2. Algebraically determine the initial velocity required in a roller coaster design if the velocity will be 26 m/s at the bottom of a vertical drop of 34 m (Acceleration due to gravity in SI units is 9.8 m/s 2).

Answer:

c = 18

Step-by-step explanation:

expand the factor in j(x) and compare like terms with k(x)

j(x) = 2(x + 3)² ← expand factor using FOIL

    = 2(x² + 6x + 9) ← distribute parenthesis by 2

   = 2x² + 12x + 18

compare to k(x) = 2x² + 12x + c

the 2 expressions are equivalent when c = 18

The zeros of the polynomial function f(x) are 3, -[(2 -i√2) / 3] and  -[(2 + i√2) / 3]

To find the zeros of the polynomial function f(x), we need to find the values of x for which f(x) = 0.

We can start by factoring out a common factor of 3x^2 from the polynomial:

f(x) = 3x^3 - 5x^2 - 10x - 6

f(x) = 3x^2(x - 5/3) - 2(5x + 3)

Now, we can set each factor equal to zero and solve for x:

3x^2(x - 5/3) - 2(5x + 3) = 0

3x^3 - 5x^2 - 10x - 6 = 0

x = 3, -[(2 -i√2) / 3],  -[(2 + i√2) / 3]

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The statement that "when dealing with fixed effects, a de-meaned model approach is superior to the lsdv approach because the de-meaned model gives us more accurate coefficient estimates" is true.

Fixed effects are a type of regression technique used to understand the effect of a single variable across different groups (such as across countries, across states, across individuals, etc.) by comparing the variance of the variable within each group to the variance of the variable across all groups.

When conducting a fixed effects regression, researchers must choose between the least squares dummy variable (LSDV) and de-meaned model approaches.Both models produce unbiased estimates of the coefficients, but the LSDV model has an advantage in that it provides more flexibility when dealing with interactions between fixed effects and time-varying covariates. On the other hand, the de-meaned model is preferred for reasons of computational efficiency and it can handle certain types of data that the LSDV model cannot (such as panel data with a small number of time periods).

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1. Definition of Congruence:∠ RSU ≅ ∠ VST

2. Definition of Congruence: m∠ RSU = m∠ VST

3. Angle Addition Postulate: m∠ RSU + m∠ USV = m/RSV

4. Angle Addition Postulate: m∠ VST + m∠  USV = m∠ UST

5. Transitive Property: m∠ RSU + m∠ USV = m∠ UST

6. Substitution Property: m∠ RSV = m∠ UST

7. Definition of Congruence: ∠ RSV ≅ ∠ UST

In mathematics, a relation R on a set X is said to have a transitive property if, for all elements a, b, c in X, whenever R relates a to b and b to c, then R also relates a to c.

This is shown in the case of number 6 where :

m∠ RSU + m∠ USV = m∠ UST

The transitive property is also shown in  equality states where all values of a, b, and c, if a = b, and b = c, then a = c.

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Answer:

Let x be the length of the base of the triangle, then the height h is given by h = 2x - 3 (since the height is 3 inches less than twice the length of the base).

The area of a triangle is given by the formula A = (1/2)bh, where b is the base and h is the height. We are given that the total area of the triangle is 7 square inches, so we can write:

(1/2)(x)(2x - 3) = 7

Multiplying both sides by 2 to eliminate the fraction, we get:

x(2x - 3) = 14

Expanding the left side, we get:

2x^2 - 3x = 14

Subtracting 14 from both sides, we get:

2x^2 - 3x - 14 = 0

We can now use the quadratic formula to solve for x:

x = (-b ± sqrt(b^2 - 4ac))/(2a)

where a = 2, b = -3, and c = -14. Plugging in these values, we get:

x = (-(-3) ± sqrt((-3)^2 - 4(2)(-14)))/(2(2))

= (3 ± sqrt(169))/4

= (3 ± 13)/4

Taking the positive value for x (since the length of the base must be positive), we get:

x = (3 + 13)/4

= 4

Therefore, the length of the base is 4 inches. To find the height h, we can use the formula h = 2x - 3:

h = 2(4) - 3

= 5

So the height of the triangle is 5 inches.

Parker is wrong, as the volume of the cube container is obtained as [tex]\frac{1}{512} t^9u^{12}[/tex].

What is volume?

Each thing in three dimensions takes up some space. The volume of this area is what is being measured. The space occupied within an object's borders in three dimensions is referred to as its volume. It is sometimes referred to as the object's capacity.

The volume of a cube is given by the formula V = s³, where s is the length of one side of the cube.

In this case, the length of the container is given as [tex]\frac{1}{8} t^3u^4[/tex].

To find the volume of the cube, we need to cube this length -

[tex]V=\big(\frac{1}{8} t^3u^4\big)^3[/tex]

Simplifying this expression using the rules of exponents, we get -

[tex]V=\big(\frac{1}{8}\big)^3 \times (t^3)^3 \times (u^4)^3[/tex]

[tex]V =\frac{1}{512} t^9u^{12}[/tex]

So, the actual volume of the container is [tex]\frac{1}{512} t^9u^{12}[/tex], not [tex]\frac{1}{12} t^9u^{12}[/tex] as Parker said.

Therefore, Parker is incorrect.

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Answer:

3 students will retake the test

Step-by-step explanation:

20% of 150 students is 30

(20÷100)×150=30 students

Now we need to find the 10% of the 30 students who failed.

(10÷100)×30=3 students

we can check this by Subtracting that 20% who failed from 100%(the total in class). So that will be 80%. Now that 80% is the number of students who passed.

And in order to find the number of students who passed we say:

(80÷100)×150= 120

all you need to do now is, subtract that 120 from 150, then find the 10% of the answer.

The value of angle E and angle F are both 22.8°

Circle theorem includes the concept of tangents, sectors, angles, the chord of a circle and proofs. A circle is the locus of all points in a plane which are equidistant from a fixed point.

The theorem that states that the angle at the center is twice the angle at the circumference is applied in this scenario.

angle P = angle of arc DG = 45.6°

therefore angle E = 1/2 angle of arc DG

= 1/2 × 45.6

= 22.8°

angle E = angle F = 22.8° . This because angle in the same segment are equal.

therefore angle E = angle F = 22.8°

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Given:-

[tex] \: [/tex]

[tex] \: [/tex]

Solution:-

[tex] \underline{ \: \sf{[a = 3.8] \: }}[/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \textsf{put the value of a = 3.8}[/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━━━━

[tex] \underline{ \sf \: [a = 7]\: }[/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \textsf{ \: put the value of a = 7 \: }[/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━━━━

[tex] \underline{ \sf{[a = -7.2]}}[/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \textsf{ \: put the value of a = -7.2 \: }[/tex]

[tex] \: [/tex]

[tex] \: [/tex]

[tex] \: [/tex]

━━━━━━━━━━━━━━━━━━━━━━━

hope it helps! :)

The area of the given trapezoid above would be = 1.2m²

A trapezoid is defined as a quadrilateral that has four sides with a pair of parallel sides.

To calculate the area of the trapezoid the formula below should be used. That is;

Area = 1/2 (a+b) h

where;

a= 1.7m

b = 0.7m

h = sin∅ = opposite/hypotenuse

where;

opposite = ?

hypotenuse =1.4

sin 45° = h/1.4

h= 0.707106781 ×1.4

h = 1m

Area= 1/2 (1.7+0.7) × 1

= 1.2 m²

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The result would be (C) 27 if the above statement is accurate.

Three categories of integers exist: Zero (0) (0) Good integers (Natural numbers) Integer Negatives (Additive inverse of Natural Numbers).

As the two numbers are consecutive, we will refer to the smaller integer as "x" and the larger one as "x + 1".

In accordance with the issue, we have:

2x + (1/2)(x + 1) = 33

To eliminate the fraction, multiply everything by 2 and you obtain the following:

4x + x + 1 = 66

If we simplify, we get:

5x = 65

When we multiply both parts with 5, we get:

x = 13

Hence, 13 is the smaller number while 14 is the larger number.

The two integers' total is:

13 + 14 = 27

Hence, the response is (C) 27.

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Answer:

What do you mean?

Step-by-step explanation:

Some questions are new and have not been answered yet.

Aiden's estimated the length of a piece of rope as 8.5 inches, while its actual length is 7.1 inches. Therefore, the percent error of Aiden's estimate is 11.3%.

To calculate the percent error, you first need to find the difference between the actual length and the estimated length. Subtract 7.1 inches from 8.5 inches and you get 1.4 inches. This is the difference between the two lengths.
Next, divide the difference by the actual length and multiply by 100. The equation is: (difference/actual length) * 100. So, (1.4/7.1)*100 = 11.3%. Therefore, the percent error of Aiden's estimate is 11.3%.
It is important to be accurate when making measurements and estimates. A small difference in numbers can lead to a large error in the final result. Knowing the percent error can help you to improve your measurements and estimates and achieve greater accuracy.

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Answer:

Trinominal

Step-by-step explanation:

By definition, Trinominals are those expressions having 3 values, in this case, x^2, x, and the constant 6 are the values.

hope it helps.

The area of the square with given side 18 cm is 324 square cm.

The equation for a square's surface area A square is a quadrilateral with four equal sides and four right angles, which is how Side x Side is obtained. By multiplying one side's length by the other side's length, which is likewise the same length, one may get the area of a square, which is the amount of space within the square. As a result, Side x Side is the formula for calculating a square's area.

Given that the length of the side of square = 18cm.

The area of the square is given as:

A = (s)(s)

Substituting the value we have:

Area = 324 square cm

Hence, the area of the square is 324 square cm.

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Answer: x=120

Step-by-step explanation:

If it is a regular polygon then the sum of internal angles is 180 degrees. Each angle should be 60. Since a straight line is 180, 180 - 60 would equal x!

Answer: 120

Step-by-step explanation: A triangle has a sum of 180°. A regular polygon has equal angles, so you divide 180 by 3, getting 60. Next x is supplementary to the adjacent angle, so you must do 180-60, yielding 120° as x.

The height of the rectangular prism of aquarium is 8 inches.

A right rectangular prism is a six-sided, three-dimensional geometric figure. The term "rectangular parallelepiped" is another name for it. Three pairs of congruent faces, each pair of which is parallel to the other, make up a right rectangular prism. In addition, it contains four vertices and three pairs of congruent edges. In a right rectangular prism, the opposing faces are parallel and congruent rectangles, and the angles separating the faces are all right angles. Multiplying the prism's length, width, and height yields the volume of a right rectangular prism.

Given that, the length is 19 in and width is 9 in and it can hold 1368 cubic inches of water.

The volume of a rectangle is given as:

V = lwh

Substitute the values:

1368 = 19(9)(h)

h = 1368/(19*9) ≈ 8 inches

Hence, the height of the rectangular prism of aquarium is 8 inches.

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The mathematician should expect to see the letter e appear 143 times in the 1,100 character email message.

It is a ratio of the count of the specific outcome or event to the total number of observations in the sample or population. Proportion is often expressed as a percentage or fraction and can be used to make inferences about the population based on the sample.

If the letter e appears 13 times out of the first 100 characters, we can assume that the proportion of e's to the total characters is constant throughout the entire message.

Let x be the number of times the letter e appears in the 1,100 character message. Then, we can set up a proportion:

13/100 = x/1100

Solving for x, we get:

x = (13/100) * 1100 = 143

Therefore, the mathematician should expect to see the letter e appear 143 times in the 1,100 character email message.

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[tex]~~~~~~~~~~~~\textit{distance between 2 points} \\\\ P(\stackrel{x_1}{19}~,~\stackrel{y_1}{10})\qquad Q(\stackrel{x_2}{26}~,~\stackrel{y_2}{13})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ PQ=\sqrt{(~~26 - 19~~)^2 + (~~13 - 10~~)^2} \implies PQ=\sqrt{( 7 )^2 + ( 3 )^2} \\\\\\ PQ=\sqrt{ 49 + 9 } \implies PQ=\sqrt{ 58 }\implies PQ\approx 7.62 \\\\[-0.35em] ~\dotfill[/tex]

[tex]~~~~~~~~~~~~\textit{middle point of 2 points } \\\\ P(\stackrel{x_1}{19}~,~\stackrel{y_1}{10})\qquad Q(\stackrel{x_2}{26}~,~\stackrel{y_2}{13}) \qquad \left(\cfrac{ x_2 + x_1}{2}~~~ ,~~~ \cfrac{ y_2 + y_1}{2} \right) \\\\\\ \left(\cfrac{ 26 +19}{2}~~~ ,~~~ \cfrac{ 13 +10}{2} \right) \implies \left(\cfrac{ 45 }{2}~~~ ,~~~ \cfrac{ 23 }{2} \right)\implies \stackrel{ \textit{\LARGE M} }{\left(22\frac{1}{2}~~,~~11\frac{1}{2} \right)}[/tex]

to get the equation of any straight line, we simply need two points off of it, let's use those two in the picture below

[tex](\stackrel{x_1}{2}~,~\stackrel{y_1}{5})\qquad (\stackrel{x_2}{4}~,~\stackrel{y_2}{7}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{\textit{\large rise}} {\stackrel{y_2}{7}-\stackrel{y1}{5}}}{\underset{\textit{\large run}} {\underset{x_2}{4}-\underset{x_1}{2}}} \implies \cfrac{ 2 }{ 2 } \implies 1[/tex]

[tex]\begin{array}{|c|ll} \cline{1-1} \textit{point-slope form}\\ \cline{1-1} \\ y-y_1=m(x-x_1) \\\\ \cline{1-1} \end{array}\implies y-\stackrel{y_1}{5}=\stackrel{m}{ 1}(x-\stackrel{x_1}{2}) \\\\\\ y-5=x-2\implies {\Large \begin{array}{llll} y=x+3 \end{array}}[/tex]

The first quartile is 6, the median is 7.5 and the third quartile is 8.5. The box plot has been attached below.

What is a median?

A data set can be described more accurately than its average by using the median, which is the midpoint of an ordered, ascending or descending list of integers.

The data is given as 5.5, 6, 6, 6.5, 7, 7.5, 8, 8, 8.5, 9, 9.

From this, we get the first quartile to be 6.

The median is 7.5 and the third quartile is 8.5.

The box plot of the data has been plotted and attached below.

Hence, the required solution has been obtained.

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Question: There are 11 dancers in a performance. Their ages (in years) are: 5.5, 6, 6, 6.5, 7, 7.5, 8, 8, 8.5, 9, 9.

1. Determine the first quartile, median, and third quartile of the dancers' ages.

2. Drag the movable points to label them on the box plot.

The graph's scales could be represented by the x-axis, which could be labelled from 0 to 12 with a scale of 1, and the y-axis, which could be labelled from 0 to 100 with a scale of 10.

The horizontal scale used in a graph is called the X axis. Measurements on the coordinate plane are made using it as a reference line. The position of an object on that plane is described by its distance from the x and y axes.

All potential data that can be graphed is represented by the numbers on the X and Y axes. Each number here is different by ten. Scale is the name for this. On a coordinate grid, scale is the distance between each square.

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Measure of ∠1 = 126 deg, Measure of ∠4 = 54 deg, and measure of ∠7 = 126 deg.

∵ Lines p and q are parallel, and t is transversal,

∴ ∠3 and ∠7 form pair of corresponding angles,

⇒ ∠3 and ∠7 are equal

∴∠7=126 deg.

Also, ∠3 and ∠1 are vertically opposite angles,

∴ ∠3 and ∠1 are equal.

⇒ ∠1=126

Again, as t is a straight line and line p intersects it,

∴ ∠3 and ∠4 form linear pair.

⇒ ∠3 and ∠4 are complementary.

⇒ ∠3+∠4=180

⇒ ∠4+126=180

⇒ ∠4=180-126

⇒ ∠4=54

Hence, measure of ∠1 is 126 deg, that of ∠4 is 54 deg, and that of ∠7 is 126 deg.

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Answer:

x = - 4 , x = 4

Step-by-step explanation:

the expression is undefined if the denominator equals zero

equate the denominator to zero and solve for x

x² - 16 = 0 ( add 16 to both sides )

x² = 16 ( take square root of both sides )

x = ± [tex]\sqrt{16}[/tex] = ± 4

that is the expression is undefined when x = - 4 or x = 4

The correct sequence of transformations is: reflect across the y-axis and then translate 9 units to the right.

In everyday parlance, transformation refers to a mathematical function. A transformation is defined as the invertible function from any set X to its own set X or any other set Y. The transformation for any term may merely signal that the geometric component of this particular function is being studied.

The correct sequence of transformations that could have been used to transform triangle ABC to produce triangle A′′B′′C′′ is:

Triangle ABC was reflected across the y-axis and then translated 9 units right.

To see why, let's compare the coordinates of the corresponding vertices of both triangles:

A (-6, 2) ---> A'' (6, -5)

B (-3, 6) ---> B'' (3, -1)

C (-3, 2) ---> C'' (3, -5)

If we reflect triangle ABC across the y-axis, we obtain a new triangle A'B'C' with vertices:

A' (6, 2)

B' (3, 6)

C' (3, 2)

Then, if we translate triangle A'B'C' 9 units to the right, we obtain triangle A''B''C'':

A'' (6+9, 2) = (15, 2)

B'' (3+9, 6) = (12, 6)

C'' (3+9, 2) = (12, 2)

Which has the same coordinates as triangle A''B''C'' given in the problem statement. Therefore, the correct sequence of transformations is: reflect across the y-axis and then translate 9 units to the right.

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An expression for the difference between their earnings is 2G - G.

In order to write a linear equation to describe this situation, we would assign variables to John's earnings and Grace's earnings respectively, and then translate the word problem into a linear equation as follows:

Since John earns twice as much as his wife Grace, a linear equation that models the situation is given by:

J = 2G

100 = 50

For the difference between their earnings, we have the following:

Difference = J - G

Difference = 2G - G

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Answer:

A. Sample space: 1, 2, 3, 4, 5, 6

D. P(6) = 1/6

Step-by-step explanation:

Option B and C are not possible because the probability of rolling any particular number on a fair six-sided cube is always 1/6. Option E describes the outcomes of 10 trials, which is not the probability model.

Answer:

The equation in slope-intercept form of the line perpendicular to y = -3x + 1 and passing through (2, 2) is y = (1/3)x + 4/3.

Step-by-step explanation:

To find the equation of the line perpendicular to y = -3x + 1 and passing through the point (2, 2), we first need to determine the slope of the line we are looking for.

The slope of any line perpendicular to y = -3x + 1 will be the negative reciprocal of the slope of y = -3x + 1. The slope of y = -3x + 1 is -3, so the slope of the line we want is the negative reciprocal of -3, which is 1/3.

So the slope of the line we want is 1/3. We also know that the line passes through the point (2, 2). We can now use the point-slope form of the equation of a line to find the equation of the line we want:

y - y1 = m(x - x1)

where m is the slope of the line, and (x1, y1) is the point the line passes through.

Substituting m = 1/3, x1 = 2, and y1 = 2, we get:

y - 2 = (1/3)(x - 2)

Multiplying both sides by 3, we get:

3y - 6 = x - 2

Subtracting x from both sides, we get:

-x + 3y = 4

So the equation of the line perpendicular to y = -3x + 1 and passing through the point (2, 2) is -x + 3y = 4, which is in standard form. We can also write it in slope-intercept form:

3y = x + 4

y = (1/3)x + 4/3

So the equation of the line we want in slope-intercept form is y = (1/3)x + 4/3.

Hope this helps you! Sorry if it doesn't. If you need more help, ask me! :]