You have 100 quarters in a jar. One of the quarters is double sided (heads). You pick out a random quarter and flip it 7 times, and get all heads. what is the probability you picked the double sided quarter? Then, given that you flipped it 7 times with all heads, what is the probability that you'll get heads on the 8th flip?

The value of  X  is 55.

The detailed answer explains how to solve the equation for x by following a step-by-step process.

Solve the equation for x:

Answer:  

The probability that the student will get at least 4 of the questions​ right is 0.0823044.

Step-by-step explanation:

For each question we have 3 choices. So,total choices will be :

[tex]3\times3\times3\times3\times3\times3=729[/tex]

Getting 4 correct means, 4 corrects and two wrongs

Now, as there are 3 answer choices, out of which only one will be correct, so 2/3 is the probability if a question is answered wrong.

And 1/3 is the probability if a question is answered correctly.

Hence, we can consider this probability :

[tex]P=(2/3)*(2/3)*(1/3)*(1/3)*(1/3)*(1/3)[/tex] = 4/729

=> P = 0.00548696

We can select any combination of 2 from 6 for being wrong, so we will multiply P by (6,2)=6!/(2!*4!) = 15

So the answer is P*15 =[tex]0.00548696*15=0.0823044[/tex]

The probability that the student will get at least 4 of the questions​ right is 0.0823044.

With 3 choices per question, the probability of getting at least 4 out of 6 questions correct is approximately 0.0823044

1: Total Choices

Each question has 3 possible answers.

So, the total choices for 6 questions would be 3 raised to the power of 6 (3^6).

2: Probability of Getting 4 Correct and 2 Wrong

Getting 4 correct and 2 wrong means selecting 4 correct answers out of 6 questions.

The probability of a question being answered correctly is 1/3, and the probability of being answered incorrectly is 2/3.

So, the probability of getting 4 correct and 2 wrong is calculated using combinations (6 choose 4) multiplied by (1/3)^4 multiplied by (2/3)^2.

3: Calculate Probability

(6 choose 4) is the number of ways to choose 4 correct answers out of 6 questions, which is 15.

The probability (P) is then calculated as 15 multiplied by (1/3)^4 multiplied by (2/3)^2.

4: Multiply by Number of Combinations

Since there are 15 ways to choose 4 correct answers out of 6 questions, multiply the probability by 15.

So, the probability that the student will get at least 4 of the questions right is approximately 0.0823044.

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Answer:

False

Step-by-step explanation:

Given that a researcher testing the effects of two treatments for anxiety computed a 95% confidence interval for the difference between the mean of treatment 1 and the mean of treatment 2

Also that  confidence interval includes zero.

When confidence interval includes zero, we need not reject the null hypothesis since null hypothesis claims that difference =0

When confidence interval includes 0 it confirms that there is no difference and hence null hypothesis should be accepted.

Final answer:

The statement is false; if a 95% confidence interval for the difference between two treatment means includes zero, it indicates no significant difference, implying the null hypothesis cannot be rejected.

Explanation:

If a researcher testing the effects of two treatments for anxiety computed a 95% confidence interval for the difference between the mean of treatment 1 and the mean of treatment 2, and this confidence interval includes the value of zero, the correct interpretation is false regarding the statement that she should reject the null hypothesis that the two population means are equal. A confidence interval that contains zero indicates that the difference between the two treatments could be zero, suggesting there is no significant difference between the two treatments.

Therefore, there is insufficient evidence to reject the null hypothesis, and it is retained.

Understanding confidence intervals is crucial in hypothesis testing. A 95% confidence interval includes the true mean 95% of the time if the same experiment is repeated under the same conditions. Including zero in this interval suggests that the effect of the treatments could be negligible, meaning we cannot confidently claim there is a difference between the treatments based on the data provided.

Answer:

  1.  n = 40

  2.  

Step-by-step explanation:

The ordinary annuity formula can be written as ...

  PV = PMT(1 -(1+r)^-n)/r

where PMT is the payment per period, r is the interest rate per period, and n is the number of periods.

This formula can be solved explicitly for n, but not for r. Iterative or other methods can be used to find r.

__

1. Filling in the given information, we have ...

  15000 = 650(1 -1.03^-n)/0.03

  450/650 = 1 - 1.03^-n . . . . . divide by the coefficient of the stuff in parens

  1.03^-n = 4/13 . . . . . . . . . . . solve for the exponential term

  -n·log(1.03) = log(4/13) . . . . take logarithms

  n = log(13/4)/log(1.03) ≈ 39.87 . . . . . solve for n

  n ≈ 40

__

2. We can rewrite the annuity formula to make it be a function of i that is zero at the desired value of i.

  f(i) = PV -PMT(1 -(1+i)^-n)/i

If we want i as a percentage, then we can replace i with i/100 and fill in the given values to get ...

  f(i) = 9000 -500(1 -(1 +i/100)^-35)/(i/100)

  f(i) = 1000(9 -50(1 -(1 +i/100)^-35)/i) . . . . multiply the fraction by 100/100

Since we're seeking a value of f(r) that is zero, we can eliminate the factor of 1000.

  f(i) = 9 -50(1 - (1+i/100)^-35)/i

The attached graph shows the solution to f(i)=0 is near i=4.27%. As a decimal rounded to 3 decimal places, this is ...

  i ≈ 0.043

Answer:

y(x) = 1 - x

Step-by-step explanation:

Given the two parametric equations:

[tex]  x(t)=cos^{2}(6t)  [/tex]  ---(1)

[tex] sin^{2}(6t) [/tex] ----(2)

We can add eq (1) and eq (2) and consider the trigonometric identity:

[tex]  cos^{2}(6t)+sin^(6t) = 1  [/tex]

so,

[tex]   x+y=1  [/tex]

in other way we  can express this like:

[tex] y(x)=1-x [tex].

Answer:0.0206

Step-by-step explanation:

Using Binomial distribution for a sample of 20 adults

Let r denotes the no of correct answers out of 20

Probability that fewer than 6 people will guess correctly is P(r<6)

P(r<6)=P(r=0)+P(r=1)+P(r=2)+P(r=3)+P(r=4)+P(r=5)

[tex]P(r=0)=^{20}C_0\left ( 0.5\right )^{0}\left ( 0.5\right )^{20}=\left ( 0.5\right )^{20}[/tex]

[tex]P(r=1)=^{20}C_0\left ( 0.5\right )^{1}\left ( 0.5\right )^{19}=20\left ( 0.5\right )^{20}[/tex]

[tex]P(r=2)=^{20}C_0\left ( 0.5\right )^{2}\left ( 0.5\right )^{18}=190\left ( 0.5\right )^{20}[/tex]

[tex]P(r=3)=^{20}C_0\left ( 0.5\right )^{3}\left ( 0.5\right )^{17}=1140\left ( 0.5\right )^{20}[/tex]

[tex]P(r=4)=^{20}C_0\left ( 0.5\right )^{4}\left ( 0.5\right )^{16}=4845\left ( 0.5\right )^{20}[/tex]

[tex]P(r=5)=^{20}C_0\left ( 0.5\right )^{5}\left ( 0.5\right )^{15}=15,504\left ( 0.5\right )^{20}[/tex]

[tex]P(r<6)=\left ( 0.5\right )^{20}\left [ 1+20+190+1140+4845+15504\right ][/tex]

[tex]P(r<6)=\left ( 0.5\right )^{20}\times 21,700[/tex]

P(r<6)=0.02069

Answer:

After 25 years the amount will be $78099.34.

Step-by-step explanation:

The compound interest formula is ;

[tex]A=p(1+r/n)^{nt}[/tex]

Where p = 22000

r = 5.1% or 0.051

n = 4

t = 25

So, putting the values in formula we get;

[tex]A=22000(1+0.051/4)^{100}[/tex]

[tex]A=22000(1.01275)^{100}[/tex]

A = $78099.34

Therefore, after 25 years the amount will be $78099.34.

Answer: 95.45 %

Step-by-step explanation:

Given : The distribution is bell shaped , then the distribution must be normal distribution.

Mean : [tex]\mu=\ 16[/tex]

Standard deviation :[tex]\sigma= 1.5[/tex]

The formula to calculate the z-score :-

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x = 13

[tex]z=\dfrac{13-16}{1.5}=-2[/tex]

For x = 19

[tex]z=\dfrac{19-16}{1.5}=2[/tex]

The p-value = [tex]P(-2<z<2)=P(z<2)-P(z<-2)[/tex]

[tex]0.9772498-0.0227501=0.9544997\approx0.9545[/tex]

In percent, [tex]0.9545\times100=95.45\%[/tex]

Hence, the percentage of data lie between 13 and 19 = 95.45 %

Answer:

n = 23

x = 17

p = 0.50

Step-by-step explanation:

For a binomial experiment we have the following variables:

1) Number of trials or Sample size:

The number of trials is represented by n. In the given scenario 23 adults were asked about their favorite pizza, so the number of trials in this will be 23. Thus

n = 23

2) Number of success

The number of success is denoted by x. Number of success indicates that how many trials resulted in the favorable outcome. In the given case, choosing a sausage pizza is a success. Since 17 adults chose the sausage, so

x = 17

3) Probability of success on single trial

This is represented by p. It is stated that 50% adults say sausage is their favorite pizza. So,

p = 50% = 0.50

In a binomial probability experiment, n represents the size of the random sample, p represents the probability of success, and x represents the number of successes. In this given scenario, n=23, p=0.50 and x=17.

In a binomial probability experiment, the parameters n, p, and x are designated as follows: 'n' is the size of the random sample which in this case is 23. 'p' is the probability of success on a single trial, here it is the percentage of adults who indicated that sausage is their favorite pizza, which in decimal form is 0.50. 'x' is the number of successes, in this scenario, the number of adults in the sample of 23 who prefer sausage on their pizza, which is 17. So, in this experiment, n=23, p=0.50 and x=17. Success in this context is defined as an individual person preferring sausage on their pizza.

To ensure the binomial experiment is valid, and can be approximated by a normal distribution, the quantities np and nq (where q is 1-p, the probability of failure) must both be greater than five (np > 5 and nq > 5). In this case, np = 23*0.50 = 11.5 and nq = 23*0.50 = 11.5, thus the experiment is valid.

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Final answer:

To find Greg's straight-line distance from home to the cafe, we calculate his north-south and east-west displacements in feet using a 660 feet block length. Then, applying the Pythagorean theorem, we determine the hypotenuse, which represents the straight-line distance. The calculated straight-line distance is approximately 3847.11 feet.

Explanation:

To calculate the straight-line distance that Greg would travel from his starting point to the cafe, we can use the Pythagorean theorem. Initially, Greg walks 7 blocks north and 3 blocks east, and then from his office to the cafe, he walks 2 blocks south and 6 blocks west. Considering that every block is 660 feet, we can determine his total displacement in the north-south direction and the east-west direction.

First, we find the net blocks traveled north-south: 7 blocks north - 2 blocks south = 5 blocks north. Then, we find the net blocks traveled east-west: 3 blocks east - 6 blocks west = 3 blocks west. Since the blocks are 660 feet each, we convert blocks into feet:

Using the Pythagorean theorem (a2 + b2 = c2), where 'a' and 'b' are the legs of the right triangle and 'c' is the hypotenuse representing the straight-line distance, we calculate:

We then plug these values into the equation:

c2 = 33002 + 19802
 = 10890000 + 3920400
 = 14810400

We find the square root of 14810400 to get the straight-line distance (c), which is:

c = √14810400 ≈ 3847.11 feet

So, the straight-line distance from Greg's home to the cafe is approximately 3847.11 feet.

Answer:

[tex]\frac{1}{35}[/tex]

Step-by-step explanation:

On a single roll of a pair of dice. When a pair of dice are rolled the possible outcomes are as follows:

(1,1)         (1,2)          (1,3)  (1,4)  (1,5)  (1,6)

(2,1)  (2,2)  (2,3)  (2,4)  (2,5)  (2,6)

(3,1)  (3,2)  (3,3)  (3,4)  (3,5)  (3,6)

(4,1)  (4,2)  (4,3)  (4,4)  (4,5)  (4,6)

(5,1)  (5,2)  (5,3)  (5,4)  (5,5)  (5,6)

(6,1)  (6,2)  (6,3)  (6,4)  (6,5)  (6,6)

The number of outcomes that gives us 12 are (6,6). There is only one outcome that gives us sum 12.

Total outcomes = 36

Odd against favor = [tex]\frac{non \ favorable\ outcomes}{favorable \ outcomes}[/tex]

Number of outcomes of getting sum 12 is 1

Number of outcomes of not getting sum 12 is 36-1= 35

odds against rolling a sum of 12= [tex]\frac{1}{35}[/tex]

Final answer:

A detailed explanation of the odds against rolling a sum of 12 on a pair of dice.

Explanation:

On a single roll of a pair of dice, the odds against rolling a sum of 12 are:

There is only one way to roll a 12, which is by getting a 6 on each die.

The probability of rolling a 6 on one die is 1/6 or approximately 0.166.

The probability of rolling a 12 on both dice is (1/6) * (1/6) = 1/36, which is about 2.8%.

Answer: 0.357

Step-by-step explanation:

Given : The number of Hatfield  = 6

The number of McCoys = 2

The number of companies = 8

The number of construction jobs -3

Now, the required probability is given by :-

[tex]\dfrac{^6C_3\times^2C_0}{^8C_3}\\\\=\dfrac{\dfrac{6!}{3!(6-3)!}}{\dfrac{8!}{3!(8-3)!}}=0.357142857143\approx0.357[/tex]

Hence, the probability that all 3 jobs go to Hatfields =0.357

Rolle's theorem works for a function [tex]f(x)[/tex] over an interval [tex][a,b][/tex] if:

This is our case: [tex]f(x)[/tex] is a polynomial, so it is continuous and differentiable everywhere, and thus in particular it is continuous and differentiable over [0,4].

Also, we have

[tex]f(0)=7=f(4)[/tex]

So, we're guaranteed that there exists at least one point [tex]c\in(a,b)[/tex] such that [tex]f'(c)=0[/tex].

Let's compute the derivative:

[tex]f'(x)=3x^2-2x-12[/tex]

And we have

[tex]f'(x)=0 \iff x= \dfrac{1\pm\sqrt{37}}{3}[/tex]

In particular, we have

[tex]\dfrac{1+\sqrt{37}}{3}\approx 2.36[/tex]

so this is the point that satisfies Rolle's theorem.

The number C that satisfies the conclusion of Rolle's Theorem on the interval [0, 4] is: [tex]\[ c = \frac{1 + \sqrt{37}}{3} \][/tex]

To verify that the function [tex]\( f(x) = x^3 - x^2 - 12x + 7 \)[/tex] satisfies the three hypotheses of Rolle's Theorem on the interval [0, 4] and then to find all numbers c that satisfy the conclusion of Rolle's Theorem, follow these steps:

1. The function f is continuous on the closed interval [a, b]:

  - [tex]\( f(x) = x^3 - x^2 - 12x + 7 \)[/tex] is a polynomial, and polynomials are continuous everywhere.

  - Therefore, f is continuous on [0, 4].

2. The function f is differentiable on the open interval (a, b):

  - Again, [tex]\( f(x) = x^3 - x^2 - 12x + 7 \)[/tex] is a polynomial, and polynomials are differentiable everywhere.

  - Therefore, f is differentiable on (0, 4).

3. f(a) = f(b) :

  - Calculate [tex]\( f(0) \)[/tex] and f(4):

 [tex]\[ f(0) = 0^3 - 0^2 - 12 \cdot 0 + 7 = 7 \] \[ f(4) = 4^3 - 4^2 - 12 \cdot 4 + 7 = 64 - 16 - 48 + 7 = 7 \][/tex]

  - Therefore,  f(0) = f(4) = 7 .

Since all three hypotheses are satisfied, by Rolle's Theorem, there exists at least one number c in (0, 4) such that  f'(c) = 0 .

Finding c

1. Compute the derivative of f:

[tex]\[ f(x) = x^3 - x^2 - 12x + 7 \] \[ f'(x) = 3x^2 - 2x - 12 \][/tex]

2. Set the derivative equal to zero and solve for x:

[tex]\[ f'(x) = 3x^2 - 2x - 12 = 0 \][/tex]

  Solve the quadratic equation using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where  a = 3,  b = -2 , and c = -12 :

[tex]\[ x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4 \cdot 3 \cdot (-12)}}{2 \cdot 3} \] \[ x = \frac{2 \pm \sqrt{4 + 144}}{6} \] \[ x = \frac{2 \pm \sqrt{148}}{6} \] \[ x = \frac{2 \pm 2\sqrt{37}}{6} \] \[ x = \frac{1 \pm \sqrt{37}}{3} \][/tex]

3. Check which solutions are in the interval (0, 4):

  - For [tex]\( x = \frac{1 + \sqrt{37}}{3} \)[/tex]:

 [tex]\[ \frac{1 + \sqrt{37}}{3} \approx \frac{1 + 6.08}{3} \approx \frac{7.08}{3} \approx 2.36 \][/tex]

  - For [tex]\( x = \frac{1 - \sqrt{37}}{3} \)[/tex]:

[tex]\[ \frac{1 - \sqrt{37}}{3} \approx \frac{1 - 6.08}{3} \approx \frac{-5.08}{3} \approx -1.69 \][/tex]

    - This solution is not in the interval (0, 4).

Answer: AD = 7.5

Step-by-step explanation: A midpoint is halfway between 2 points, which is AB. AB = 15. To find AD, which is half of the line, divide 15 by 2.

15/2 = 7.5

AD is 7.5

Answer:

AD = 7.5 units

Step-by-step explanation:

It is given in the question, a segment AB having measure = 15 units

If D is the midpoint of the segment AB, then we have to find the measure of segment AD.

Since D is the midpoint of AB then length of segment AD = [tex]\frac{1}{2}\times AB[/tex]

= [tex]\frac{1}{2}\times 15[/tex]

= 7.5 units

Therefore, AD = 7.5 units will be the answer.

Answer:[tex]\theta =\arctan \mu [/tex]

Step-by-step explanation:

we know force sin component would oppose the weight  of object thus normal reaction will not be W rather it would be

[tex]N=W-Fsin\theta [/tex]

therefore force cos component will balance the friction force

F[tex]cos\theta[/tex] =[tex]\left ( \mu N\right )[/tex]

F[tex]cos\theta[/tex] =[tex]\left ( \mu \left ( W-Fsin\theta \right )\right )[/tex]

F=[tex]\frac{\mu W}{cos\theta +\mu sin\theta}[/tex]

F will be smallest when [tex]cos\theta +\mu sin\theta[/tex] will be maximum

and it will be maximum when we differentiate it to get

[tex]\theta =\arctan \mu[/tex]

The magnitude of the force, F, is varies with the angle the rope makes

with the plane according to the given equations.

F will be smallest when [tex]\underline{\theta \ is \ arctan (\mu)}[/tex].

Reason:

The given parameters are;

Angle the rope makes with the plane = θ

The magnitude of the force is, [tex]F = \dfrac{ \mu \cdot W}{\mu \cdot sin(\theta) +cos(\theta) }[/tex]

The value of θ for which the value of F is smallest.

Solution;

When, F is smallest, we have;

[tex]\dfrac{dF}{d \theta} = \dfrac{d}{d\theta} \left(\dfrac{ \mu \cdot W}{\mu \cdot sin(\theta) +cos(\theta) } \right) = \dfrac{-\mu \cdot W \cdot (\mu \cdot cos(\theta) -sin(\theta))}{\left( \mu \cdot sin(\theta) +cos(\theta) \right)^2} = 0[/tex]

Therefore;

-μ·W·(μ·cos(θ) - sin(θ))

μ·cos(θ) = sin(θ)

By symmetric property, we have;

sin(θ) = μ·cos(θ)

[tex]\mathbf{\dfrac{sin(\theta)}{cos(\theta)} = tan (\theta) = \mu}[/tex]

Which gives;

θ = arctan(μ)

Therefore;

F, will be smallest when [tex]\underline{\theta = arctan (\mu)}[/tex].

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Question; The given equation of the magnitude of the force in relation to the angle the rope makes with the plane, θ, is presented as follows;

[tex]F = \dfrac{ \mu \cdot W}{\mu \cdot sin(\theta) +cos(\theta) }[/tex]

Answer:

296 cubic feet

Step-by-step explanation:

First and foremost, you have to have everything in either feet or inches.  Right now they are in both.  Since the answer is asked for in feet, let's convert everything to feet.  The width is already in feet, so that's good.

However, even though the length is 49 feet, we still have to convert the 4 inches part of that to feet.  Using the fact that there are 12 inches in a foot:

[tex]4in.*\frac{1ft}{12in.}=\frac{1}{3}ft[/tex] so we have

[tex]49\frac{1}{3}ft[/tex]

Convert that to improper to make the multiplication easier in the end:

[tex]49\frac{1}{3}=\frac{148}{3}ft[/tex]

Now we have to convert the 8 inches to feet using the same reasoning:

[tex]8in.*\frac{1ft}{12in.}=\frac{2}{3}ft[/tex]

Now everything is in terms of feet.  The volume is found by multiplying length times width times height:

[tex](\frac{148}{3} )(\frac{9}{1})(\frac{2}{3})=  \frac{2664}{9}ft[/tex]

Divide that and it comes out to an even 296 cubic feet

The [tex]n[/tex]-th term in the series is 6 multiplied by the [tex](n-1)[/tex]-th power of 5/6:

[tex]a_1=6=6\left(\dfrac56\right)^{1-1}[/tex]

[tex]a_2=5=6\left(\dfrac56\right)^{2-1}[/tex]

[tex]a_3=\dfrac{25}6=6\left(\dfrac56\right)^{3-1}[/tex]

and so on.

[tex]\displaystyle\sum_{n=1}^\infty6\left(\frac56\right)^{n-1}[/tex]

Consider the [tex]N[/tex]-th partial sum,

[tex]S_N=\displaystyle\sum_{n=1}^N6\left(\frac56\right)^{n-1}[/tex]

[tex]S_N=6\left(1+\dfrac56+\cdots+\dfrac{5^{N-2}}{6^{N-2}}+\dfrac{5^{N-1}}{6^{N-1}}\right)[/tex]

Multiplying both sides by 5/6 gives

[tex]\dfrac56S_N=6\left(\dfrac56+\dfrac{5^2}{6^2}+\cdots+\dfrac{5^{N-1}}{6^{N-1}}+\dfrac{5^N}{6^N}\right)[/tex]

and substracting this from [tex]S_N[/tex] gives

[tex]\dfrac16S_N=6\left(1-\dfrac{5^N}{6^N}\right)[/tex]

[tex]S_N=36\left(1-\left(\dfrac56\right)^N}\right)[/tex]

As [tex]N\to\infty[/tex], it's clear that the sum converges to 36.

The geometric series in the question is convergent with a common ratio of 5/6. Using the formula for the sum of an infinite geometric series, the sum of the series is found to be 36.

In mathematics, specifically in series, determining whether a geometric series is convergent or divergent is centered around the common ratio value. In terms of this particular series: 6 + 5 + 25/6 + 125/36 + ..., the common ratio is 5/6. Given this common ratio, it's clear that it falls between -1 and 1. Hence, this geometric series is convergent.

Once we establish it is a convergent series, we can calculate its sum using the formula for the sum of an infinite geometric series: S = a / (1 - r), where 'a' is the first term and 'r' is the common ratio. Inserting the respective values a = 6 and r = 5/6, we get: S = 6 / (1 - 5/6) = 36. Hence, the sum of this infinite geometric series is 36.

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ANSWER

[tex]x = \frac{3}{2} \: or \: x = - 3[/tex]

EXPLANATION

We want to find the roots of the parabola with equation:

[tex]2 {x}^{2} + 5x - 9 = 2x[/tex]

We need to write this in the standard quadratic equation form.

We group all terms on the left to get:

[tex]2 {x}^{2} + 5x - 2x - 9 = 0[/tex]

We simplify to get:

[tex]2 {x}^{2} +3x- 9 = 0[/tex]

We now compare to:

[tex]a {x}^{2} + bx + c = 0[/tex]

[tex] \implies \: a = 2 , \: \: b = 3 \: \: and \: c=- 9[/tex]

[tex] \implies ac = 2 \times - 9 = - 18[/tex]

The factors of -18 that sums up to 3 are -3, 6.

We split the middle term with these factors to get:

[tex]2 {x}^{2} +6x - 3x- 9 = 0[/tex]

Factor by grouping:

[tex]2x(x + 3) -3(x + 3) = 0[/tex]

Factor again to obtain:

[tex](2x - 3)(x + 3) = 0[/tex]

Apply the zero product principle to get:

[tex]2x - 3 = 0 \: or \: x + 3 = 0[/tex]

[tex] \implies \: x = \frac{3}{2} \: or \: x = - 3[/tex]

Answer:  The required solution is

[tex]y=(-2+t)e^{-5t}.[/tex]

Step-by-step explanation:   We are given to solve the following differential equation :

[tex]y^{\prime\prime}+10y^\prime+25y=0,~~~~~~~y(0)=-2,~~y^\prime(0)=11~~~~~~~~~~~~~~~~~~~~~~~~(i)[/tex]

Let us consider that

[tex]y=e^{mt}[/tex] be an auxiliary solution of equation (i).

Then, we have

[tex]y^prime=me^{mt},~~~~~y^{\prime\prime}=m^2e^{mt}.[/tex]

Substituting these values in equation (i), we get

[tex]m^2e^{mt}+10me^{mt}+25e^{mt}=0\\\\\Rightarrow (m^2+10y+25)e^{mt}=0\\\\\Rightarrow m^2+10m+25=0~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~[\textup{since }e^{mt}\neq0]\\\\\Rightarrow m^2+2\times m\times5+5^2=0\\\\\Rightarrow (m+5)^2=0\\\\\Rightarrow m=-5,-5.[/tex]

So, the general solution of the given equation is

[tex]y(t)=(A+Bt)e^{-5t}.[/tex]

Differentiating with respect to t, we get

[tex]y^\prime(t)=-5e^{-5t}(A+Bt)+Be^{-5t}.[/tex]

According to the given conditions, we have

[tex]y(0)=-2\\\\\Rightarrow A=-2[/tex]

and

[tex]y^\prime(0)=11\\\\\Rightarrow -5(A+B\times0)+B=11\\\\\Rightarrow -5A+B=11\\\\\Rightarrow (-5)\times(-2)+B=11\\\\\Rightarrow 10+B=11\\\\\Rightarrow B=11-10\\\\\Rightarrow B=1.[/tex]

Thus, the required solution is

[tex]y(t)=(-2+1\times t)e^{-5t}\\\\\Rightarrow y(t)=(-2+t)e^{-5t}.[/tex]

Answer:

The dimensions of the package is [tex]r=\frac{48}{\pi}\ \text{and} \ h=48[/tex].

Step-by-step explanation:

Consider the provided information.

As it is given that, cylindrical package to be sent by a postal service can have a maximum combined length and girth is 144 inches.

Therefore,

144 = 2[tex]\pi[/tex]r + h

144-2[tex]\pi[/tex]r = h

The volume of a cylindrical package can be calculated as:

[tex]V=\pi r^{2}h[/tex]

Substitute the value of h in the above equation.

[tex]V=\pi r^{2}(144-2\pi r)[/tex]

Differentiate the above equation with respect to r.

[tex]\frac{dV}{dr}=2\pi r(144-2\pi r)+\pi r^{2}(-2\pi)[/tex]

[tex]\frac{dV}{dr}=288\pi r-4{\pi}^2 r^{2}-2{\pi}^2 r^{2}[/tex]

[tex]\frac{dV}{dr}=288\pi r-6{\pi}^2 r^{2}[/tex]

[tex]\frac{dV}{dr}=-6\pi r(-48+\pi r)[/tex]

Substitute [tex]\frac{dV}{dr}=0[/tex] in above equation.

[tex]0=-6\pi r(-48+\pi r)[/tex]

Therefore,

[tex]0=-48+\pi r[/tex]

[tex]r=\frac{48}{\pi}[/tex]

Now, substitute the value of r in 144-2[tex]\pi[/tex]r = h.

[tex]144-2\pi\frac{48}{\pi}=h[/tex]

[tex]144-96=h[/tex]

[tex]48=h[/tex]

Therefore the dimensions of the package should be:

[tex]r=\frac{48}{\pi}\ \text{and} \ h=48[/tex]

This is about optimization problems in mathematics.

Dimensions; Height = 48 inches; Radius =  48/π inches

Girth is same as perimeter which is circumference of the circular side.

Thus; Girth = 2πr

2πr + h = 144

h = 144 - 2πr

Volume of cylinder; V = πr²h

Let us put 144 - 2πr for h to get;

V = πr²(144 - 2πr)

V = 144πr² - 2π²r³

Differentiating with respect to r gives;

dV/dr = 288πr - 6π²r²

Thus; 288πr - 6π²r² = 0

Add 6π²r² to both sides to get;

288πr = 6π²r²

Rearranging gives;

288/6 = (π²r²)/πr

48 = πr

r = 48/π inches

h = 144 - 2π(48/π)

h = 144 - 96

h = 48 inches

Read more at; https://brainly.com/question/12964195

Final answer:

In the political party's survey example, the population is all registered voters in the community, the sample is the 1500 randomly selected voters, and the situation is an example of a survey with nonresponse bias.

Explanation:

Understanding Population and Sample in Surveys

When it comes to surveys, it's important to differentiate between a population and a sample. For the political party's mail survey:

The population refers to all registered voters in the community since they are the entire group of individuals the survey is designed to understand and represent.

The sample is the 1500 randomly selected voters who received the questionnaire, as they are a manageable number intended to represent the larger population of all registered voters in the community.

This example is of a survey containing nonresponse bias, because a significant portion of the surveys were not returned, which could skew the results and not accurately represent the overall population.

Nonresponse is an issue that affects the reliability and accuracy of survey results because those who do not respond could have systematically different views from those who do. This is an example of nonsampling error, as the error arises not from the method of selecting the sample but from the lack of responses.

Final answer:

The population in the survey is all registered voters in the community, and the sample is the 1500 randomly selected voters. This scenario is an example of a survey containing nonresponse bias.

Explanation:

In the scenario described, the population is Option C) all registered voters in this community, since the study seeks to understand an attribute (opinion on the president's job performance) of this entire group. The sample is Option B) the 1500 randomly selected voters receiving the questionnaire, as they are the portion of the population supposed to represent the larger group's opinions. The issue described is an example of B) a survey containing nonresponse bias, which occurs when the subset of the sample that responds (480 returned surveys) is different in some way from those who do not respond, which can potentially skew the survey results. Finally, nonresponse bias is a critical challenge in survey methods since response rates can affect the representativeness of the sample and hence the accuracy of the survey's conclusions.

Answer:

Step-by-step explanation:

We know that X-Y = X∩Y'

Using it ,we get

A ∩(B∩C') which can be written as (A∩B)∩C' or (A∩B) - C

And right hand side is

(A∩B)-(B∩C) =B ∩(A-C) = B∩(A∩C') = A∩B∩C'

Since both left and right side both leads to same expression A∩B∩C'

Therefore both are equal.

Answer:

B. $271

Step-by-step explanation:

Given,

Present value of the loan, PV = $ 6000,

Annual rate of interest = 8 % = 0.08,

So, the monthly rate of interest, r = [tex]\frac{0.08}{12}[/tex],

Also, time = 2 years,

So, the total number of months, n = 24,

Hence, the monthly payment would be,

[tex]A=\frac{PV(r)}{1-(1+r)^{-n}}[/tex]

[tex]=\frac{6000(\frac{0.08}{12})}{1-(1+\frac{0.08}{12})^{-24}}[/tex]

[tex]=\$271.363748737[/tex]

[tex]\approx \$271[/tex]

Option B is correct.

Answer: There is 100 ounces of 35% salt solution and 100 ounces of water.

Step-by-step explanation:

Since we have given that

Percent of salt in a solution = 35%

Percent of salt in a mixture = 20%

Number of ounces of a mixture = 175 ounces

We need to find the number of ounces of water and salt as well as .

We would use "Mixture and Allegation":

      Salt                       Water

      35%                           0%

                       20%

---------------------------------------------------------------

20% - 0%          :                35% - 20%

   20%              :                     15%

   4                    :                      3

So, Ratio of salt and water in the mixture is 4 : 3.

So, Number of ounces of salt in the mixture is given  by

[tex]\dfrac{4}{7}\times 175\\\\=100\ ounces[/tex]

Number of ounces of water in the mixture is given by

[tex]\dfrac{3}{7}\times 175\\\\=75\ ounces[/tex]

Hence, there is 100 ounces of 35% salt solution and 100 ounces of water.

Final answer:

To make a 175-ounce mixture with 20% salt, the scientist should mix 75 ounces of water with 100 ounces of the 35% salt solution.

Explanation:

The student is asking for help with a typical mixture problem in algebra that involves determining the amounts of two different concentrations in order to create a mixture with a desired concentration. To solve this, we can set up two equations, one based on the total volume of the mixture and one based on the total amount of salt.

Let x be the amount of water (0% salt) and y be the amount of the 35% salt solution. The total volume should be 175 ounces, so we have:

Equation 1: x + y = 175

The total amount of salt in the solution must be 20% of 175 ounces, which is 35 ounces. So for the salt amount, we have:

Equation 2: 0.35y = 35

Solving Equation 2 gives us y = 100 ounces for the 35% solution. Substituting y in Equation 1, we get x = 75 ounces for the water. Therefore, the scientist should mix 75 ounces of water with 100 ounces of the 35% salt solution to obtain 175 ounces of a 20% salt mixture.

Answer:  The correct option is (C) 38.

Step-by-step explanation:  Given that a polyhedron has 20 vertices and 20 faces.

We are to find the number of edges of the polyhedron using Euler's formula.

Euler's formula :

For any polyhedron, the number of vertices and faces together is exactly two more than the number of edges.

Mathematically, V − E + F = 2, where V, E and F represents the number of vertices, number of edges and number of faces of the polyhedron.

For the given polyhedron, we have

number of vertices, V = 20,

number of faces, F = 20

and

number of edges, E = ?

Therefore, from Euler's formula

[tex]V-E+F=2\\\\\Rightarrow 20-E+20=2\\\\\Rightarrow 40-E=2\\\\\Rightarrow E=40-2\\\\\Rightarrow E=38.[/tex].

Thus, the required number of edges of the given polyhedron is 38.

Option (C) is CORRECT.

Using Euler's formula V - E + F = 2 for a polyhedron with 20 vertices and 20 faces, we find that the number of edges (E) is 38.

The question pertains to finding the number of edges of a polyhedron using Euler's formula, which states V - E + F = 2, where V is the number of vertices, E is the number of edges, and F is the number of faces. Given that the polyhedron has 20 vertices (V = 20) and 20 faces (F = 20), we can rearrange Euler's formula to solve for the number of edges (E): E = V + F - 2. Plugging in the values we get E = 20 + 20 - 2, which simplifies to E = 38. Therefore, the polyhedron has 38 edges.

Answer:

C. Must be 1

Step-by-step explanation:

The correlation coefficient of an equation represents the relation between the  two variable ( dependent and independent )

It lies between -1 to 1,

If an equation has strongly positive correlation then the value of correlation coefficient is 1,

If there is no relation then the value of correlation is 0,

If there is strongly negative relation then the value of correlation coefficient is -1,

Here, the equation is,

y = 2 + 3x

Since, the value of y is increasing with increasing the value of x,

We can say that there is strong positive relation between the variables x and y,

Hence, by the above statements,

The correlation coefficient between y and x in this data set must be 1.

Option 'C' is correct.

Answer:

Ella got married on a Wednesday.

Step-by-step explanation:

Let's solve this problem by understanding the following:

Each week is composed by 7 days: Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, and Sunday.

So 1 week = 7 days;

Because the married day was on the 9074th day of her life, we can find the number of weeks that 9074 days represent:

[tex]\frac{1 week}{7 days} * 9074 days = 1296.285714 weeks[/tex]

This means that 9074 days represent 1296.285714 weeks, which can be interpreted as 1296 entire weeks and a fraction of a week (0.285714).

Now let's calculate how many days 0.285714 weeks represent:

[tex]\frac{7 days}{1 week} * 0.285714 weeks = 2 days[/tex]

This means that 9074 days are actually 1296 weeks and 2 days, because Ella was born on a Monday, and because after 7 days (1 week) it is Monday again, after 1296 weeks it is Monday, but as we also calculated 2 extra days, then the married day is two days after a Monday, that is a Wednesday.

In conclusion, Ella got married on a Wednesday.

Answer:

if x=-1 then its is NOT in the domain of h.

Step-by-step explanation:

Domain is the set of values for which the function is defined.

we are given the function

h(x) = x + 1 / x^2 + 2x + 1

h(x) = x+1 /x^2+x+x+1

h(x) = x+1/x(x+1)+1(x+1)

h(x) = x+1/(x+1)(x+1)

h(x) = x+1/(x+1)^2

So, the function h(x) is defined when x ≠ -1

Its is not defined when x=-1

So, if x=-1 then its is NOT in the domain of h.

Answer: [tex]x=-1[/tex]

Step-by-step explanation:

Given the function h(x):

[tex]h(x)=\frac{x+1}{ x^2 + 2x + 1}[/tex]

The values that are not in the domain of this function are those values that  make the denominator equal to zero.

Then, to find them, you can make the denominator equal to zero and solve for "x":

[tex]x^2 + 2x + 1=0\\\\(x+1)(x+1)=0\\\\(x+1)^2=0\\\\x=-1[/tex]

Answer:1,2,3

Step-by-step explanation:

F(x)=[tex]x^{3}[/tex]+2[tex]x^2[/tex]-[tex]5x[/tex]-[tex]6[/tex]=0

disintegrating 2[tex]x^2[/tex] to [tex] x^2[/tex] + [tex]x^2[/tex]

[tex]x^{3}[/tex]+[tex]x^2[/tex]+[tex]x^2[/tex]-[tex]5x-6[/tex]=0

[tex]x^2[/tex][tex]\left ( x+1\right )[/tex]+[tex]x^2[/tex]-5x-6=0

[tex]x^2[/tex][tex]\left ( x+1\right )[/tex]+[tex]x^2[/tex]-6x+x-6=0

[tex]x^2[/tex][tex]\left ( x+1\right )[/tex]+[tex]\left (x-6 \right )[/tex][tex]\left ( x+1\right )[/tex]=0

[tex]\left ( x+1\right )[/tex][tex]\left ( x^2+x-6\right )[/tex]=0

[tex]\left ( x+1\right )[/tex][tex]\left ( x^2+3x-2x-6\right )[/tex]=0

[tex]\left ( x+1\right )[/tex][tex]\left ( x+3\right )[/tex][tex]\left ( x-2\right )[/tex]=0

Answer: 0.4

Step-by-step explanation:

Given : The number of candidates are equally qualified for President = 5

The number of candidates are members of a minority group =2

Since , to avoid bias in the selection of the candidate, the company decides to select the president by lottery. Here the chances of each candidates is same.

The probability one of the minority candidates is hired is given by :-

[tex]\text{P(Minority)}=\dfrac{\text{Number of minority candidates}}{\text{Total candidates}}\\\\=\dfrac{2}{5}=0.4[/tex]

Hence, the  probability one of the minority candidates is hired =0.4

Final answer:

The probability that one of the minority candidates is hired is 0.4, or 40%, since there are 2 minority candidates out of a total of 5 candidates.

Explanation:

Since there are five equally qualified candidates and two of them are minority candidates, we can calculate the probability by considering the ratio of the number of minority candidates to the total number of candidates.

The probability (P) that one of the minority candidates is hired can be calculated using the formula P = (Number of Favorable Outcomes) / (Total Number of Possible Outcomes).

Here, the Number of Favorable Outcomes is 2 (since there are two minority candidates) and the Total Number of Possible Outcomes is 5 (since there are five candidates in total).

So, the probability is P = 2/5.

Let's compute this:

P = 2/5

P = 0.4

Therefore, the probability that one of the minority candidates will be hired is 0.4, or 40%.