A student guesses on every question of a​ multiple-choice test that has 6 ​questions, each with 3 possible answers. What is the probability that the student will get at least 4 of the questions​ right?

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

a)

[tex]6\cdot5\cdot2\cdot2=120[/tex]

b)

[tex]4\cdot5\cdot2\cdot2 +2\cdot5\cdot1\cdot1=80+10=90[/tex]

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).

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: 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%.

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:

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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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:

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:

131ft is the amount of fencing material needed

Step-by-step explanation:

Perimeter is the distance around a shape: we have to sum all the distances

P = d1 + d2 + d3

P = 34 ft + 30 ft + 67 ft = 131 ft

Answer:

D. 131 ft.

Step-by-step explanation:

If Sarah is planning to fence in her backyard garden and one side of the garden is 34 feet long, another side is 30 feet long, and the third side is 67 feet long. The perimeter of Sarah’s garden to determine the amount of fencing material needed is 131 feet.

The question asks for the probability that the sample mean of income differs from the true mean by over 42 dollars for a sample size of 412. We use the Central Limit Theorem to approach this problem, calculating the standard error and Z-score to refer to the standard normal distribution to find the associated probability.

This question pertains to the field of statistics, more specifically, the Central Limit Theorem. The Central Limit Theorem says that if we take many samples from a population, the distribution of the sample means will approximate a normal distribution, regardless of the shape of the population distribution. We can use this theorem to calculate a probability related to the sample mean.

In this case, the population mean (μ) is a per capita income of 24,787 dollars and the population variance (σ²) is 169,744 dollars. We're asked to find the probability that the sample mean would differ from the true mean by more than 42 dollars if a sample of 412 persons is randomly selected.

The standard error of the sample mean is calculated by σ / sqrt(n), where σ is the standard deviation (sqrt(σ²)), and n is the sample size (412). After finding the standard error, we will calculate the Z-score of 42, which is the number of standard errors 42 is away from the mean. Calculating the Z-score is achieved by z = (X - μ) / SE, where X is the value of 42.

We can refer the calculated Z-score to the standard normal distribution to find the associated probability. However, since we are looking for the probability of a difference greater than 42, we want the probabilities in the tails of the distribution beyond our calculated Z-score, so it would be 1 minus this value.

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Answer:

r(q(-4)) = -10

Step-by-step explanation:

q(x) = -x - 2

r(x) = -2x^2 -2

r(q(-4))

First find q(-4)

Let x=-4

q(-4) = -(-4) - 2

q(-4) = +4 -2 = 2

q(-4) =2

We substitute this value in for x in r(x)

r(2) = -2(2)^2 -2

     = -2 (4) -2

     = -8 -2

     = -10

For this case we have the following functions:

[tex]q (x) = - x-2\\r (x) = - 2x ^ 2-2[/tex]

We must find [tex]r (q (x))[/tex]. So:

We substitute [tex]q (x)[/tex]in [tex]r (x).[/tex]

[tex]r (q (x)) = - 2 (-x-2) ^ 2-2[/tex]

Now we substitute[tex]x = -4[/tex]

[tex]r (q (-4)) = - 2 (- (- 4) -2) ^ 2-2\\r (q (-4)) = - 2 (+ 4-2) ^ 2-2\\r (q (-4)) = - 2 (2) ^ 2-2\\r (q (-4)) = - 2 (4) -2\\r (q (-4)) = - 8-2\\r (q (-4)) = - 10[/tex]

Answer:

[tex]r (q (-4)) = - 10[/tex]

Answer:

5.92 per cheesecake.

Step-by-step explanation:

The initial number cherry cheesecakes = 280,

Also, the cost of each cheesecakes = $ 2.51,

So, the total cost price = 280 × 2.51 = $ 702.8,

Markup = 65 %,

Thus, the total selling cost = 702.8 + 65% of 702.8 = $ 1159.62,

Now, the spoilage rate = 30 %,

So, the total new number of cheesecakes = 280 - 30% of 280 = 280 - 84 = 196,

Hence, the selling cost of each cheesecake = [tex]\frac{1159.62}{196}[/tex]

[tex]=\$ 5.91642857143[/tex]

[tex]\approx \$ 5.92[/tex]

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:

Step-by-step explanation:

A) Suppose that we have the complex numbers

[tex]z= x + iy \quad \text{and} \quad \\\\ \tilde{z}=\tilde{x} + i \tilde{y}[/tex]

Remember that to sum complex numbers, we sum the real parts of the two numbers to get the real part and the imaginary parts of the two numbers to get the imaginary part. Hence,  

[tex]z+\tilde{z} = (x + i y) + (\tilde{x} + i \tilde{y}) = (x + \tilde{x})+i (y+\tilde{y})[/tex]

On the other hand, if we sum the matrix visualizations of [tex]z \quad \text{and} \quad \tilde{z}[/tex] we get

[tex]\left[\begin{array}{cc}x &y\\-y&x\end{array}\right] + \left[\begin{array}{cc}\tilde{x}&\tilde{y}\\ -\tilde{y}&\tilde{x}\end{array}\right] = \left[\begin{array}{cc}x + \tilde{x}& y + \tilde{y}\\-(y+\tilde{y})&x+\tilde{x}\end{array}\right][/tex]

which is the matrix visualization of [tex]z + \tilde{z}[/tex].

To multiply two complex numbers, we use the distributive law to multiplly and then separete the real part from the imaginary part

[tex]z \cdot \tilde{z}= (x + iy) \cdot (\tilde{x} + i \tilde{y})=(x \tilde{x} + i x \tilde{y} + i \tilde{x} y - y\tilde{y} ) = (x\tilde{x}-y\yilde{y})+i(x\tilde{y}+\tilde{x}y)[/tex]

Again, if we multiply the matrix visualizations of [tex]z \quad \text{and} \quad \tilde{z}[/tex] we get

[tex]\left[\begin{array}{cc}x&y\\-y&x\end{array}\right]\left[\begin{array}{cc}\tilde{x}&\tilde{y}\\-\tilde{y}&\tilde{x}\end{array}\right] = \left[\begin{array}{cc}x\tilde{x}-y\tilde{y}&x\tilde{y}+y\tilde{x}\\-y\tilde{x}-x\tilde{y}&x\tilde{x}-y\tilde{y}\end{array}\right][/tex]

which is the matrix viasualization of [tex]z\cdot\tilde{z}.[/tex]

B)  Since the usual matrix operations are consisten with the usual addition and multiplication rules in the complex numbers, we can use them to find the multiplicative inverses of a complex number [tex]z=x+iy[/tex].

We are looking for the complex number [tex]z^{-1}=(x+iy)^{-1}[/tex] which in terms of matrices is equivalent to find the matrix

[tex]\left[\begin{array}{cc}x&y\\-y & x\end{array}\right]^{-1}= \dfrac{1}{x^{2}+y^{2}} \left[\begin{array}{ccc}x&-y\\y&x\end{array}\right][/tex]    

Hence,

[tex]z^{-1}=\dfrac{1}{x^2 +y^2} (x-iy)=\dfrac{1}{|z|^2}(x-iy)[/tex]

Answer:

210

Step-by-step explanation:

In 2005 the employee was tripple of 1995 which is 50×3=150. So number of employees in 2005=150.

In 2015 number of employees were 40% of employees in 2005.

40% of 150 =150×40÷100 = 60,

Therefore number of employees in 2015 =150+60= 210 employees.

Answer:

210 employees in 2015

Step-by-step explanation:

In order to solve this we first have to calculate the number of employees that there were in 2005, since the problem says that they tripled their employees from 1995, that meas three times 50, meaning 150 employees in 2005, if by 2015 the number of employees had increase in 40% we just do a simple rule of three, 150 being the 100% and trying to calculate the 140%:

[tex]\frac{150}{100}= \frac{x}{15} \\x=\frac{150*15}{100}\\ x= 210[/tex]

So now we know that there were 210 employees in 2015.

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.

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: 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 %

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 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: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: 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:

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:

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.

To evaluate the triple integral, set up the limits of integration based on the given bounds. The parabolic cylinder and the planes define the limits for x, y, and z. Evaluate the triple integral using appropriate integration techniques.

To evaluate this triple integral, we need to set up the limits of integration based on the given bounds of the parabolic cylinder and the planes. The parabolic cylinder is defined by the equation z = 25 - y^2, so we need to find the limits for x, y, and z. Since the plane z = 0 is a boundary, the lower limit of integration for z is 0. The other boundary planes, x = 5 and x = -5, set the limits for x. For y, we need to find the limits based on the parabolic cylinder equation. By rearranging the equation, we have y = ±sqrt(25-z), which gives us the upper and lower limits for y.

Now, we can set up the triple integral as follows:

∫-55 ∫-sqrt(25-z)sqrt(25-z) ∫025-y^2 x^6e^y dzdydx

We can then evaluate this triple integral using the appropriate integration techniques, such as using the power rule and integration by parts for the x^6e^y term.

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To evaluate the triple integral, we need to find the limits of integration for each variable. Given the given bounds, the limits for x and y can be determined, and then the integral can be solved using standard techniques.

To evaluate the triple integral, we need to find the limits of integration for each variable.

Given that E is bounded by the parabolic cylinder z = 25 - y^2 and the planes z = 0, x = 5, and x = -5, the limits for y will be -√(25-x^2) to √(25-x^2), the limits for x will be -5 to 5, and the limits for z will be 0 to 25 - y^2.

The integral becomes: ∫∫∫ x^6e^y dz dx dy, with the limits as mentioned above. You can now solve this triple integral using standard integration techniques.

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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:

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:

wow that school has alot of test don't it lol

Step-by-step explanation:

anyway what's the question

After calculating the Z-scores for the student's performance in each class, we found that she performed closer to the top of her class in statistics. Thus, she did relatively better on the statistics exam compared to the chemistry exam.

To determine relative performance in each class, we need to calculate the number of standard deviations the student's score is from the mean (also known as a

Z-score

).

For the chemistry exam, where the mean was 90 and the standard deviation was 64, the student's Z-score is: (102 - 90) / 64 = 0.1875.

For the statistics exam, with a mean of 70 and standard deviation of 16, the student's Z-score is: (77 - 70) / 16 = 0.4375.

Comparing these Z-scores, we can conclude that the student did relatively better in the statistics class because her Z-score (0.4375) was higher there than in her chemistry class (0.1875). This indicates she performed closer to the top of her class in statistics. Therefore, the answer would be selection C: the student did relatively better on the statistics exam than on the chemistry exam, compared to the other students in the two classes.

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