Prove that x^3+15x^2+74x+120 is always divisible by 6. Please

The unit rate of rainfall for 72 hours of rain will be equal to u = 8 inches/hour.

The four basic mathematical operations are the addition, subtraction, multiplication, and division of two or even more integers. Among them is the examination of integers, particularly the order of actions, which is crucial for all other mathematical topics, including algebra, data organization, and geometry.

As per the provided data in the question,

Total time, t = 72 hours and,

The total amount of water in inches, w = 9 inches.

Then unit rate, u will be,

u = 72/9

u = 8 inches/hour.

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Answer: 6 and 7

Step-by-step explanation:

Find the square root of 42 = 6.48

Closest whole numbers to 6.48 are 6 and 7

Yes, The relation represent a function.

What is Function?

A relation between a set of inputs having one output each is called a function.

Given that;

The points are,

{ (- 12, - 21) , (- 21, - 22) , (22, - 12) , (21, 22) , (17, -22) } ​

Now,

Since, The points are,

{ (- 12, - 21) , (- 21, - 22) , (22, - 12) , (21, 22) , (17, -22) } ​

Clearly, All inputs having exactly one output.

Thus, The relation represent a function.

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Answer: y = 40, slope is 0, the line is horizontal

Step-by-step explanation:

When chemist mixes Chemical A and Chemical B to make a solution

Part A

When the quantity of chemical B = 48 ml, then the quantity of chemical A = 4 ml

Part B

When the quantity of chemical A = 9 ml, then the quantity of chemical B = 108 ml

A chemist mixes Chemical A and Chemical B to make a solution

She uses the same ratio of chemicals for each batch

The solution A and B are in linear relationship

y = kx

Where y is the quantity of chemical A

x is the quantity of chemical B

k is the constant

Substitute the values in the table and find k

36 = k × 3

k = 36 /3

k = 12

Part A

When chemical B = 48 ml

48 = 12 × x

x = 48 / 12

x = 4 ml

Part B

When chemical A = 9 ml

y = 12 × 9

y = 108 ml

Therefore, when Chemical B = 48 ml, chemical B is 4ml . When chemical A = 9 ml, Chemical B = 108 ml

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I think that it is true because it is asking for a summary of the data and finding the mean and standard deviation of the speeds and weight would give you the summary of all the data.

The standard deviation is a statistic that expresses how much variance or dispersion there is in a group of numbers. While a high standard deviation suggests that the values are dispersed throughout a larger range, a low standard deviation suggests that the values tend to be near to the established mean. The term "standard deviation" (or "") refers to a measurement of the data's dispersion from the mean. A low standard deviation indicates that the data are grouped around the mean, whereas a high standard deviation shows that the data are more dispersed.

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The required 90% confidence interval for population mean age is (22.35,23.45)

Given,

Sample size, n = 20

Sample mean, μ = 22.9 years

Standard deviation, σ = 1.5 years

Significance level, α = 0.05

We have to find the 90% confidence interval;

Here,

x ± z(critical) σ/√n

Substituting the values;

z(critical) at  α₀.₀₅ = ±1.64

22.9 ± 1.64 (1.5/√20)

= 22.9 ± 0.55

= (22.35, 23.45)

That is,

(22.35,23.45) is the required 90% confidence interval for population mean age.

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The 96th term of the given arithmetic sequence is -220

From the question, we are to determine the 96th term of the given arithmetic sequence.

The given arithmetic sequence is

-30, -32, -34

The nth term of an arithmetic sequence is given by

aₙ = a + (n - 1)d

Where a is the first term

and d is the common difference

From the sequence,

a = -30

d = a₂ - a₁ = -32 - (-30)

d = -32 + 30

d = -2

Thus,

The 96th term is

a₉₆ = -30 + (96 -1)-2

a₉₆ = -30 + (95)-2

a₉₆ = -30 + (-190)

a₉₆ = -30 - 190

a₉₆ = -220

Hence, the 96th term is -220

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The required relationship between weight and number of months is given as y = 6x + 90.

As of the given data, assume the relationship on the graph,
Initially ordered pair (0, 90) and instantaneous ordered pair (8, 138)

The slope of the line is a tangent angle made by line with horizontal. i.e. m =tanx where x in degrees.

Here,
The slope of the relationship,

M = (y₂ - y₁) /  (x₂ - x₁)

m = 138 - 90 / 8  - 0
m = 48 / 6
m = 6

Now,
Equation of the relationship,

y - y₁ = m (x - x₁)
y - 90 = m{x - 0}
y = 6x + 90

Thus, y = 6x + 90 is the needed relationship between weight and the number of months.

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The confidence interval for the given data be,

(144.83 , 155.17).

On Subtracting 1 from your sample size, we get

150 – 1 = 149.

On Subtracting confidence level from 1, and then divide by two.

(1 – .95) / 2 = 0.025

Now, df = 149 and α = 0.025

from the table at df = 149 we got 2.262.

Divide your sample standard deviation by the square root of your sample size.

28 / √(150) = 2.28

Now, 2.262 × 2.28 = 5.17

So, the confidence interval be,

(150 - 5.17 , 150 + 5.17) = (144.83 , 155.17)

Hence, the confidence interval for the given ANOVA test to calculate the average typing speed be, (144.83 , 155.17).

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Answer:

x = 0

Step-by-step explanation:

1. Distribute

2. Add the numbers

3. Rearrange terms

4. Subtract 19 from both sides

5. Simplify

6. Add 5x to both sides

7. Simplify

8. Divide both sides by the same factor

9. Simplify

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Step-by-step explanation:

The perimeter of 2nd right isosceles triangle is 16(√2 + 2).

If a is the equal sides of a right isosceles triangle,

Then its hypotenuse is √2 a and perimeter is √2a (√2  + 1 )

The perimeter of 1st right isosceles triangle is 16(√2 + 1)

= √2a (√2  + 1 )

= 16 (√2  + 1 )

a = 8√2

The hypotenuse of 1st triangle = √2  a

= √2 * 8√2

= 16cm

Let A be the equal sides of 2nd right isosceles triangle

Then its hypotenuse is √2 a and perimeter is √2a (√2  + 1 )

According to the question,

A = 16 cm

The perimeter of 2nd right isosceles triangle is √2 * 16(√2 + 1)

= 16(√2 + 2)

Hence, The perimeter of 2nd right isosceles triangle is 16(√2 + 2).

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Answer:

m<D = 108

Step-by-step explanation:

m<ACD = 28 degrees. This is because if a transversal cuts through two parallel lines, Alternate Interior Angles are Congruent.

m<CAD = 44 degrees. This is because if a transversal cuts through two parallel lines, Alternate Interior Angles are Congruent. (Same reasoning.)

Now, since the sum of all angles in a triangle is 180 degrees, we can use the two angles that we know as well as angle D to add up to 180 and figure out how much D is. (All of the angles in triangle ADC)

<CAD + <ACD + <D = 180

Substitute the values in.

44 + 28 + <D = 180

72 + <D = 180

Subtract on both sides

<D = 108

The measure of angle D is 108 degrees.

The first set of steps solving 5x-y = 4 to get y = 5x-4 is correct.

The second set of steps are incorrect since it involves circular logic. Notice we're plugging y = 5x-4 back into the original equation we just solved. It would be like plugging an extension cord into itself, and somehow expecting electricity to magically come out of nowhere.

Instead we should plug that y expression into the first equation and solve for x like so:

8x+2y = -12

8x+2(5x-4) = -12

8x+10x-8 = -12

18x-8 = -12

18x = -12+8

18x = -4

x = -4/18

x = -2/9

Then we use this to find the value of y.

y = 5x-4

y = 5(-2/9)-4

y = -10/9-4

y = -10/9 - 36/9

y = (-10-36)/9

y = -46/9

In summary we have the solution (x,y) = ( -2/9, -46/9 )

The equation of the line perpendicular to given equation is 3x - y + 2 = 0

what is perpendicular line?

A straight line that forms a right angle of 90 degrees with another line is known as a perpendicular in mathematics. Alternatively stated, two lines are perpendicular to one another if they cross at a right angle.

solution:

The slope for the perpendicular line is - ( 1/m )

Then the given line has the slope y = (1/3)x - 2 is 1/3

substitute 1/3 in perpendicular equation, Then the slope for the perpendicular line is - ( 1/m ) is -3

The line equation is y = mx +c, ( -2, -4 )

Then c = -2

substitute slope for the perpendicular line is - ( 1/m ) is -3 and c = -2 in line equation

then the equation is 3x - y + 2 = 0

Therefore the equation of line perpendicular to given equation is 3x - y + 2 = 0

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The 91st term in the sequence is 1,1.01,1.02 is 1.9.

An arithmetic sequence is a set of numbers that are ordered and have a common difference between each consecutive term. In the arithmetic sequence 3, 9, 15, 21, 27, for example, the common difference is 6. An arithmetic progression is another name for an arithmetic sequence. We use the formula for finding the nth term to find a specific term in an arithmetic sequence. Step 1: An arithmetic sequence's nth term is given by a = a + (n - 1)d.

Based on the information given, it should be noted this is an arithmetic sequence.

The formula t calculate the nth term will be:

= a + (n - 1)d

where

a = first term = 1

n = 91

d = common difference = 0.01

The 91st term will be:

= a + (n - 1)d

= 1 + (91 - 1) × 0.01

= 1 + (90 × 0.01)

= 1 + 0.9

= 1.9

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The key reason why this investigation's findings are erroneous is sampling bias. In statistics, sampling bias refers to a bias during which a sample is constructed in a way that makes some members of the intended population less likely than others to be included.

A private university's 100 students, 50 men, and 50 women were randomly chosen to participate in a study on how American college students felt about commuting. They were questioned about their daily commuting distance (in minutes) to college. The key reason why this investigation's findings are erroneous is sampling bias.

When certain individuals of a population are consistently more likely to be chosen in a sample than others, this is known as sampling bias. In the medical sciences, it is also known as ascertainment bias. Because sampling bias jeopardizes external validity, particularly population validity, it restricts the generalizability of findings.

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Answer:

it is b 5/7

Step-by-step explanation:

bc you rise 5 and run 7

hope this helps

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The fraction of table filled is 285/301 or in decimal as 0.946

Fractions are expressed in mathematics as a number value that defines a part of a whole or group of objects. Fraction is a component or sector of a given quantity taken from the whole, which might be a number, a specified value, or an item.

In a given fraction, the value on the top is called the numerator, while the value on the bottom is known as the denominator. The numerator defines the number of equal parts taken, whereas the denominator defines the total number of equal parts in a whole.

In this question, we have to add the total number of attendees first

212 + 89 = 301

But each seat except 1 is filled by 16 people

Let's subtract 16 from the total number

301 - 16  = 285

The fraction of table filled is  285 ÷ 301 = 285/301 = 0.946

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[tex]\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\\ P=\textit{initial amount}\dotfill &20000\\ r=rate\to 10\%\to \frac{10}{100}\dotfill &0.1\\ t=years\\ \end{cases} \\\\\\ A=20000(1 - 0.1)^{t} \implies A=20000(0.9)^t[/tex]

how long till it's $7000?

[tex]\qquad \textit{Amount for Exponential Decay} \\\\ A=P(1 - r)^t\qquad \begin{cases} A=\textit{current amount}\dotfill &7000\\ P=\textit{initial amount}\dotfill &20000\\ r=rate\to 10\%\to \frac{10}{100}\dotfill &0.1\\ t=\textit{elapsed time} \end{cases} \\\\\\ 7000=20000(1 - 0.1)^{t} \implies \cfrac{7000}{20000}=0.9^t\implies \cfrac{7}{20}=0.9^t[/tex]

[tex]\log\left( \cfrac{7}{20} \right)=\log(0.9^t)\implies \log\left( \cfrac{7}{20} \right)=t\log(0.9) \\\\\\ \cfrac{\log\left( \frac{7}{20} \right)}{\log(0.9)}=t\implies 9.96\approx t \qquad \textit{about 9 years and 11 months}[/tex]

Check the picture below, so the parabola looks more or less like so.

keeping in mind the vertex is always half-way between the focus point and the directrix, let's also notice the "p" distance.

[tex]\textit{vertical parabola vertex form with focus point distance} \\\\ 4p(y- k)=(x- h)^2 \qquad \begin{cases} \stackrel{vertex}{(h,k)}\qquad \stackrel{focus~point}{(h,k+p)}\qquad \stackrel{directrix}{y=k-p}\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix}\\\\ \stackrel{"p"~is~negative}{op ens~\cap}\qquad \stackrel{"p"~is~positive}{op ens~\cup} \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\begin{cases} h=-4\\ k=2\\ p=-1 \end{cases}\implies 4(-1)(y-2)=( ~~ x-(-4) ~~ )^2\implies -4(y-2)=(x+4)^2 \\\\\\ y-2=-\cfrac{1}{4}(x+4)^2\implies {\Large \begin{array}{llll} y=-\cfrac{1}{4}(x+4)^2 + 2 \end{array}}[/tex]

Step-by-step explanation:

390 ft² (aka 390 ft. squared)

bh times 1/2

13 times 5 divided by two is 37.5 but sine you have two just skip the division and say 75

and then multiply 12 by 10 for the square and get 120

(120+75)*2=390

so the answer is 390 ft. squared

The minimum power input needed to drive this escalator is

15.59 kw.

What is escalator?

A moving staircase that transports people between floors of a building or other structure is called an escalator. It is made up of a motor-driven chain of separately linked steps on a track that rotates on two other tracks, keeping them horizontal.

Given:

[tex]N_p = 50\\m_p = 7.5 kg\\V = 0.6m/s\\[/tex]

θ = 45°

Total mass = 50 x 75 = 3750 kg

The energy required = Change in potential energy

ΔE = mg ΔZ

Power = ΔE/t = mg(ΔZ/t)

ΔZ/t = vsinθ = 0.6 sin(45°) = 0.4247

Power = mg x 0.4243 = 3750 x 9.8 / 100 x 0.4243

P = 15.59 kw

Hence,  the minimum power input needed to drive this escalator is

15.59 kw.

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Answer:

16.2

Step-by-step explanation:

use long division so:

The probability he or she selects a management major is 0.176. This is estimated using empirical probability.

Probability is a branch of mathematics that deals with numerical probabilities of events. It is expressed in a range from 0 to 1, not below or above this range. The probability formula is P(E) = (number of favorable outcomes)÷(total number of outcomes).

a) Here, given the total number of outcomes is 34. The favorable outcome is 6. Therefore, the probability he or she selecting the management major is,

P(E)=6/34=0.176.

b) The empirical probability employs numerous trials to calculate likelihood using real observed frequencies. It is an experimental probability that is based on previous or past data.

In the given question, we used past data to estimate the probability. Therefore, it is empirical.

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a) scatter diagram made using the provided data is attached.

b) According to the scatter diagram, the number of defective parts discovered rises with line speed.

c) Y = 27.5 - 0.3 X is the regression model.

What does statistics mean in plain English? a field of mathematics concerned with the gathering, examination, presentation, and interpretation of vast amounts of numerical data. a compilation of numerical data

20 faulty pieces are anticipated to be present when the line speed is 25 feet per minute.

Detailed explanation:

The given information is displayed as follows:

Line Speed;                   20,   20,    30,    30,    40,    40,    50,     50

Number of Defective;   23,    21,     19,     16,     15,     17,     14,       11  

Parts located

a. The scatter diagram made with Microsoft Excel is attached.

b. The scatter figure appears to show a relationship between increased line speed and the quantity of detected defective parts.

c. The following equation is given for linear regression using the least squares approach;

Y = a + b·X

Where;

b= N∑XY - (∑X)(∑Y)/N∑X^2 - (∑X)^2

a= ∑Y - b∑X / N

We have data from Microsoft Excel;

∑X = 280, ∑Y = 136, ∑X² = 10,800, ∑XY = 4,460, (∑X)² = 78,400, N = 8

By entering the values, we obtain;

b = (8×4,460 - 280×136)/(8×10,800 - 78,400) = -0.3

a = (136 - (-0.3)×280)/8 = 27.5

As a result, we have the following linear regression;

Y = 27.5 - 0.3·X

Consequently, we have when the line speed is 25 feet per minute;

Y = 27.5 - 0.3 × 25 = 20

When the line speed is 25 feet per minute, Y = 20 faulty components should be expected to be identified.

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Answer: w = 7

Step-by-step explanation:

Isolate the variables by dividing each side with the constants.

Hope this helps!

The dimension of the rectangular pen that maximizes the area is 61,250ft.

What is a dimension?

Something's size, including its breadth, height, and depth: She took exact measurements of the room's dimensions in one direction, like the distance from the ceiling to the floor. The room was surprisingly small, in terms of length, width, and height.

Here, we have

L+2W = 700

Area = A= LW = (700-2W)W = 700W -2W²

Take the derivative, set it equal to zero, and solve for W

-4W + 700 = 0

W = 175 feet

L = 700-2(175) = 350 feet

350 by 175 ft maximizes area = 61,250 square feet

Hence, the dimension of the rectangular pen that maximizes the area is 61,250ft.

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The probability of getting a prime number of heads given that heads occurs on at least 6 of the tosses is 0.0078

The number of observations n is 7

Here the success is heads

The probability of heads is 1/2

We need to find the the probability of getting a prime number of heads.

Heads occurs on at least 6 of the tosses

prime number greater than equal to 6 and less than equal to 7 is 7

So, X = 7

P(X = 7) = 7C7(1/2)⁷(1/2)⁰

P(X = 7) =  1 x (1/2)⁷ x

P(X =7 ) = 1/128

= 0.0078

Therefore, the probability of getting a prime number of heads given that heads occurs on at least 6 of the tosses is 0.0078

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