30POINTS
Factor completely.
3x^5 - 75x^3 =

The final function that represents the given transformations is:

f(x) = -log(x + 2) - 3

What is parent function?

In mathematics, a parent function is a basic function from which other functions can be derived through a set of transformations. It serves as a starting point for understanding more complex functions by showing the most basic characteristics of the function.

Starting with the parent function f(x) = log(x), the given transformations are:

Horizontal shift to the left by 2 units: The graph has been shifted 2 units to the left, which means that the argument of the logarithmic function should be (x + 2) instead of "x". So the function becomes:

f(x) = log(x + 2)

Reflection about the x-axis: The graph is the opposite of the parent function, which means it has been reflected about the x-axis. This changes the sign of the function, so we get:

f(x) = -log(x + 2)

Vertical shift down by 3 units: The graph has been shifted down by 3 units, which means that the entire function has been shifted downward by 3 units. So we get:

f(x) = -log(x + 2) - 3

Therefore, the final function that represents the given transformations is:

f(x) = -log(x + 2) - 3

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When two objects are massive and close to each other, the gravitational force between them is strong.

Gravity is a fundamental force of nature that exists between any two objects in the universe. The strength of the gravitational force depends on two factors: the masses of the two objects and the distance between them.

The larger the masses of the two objects, the greater the gravitational force between them. Likewise, the closer the objects are to each other, the stronger the gravitational force.

This relationship is described by Newton's Law of Gravitation, which states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them.

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Answer:

h(x) = (x^2 - 64)

Step-by-step explanation:

If the solutions to h of x = 0 are x = -8 and 8, then h(x) must be a quadratic function with roots at x = -8 and x = 8.

One way to write the quadratic function that satisfies these conditions is:

h(x) = (x+8)(x-8)

where a is some non-zero constant that determines the shape of the parabola.

Expanding the above equation, we get:

h(x) = (x^2 - 64)

Therefore, one possible quadratic function that could represent h is:

h(x) = (x^2 - 64)

where a is any non-zero constant.

The probability that more than a fifth (i.e. 6 or more) of the 10 buyers would prefer green is 0.3981.

To conduct a study of the color preferences of new car buyers, a researcher wishes to conduct a research. Suppose that 30% of the population prefers green. If ten buyers are selected at random, what is the probability that more than a fifth of the buyers will prefer green?

The random variable X represents the number of buyers who prefer green. When n = 10 and p = 0.3, this is a binomial probability experiment. P(X > 2) is the probability that more than a fifth of the buyers would prefer green.P(X > 2) = 1 - P(X ≤ 2)P(X > 2) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)]P(X > 2) = 1 - [C(10,0) (0.3)^0 (0.7)^10 + C(10,1) (0.3)^1 (0.7)^9 + C(10,2) (0.3)^2 (0.7)^8]P(X > 2) = 1 - [1(0.7)^10 + 10(0.3)(0.7)^9 + 45(0.3)^2 (0.7)^8]P(X > 2) = 0.2743

Therefore, the probability that more than a fifth of the buyers would prefer green is 0.2743, rounded to four decimal places.

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The binomial is (a² + 4)

We can rewrite this as follows:

[tex]-a^2*(a^2 + 4) + (a^2 + 4)[/tex]

You can think of the a squared plus 4 thing as a common factor because it appears in both terms, then we can rewrite this as:

[tex]-a^2*(a^2 + 4) + (a^2 + 4)\\-a^2*( a^2 + 4 + 1)\\-a^2*(a^2 + 5)[/tex]

That is the expression factorized.

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the area of the equilateral triangle using the formula for the area of a regular polygon: 43.5 units squared

The area of a customary polygon is given as:

[tex]=\frac{1}{2} ap[/tex]

'a' is the apothem and 'p' is the edge.

The edge of the equilateral triangle is 3×10=30 units.

The apothem is given by:

[tex]a= \frac{8}{\sqrt[2]{3} }\\ \\a= \frac{10}{\sqrt[2]{3} } \\\\a=2.9[/tex]

Consequently, the area is

1/2 * 2.9 * 30 = 43.5

On the off chance that all the polygon sides and inside points are equivalent, they are known as ordinary polygons. Instances of standard polygons are square, equilateral triangle, and so forth. In standard polygons, not exclusively are the sides consistent however the points are as well. That means they are equiangular.

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the complete question is:

Bianca calculated the height of the equilateral triangle with side lengths of 10. tangent (30) = StartFraction 5 Over h EndFraction An equilateral triangle with side lengths of 10 is shown. A bisector is drawn to split the side into 2 equal parts and splits the angle into 2 30 degree segments. Then, she used the formula for area of a triangle to approximate its area, as shown below. A = one-half b h. = one-half (10) (8.7). = 43.5 units squared. Calculate the area of the equilateral triangle using the formula for area of a regular polygon, and compare it to Bianca's answer. The apothem, rounded to the nearest tenth, is units. The perimeter of the equilateral triangle is units. Therefore, the area of the equilateral triangle is , or approximately 43.5 units2. The calculated areas are .

Answer:

  6 inches

Step-by-step explanation:

You want to know the radius of a cylinder with volume 72π in³ and height 2 in.

The given numbers can be put in the given formula, and the equation solved for r.

  V = πr²h

  72π in³ = πr²(2 in)

  36 in² = r²

  r = √(36 in²) = 6 in

The radius of the cylinder is 6 inches.

Answer:

The radius of the hubcap is half of its diameter, which is:

r = d/2 = 26/2 = 13 cm

The area of the hubcap, in square centimeters, is given by the formula:

A = πr²

Substituting the value of r, we get:

A = 3.14 x 13²

A = 3.14 x 169

A = 530.66

Therefore, the area of the hubcap is approximately 530.66 square centimeters.

Step-by-step explanation:

I did this question and the correct answer is 169π.

Step-by-step explanation:

When you take a strip of paper, put two half twists in it, and glue the ends together and then cut it lengthwise along the center core line, it results in two interlocked loops.What you get is two interlocked loops when you cut it lengthwise along the center core line. This occurs because the paper strip has been twisted twice. The twist that was made in the paper strip produces two loops that are intertwined after it is cut down the center line.Paper is quite a flexible substance. When it's twisted, it retains its new shape. Because of this, when you twist a strip of paper, it retains its new form. When the strip is bent and connected to form a loop, the twisted part becomes embedded in the paper's core. This produces two loops that are interconnected. When the loop is sliced down the middle, the two loops remain interconnected since the twists in the strip have become embedded in the core.

Answer:

D (D)

Step-by-step explanation:

We can know from the question that each face has two A, two B, and two C, two D, and two E. And these three faces are the sides of the triangular pyramid (Each face vertex is D).

∴ We can get the picture of the fourth face.

∴ There are two A, two B, two C, one D, and two E in the picture.

∴ The middle of the fourth face is D.

*See the attachment*

Answer:

D

Step-by-step explanation:

Answer: 56

Step-by-step explanation:

I guess and checked all the numbers that add up to 11

The statement is False. At a significance level of alpha equal to .05, if your model has a p-value close to 0 for the overall f test, but none of the individual variables in the model have a p-value <0.05, then you do not have an issue with multi-collinearity. Therefore the given statement is false.

A p-value is a measure of the strength of evidence against the null hypothesis. The smaller the p-value, the stronger the evidence against the null hypothesis.

When the p-value is less than or equal to the significance level alpha, then we reject the null hypothesis. In general, when the overall F-test has a low p-value and none of the individual variables in the model have a p-value less than 0.05, it is an indication of multi-collinearity.

Multi-collinearity is a situation in which two or more independent variables in a multiple regression model are highly correlated with each other. It can lead to unstable estimates of the regression coefficients and make it difficult to interpret the results of the model.

In conclusion, at a significance level of alpha equal to .05, if your model has a p-value close to 0 for the overall f test, but none of the individual variables in the model have a p-value <0.05, then you do not have an issue with multi-collinearity. The statement is false.

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Answer:

Step-by-step explanation:

Answer:

D) x=20

Step-by-step explanation:

On the spinner we have numbers from 0 to 9.

It is 10 numbers and we hit only 0,1, 2 and 3 (4 f them)

Lateya have propability [tex]\frac{4}{10}[/tex] to hit those numbers.

[tex]\frac{4}{10} *50=4*5=20[/tex]

The given statement, "the margin-of-error refers to the number of percentage points by which survey results may vary" is true because the margin of error is an important factor to consider when interpreting the results of a poll.

In a poll, the margin of error refers to the amount of random sampling error that is expected to be present in the results of the survey. It is a measure of the statistical uncertainty associated with using a sample of a population to make inferences about the opinions or characteristics of the entire population.

The margin of error is typically reported as a plus or minus percentage, such as "plus or minus 3 percentage points," and it represents the range of values within which the true population parameter is likely to fall with a certain level of confidence. For example, if a poll has a margin of error of plus or minus 3 percentage points with a 95% level of confidence, this means that if the poll were conducted 100 times, the results would be expected to fall within plus or minus 3 percentage points of the true population parameter in 95 of those surveys.

Therefore, the margin of error is an important factor to consider when interpreting the results of a poll and helps to provide a measure of the precision and reliability of the survey findings.

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Answer:

Step-by-step explanation:

1. -(5/x^6)

2. 4398046511104

3. 65536

4. 1/100

5. 1/5764801

If the budget were to increase to $3000, allowing us to survey many more people from a simple random sample, the test statistic would likely be more accurate and reliable.

As per the given scenario,

if we have a limited budget of $500, we can only survey 50 people from a simple random sample. In order to study the relationship between social class and voting preferences, the budget must be increased to $3000.

On increasing the budget, we can survey more people from a simple random sample, which would lead to a better representation of the population.
Due to the small budget, the test statistic would be less accurate as only a small number of people are being surveyed. However, if the budget increases to $3000, it would allow for a larger sample size and therefore the test statistic would be more accurate.

This would result in a narrower confidence interval and a higher level of confidence in the accuracy of the results.
Moreover, with a larger budget, it would be possible to use more sophisticated sampling techniques such as stratified random sampling, which could further increase the accuracy of the test statistic.

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Red-green color blindness is controlled by an x-linked gene in humans. The allele that causes color blindness is rare and recessive to the allele for normal vision. A man and woman both with normal vision had color-blind fathers. If this man and woman have a child, Therefore, the probability of a child being color-blind is 25%.

Red-green color blindness is an X-linked recessive disorder. This means that the gene for the disorder is found on the X chromosome. Because males have only one X chromosome (inherited from their mother), they will develop color blindness if they inherit the allele from their mother. On the other hand, females need to inherit two copies of the allele, one from each parent, to develop the disorder.A man with normal vision will have the genotype XY, and the woman will have the genotype XX.

Each child of the couple will receive one X chromosome from the mother and one Y chromosome from the father. The probability of the child inheriting the allele for color blindness from its father is 0 because he has no alleles for color blindness.

The probability of the child inheriting the allele for color blindness from its mother is 1/2 because the mother's X chromosome has a 1/2 probability of carrying the allele. Because the mother is a carrier, the probability of the child receiving the allele is 1/2.

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Answer:

Step-by-step explanation:

We can simplify the expression by first finding the common denominator for the fractions in each term, which is 45:

4/9 x 18/5 + 14/5 x 2/9 - 8/18

= (418)/(95) + (142)/(59) - 8/18 (multiplying to get the common denominator)

= 72/45 + 28/45 - 8/18 (simplifying each term)

= (72 + 28)/45 - 8/18 (adding the first two terms)

= 100/45 - 4/9 (simplifying)

= (1001)/(451) - (45)/(95) (multiplying by 1 in the form of 5/5)

= 100/45 - 20/45 (simplifying)

= 80/45 (final answer, which can be simplified to 16/9)

Therefore, 4/9 x 18/5 + 14/5 x 2/9 - 8/18 = 16/9.

These equatiοns represent the fact that the slοpes and y-intercepts οf the lines fοrming the sides οf the triangle are related, and that the lines intersect at the vertices οf the triangle.

An equatiοn is a mathematical statement that asserts that twο expressiοns are equal. It cοnsists οf twο sides, a left-hand side and a right-hand side, cοnnected by an equal sign "=". The expressiοns οn either side οf the equal sign can be numbers, variables, οr cοmbinatiοns οf bοth.

Here,

Part a: If the three sides οf the triangle are denοted by a, b, and c, then we can write the fοllοwing system οf equatiοns:

a + b = c

b - a = 3

These equatiοns represent the fact that the sum οf twο sides οf a triangle is equal tο the third side (a + b = c), and that the difference between twο sides is equal tο a given value (b - a = 3).

Part b: If the cοοrdinates οf the vertices οf the triangle are given by (x₁, y₁), (x₂, y₂), and (x₃, y₃), then we can write three equatiοns representing the three sides οf the triangle. Fοr example, the equatiοn οf the line passing thrοugh (x₁, y₁) and (x₂, y₂) is given by:

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

where (x, y) are the cοοrdinates οf any pοint οn the line. Similarly, we can write equatiοns fοr the οther twο sides οf the triangle, and οbtain a system οf three linear equatiοns.

Part c: If the equatiοns οf the three sides οf the triangle are given by Ax + By + C = 0, Dx + Ey + F = 0, and Gx + Hy + I = 0, then we can write the fοllοwing system οf equatiοns:

A/Bx + C/B = -(A/By + C/B) (frοm the first equatiοn)

D/Ex + F/E = -(D/Ey + F/E) (frοm the secοnd equatiοn)

G/Hx + I/H = -(G/Hy + I/H) (frοm the third equatiοn)

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Complete question:

The lines that form the three sides of the triangle can be grouped into three different systems of two linear equations. Create 3 equations using different methods.

The interquartile range (IQR) of the given data set is 16.

To find the interquartile range (IQR), we first need to calculate the first and third quartiles of the data set:

Arrange the data set in order from smallest to largest:

22, 26, 28, 30, 32, 35, 35, 42, 45, 49, 50

Find the median (middle value) of the lower half of the data set (the first quartile, Q1):

22, 26, 28, 30, 32, 35

The median of this set is 29, which is halfway between 28 and 30.

Find the median (middle value) of the upper half of the data set (the third quartile, Q3):

35, 42, 45, 49, 50

The median of this set is 45.

Calculate the IQR by subtracting Q1 from Q3:

IQR = Q3 - Q1

   = 45 - 29

   = 16

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Answer:

6

Step-by-step explanation:

imagine that EF continues through F and ends on AB. this point, that is on segment AB, is what i will call G.

AFG is similar to CFE.

the length of segment GF is 18-4=14.

now we know 2 side lengths of triangle GAF: FA is 21 and FG is 14.

FE is similar to GF with a factor of 4/14 = 2/7.

FE is similar to FC with this same ratio: (2/7) * 21 = 6

therefore, the length of FC is 6.

[tex]\textit{volume of a cylinder}\\\\ V=\pi r^2 h~~ \begin{cases} r=radius\\ h=height\\[-0.5em] \hrulefill\\ h=19\\ V=4100 \end{cases}\implies 4100=\pi r^2(19)\implies \cfrac{4100}{19\pi }=r^2 \\\\\\ \sqrt{\cfrac{4100}{19\pi }}=r\implies 8.29\approx r[/tex]

Answer:

[tex]y = 6x - 4[/tex]

Step-by-step explanation:

To find the equation of the line that passes through the given points (-1, -10) and (3, 14), first find the slope by inputting the points into the slope formula.

[tex]\boxed{\begin{minipage}{8cm}\underline{Slope Formula}\\\\Slope $(m)=\dfrac{y_2-y_1}{x_2-x_1}$\\\\where $(x_1,y_1)$ and $(x_2,y_2)$ are two points on the line.\\\end{minipage}}[/tex]

Let (x₁, y₁) = (-1, -10)

Let (x₂, y₂) = (3, 14)

Therefore, the slope of the equation is:

[tex]\implies m=\dfrac{14-(-10)}{3-(-1)}=\dfrac{14+10}{3+1}=\dfrac{24}{4}=6[/tex]

Now that we have determined the slope of the line, we can input the found slope and either of the two given points into the point-slope formula to create the equation of the line.

[tex]\boxed{\begin{minipage}{5.8 cm}\underline{Point-slope form of a linear equation}\\\\$y-y_1=m(x-x_1)$\\\\where:\\ \phantom{ww}$\bullet$ $m$ is the slope. \\ \phantom{ww}$\bullet$ $(x_1,y_1)$ is a point on the line.\\\end{minipage}}[/tex]

Substituting m = 6 and (x₁, y₁) = (-1, -10) into the point-slope formula:

[tex]\implies y-(-10)=6(x-(-1))[/tex]

Simplifying:

[tex]\implies y+10=6(x+1)[/tex]

[tex]\implies y+10=6x+6[/tex]

[tex]\implies y=6x-4[/tex]

Therefore, the equation of the line that passes through (-1,-10) and (3,14) is:

[tex]\boxed{y = 6x - 4}[/tex]

Answer:

6x - y - 4 = 0

Step-by-step explanation:

To find:-

Solution:-

We are here given two lines and we are interested in finding out the equation of the line passing through the given points.

Firstly here we will find out the slope of the line as ,

[tex]:\implies \sf m =\dfrac{y_2-y_1}{x_2-x_1}\\[/tex]

Now on substituting the respective values, we have;

[tex]:\implies \sf m =\dfrac{-10-14}{-1-3} \\[/tex]

[tex]:\implies \sf m = \dfrac{-24}{-4}\\[/tex]

[tex]:\implies \sf\pink{ m = 6}\\[/tex]

Now we can use the point slope form of the line to find out the equation of the line . The point slope form of the line is ,

[tex]:\implies \sf \pink{ y - y_1 = m(x_2-x_1)}\\[/tex]

Take any one of the given points for (x₁,y₁) . Here i am taking (3,14) .

Now finally substitute the respective values,

[tex]:\implies \sf y - 14= 6(x -3 )\\[/tex]

[tex]:\implies \sf y- 14= 6x - 18 \\[/tex]

[tex]:\implies \sf 6x - y - 18 + 14 = 0\\[/tex]

[tex]:\implies \sf\pink{ 6x - y -4 = 0 }\\[/tex]

Hence the required equation of the line in standard form is 6x - y - 4 = 0 .

Creating a rule y=f(x) by applying transformations to the fundamental function f(x) = √x  The rule that would produce the given graph is f(x) = 2√(x-1) + 1.

To transform the basic square root function f(x) = √x to the given graph, we can use the following transformations:

1. Horizontal shift to the right by 1 unit: f(x-1) = √(x-1)

This will move the point (1, 1) on the basic function to (2, 1) on the new graph.

2. Vertical stretch by a factor of 2: 2f(x-1) = 2√(x-1)

This will result in a two-fold vertical stretch of the graph.

3. Vertical shift upward by 1 unit: 2f(x-1) + 1 = 2√(x-1) + 1

This will move the entire graph upward by 1 unit.

Therefore, the rule that would produce the given graph is f(x) = 2√(x-1) + 1.

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Answer:

Step-by-step explanation:

(a) A slice through the center, parallel to a side:

To visualize this slice, imagine cutting the rectangle horizontally, exactly in the middle, so that you end up with two smaller rectangles stacked on top of each other. The resulting cross section will be a rectangle with the same width as the original rectangle, but only half of its length.

(b) A slice that cuts through the left top edge and the right bottom edge:

To visualize this slice, imagine cutting the rectangle diagonally from the top left corner to the bottom right corner. The resulting cross section will be a right triangle, with one of its legs being half of the original rectangle's width, and the other leg being half of its length.

(c) A slice that cuts across a corner through three sides of the rectangle:

To visualize this slice, imagine cutting the rectangle diagonally from one of the top corners to the opposite bottom corner, and then continuing the cut along one of the sides of the rectangle, until you reach the middle of that side. The resulting cross section will be a trapezoid, with one of its parallel sides being half of the original rectangle's width, and the other parallel side being half of its length. The two non-parallel sides will be angled.

A grouped frequency table (grouped frequency conveyance) is an approach to coordinating an enormous arrangement of information into additional reasonable gatherings.

The grouped frequency table

________________________________________________________

 Width (in mm) C.F. Class Interval Frequency ________________________________________________________

 ≥ 20 50 20 - 30 50 - 44

= 6 ≥ 30 44 30 - 40 44 - 28

= 16

 ≥ 40 28 40 - 50 28 - 20

= 8

 ≥ 50 20 50 - 60 20 - 15

= 5

 ≥ 60 15 60 - 70 15 - 0

= 15

_____________________________________________________

                                                                                          ∑f = 50

The gatherings that we arrange the mathematical information into are called class intervals. They can have something similar or different class widths and should not cover.

The principal segment is for the gatherings where we compose the class intervals.

The center section is for the count marks.

The last segment is the frequency section where we can include the count marks.

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the complete question is:

The widths of 50 leaves of a plant were measured in mm and their cumulative frequency distribution is shown in the following table. Make frequency distribution table for this . CLASS FREQUENCY >20 50 >30 44 >40 28 >50 20 ​>60 15

Answer:

the star , the down arrow (3rd one), the stop sign

The skier needs to go at least 65.4 miles/hour to beat the person in first place.

What is trigonometry?

It is the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.

To solve this problem, we need to use trigonometry to find the horizontal distance the skier needs to cover in order to beat the person in first place, and then use this distance and the time it took the first-place skier to calculate the required speed.

A represents the skier's starting point, B represents the end of the slope where the person in first place finished, and C represents the top of the mountain. The angle of depression is labeled θ and is equal to 60 degrees.

Using trigonometry, we can calculate the horizontal distance AB using the tangent function:

tan θ = opposite/adjacent

where opposite = h = 10000 ft (the height of the mountain) and adjacent = AB (the horizontal distance we want to find).

Substituting θ = 60 degrees and h = 10000 ft, we get:

tan 60 = AB/10000

√3 = AB/10000

AB = 10000√3 ≈ 17321.17 ft

So the skier needs to cover a horizontal distance of about 17321.17 feet in order to beat the person in first place.

To find the required speed, we need to convert this distance to miles and divide by the time it took the first-place skier (3 minutes):

17321.17 ft = 17321.17/5280 miles ≈ 3.28 miles

Speed = distance/time = 3.28 miles / 3 minutes = 1.09 miles/minute

To convert miles/minute to miles/hour, we need to multiply by 60 (the number of minutes in an hour):

Speed = 1.09 miles/minute x 60 minutes/hour ≈ 65.4 miles/hour

Therefore, the skier needs to go at least 65.4 miles/hour to beat the person in first place.

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