A differential equation is given. Classify it as an ordinary differential equation​ (ODE) or a partial differential equation​ (PDE), give the​ order, and indicate the independent and dependent variables. If the equation is an ordinary differential​ equation, indicate whether the equation is linear or nonlinear.

5 (d^2x/dt^2) + 4 (dx/dt) + 9x = 2 Cos 3t

Answer:

The sample space = {SSS, SSF, SFS, SFF, FSS, FSF, FFS, FFF}

Step-by-step explanation:

When we say sample space, we mean the list of all possible outcome from an event. For this even of sales representative presenting at homes., only two outcome is possible. Whether:

1.  the home(s) buys his product (S)

2. the home(s) did not buy his product (F).

Thus from three (3) homes, that will be:

==> [tex]2^{3}[/tex] = 2*2*2 = 8 possible outcomes.

The sample space for the experiment where a sales representative makes presentations at three homes with each home resulting in either a sale (S) or no sale (F) consists of 8 possible outcomes: SSS, SSF, SFS, SFF, FSS, FSF, FFS, and FFF.

To find the sample space for the experiment where a sales representative makes presentations about a product in three homes per day, and in each home, there may be a sale (denoted by S) or there may be no sale (denoted by F), we have to consider all the possible outcomes. Each home has two possible outcomes, meaning that the total sample space consists of 23 = 8 possible combinations for the three presentations.

The sample space S can be written as:

Each combination represents one possible outcome for the day's sales presentations.

Answer:

The variance is 2.5.

Step-by-step explanation:

Let X = number of visitors in a museum.

The random variable X has an average of  5 visitors per 2 minutes.

Then in 1 minute the average number of visitors is, [tex]\frac{5}{2} =2.5[/tex]

The random variable X follows a Poisson distribution with parameter λ = 2.5.

The variance of a Poisson distribution is:

[tex]Variance=\lambda[/tex]

The variance of this distribution is:

[tex]V(X)=\lambda=2.5[/tex]

Thus, the variance is 2.5.

Answer:

b. 0.6592 and 0.7044

Step-by-step explanation:

In a sample with a number n of people surveyed with a probability of a success of [tex]\pi[/tex], and a confidence interval [tex]1-\alpha[/tex], we have the following confidence interval of proportions.

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which

z is the zscore that has a pvalue of [tex]1 - \frac{\alpha}{2}[/tex].

For this problem, we have that:

A survey of an urban university showed that 750 of 1,100 students sampled attended a home football game during the season. This means that [tex]n = 1100, p = \frac{750}{1100} = 0.6818[/tex]

90% confidence interval

So [tex]\alpha = 0.1[/tex], z is the value of Z that has a pvalue of [tex]1 - \frac{0.1}{2} = 0.95[/tex], so [tex]Z = 1.645[/tex].

The lower limit of this interval is:

[tex]\pi - z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.6818 - 1.645\sqrt{\frac{0.6818*0.3182}{1100}} = 0.6592[/tex]

The upper limit of this interval is:

[tex]\pi + z\sqrt{\frac{\pi(1-\pi)}{n}} = 0.6818 + 1.645\sqrt{\frac{0.6818*0.3182}{1100}} = 0.7044[/tex]

So the correct answer is:

b. 0.6592 and 0.7044

Answer:

For the first case we are going to assume that the order matters, on this case 6, H is not the same as H,6

The sampling space denoted by S and is given by:

S ={(1,H), (2,H),(3,H),(4,H),(5,H),(6,H),

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

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

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

If we consider that (5,H) is equal to (H,5) "order no matter" then we will have just 12 elements in the sampling space:

S ={(1,H), (2,H),(3,H),(4,H),(5,H),(6,H),

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

Step-by-step explanation:

By definition the sample space of an experiment "is the set of all possible outcomes or results of that experiment".

For the case described here: "Toss a coin and a six-sided die".

Assuming that we have a six sided die with possible values {1,2,3,4,5,6}

And for the coin we assume that the possible outcomes are {H,T}

For the first case we are going to assume that the order matters, on this case 6, H is not the same as H,6

The sampling space denoted by S and is given by:

S ={(1,H), (2,H),(3,H),(4,H),(5,H),(6,H),

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

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

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

If we consider that (5,H) is equal to (H,5) "order no matter" then we will have just 12 elements in the sampling space:

S ={(1,H), (2,H),(3,H),(4,H),(5,H),(6,H),

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

Answer:

the probability that Ron will get hit by a pitch exactly once is 36.71%

Step-by-step explanation:

The random variable X= number of times Ron is hits by pitches in 23 plate appearances  follows ,a binomial distribution. Where

P(X=x) = n!/(x!*(n-x)!)*p^x*(1-p)^x

where

n= plate appearances =23

p= probability of being hit by pitches = 21/602

x= number of successes=1

then replacing values

P(X=1) = 0.3671 (36.71%)

The probability that Ron will get hit by a pitch exactly once in his 23 playoff plate appearances, given his regular season hit rate, is approximately 0.37 or 37%.

The subject of this problem is probability; it's asking us to calculate the chances of a specific event happening. It is given that during the regular season, Ron was hit by pitches 21 times out of 602 plate appearances. Thus, the probability of him getting hit by a pitch is 21/602, or approximately 0.035.

In the playoffs, he has 23 plate appearances. We want to find the probability that he gets hit exactly once. This is a binomial probability problem, using the formula:

P(X=k) = C(n, k) * (p^k) * ((1-p)^(n-k))

Where n is the number of trials (plate appearances), k is the number of successes we want (getting hit by the pitch), p is the success probability, and C(n, k) is the combination operator. Substituting the given values:

P(X=1) = C(23, 1) * (0.035^1) * ((1-0.035)^(23-1))

Performing this calculation gives a pitch hitting probability of about 0.37 or 37%, which means Ron is likely to be hit by one pitch during the 23 plate appearances in the playoffs.

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Answer:

a) 4.5

b) x = 11.2, s = 4.65

c) 93.33%                                                

Step-by-step explanation:

We are given he following data in the question:

8, 12, 11, 15, 14, 10, 8, 3, 8, 7, 21, 12, 9, 19, 11

a) Estimation of standard deviation using range

Sorted data: 3, 7, 8, 8, 8, 9, 10, 11, 11, 12, 12, 14, 15, 19, 21

Range = Maximum - Minimum = 21 - 3 = 18

Range rule thumb:

[tex]s = \dfrac{\text{Range}}{4} = \dfrac{18}{4} = 4.5[/tex]

b) Mean and standard deviation

Formula:

[tex]\text{Standard Deviation} = \sqrt{\displaystyle\frac{\sum (x_i -\bar{x})^2}{n-1}}[/tex]  

where [tex]x_i[/tex] are data points, [tex]\bar{x}[/tex] is the mean and n is the number of observations.  

[tex]Mean = \displaystyle\frac{\text{Sum of all observations}}{\text{Total number of observation}}[/tex]

[tex]Mean =\displaystyle\frac{168}{15} = 11.2[/tex]

Sum of squares of differences = 302.4

[tex]S.D = \sqrt{\dfrac{302.4}{14}} = 4.65[/tex]

c)  fraction of the scores actually lie in the interval x ± 2s

[tex]x \pm 2s = 11.2 \pm 2(4.65) = (1.9,20.5)[/tex]

Since 14 out of 15 entries lie in this range, we can calculate the percentage as,

[tex]\dfrac{14}{15}\times 100\% = 93.33\%[/tex]

Answer:

part 1) [tex]13[/tex]

part 2) [tex]\frac{119}{45}[/tex]

part 3) [tex]2[/tex]

part 4) [tex]\frac{1,958,309}{128}[/tex]

part 5) [tex]4\ yd^2[/tex]

Step-by-step explanation:

The complete question in the attached figure

we know that

Applying PEMDAS

P ----> Parentheses first  

E -----> Exponents (Powers and Square Roots, etc.)  

MD ----> Multiplication and Division (left-to-right)  

AS ----> Addition and Subtraction (left-to-right)

Part 1) we have

[tex]\frac{2}{3}(6)+\frac{3}{4}(12)[/tex]

Remember that when multiply a fraction by a whole number, multiply the numerator of the fraction by the whole number and maintain the same denominator

so

[tex]\frac{12}{3}+\frac{36}{4}[/tex]

[tex]4+9=13[/tex]

Part 2) we have

[tex]2\frac{1}{3}(3\frac{2}{5}:3)[/tex]

Convert mixed number to an improper fraction

[tex]2\frac{1}{3}=2+\frac{1}{3}=\frac{2*3+1}{3}=\frac{7}{3}[/tex]

[tex]3\frac{2}{5}=3+\frac{2}{5}=\frac{3*5+2}{5}=\frac{17}{5}[/tex]

substitute

[tex]\frac{7}{3}(\frac{17}{5}:3)[/tex]

Solve the division in the parenthesis (applying PEMDAS)

[tex]\frac{7}{3}(\frac{17}{15})[/tex]

[tex]\frac{119}{45}[/tex]

Part 3) we have

[tex]\frac{7}{8}:(1\frac{1}{4}:4)[/tex]

Convert mixed number to an improper fraction

[tex]1\frac{1}{4}=1+\frac{1}{4}=\frac{1*4+1}{4}=\frac{5}{4}[/tex]

substitute

[tex]\frac{7}{8}:(\frac{5}{4}:4)[/tex]

Solve the division in the parenthesis (applying PEMDAS)

[tex]\frac{7}{8}:(\frac{5}{16})[/tex]

Multiply in cross

[tex]\frac{80}{40}=2[/tex]

Part 4) we have

[tex]18:(\frac{2}{3})^2+25:(\frac{2}{5})^7[/tex]

exponents first

[tex]18:(\frac{4}{9})+25:(\frac{128}{78,125})[/tex]

Solve the division

[tex](\frac{162}{4})+(\frac{1,953,125}{128})[/tex]

Find the LCD

LCD=128

so

[tex]\frac{32*162+1,953,125}{128}[/tex]

[tex]\frac{1,958,309}{128}[/tex]

Part 5) Find the area of triangle

The area of triangle is equal to

[tex]A=\frac{1}{2}(b)(h)[/tex]

substitute the given values

[tex]A=\frac{1}{2}(6)(1\frac{1}{3})[/tex]

Convert mixed number to an improper fraction

[tex]1\frac{1}{3}=1+\frac{1}{3}=\frac{1*3+1}{3}=\frac{4}{3}[/tex]

substitute

[tex]A=\frac{1}{2}(6)(\frac{4}{3})=4\ yd^2[/tex]

Answer:

3 and 4

Step-by-step explanation:

Well to start we have to know that they are asking us, a factorized form of a quadratic expression

a quadratic expression is of the form

ax ^ 2 + bx + c

Now the factored form is as follows            

 a ( x - x1 )  ( x - x2 )

Next, let's look at each of the options

In this case we lack a term with x since if we solve we have a linear equation

1.    5(x+9)

  5x + 45

In this case if we pay attention they are being subtracted instead of multiplying, so we will not get a quadratic function

2.    (x+4) - (x+6)

       -2

In this case we have everything we need, now let's try to solve

3.     (x-1) (x-1)

      x^2 - x - x + 1

      x^2 - 2x + 1       quadratic function

In this case we have everything we need, now let's try to solve

4.     (x-3) (x+2)

        x^2 -3x +2x -6

         x^2 -x - 6    quadratic function

In this case we have a quadratic function but we do not have it in its factored form since we can observe the x ^ 2

5.     x^2 + 8x

Answer:

There is a 50% probability that its total life will exceed 20,000 miles.

Step-by-step explanation:

To solve this question, we use the following formula:

[tex]P(A|B) = \frac{P(A \cap B)}{P(B)}[/tex]

In which P(A|B) is the probability of A happening, given that B has happened, [tex]P(A \cap B)[/tex] is the probability of A and B happening, and P(B) is the probability of B happening.

In this problem, we want:

The probability of the total life of the car battery exceeding 20,000 miles, given that it exceeded 10,000 miles.

[tex]P(A \cap B)[/tex] is the probability of exceeding 20,000 and 10,000 miles. It is the same as the probability of exceeding 20,000 miles(If it exceeded 20,000 miles, necessarily it will have exceeded 10,000 miles). So [tex]P(A \cap B) = 0.4[/tex]

P(B) is the probability of exceeding 10,000 miles. So [tex]P(B) = 0.8)[/tex]

So

[tex]P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{0.4}{0.8} = 0.5[/tex]

There is a 50% probability that its total life will exceed 20,000 miles.

Final answer:

If a new car battery is still working after 10,000 miles, the probability that its total life will exceed 20,000 miles is 0.5 or 50%.

Explanation:

The question pertains to conditional probability, which is the probability of an event occurring given that another event has already occurred. Here, we are asked to find the probability that a new car battery will exceed 20,000 miles given that it has already functioned for more than 10,000 miles. This question essentially requires us to calculate conditional probability.

Given:

To find the conditional probability that its total life will exceed 20,000 miles given it has already worked for over 10,000 miles (P(B|A)), we use the formula:

P(B|A) = P(B & A) / P(A)

However, since any battery that has functioned for more than 20,000 miles must have also functioned for more than 10,000 miles, P(B & A) = P(B), hence:

P(B|A) = 0.4 / 0.8 = 0.5

Therefore, if a new car battery is still working after 10,000 miles, the probability that its total life will exceed 20,000 miles is 0.5 or 50%.

Answer:

The question is lacking the image of the histogram, but the file attachment to this answer contains the complete question and histogram image.

The percentage of the rats that navigated the maze in less than 5.5 minutes is 84%

Step-by-step explanation:

First of all let us compute the total frequency which represents the total number of rats in the experiment.

from the information given in the question, we are told that the total number of rats involved in the experiment are 25 rats and this makes up the total frequency.

To calculate the percentage of rats that navigated the maze in less than 5.5 minutes, you will first of all need to identify the 5.5 minute point on the histogram and add all the frequencies below that point. This gives the total number of rat that navigated in less than 5.5 minutes. These frequencies from 0 to <5.5 are; 3, 8, 6, and 4. And the total is given as the sum of these frequencies shown below:

Total number of rats that navigated in less than 5.5 minutes = 3 + 8 + 6 + 4 = 21. Hence, a total of 21 rats navigated the maze in less than 5.5 minutes.

Now to find the percentage of the number of rats that navigated in less than 5.5 minutes, we have to find out what percentage of 25 (total number of rats) is 21 (number of rats that navigated in less than 5.5 minutes). This is calculated thus:

[tex]\frac{21}{25}[/tex] × 100 = 0.84 × 100 = 84 %.

Therefore 84% of the total number of rats navigated the maze in less than 5.5 minutes in the experiment.

To find the percentage of rats that navigated the maze in less than 5.5 minutes, locate the bar on the histogram that represents that interval and calculate the percentage based on the number of rats in that interval compared to the total number of rats.

The histogram shows the times required for 25 rats to navigate a maze. To find the percentage of rats that navigated the maze in less than 5.5 minutes, we need to look at the data on the histogram. The histogram is divided into various time intervals, and the bars represent the number of rats that fall into each interval. You need to locate the bar that corresponds to the interval less than 5.5 minutes and calculate the percentage of rats represented by that bar.

Let's say the bar for the interval less than 5.5 minutes represents 10 rats. To calculate the percentage, we divide the number of rats in that interval (10) by the total number of rats (25) and multiply by 100:

(10/25) x 100 = 40%

Therefore, 40% of the rats navigated the maze in less than 5.5 minutes.

Answer:

There is no enough evidence to claim that the average cost of accesories is different from $3,000.

Step-by-step explanation:

The significance level for this test is α=0.05.

The classical method is based on regions of rejection of acceptance, according to the sample parameter. In this case, the standard deviation of the population is unknown.

The null and alternative hypothesis are:

[tex]H_0: \mu=3000\\\\ H_a: \mu\neq 3000[/tex]

This is a two-tailed test, with significance level of 0.05.

The t-value for this sample is:

[tex]t=\frac{x-\mu}{s/\sqrt{N}} =\frac{3256-3000}{2300/\sqrt{50}}=\frac{256}{325}=0.787[/tex]

The degrees of freedom are:

[tex]df=n-1=50-1=49[/tex]

For df=49 and α=0.05 (two-tailed test), the critical values are [tex]|t|>2.009[/tex], so the value t=0.787 is within the acceptance region.

The null hypothesis can not be rejected.

Answer:

If k = −1 then the system has no solutions.

If k = 2 then the system has infinitely many solutions.

The system cannot have unique solution.

Step-by-step explanation:

We have the following system of equations

[tex]x - 2y +3z = 2\\x + y + z = k\\2x - y + 4z = k^2[/tex]

The augmented matrix is

[tex]\left[\begin{array}{cccc}1&-2&3&2\\1&1&1&k\\2&-1&4&k^2\end{array}\right][/tex]

The reduction of this matrix to row-echelon form is outlined below.

[tex]R_2\rightarrow R_2-R_1[/tex]

[tex]\left[\begin{array}{cccc}1&-2&3&2\\0&3&-2&k-2\\2&-1&4&k^2\end{array}\right][/tex]

[tex]R_3\rightarrow R_3-2R_1[/tex]

[tex]\left[\begin{array}{cccc}1&-2&3&2\\0&3&-2&k-2\\0&3&-2&k^2-4\end{array}\right][/tex]

[tex]R_3\rightarrow R_3-R_2[/tex]

[tex]\left[\begin{array}{cccc}1&-2&3&2\\0&3&-2&k-2\\0&0&0&k^2-k-2\end{array}\right][/tex]

The last row determines, if there are solutions or not. To be consistent, we must have k such that

[tex]k^2-k-2=0[/tex]

[tex]\left(k+1\right)\left(k-2\right)=0\\k=-1,\:k=2[/tex]

Case k = −1:

[tex]\left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&-1-2\\0&0&0&(-1)^2-(-1)-2\end{array}\right] \rightarrow \left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&-3\\0&0&0&-2\end{array}\right][/tex]

If k = −1 then the last equation becomes 0 = −2 which is impossible.Therefore, the system has no solutions.

Case k = 2:

[tex]\left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&2-2\\0&0&0&(2)^2-(2)-2\end{array}\right] \rightarrow \left[\begin{array}{ccc|c}1&-2&3&2\\0&3&-2&0\\0&0&0&0\end{array}\right][/tex]

This gives the infinite many solution.

We use matrix row reduction to determine the values of k that result in no solution, a unique solution, or infinitely many solutions in the system of equations.

To determine the values of k for which the system of equations has no solution, a unique solution, or infinitely many solutions, we will use the concept of matrix row reduction. First, let's rewrite the system of equations in augmented matrix form:

[1 -2 3 2 | 0] [2 1 1 -1 | 0] [2 -1 4 -k^2 | 0]

Performing row reduction on this augmented matrix, we can find the values of k where each situation occurs. If there is a row of 0's followed by a non-zero constant (in the rightmost column), then the system has no solution. If the row reduction yields a matrix with a non-zero row followed by zeroes (except for the last row), then the system has infinitely many solutions. Otherwise, the system has a unique solution.

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Answer:

a. Current: 5 units/hour. Previous: 4.4 units/hour

b. Increase

Step-by-step explanation:

a. Current period productivity is 200 / 40 = 5 units/hour

Previous period productivity is 132 / 30 = 4.4 units/hour

b. As this week's productivity = 5 units/hours which is larger than last week's productivity = 4.4 units/hour. The worker's productivity for this week has increased.

Answer:

Option C) 0.476

Step-by-step explanation:

We are given the following in the question:

1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1

where 1 means that Sam will get a hit and 0 mean Sam will be out.

Total number of outcomes = 42

Number of times Sam will get a hit = n(1) = 20

Number of times Sam will be out = n(0) = 22

We have to find the probability that Sam gets a hit.

Formula:

Thus, 0.476 is the probability that Sam will get a hit.

The probability that to gets a hit is 0.476.

What is mean by Probability?

The term probability refers to the likelihood of an event occurring.

Given that;

Swinging Sammy Skor's batting prowess was simulated to get an estimate of the probability that Sammy will get a hit.

Let Hit = 1 and Out = 0

Now,

Total number of outcomes = 42

And, Number of times Sammy get a hit = 20

Number of times Sammy will be out = 22

Since, The probability to get a hit is defined as;

Probability = Number of times he get hit / Total number of outcomes

                = 20 / 42

                = 0.476

Thus, The probability that to gets a hit is 0.476.

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Answer:

1.33 s

Step-by-step explanation:

Given:

Δy = -5 ft

v₀ = 17.5 ft/s

a = -32 ft/s²

Find: t

Δy = v₀ t + ½ at²

-5 = 17.5 t + ½ (-32) t²

0 = -16t² + 17.5t + 5

0 = 32t² − 35t − 10

t = [ 35 ± √((-35)² − 4(32)(-10)) ] / 64

t = (35 ± √2505) / 64

t = 1.33

Amy has 1.33 seconds to react before the volleyball hits the ground.

Amy has to react to the volleyball based on its initial upward velocity and the height at which it was hit, using gravitational equations to calculate the time before it hits the ground.

Patricia serves the volleyball to Amy with an upward velocity of 17.5ft/s. The ball is 5 feet above the ground when she strikes it. To determine how long Amy has to react before the volleyball hits the ground, we can use the equations of motion under gravity. Assuming the acceleration due to gravity (g) is 32.2ft/s2 (downward), the time (t) for the volleyball to reach the ground can be found by solving the quadratic equation derived from the formula:

h = v0t - (1/2)gt2

where h is the height above the ground (5 feet), v0 is the initial velocity (17.5ft/s), and t is the time in seconds. One can use the quadratic formula to solve for t. However, to provide a concrete example and simplify the calculation for the purpose of this answer, we would use a calculator or other computational tools to solve for t numerically, remembering to consider the positive root that makes physical sense. One would typically find a time in the range of a second or slightly more for this scenario.

Answer:

d. skewed right

Step-by-step explanation:

The shape of the given distribution is rightly skewed. For a symmetric distribution mean and median are equal and if mean is greater than median then the distribution is rightly skewed and if mean is less than median then the distribution is skewed left.

In the given distribution mean is greater than median and so the given distribution is skewed right.

To determine if the nutrition label is accurate, a hypothesis test can be conducted using the provided sample data. Calculating the z-score and finding the p-value will determine if there is evidence that the label is inaccurate.

In order to determine if there is evidence that the nutrition label does not provide an accurate measure of calories in the bags of potato chips, we can conduct a hypothesis test using the given sample data. Let's state the null and alternative hypotheses:

Null Hypothesis (H0): The nutrition label provides an accurate measure of calories in the bags of potato chips.

Alternative Hypothesis (Ha): The nutrition label does not provide an accurate measure of calories in the bags of potato chips.

To test these hypotheses, we can calculate the z-score using the formula:

z = (sample mean - population mean) / (population standard deviation / sqrt(sample size))

In this case, the population mean is 130 calories (as stated on the nutrition label), the sample mean is 134 calories, the population standard deviation is 17 calories (as given), and the sample size is 35 bags (as given). Plugging in these values, we can calculate the z-score.

Once we have the z-score, we can find the p-value associated with it from a standard normal distribution table or using statistical software. If the p-value is less than the chosen significance level (usually 0.05), we reject the null hypothesis and conclude that there is evidence that the nutrition label does not provide an accurate measure of calories in the bags of potato chips.

Without knowing the calculated p-value, we cannot make a decision based on the given random sample.

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The cone [tex]z=\sqrt{x^2+y^2}[/tex] and the sphere [tex]z=\sqrt{4-x^2-y^2}[/tex] intersect in a circle of radius [tex]\sqrt 2[/tex] in the plane [tex]z=\sqrt2[/tex]:

[tex]\sqrt{x^2+y^2}=\sqrt{4-x^2-y^2}\implies 2x^2+2y^2=4\implies x^2+y^2=2[/tex]

[tex]\implies z=\sqrt{x^2+y^2}=\sqrt2[/tex]

In Cartesian coordinates, the volume is then given by the integral

[tex]\displaystyle\int_{-\sqrt2}^{\sqrt2}\int_{-\sqrt{2-x^2}}^{\sqrt{2-x^2}}\int_{\sqrt{x^2+y^2}}^{\sqrt{4-x^2-y^2}}\mathrm dz\,\mathrm dy\,\mathrm dx[/tex]

In cylindrical coordinates, the integral is

[tex]\displaystyle\int_0^{2\pi}\int_0^{\sqrt2}\int_r^{\sqrt{4-r^2}}r\,\mathrm dz\,\mathrm dr\,\mathrm d\theta[/tex]

For the given problem, the volume of the structure can be calculated using both Cartesian and cylindrical coordinates. For Cartesian coordinates, the coordinates represents as  [tex]z=x^2+y^2[/tex]and  [tex]x^2+y^2+z^2=4.[/tex]For cylindrical coordinates, equations represent as [tex]z=r^2[/tex]and [tex]r^2+z^2=4[/tex]. Both integrations describe the volume of the structures.

In the given problem, the volume contains two geometric shapes: the cone and the sphere. The volume of the shape can be calculated using both Cartesian and cylindrical coordinates.

Using Cartesian coordinates, first describe cone and sphere as  [tex]z=x^2+y^2[/tex] and [tex]x^2+y^2+z^2=4[/tex] respectively. Define the volume by double integration:

∫∫ D (4 - z) dxdy

Where D is the region in the xy-plane bounded by the projection of the volume.

Using cylindrical coordinates, we represent the figures as [tex]z=r^2[/tex]and  [tex]r^2+z^2=4.[/tex] The volume integral in cylindrical coordinates is then given by:

∫ (from 0 to 2pi) ∫ (from 0 to √2) ∫ (from  [tex]r^2 \ to \ 2-r^2[/tex]) rdzdrdθ

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Answer:

a. Number of heads

b.

x      p(x)

0       1/8

1         3/8

2       3/8

3        1/8

Step-by-step explanation:

a)

A coin is flipped three times and the number of heads are counted.

We are interested in counting heads so, a random variable X is the number of heads appears on a coin.

b)

The sample space for flipping a coin three times is

S={HHH,HHT,HTH,HTT,THH,THT,TTH,TTT}

n(S)=8

The random variable X (number of heads) can take values 0,1,2 and 3 .

0 head={TTT}

P(0 heads)=P(X=0)=1/8

1 head={HTT,THT,TTH}

P(1 head)= P(X=1)=3/8

2 heads= {HHT,HTH,THH}

P(2 heads)=P(X=2)=3/8

3 heads={HHH}

P(3 heads)=1/8

The probability distribution for number of heads can be shown as

x      p(x)

0       1/8

1         3/8

2       3/8

3        1/8

The random variable is the number of heads obtained when flipping a coin three times. The probability distribution for the number of heads can be found using the binomial probability formula.

a) The random variable in this experiment is the number of heads obtained when flipping a coin three times. It can take on the values 0, 1, 2, or 3.

b) To write the probability distribution for the number of heads, we need to determine the probability of getting 0, 1, 2, or 3 heads. Since each coin flip is an independent event, we can use the binomial probability formula to calculate these probabilities.

For example, the probability of getting exactly 2 heads can be calculated as: P(X = 2) = (3 choose 2) * (0.5^2) * (0.5^1) = 3 * 0.25 * 0.5 = 0.375.

The probability distribution for the number of heads is:
X = 0, P(X = 0) = (3 choose 0) * (0.5^0) * (0.5^3) = 1 * 1 * 0.125 = 0.125
X = 1, P(X = 1) = (3 choose 1) * (0.5^1) * (0.5^2) = 3 * 0.5 * 0.25 = 0.375
X = 2, P(X = 2) = (3 choose 2) * (0.5^2) * (0.5^1) = 3 * 0.25 * 0.5 = 0.375
X = 3, P(X = 3) = (3 choose 3) * (0.5^3) * (0.5^0) = 1 * 0.125 * 1 = 0.125

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Final answer:

To express Δy in terms of Δx, we can use the concept of slope. The formula that expresses Δy in terms of Δx is Δy = (2.5 - y1) / (1.5 - x1) * Δx.

Explanation:

To express Δy in terms of Δx, we can use the concept of slope. The formula for slope is:

Slope = (Δy) / (Δx)

To find the slope between two points, we can use the formula:

Slope = (y2 - y1) / (x2 - x1)

In this case, if y = 2.5 when x = 1.5, we can substitute these values into the formula and simplify:

Slope = (2.5 - y1) / (1.5 - x1)

Since we are only interested in expressing Δy in terms of Δx, we can solve for Δy:

Δy = Slope * Δx

Therefore, the formula that expresses Δy in terms of Δx is:

Δy = (2.5 - y1) / (1.5 - x1) * Δx

If we have y = mx + b and we only know one point (1.5, 2.5), we need a second point or more context for an exact equation. The change in y, Δy, with respect to a change in x, Δx, is found using: Δy = m * Δx.

To express Δy in terms of Δx, you can use the concept of derivatives and the definition of a linear function. Here's a step-by-step solution:

Given information: We know that "y = 2.5" when "x = 1.5."

Setting up the function: Let's assume that the relationship between y and x is linear. In a linear function, the rate of change of y with respect to x is constant. We can write the linear equation in the form: y = mx + b, where m is the slope and b is the y-intercept.

Finding the slope (m): Since linear functions have a constant slope, we need to calculate m. If we assume that y changes by some amount Δy when x changes by Δx, then the slope (m) can be represented as: m = Δy / Δx.

Using the derivative: For a linear equation, dy/dx = m. Therefore, Δy = m * Δx.

Given the specific solution: In the problem, we were given a point (x, y) = (1.5, 2.5). However, we need another point or more information to determine the exact form of the function y in terms of x. Without additional information, we cannot definitively determine the slope.

Assuming a direct variation: In simple cases, we might assume a direct variation (y = kx), but this requires more context. Based on the provided hint, if we use the ratio y/x = k, we can set up an initial formula to start with.

Answer:

a. $1773.82

b. $1592.1

Step-by-step explanation:

1. If he pays k dollar in the first year, then the amount that he owned without interest is

7277 - k

The amount that he owned including interest of 11% in the 2nd year is

(7277 - k)*1.11 or 7277*1.11 - 1.11k

After 2nd year and paying k then the amount he owned (without interest is)

(7277 - k)*1.11 - k

With interest

[(7277 - k)*1.11 - k]1.11 or [tex]7277*1.11^2 - 1.11^2k - 1.11k[/tex]

So after 5 years

[tex]7277*1.11^5 - (1.11^5 + 1.11^4 + 1.11^3 + 1.11^2 +1.11)k[/tex]

[tex]12262.17 - 6.91 k[/tex]

Since he's dept-free after 5 year then

[tex]12262.17 - 6.91 k = 0[/tex]

[tex]k = 12262.17 / 6.91 = 1773.82[/tex] dollar

2. The total amount he would have to pay over 5 years is 5k = 5*1773.82 = 8869.1

So the interest we has to pay over 5 years is the total subtracted by the principal, which is 8869.1 - 7277 = 1592.1 dollar

Answer: area = 5,456

Answer:

5456 sq. inches

Step-by-step explanation:

Let width = w,

Then length = w + 26

Perimeter = 2[length + width]

2[(w +26) + w] = 2[2w + 26] = 4w + 52

Perimeter given is 300

So, 4w + 52 = 300

4w = 248

w = 248/4 =62

length = 62 + 26 = 88

Area = length × width

Area = 88 × 62 = 5456 sq. inches

Answer:

[tex] x= (\frac{2V}{7})^{1/3}[/tex]

[tex] y= (\frac{2V}{7})^{1/3}[/tex]

[tex] z = \frac{V}{xy}= \frac{V}{(\frac{2V}{7})^{2/3}}[/tex]

Step-by-step explanation:

This is a minimization problem.

For this case we assume that we have a box and the volume is given by:

[tex] V = xyz[/tex]   (1)

For this case we know that slate costs seven times as much (per unit area) as glass so then 7xy this value and if we find the cost function like this:

[tex] C(x,y,z) = 2yz+ 2xz + 7xy[/tex]

If we solve z from equation (1) we got:

[tex] z= \frac{V}{xy}[/tex]   (2)

So then we can replace equation (2) into the cost equation and we got:

[tex] C(x,y,V/xy)= 2y (\frac{V}{xy}) +2x(\frac{V}{xy})+ 7xy[/tex]

And with this we have a function in terms of two variables x and y.

We can simplify the last equation and we got:

[tex] C(x,y,V/xy)= \frac{2V}{x} +\frac{2V}{y} + 7xy[/tex]

In order to solve the problem for the dimensions we can take the partial derivates respect to x and y and we got:

[tex] C_x = -\frac{2V}{x^2} +7y =0[/tex]

[tex] C_y = -\frac{2V}{y^2} +7x =0[/tex]

We can set the last two equations equal since are equal to 0 and we got:

[tex]  -\frac{2V}{x^2} +7y =-\frac{2V}{y^2} +7x [/tex]

And the only possible solution for this case is [tex] x=y[/tex]

So then if we use x=y for the partial derivate of x we have:

[tex] C_x (x,y=x) = -\frac{2V}{x^2} +7x =0[/tex]

And solving for x we got:

[tex] \frac{2V}{x^2} =7x[/tex]

[tex] 7x^3 = 2V[/tex]

[tex] x= (\frac{2V}{7})^{1/3}[/tex]

And analogous we can do the same thing for the partial derivate of y and we got:

[tex] C_y (x=y,y) = -\frac{2V}{y^2} +7y =0[/tex]

And solving for x we got:

[tex] \frac{2V}{y^2} =7y[/tex]

[tex] 7y^3 = 2V[/tex]

[tex] y= (\frac{2V}{7})^{1/3}[/tex]

And for z we can replace and we got:

[tex] z = \frac{V}{xy}= \frac{V}{(\frac{2V}{7})^{2/3}}[/tex]

So then the dimensions in order to minimize the cost would be:

[tex] x= (\frac{2V}{7})^{1/3}[/tex]

[tex] y= (\frac{2V}{7})^{1/3}[/tex]

[tex] z = \frac{V}{xy}= \frac{V}{(\frac{2V}{7})^{2/3}}[/tex]

Answer:

The answer to your question is below

Step-by-step explanation:

Inequality 1

                        7(x + 2) - 8 ≥ 13

                        7x + 14 - 8 ≥ 13

                        7x + 6 ≥ 13

                         7x ≥ 13 - 6

                         7x ≥ 7

                           x ≥ 7/7

                           x ≥ 1

Inequality 2

                       8x - 3 < 4x - 3

                        8x - 4x < - 3 + 3

                               4x < 0

                                 x < 0 / 4

                                 x < 0

Interval notation   (-∞ , 0) U [1, ∞)

See the graph below

Answers: 0.286

Explanation:

Let E → major in Engineering

Let S → Play club sports

P (E) = 28% = 0.28

P (S) = 18% = 0.18

P (E ∩ S ) = 8% = 0.08

Probability of student plays club sports given majoring in engineering,

P ( S | E ) = P (E ∩ S ) ÷ P (E) = 0.08 ÷ 0.28 = 0.286

To find the probability that a student plays club sports given that they major in engineering, use conditional probability.

To find the probability that a student plays club sports given that they major in engineering, we need to use conditional probability.

The formula for conditional probability is:

P(A|B) = P(A ∩ B) / P(B)

In this case, A represents playing club sports and B represents majoring in engineering. We are given that P(A ∩ B) = 8% and P(B) = 28%.

Plugging these values into the formula, we get:

P(A|B) = 8% / 28% = 0.2857

So the probability that a student plays club sports given that they major in engineering is approximately 0.2857, or 28.57%.

Answer:

[tex]$ \frac{\textbf{1.55}}{\textbf{1}} \hspace{1mm} \textbf{=} \hspace{1mm} \frac{\textbf{x}}{\textbf{3.5}} $[/tex]

Step-by-step explanation:

Let us assume that the total cost of the grapes = x $.

Given that it costs 1.55 $ for one pound. We are asked to determine how much it would cost for 3.5 pounds.

note that for one pound you pay 1.55 so how much should you pay for 3.5 pounds? Clearly, the cost increases with the increase in weight of the item.

This is equal to [tex]$ \frac{1.55}{1} \times 3.5 = x $[/tex]

[tex]$ \iff \frac{1.55}{1} = \frac{x}{3.5} $[/tex]

Hence, the answer.

Answer:

(9 weeks 5 days) - (1 week 6 days) =  55 days

Step-by-step explanation:

Answer:

Step-by-step explanation:

Answer:

132 history textbooks, 87 chemistry textbooks

Step-by-step explanation:

[tex]C + H = 219\\C = H - 45\\[/tex]

[tex]1. H - 45 + H = 219\\2. 2H = 264\\3. H = 132[/tex]

H = 132,

C = 132 - 45 = 87

Answer: 87 Chemistry and 132 history textbooks.

Step-by-step explanation:

Let x represent the number of chemistry textbooks that was sold.

Let y represent the number of history textbooks that was sold.

The textbook store sold a combined total of 219 history and chemistry textbooks in a week. This means that

x + y = 219 - - - - - - - - - - - -1

The number of chemistry textbooks sold was 45 less than the number of history textbooks sold. This means that

x = y - 45

Substituting x = y - 45 into equation 1, it becomes

y - 45 + y = 219

2y = 219 + 45 = 264

y = 264/2 = 132

x = y - 45 = 132 - 45

x = 87

Answer:

S ={(1+1=2), (1+2=3), (1+3=4), (1+4=5), (1+5=6), (1+6=7),

     (2+1=3), (2+2=4),(2+3=5),(2+4=6),(2+5=7),(2+6=8),

     (3+1=4), (3+2=5),(3+3=6),(3+4=7),(3+5=8),(3+6=9),

     (4+1=5), (4+2=6),(4+3=7),(4+4=8),(4+5=9),(4+6=10),

     (5+1=6), (5+2=7),(5+3=8),(5+4=9),(5+5=10),(5+6=11),

     (6+1=7), (6+2=8),(6+3=9),(6+4=10),(6+5=11),(6+6=12)}

Step-by-step explanation:

By definition the sample space of an experiment "is the set of all possible outcomes or results of that experiment".

For the case described here: "Toss a six-sided die twice and record the sum of the results".

Assuming that we have a six sided die with possible values {1,2,3,4,5,6}

The sampling space denoted by S and is given by:

S ={(1+1=2), (1+2=3), (1+3=4), (1+4=5), (1+5=6), (1+6=7),

     (2+1=3), (2+2=4),(2+3=5),(2+4=6),(2+5=7),(2+6=8),

     (3+1=4), (3+2=5),(3+3=6),(3+4=7),(3+5=8),(3+6=9),

     (4+1=5), (4+2=6),(4+3=7),(4+4=8),(4+5=9),(4+6=10),

     (5+1=6), (5+2=7),(5+3=8),(5+4=9),(5+5=10),(5+6=11),

     (6+1=7), (6+2=8),(6+3=9),(6+4=10),(6+5=11),(6+6=12)}

The possible values for the sum are 2,3,4,5,6,7,8,9,10,11,12

Answer:

addition tioin multiplication

Step-by-step explanation:

Using the given data, the coefficient of variation is 30% which matches option b.

Complete Question :

The hourly wages of a sample of 130 system analysts are given below. mean = 60 range = 20 mode = 73 variance = 324 median = 74. The coefficient of variation equals a. 0.30%. b. 30% O c. 5.4% d. 54%.