5. Find all three cube roots of the the complex number z = 473 + 4i, and plot
them in the complex plane.
Verify the identity​

Answer:

a) X={BBB;BBG:BGG:GGG}

b) P(BBB)=0.125

c) P(BBB)=0.133

Step-by-step explanation:

The sample space states all the possible values that the random variable can take. In this case, the order does not matter, so the possible combinations for the random variable are:

X={BBB;BBG:BGG:GGG}

The probability p of having a boy is p=0.5, as it is equally likely to have a girl or a boy.

The probability that all 3 children are boys can be calculated multiplying 3 times the probability of having a boy. That is:

[tex]P(X=BBB)=p\cdot p\cdot \cdot p =p^3=0.5^3=0.125[/tex]

In the case that the chance of having a boy is p'=0.51, the probabiltity of having 3 boys become:

[tex]P(X=BBB)=p'^3=0.51^3=0.133[/tex]

Answer:

[tex]270[/tex]

Step-by-step explanation:

[tex]120 - ( - 150) \\ 120 + 150 \\ = 270[/tex]

The distance between 120 and -150 on a number line is calculated by subtracting the smaller number from the larger number. Given that -150 is negative, the subtraction turns into addition, leading to a distance of 270.

To find the distance between 120 and -150 on a number line, you need to take the following steps:

Therefore, the distance between 120 and -150 on a number line is 270

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Answer:

what is the question?

Step-by-step explanation:

There are five successful outcomes for choosing an even number between 0 and 9, which are 0, 2, 4, 6, and 8.

Hernando is choosing a random number between 0 and 9. The even numbers in this range are 0, 2, 4, 6, and 8. To determine the number of successful outcomes for choosing an even number, we simply count the even numbers listed.

There are five even numbers between 0 and 9 which are: 0, 2, 4, 6, and 8. Therefore, there are five successful outcomes for choosing an even number.

Answer:

[tex]a = \frac{w + 4 - 3b}{6} [/tex]

Step-by-step explanation:

[tex]w = 3(2a + b) - 4 \\ w + 4 = 6a + 3b \\ w + 4 - 3b = 6a \\ \frac{w + 4 - 3b}{6} = \frac{6a}{6} \\ a = \frac{w + 4 - 3b}{6} [/tex]

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Answer:

y = 3/5x + (-3)

Step-by-step explanation:

because slope intercept form is y = mx + b

and m = 3/5

and b = -3

so you plug them in to the formula and there's your answer

The equation in slope-intercept form for the line with a slope of 3/5 and a y-intercept of -3 is y = (3/5)x - 3.

We have,

The slope-intercept form of a linear equation is given by y = mx + b, where m is the slope and b is the y-intercept.

Slope (m) = 3/5

Y-intercept (b) = -3

Plug in the values of the slope and y-intercept into the equation.

y = (3/5)x - 3

Therefore,

The equation in slope-intercept form for the line with a slope of 3/5 and a y-intercept of -3 is y = (3/5)x - 3.

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Answer:

(A)The sample comes from a population of tires with 60,000 miles where 5% of tires are worn out.

Step-by-step explanation:

The tire manufacturer has a 60,000 mile warranty for tread life. To verify the authenticity of its claim, it picks a sample of 250 tires (out of the population of tires) that have been used for 60,000 miles.  

Given the hypothesis:

[tex]H_0[/tex]: The proportion of tires that are worn out after 60,000 miles is equal to 0.05.  

[tex]H_a[/tex]: The proportion of tires that are worn out after 60,000 miles is less than 0.05.

It is assumed that 5% of the tires in the sample are worn out and the hypothesis is formulated to prove or disprove this assumption.

Answer:

The time it will take to paint 60 banners is 12 hours.

Step-by-step explanation:

The question is:

Several students from one school painted 15 banners in 3 hours. How many hours will it take for students to paint 60 banners

Solution:

The Unitary method is a mathematical procedure to determine the value of 1 unit from n unit values.

It is provided that the time it takes to paint 15 banners is, t = 3 hours.

Then the time it should have taken to paint 1 banner is, [tex]\frac{3}{15}=\frac{1}{5}\ hours[/tex].

Compute the time it will take to paint 60 banners as follows:

Time = [tex]60\times \frac{1}{5}=12[/tex]

Thus, the time it will take to paint 60 banners is 12 hours.

Answer:

Step-by-step explanation:

This one is really hard I cannot

Answer:

Depending on the exact measurements of the sides, the way to find perimeter would be to add up all the measurements along the sides. If this shape were only two sides, which is impossible, the perimeter would be 10 + 19, or 29.

Step-by-step explanation:

Answer:

you have to stretch the shape out and then add them

Step-by-step explanation:

so 10+19+10+19=58

Answer:

[tex]10.08-1.64\frac{0.1}{\sqrt{6}}=10.013[/tex]    

[tex]10.08+1.64\frac{0.1}{\sqrt{6}}=10.147[/tex]    

So on this case the 90% confidence interval would be given by (10.013;10.147)    

Step-by-step explanation:

Previous concepts

A confidence interval is "a range of values that’s likely to include a population value with a certain degree of confidence. It is often expressed a % whereby a population means lies between an upper and lower interval".

The margin of error is the range of values below and above the sample statistic in a confidence interval.

Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".

[tex]\bar X[/tex] represent the sample mean for the sample  

[tex]\mu[/tex] population mean (variable of interest)

[tex]\sigma =0.1[/tex] represent the population standard deviation

n represent the sample size  

Solution to the problem

The confidence interval for the mean is given by the following formula:

[tex]\bar X \pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}[/tex]   (1)

In order to calculate the mean and the sample deviation we can use the following formulas:  

[tex]\bar X= \sum_{i=1}^n \frac{x_i}{n}[/tex] (2)  

[tex]s=\sqrt{\frac{\sum_{i=1}^n (x_i-\bar X)}{n-1}}[/tex] (3)  

The mean calculated for this case is [tex]\bar X=10.08[/tex]

Since the Confidence is 0.90 or 90%, the value of [tex]\alpha=0.1[/tex] and [tex]\alpha/2 =0.05[/tex], and we can use excel, a calculator or a tabel to find the critical value. The excel command would be: "=-NORM.INV(0.05,0,1)".And we see that [tex]z_{\alpha/2}=1.64[/tex]

Now we have everything in order to replace into formula (1):

[tex]10.08-1.64\frac{0.1}{\sqrt{6}}=10.013[/tex]    

[tex]10.08+1.64\frac{0.1}{\sqrt{6}}=10.147[/tex]    

So on this case the 90% confidence interval would be given by (10.013;10.147)    

Answer: the t-test statistic is - 1.77

Step-by-step explanation:

For the null hypothesis,

H 0: μ = 2600

For the alternative hypothesis,

H a: μ ≠ 2600

This is a two tailed test

Since no population standard deviation is given, the distribution is a student's t.

Since n = 50,

Degrees of freedom, df = n - 1 = 50 - 1 = 49

t = (x - µ)/(s/√n)

Where

x = sample mean = $2450

µ = population mean = $2600

s = samples standard deviation = $600

Therefore,

t = (2450 - 2600)/(600/√50) = - 1.77

The t-test statistic calculated for the given data comparing the mean Pell grant award at San Jose State University to the national average for 2007-2008 is approximately -1.77.

The t-test statistic is calculated using the formula: t = (sample mean - population mean) / (standard deviation / sqrt(sample size)). In this case, the sample mean is the mean Pell grant award for the 50 students from San Jose State University ($2,450), the population mean is the average Pell grant award according to the National Postsecondary Student Aid Study for 2007-2008 ($2,600), the standard deviation is $600, and the sample size is 50. Substitute these values into the formula to get: t = ($2,450 - $2,600) / ($600 / sqrt(50)) = -$150 / ($600 / 7.071) = -$150 / 84.85 = -1.77. So the t-test statistic is approximately -1.77, comparing the mean Pell grant award at San Jose State University to the national average for 2007-2008.

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Answer:

By the Empirical Rule, 68% of light bulbs last between 1345 hours and 1455 hours

Step-by-step explanation:

The Empirical Rule states that, for a normally distributed random variable:

68% of the measures are within 1 standard deviation of the mean.

95% of the measures are within 2 standard deviation of the mean.

99.7% of the measures are within 3 standard deviations of the mean.

In this problem, we have that:

Mean = 1400 hours

Standard deviation = 55 hours

Using the empirical rule, what percentage of light bulbs last between 1345 hours and 1455 hours?

1345 = 1400 - 1*55

So 1345 is one standard deviation below the mean.

1455 = 1400 + 1*55

So 1455 is one standard deviation above the mean.

By the Empirical Rule, 68% of light bulbs last between 1345 hours and 1455 hours

I suppose "below the sphere" means "inside" it, or below the upper half of it. The integral is

[tex]\displaystyle\iiint_Fz\,\mathrm dV[/tex]

where

[tex]F=\{(x,y,z)\mid0\le x\le2,x\le y\le2,0\le z\le\sqrt{8-x^2-y^2}\}[/tex]

Evaluating the integral in Cartesian coordinates is straightforward enough to not require changing coordinates:

[tex]\displaystyle\int_0^2\int_x^2\int_0^{\sqrt{8-x^2-y^2}}z\,\mathrm dz\,\mathrm dy\,\mathrm dx[/tex]

[tex]=\displaystyle\frac12\int_0^2\int_x^2(8-x^2-y^2)\,\mathrm dy\,\mathrm dx[/tex]

[tex]=\displaystyle\frac12\int_0^2\left(16-8x-2x^2+\frac{4x^3-8}3\right)\,\mathrm dx=\boxed{\frac{16}3}[/tex]

The problem involves calculating a triple integral of a function over a specified region. The function is transformed into spherical coordinates considering the given boundaries. Afterwards, we solve the triple integral by integrating with respect to the spherical coordinates.

The given function in the question you're asking about belongs to triple integration, a concept in calculus that extends the idea of one-dimensional integration to three dimensions. In the context of the problem, your function, f(x, y, z) = z, is defined over the region F below the sphere x2 + y2 + z2 = 8 and above the triangular region in the xy-plane bounded by x = 0, y = 2, and y = x.

Firstly, let's convert the coordinates to spherical coordinates. This is appropriate especially when dealing with volumes of spheres or parts of spheres. If we translate these conditions into spherical coordinates, where:

The triple integral thus becomes:

∫02π∫0π/4∫02 ρcosϕ * ρ2sinϕ dρ dϕ dθ.

Note that ρ2sinϕ in the equation above is the Jacobian determinant that arises from the transformation to spherical coordinates. Proceed to solve the triple integral above by integrating with respect to ρ, then ϕ, and finally θ.

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Answer:

Step-by-step explanation:

Given that,

Clara has a list of 720 names

She decided to survey 30 names from the sample.

It is very straight forward, there are 30 names in Clara sample

But this sample can be use to generalize the overall population of 720 students if they are given equal chances of being selected maybe by taking a survey or picking their names randomly from a box

If it not a biased sample then we can use the survey to generalized the overall population

So, there are 720 names in Clara's sample

Answer:

d) 300 times for the first game and 30 times for the second

Step-by-step explanation:

We start by noting that the coin is fair and the flip of a coin has a probability of 0.5 of getting heads.

As the coin is flipped more than one time and calculated the proportion, we have to use the sampling distribution of the sampling proportions.

The mean and standard deviation of this sampling distribution is:

[tex]\mu_p=p\\\\ \sigma_p=\sqrt{\dfrac{p(1-p)}{N}}[/tex]

We will perform an analyisis for the first game, where we win the game if the proportion is between 45% and 55%.

The probability of getting a proportion within this interval can be calculated as:

[tex]P(0.45<x<0.55)=P(z_L<z<z_H)[/tex]

referring the z values to the z-score of the standard normal distirbution.

We can calculate this values of z as:

[tex]z_H=\dfrac{p_H-\mu_p}{\sigma_p}=\dfrac{(p_H-p)}{\sqrt{\dfrac{p(1-p)}{N}}}=\sqrt{\dfrac{N}{p(1-p)}}*(p_H-p)>0\\\\\\z_L=\dfrac{p_L-\mu_p}{\sigma_p}=\dfrac{p_L-p}{\sqrt{\dfrac{p(1-p)}{N}}}=\sqrt{\dfrac{N}{p(1-p)}}*(p_L-p)<0[/tex]

If we take into account the z values, we notice that the interval increases with the number of trials, and so does the probability of getting a value within this interval.

With this information, our chances of winning increase with the number of trials. We prefer for this game the option of 300 games.

For the second game, we win if we get a proportion over 80%.

The probability of winning is:

[tex]P(p>0.8)=P(z>z^*)[/tex]

The z value is calculated as before:

[tex]z^*=\dfrac{p^*-\mu_p}{\sigma_p}=\dfrac{p^*-p}{\sqrt{\dfrac{p(1-p)}{N}}}=\sqrt{\dfrac{N}{p(1-p)}}*(p^*-p)>0[/tex]

As (p*-p)=0.8-0.5=0.3>0, the value z* increase with the number of trials (N).

If our chances of winnings depend on P(z>z*), they become lower as z* increases.

Then, we can conclude that our chances of winning decrease with the increase of the number of trials.

We prefer the option of 30 trials for this game.

Final answer:

For the first game where 45% to 55% heads are needed, flipping the coin 300 times is beneficial due to the law of large numbers. For the second game requiring over 80% heads, flipping only 30 times is better to have a higher chance of deviation from the theoretical probability. So the correct option is d.

Explanation:

In determining whether you would rather flip a coin 30 times or 300 times for each game, one must consider the law of large numbers, which implies that as the number of trials increases, the experimental probabilities tend to get closer to the theoretical probabilities. In this case, a fair coin has a 50% chance of landing on heads on any given flip.

For the first game, where you win if you land between 45% and 55% heads, it's advantageous to flip the coin more times. This is because, with a larger number of flips (like 300), the results are more likely to converge to the expected probability, making it more probable you'll land within that range. Hence, you should choose 300 flips for the first game.

For the second game, where you win if you land more than 80% heads, you would prefer fewer flips, such as 30. It's less likely for a fair coin to consistently land on heads as the number of flips increases, so reducing the number of flips increases the chance of a more significant deviation from the expected probability.

The correct answer, therefore, would be option d: 300 times for the first game and 30 times for the second game.

Step-by-step explanation:

[tex]hi \: your \: answer \: is \: d \\ w = {x}^{2} - 2yz \\ w - {x }^{2} = - 2yz \\ 2yz = {x }^{2} - w \\ divide \: both \: sides \: by \: 2z \\ y = \frac{ {x}^{2} - w }{2z} [/tex]

Answer:

Answer B

[tex]y = \frac{w - {x}^{2} }{2z} [/tex]

Step-by-step explanation:

[tex]w = {x}^{2} - 2yz \\ w - {x}^{2} = 2yz \\ \frac{w - {x}^{2} }{2z} = \frac{2zy}{2z} \\ \\ y = \frac{w - {x}^{2} }{2z} [/tex]

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Given:

A rhino runs at a speed of about 28 miles per hour.

We need to determine the speed in feet per second.

Converting miles to feet:

The miles can be converted to feet by multiplying 5280 with 28 miles per hour.

Thus, we have;

[tex]28\times 5280=147840[/tex]

Thus, the speed of the rhino in feet per hour is 147840

Converting hours to seconds:

The hours can be converted into seconds by dividing 147840 by 3600 (Because an hour has 60 seconds in a minute and 60 minutes in an hour)

Thus, we get;

[tex]Speed=\frac{147840}{3600}[/tex]

[tex]Speed = 41.066667[/tex]

Rounding off to the nearest whole number, we get;

[tex]Speed =41[/tex]

Therefore, the speed of the rhino is 41 feet per second.

Hence, Option B is the correct answer.

Answer:

try y=3/4x+2

Step-by-step explanation:

2 because it is the y intercept

3/4 because it is the slope

Answer:

The slope for this is 3/4. y=3/4x+2

Answer:

156

Step-by-step explanation:

Total number of games played by Ronique's team is ≤ 279(round off).

" At least represents the minimum quantity of the number . it can be more than the given condition."

According to the question,

Number of games  team won each year for the past 7 years are

31, 24, 32, 22, 34, 28, 38.

Total number of games won by team = 31 + 24 + 32+22 + 34 + 28 + 38

                                                              = 209

Team won at least  75% of the seasons they played

'x' represents the total number of seasons they played

Therefore,

75% of x ≤ 209

⇒[tex]\frac{75}{100} (x)\leq 209[/tex]

⇒ [tex]x \leq \frac{(209)(100)}{75}[/tex]

⇒ [tex]x\leq 278.666..[/tex]

⇒[tex]x\leq 279 (round off)[/tex]

Hence, total number of games played by Ronique's team is ≤ 279(round off).

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We have been given that Ben has a square garden that has sides measuring X feet, Ben wants to increase every side of his garden by 11 feet in order to plant more vegetables. So each side of garden would be [tex]X+11[/tex].

We are also told that after Ben increased his garden it will have a perimeter of 104 feet.

We know that perimeter of square is 4 times the side length. So perimeter of garden after enlargement would be [tex]4(X+11)[/tex].

Now we will equate perimeter of new garden with 104 as:

[tex]4(X+11)=104[/tex]

Therefore, the equation [tex]4(X+11)=104[/tex] can be used to solve for X.

[tex]\frac{4(X+11)}{4}=\frac{104}{4}[/tex]

[tex]X+11=26[/tex]

[tex]X+11-11=26-11[/tex]

[tex]X=15[/tex]

Therefore, the value of X is 15 feet.

Answer:

The answer is 115/1957 and in decimal form is 0.0558634

Step-by-step explanation:

Answer:

B=16

Step-by-step explanation:

If 75% is 12, 100% would be 16. B = 16!

Answer:16

Step-by-step explanation:

12/3=4 4=25%of b 4+12=16

Given one to one f(6)=3, f'(6)=5, find tangent line to y=f⁻¹(x) at (3,6)

OK, we know

f⁻¹(3)=6

The inverse function is the reflection in y=x.  So slopes, i.e. the derivative will be the reciprocal.  We know the derivative of f at 6 is 5, so the derivative of f⁻¹  at y=6 is 1/5, which corresponds to x=3.

f⁻¹ ' (3) = 1/5

That slope through (3,6) is the tangent line we seek:

y - 6 = (1/5) (x-3)

That's the tangent line.

y = x/5 + 27/5

First, find the inverse function.

Note that an inverse is a reflection; y = x

The derivative given will be the reciprocal and when f is at 6, it equals 5. So, when it is at y = 6 it is at x = 1/5 and corresponds to 3.

f^-1(3) = 1/5

The point (3, 6) is the point in which the slope goes through. We can write all this information in slope-intercept form.

y = 1/5x + 27/5

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Answer:

The IQR is 30

Step-by-step explanation:

Answer: the difference between the second and fifth element of the set after it has been ordered from least to greatest (D)

Step-by-step explanation: I just finished the quiz and got It correct

Answer:

a = 1/7, b = 0.2, c = 3/9

Step-by-step explanation:

[tex]0.2=\frac{2}{10} = \frac{1}{5}[/tex]

[tex]\frac{1}{7}[/tex]

[tex]\frac{3}{9}=\frac{1}{3}[/tex]

[tex]\frac{1}{5} > \frac{1}{7}[/tex] placing it farther along the number line, making 0.2 the red label (b)

[tex]\frac{1}{7}[/tex] is the smallest fraction that you are given, so it will be closest to 0, making it the blue label (a)

[tex]\frac{1}{3} > \frac{1}{4}[/tex] so [tex]\frac{3}{9}[/tex] is the green label (c)

Answer:

[tex]z=\frac{0.44 -0.53}{\sqrt{\frac{0.53(1-0.53)}{250}}}=-2.85[/tex]  

[tex]p_v =2*P(z<-2.85)=0.0044[/tex]  

Step-by-step explanation:

Data given and notation

n=250 represent the random sample taken

[tex]\hat p=0.44[/tex] estimated proportion of of the readers owned a particular make of car.

[tex]p_o=0.53[/tex] is the value that we want to test

[tex]\alpha=0.05[/tex] represent the significance level

z would represent the statistic (variable of interest)

[tex]p_v[/tex] represent the p value (variable of interest)  

Concepts and formulas to use  

We need to conduct a hypothesis in order to test the claim that the true porportion is equal to 0.53.:  

Null hypothesis:[tex]p=0.53[/tex]  

Alternative hypothesis:[tex]p \neq 0.53[/tex]  

When we conduct a proportion test we need to use the z statistic, and the is given by:  

[tex]z=\frac{\hat p -p_o}{\sqrt{\frac{p_o (1-p_o)}{n}}}[/tex] (1)  

The One-Sample Proportion Test is used to assess whether a population proportion [tex]\hat p[/tex] is significantly different from a hypothesized value [tex]p_o[/tex].

Calculate the statistic  

Since we have all the info requires we can replace in formula (1) like this:  

[tex]z=\frac{0.44 -0.53}{\sqrt{\frac{0.53(1-0.53)}{250}}}=-2.85[/tex]  

Statistical decision  

It's important to refresh the p value method or p value approach . "This method is about determining "likely" or "unlikely" by determining the probability assuming the null hypothesis were true of observing a more extreme test statistic in the direction of the alternative hypothesis than the one observed". Or in other words is just a method to have an statistical decision to fail to reject or reject the null hypothesis.  

The next step would be calculate the p value for this test.  

Since is a bilateral test the p value would be:  

[tex]p_v =2*P(z<-2.85)=0.0044[/tex]