(k-4)

Thus , on factorising and equating it to zero we get

(k-3)=0 or (k-4)=0

k=3 or k=4

Now , replacing k by [tex]x^{2}[/tex] ,we get

[tex]x^{2}[/tex] = 3 or  [tex]x^{2}[/tex] = 4

Further on solving ,we get

( [tex]x^{2}[/tex]-3)=0 or ( [tex]x^{2}[/tex]-4)=0

(x-[tex]\sqrt{3}[/tex])(x+ [tex]\sqrt{3}[/tex]) =0  or (x+2)(x-2)=0

x=-[tex]\sqrt{3}[/tex] ,[tex]\sqrt{3}[/tex] or x= -2,2

Thus , x=-[tex]\sqrt{3}[/tex] ,[tex]\sqrt{3}[/tex],-2,2

Hence ,the obtained values of y are the zeros of x .

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The Trigonometric Expression can be written as  :

cos⁴ x = 18(3 + 4cos2x + cos4x)

Power:

In mathematics, a power defines a base number raised to an exponent. The base is the factor multiplied by itself, and the exponent is the number of times the same base is multiplied.

Reducing / Lowering Power:

Power identities are trigonometric identities, allowing us to rewrite trigonometric expressions raised to powers in terms of simpler trigonometric expressions.

According to the question:

cos2x = 2cos²x -1

⇒ cos²x = 1/2 (1 + cos2x)

⇒ cos²2x = 1/2 (1 + cos4x)

⇒ cos⁴ x = (cos²x)²  

          = 1/4(1 + 2cos2x+cos²2x)

⇒ cos⁴ x = 1/4( 1 + 2cos2x + 1/2(1+cos4x))

⇒ cos4x = 1/8(3 + 4cos2x+cos4x)

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Answer:

p=31

Step-by-step explanation:

sum of all three angles of any triangle equal 180 degrees

2p+(p+46)+(p+10)=180

4p+56=180

    -56.  -56

4p=124

/4.   /4

p=31

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​(Option C. Students to their LRN) In this case, each student has a unique LRN, meaning that it can be used in a one-to-one function where each student is mapped to their LRN.

A one-to-one function is a type of function where each element in the domain has one and only one image in the range. In this case, each student can be mapped to their LRN (Learning Reference Number) in a one-to-one function. This means that for every student, there is a unique LRN, and for every LRN, there is a unique student. This type of function is useful for keeping track of individual student information, as each student has a unique LRN associated with them.

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The statement that correctly compares -9 and 2 is A. -9 < 2 because 9 is to the left of 2 on a number line.

In arithmetic, a number line is a depiction of a graduated straight line which functions as a graphic representation of real numbers. It is presumed that every number line point and every real number correspond to one another.

Horizontal straight lines termed number lines have integers along them spaced equally apart. All of the numbers in a series can be shown on a number line. This line extends indefinitely at both ends.

In this case, -9 is on the left of a number line since it's negative. In conclusion, the correct option is A

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The division of 4 by 6 will be 2/3 or 0.67.

We have the following expression -

4 ÷ 6

We can write -

[x] = 4 ÷ 6

We can write -

4 ÷ 6 = 4/6 = 2/3 = 0.67

Therefore, the division of 4 by 6 will be 2/3 or 0.67.

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The difference of the functions, f(x) = 3x² + 4x - 6 and g(x) = 6x³ - 5x² - 2, is: D. (f - g)(x) = -6x³ + 8x² + 4x - 4.

Finding the difference of two functions involves combining like terms together and then simplify.

Given the functions:

f(x) = 3x² + 4x - 6

g(x) = 6x³ - 5x² - 2

To find (f - g)(x), it implies that we find the difference between f(x) and g(x).

(f - g)(x) = f(x) - g(x)

Substitute

f(x) - g(x) = (3x² + 4x - 6) - (6x³ - 5x² - 2)

Open the parentheses:

f(x) - g(x) = 3x² + 4x - 6 - 6x³ + 5x² + 2

Combine like terms

f(x) - g(x) = - 6x³ + 8x² + 4x - 4

Therefore, the answer is: D. (f - g)(x) = -6x³ + 8x² + 4x - 4

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The hypothesis test that can be used to determine whether the mean annual premium in Pennsylvania is lower than the national mean annual premium is between -2.064 and 2.064.

Hypothesis testing

The standard formula to calculate the hypothesis testing is given by the z test is

Z = (X – U) / (SD / √n)

Where:

X – Sample Mean

U – Population Mean

SD – Standard Deviation

n – Sample size

Given,

Here we have the the insure website reports that the mean annual premium for automobile insurance in the united states was $1,503 in march 2014. being from Pennsylvania at that time, you believed automobile insurance was cheaper there and decided to develop statistical support for your opinion. a sample of 25 automobile insurance policies from the state of Pennsylvania showed a mean annual premium of $1,430 with a standard deviation of is $165.

Here we need to develop the a hypothesis test that can be used to determine whether the mean annual premium in Pennsylvania is lower than the national mean annual premium.

While we looking into the given question we have identified the values of

Sample mean = 1430

Sample size = 25

Population mean = 1503

Standard deviation = 164

Then the hypothesis testing is calculated as,

=> Z = (1430 – 1503) / (164 / √25)

Therefore, if we reject the null hypothesis and accept the alternative hypothesis. The z score of -2.23 is within the rejection area.

And the 2 critical values (cutoff points) are -2.064 and 2.064. Since the z score, -2.23, is outside this interval, we reject the null hypothesis. We accept the alternative hypothesis.

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It is not possible to determine the area of square 3 based on the information given. The area of a square is determined by its side length, which is not mentioned in the problem. In order to find the area of a square, you need to know the length of at least one side. The perimeter of a square, which is the total length of all its sides, is not sufficient to determine the area of the square. You would need to know the side length of square 1 in order to calculate its area.

Answer:400

Step-by-step explanation:

The ratio of tablespoons of raspberries to tablespoons of sugar should be considered as the 8 to 6.

What is ratio?

A ratio is an ordered pair of numbers a and b, written a / b where b does not equal 0. A proportion is an equation in which two ratios are set equal to each other. For example, if there is 1 boy and 3 girls you could write the ratio as: 1 : 3 (for every one boy there are 3 girls) 1 / 4 are boys and 3 / 4 are girls.

Calculation of the ratio:

Since Macy's aunt makes raspberry jam using 6 tablespoons of sugar and 8 tablespoons of raspberries.

So here the ration be like

= 8 raspberries tablespoons to 6 tablespoons of sugar

= 8 to 6

hence, the second option is correct.

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Using limits, it is found that the statement which describes the end behavior of the function f(x) is: As x approaches negative infinity, f(x) approaches positive infinity.

How we find the end behavior of a function f(x)?

It is given by the limit of f(x) as x approaches both positive infinity and negative infinity.

In this problem, the function is:

f (x) =   1/7 | x - 4 | + 3

Hence, the limits are:

[tex]\lim_{n \to -\infty} f(x) = \lim_{n \to -\infty} [1/7 | x - 4 | + 3] = \frac{1}{7} |-\infty - 4| + 3 = \infty\\\\ \lim_{n \to \infty} f(x) = \lim_{n \to -\infty} [1/7 | x - 4 | + 3] = \frac{1}{7} |\infty - 4| + 3 = \infty\\[/tex]

Approaches positive infinity in both cases, hence as x approaches negative infinity, f(x) approaches positive infinity.

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Answer:

1) 60

2)  15

3) 45

4) 105

5) 75

Step-by-step explanation:

7y + 5y = 180  A straight angle is 180  Combine like terms

12y = 180  Divide both sides by 12

y = 15

The sum of the interior angles is 180

y = 15

180 - (5y + 3y) = x

180 -(5(15) + 3(15)) = x

180 - (75 + 45) = x

180 - 120 = x

60 = x

Angle c

3y

3(15)

45

Angle DBC

7y

7(15)

105

Angle CBA

5y

5(15)

75

Answer:

Step-by-step explanation:

You have triangles with two sides and the included angle given, or all three sides given. You want the measures of the missing sides or angles.

For triangle ABC, the law of cosines tells us ...

  c² = a² +b² -2ab·cos(C)

Solving for the third side, we get ...

  c = √(a² +b² -2ab·cos(C))

Solving for the angle, we get ...

  C = arccos((a² +b² -c²)/(2ab))

The names of the sides and vertices can be permuted, as long as the angle is between the given sides.

  a = √((7² +9² -2·7·9·cos(35°)) ≈ 5.176

The measure of side 'a' is about 5.2 units.

  C = arccos((8² +7² -10²)/(2·8·7)) = arccos(13/112) ≈ 83.33°

  A = arcsin((8/10)·sin(83.33°)) ≈ 52.62°

  B = arcsin((7/10)·sin(83.33°)) ≈ 44.05°

The last two angles are found using the law of sines:

  sin(A)/a = sin(B)/b = sin(C)/c

  b = √(230² +360² -2·230·360·cos(38°)) ≈ 228 . . . . feet

The third side measures 228 feet.

__

Additional comment

A calculator or spreadsheet can be programmed with the necessary formulas so that you can avoid a lot of repetitive work when you have several of these to solve. You need to make sure angles are treated appropriately, as the default is often radians, rather than degrees.

When solving for angles from three given sides, we like to solve for the largest angle first. That avoids any ambiguity when using the Law of Sines to find the other angles.

When using the Law of Sines to find a second angle after using the Law of Cosines, it can work well to find the smallest remaining angle. (In the first problem, that would be the angle opposite the side of length 7; in the last problem, that would be the angle opposite the side of length 230.)

the matrix A, which is composed of all scalar multiples of the vector [6 3], and represents the reflection about the line L in R2.A = [[-6/7, 9/7],[9/7, 6/7]]

Let L be the line in R^2 which consists of all scalar multiples of the vector [6 3], then the equation of the line is y= (3/6)x.

We need to find the matrix A which represents the reflection about the line L.

Let (x1, y1) be any point on the line L and let (x2, y2) be its reflection about the line L.

Now, we know that the slope of the line L is 3/6, hence the slope of the normal to the line L is -6/3.

So, the equation of the normal to the line L is y = -(6/3)x. Now, we can use the point-slope form of the line equation to calculate the equation of the line passing through (x1, y1) and its reflection (x2, y2).

The line that connects (x1, y1) and (x2, y2) has the following equation:

y-y1 = (y2-y1)/(x2-x1) (x-x1)

=> y-y

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what i got when i looked it up: 2750.0005

what i got when i did it 500.0005

The system of equations y 2 3x 3 x= -2 has a solution.

A mathematical statement that has a "equal to" symbol between two expressions with equal values is called an equation. as in 3x + 5 Equals 15, for instance.

Equations come in a variety of forms, including linear, quadratic, cubic, etc.

Equations are mathematical statements with two algebraic expressions flanking the equals (=) sign on either side.

It demonstrates the equality of the relationship between the expressions printed on the left and right sides. We have LHS = RHS (left hand side = right hand side) in every mathematical equation.

To determine the value of an unknown variable that represents an unknown quantity, equations can be solved. The statement is not an equation if there is no "equal to" symbol in it.

According to our question-

y = -2*2/3+3

y= -4/3+3

now simplifying,

5/3

Hence, The system of equations y 2 3x 3 x= -2 has a solution.

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Complete question-

What is the solution to the system of equations y 2 3x 3 x?

Answer: 120 square feet

Step-by-step explanation:

Area formula of triangle:

(height of triangle x length) divided by 2

[tex]\frac{30*8}{2} =\frac{240}{2} =120[/tex]

Answer:

120 ft²

Step-by-step explanation:

[tex]\boxed{\begin{minipage}{4 cm}\underline{Area of a triangle}\\\\$A=\dfrac{1}{2}bh$\\\\where:\\ \phantom{ww}$\bullet$ $b$ is the base. \\ \phantom{ww}$\bullet$ $h$ is the height.\\\end{minipage}}[/tex]

From inspection of the given triangle:

Substitute the given base and height into the formula to find the area of the triangle:

[tex]\implies A=\dfrac{1}{2} \cdot 30 \cdot 8[/tex]

[tex]\implies A=15 \cdot 8[/tex]

[tex]\implies A=120\; \sf ft^2[/tex]

Therefore, the area of the given triangle is 120 ft².

In hyperbolic geometry, the angle sum of a triangle is always less than 180 degrees.

A lune is a wedge of a sphere with angle θ, represented by L(θ) in the proof.

α, β, and γ are the three angles of the triangle.

4πr²+4area[αβγ]=2L(α)+2L(β)+2L(γ)

2(2πr²+2area[αβγ])=2(L(α)+L(β)+L(γ))

2πr²+2area[αβγ]=L(α)+L(β)+L(γ)

At this point, we need to use a theorem that states that a lune whose corner angle is θ radians has an area of 2θr².

2πr²+2area[αβγ]=2αr²+2βr²+2γr²

2πr²+2area[αβγ]=2r²(α+β+γ)

π+area[αβγ]r²=α+β+γ

At this point, it is clear that the sum of the angles is equal to π plus the area[αβγ]r² (which cannot be zero).

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