A 6,000​-liter cistern is empty when water begins flowing into it​ (at t = ​0) at a rate​ (in L/min) given by Q`(t) = 9 √t, where t is measured in minutes.
a. How much water flows into the cistern in 1.75 hours?
b. Find and graph the function that gives the amount of water in the tank at any time t.

Answer:

C. y=x+7

Step-by-step explanation:

If you add 7 to the values on the left, you'll get the values on the right.

Answer: I think c is the answer

Step-by-step explanation:

Answer:

True

Step-by-step explanation:

The negative linear relationship means that there is inverse relationship between two variables. It means that if the independent variable increases the dependent variable decreases and if the independent decreases the dependent variable increases. It means that due to larger values of independent variable there occurs the smaller values for dependent variable.

A minimum of 10 servers is needed to keep the average time in the system under 6 minutes.

The utilization factor is the ratio of the power station's maximum demand to its rated capacity. The time that equipment is in use./ The total time that it could be in use.

Given, In a record store with a service rate of 15 per hour, customers typically enter at a rate of 1 per minute (exponential interarrivals) (exponential service times).

Customers Per Minute into record shop = 1 per minute

Per Hour = 1 * 60 = 60

Number of Customers > Per Hour/(Server)

6 > 60/ server

server > 10

Therefore, as per the utilization factor, a minimum of 10 servers should be installed to keep the average time under 6 minutes.

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The given scenario can be modeled as a queuing system, where the arrival rate (\(\lambda\)) and service rate (\(\mu\)) are given.
From the information provided:
- The arrival rate, \(\lambda\), is 1 customer per minute, hence \(\lambda = 1\).
- The service rate, \(\mu\), is 15 customers per hour. Since we are working with minutes, we need to convert this to minutes. There are 60 minutes in an hour, so \( \mu = \frac{15}{60} = 0.25 \) customers per minute.
The goal is to keep the average time in the system, \( W \), under 6 minutes. The average time in the system for a M/M/c queue (which is the kind we're dealing with since both arrival and service rates are exponentially distributed) can be calculated using the formula for the average time in a system with \(c\) servers:
\[ W = \frac{W_q + 1}{\mu} \]
where:
- \(W_q\) is the average time a customer spends waiting in the queue.
- \(\mu\) is the service rate per server.
- The term \( \frac{1}{\mu} \) is the average service time.
The queue time \(W_q\) depends on the number of servers \(c\) and the utilization factor \(\rho = \frac{\lambda}{c\mu}\). The queue time increases with the utilization factor. The formula for \( W_q \) in an M/M/c queue is complex as it involves Erlang B formulas and iterative methods to solve for different values of \(c\).
However, for the practical purposes of this problem, we can use the approximation for \( W \) without explicitly calculating \( W_q \) as:
\[ W = \frac{1}{\mu - \lambda} \]
provided that this system is stable, which occurs only if \( \lambda < c \cdot \mu \).
We need to find the minimum number of servers \( c \) such that the average time in the system \( W \) is less than 6 minutes:
\[ W < 6 \]
Using the approximate formula:
\[ \frac{1}{c\mu - \lambda} < 6 \]
Therefore:
\[ c\mu - \lambda > \frac{1}{6} \]
Solving for the number of servers \( c \):
\[ c > \frac{\lambda + \frac{1}{6}}{\mu} \]
Plugging in the values of \(\lambda\) and \(\mu\):
\[ c > \frac{1 + \frac{1}{6}}{0.25} \]
\[ c > \frac{\frac{6}{6} + \frac{1}{6}}{{0.25}} \]
\[ c > \frac{\frac{7}{6}}{0.25} \]
\[ c > \frac{7}{6} \cdot \frac{1}{0.25} \]
\[ c > \frac{7}{6} \cdot 4 \]
\[ c > \frac{7 \cdot 2}{3} \]
\[ c > \frac{14}{3} \]
As we cannot have a fraction of a server, we need to round up to the nearest whole number:
\[ c > 4.67 \]
So we need a minimum of 5 servers to keep the average time in the system under 6 minutes.

Answer:

[tex](x-4)^2+(y-7)^2=4[/tex]

z=6

Step-by-step explanation:

The equation of a circle is satisfied by every point on it. The equation of a circle of radius r and point (h,k,l) parallel to the xy-plane is:

[tex](x-h)^2+(y-k)^2=r^2[/tex]

we can just substitute values into the equation:

[tex](x-4)^2+(y-7)^2=r^2[/tex] where z=6

The radius is 2. Therefore:

[tex](x-4)^2+(y-7)^2=4[/tex]

Final answer:

The equations describing the circle with a radius of 2 that is centered at (4, 7, 6) and parallel to the xy-plane are (x - 4)² + (y - 7)² = 4 and z = 6.

Explanation:

The equation that describes a circle of radius 2 centered at (4, 7, 6) that is parallel to the xy-plane can be found using the general equation of a circle in three dimensions, considering that the circle lies in a plane parallel to the xy-plane, hence the z-coordinate will remain constant. The standard equation for a circle in two dimensions is (x - h)² + (y - k)² = r², where (h, k) is the center of the circle and r is its radius.

For a circle of radius 2, the equation becomes (x - 4)² + (y - 7)² = 2². Since the circle is parallel to the xy-plane and the center has a z-coordinate of 6, the equation does not change with respect to z, remaining constant at z = 6. Therefore, the complete set of equations describing the circle are (x - 4)² + (y - 7)² = 4 and z = 6.

Answer:

Collectively, the total sample took 9000 course credits.

Step-by-step explanation:

The mean number of course credits taken is the total number of credits which the sample took collectively divided by the size of sample. Mathematically

[tex]M = \frac{T}{N}[/tex]

In which

M is the mean number of course credits taken

T is the total number of course credits taken

N is the size of the sample.

In this problem, we have that:

[tex]M = 18, N = 500[/tex]

We have to find T. So

[tex]M = \frac{T}{N}[/tex]

[tex]18 = \frac{T}{500}[/tex]

[tex]T = 500*18[/tex]

[tex]T = 9000[/tex]

Collectively, the total sample took 9000 course credits.

Answer:

Unimodal Distribution

Mode = 6        

Step-by-step explanation:

We are given the data of hours that 9 students spend on the computer on a typical day:

1, 6, 7, 6, 8, 11, 6, 12, 15

Sorted data: 1, 6, 6, 6, 7, 8, 11, 12, 15

Mode is the most frequent observation in the given data.

Here, 6 repeats itself most frequently that is 3 times. No other observation repeats itself three times.

Thus, the mode of the data is 6.

Since, there is a single mode for the given data, the distribution of students is unimodal.

Unimodal Distribution

Mode = 6        

The sorted data is 1, 6, 6, 6, 7, 8, 11, 12, 15

Since 6 repeats 3 times so the mode should be 6

Also, there should be the single mode so the type of distribution is unimodal  

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Answer:

Every 10th product in the line is selected

Step-by-step explanation:

Convenience sampling also available sampling, or nearest in reach sampling.

it is a type of non-probability sampling that involves the sample being drawn from a population that is in reach or that is easily at hand.

example. A questionnaire being distributed to people met in a mall.

for the manufacturing company in question, the first 10 product in line were the first set of product the machine will produce (at hand).

it is normally use to test run the operation of the machine.

Convenience sampling in manufacturing is best described as selecting every 10th product in the line for testing. It is a simple, quick, and cost-effective way to identify potential issues.

In the context of manufacturing, convenience sampling represents a type of sampling where samples are chosen because they are readily available or easy to obtain. In the provided choice list, the best description of convenience sampling is 'Every 10th product in the line is selected'. This method is chosen for its simplicity and speed. While it may not provide a comprehensive result since it won't cover all the various different scenarios, it is a cost-effective and time-efficient way of identifying potential issues in machine operations.

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To find the average wingspan, the z-score for 67% of the normal distribution was found to be 0.44. Using the formula with the given standard deviation and wingspan less than 3.21 meters, the average wingspan was calculated to be approximately 3.02 meters.

In order to find the average wingspan of the birds in the albatross colony given the standard deviation of 0.43 meters and that 67% of the birds have a wingspan less than 3.21 meters, we apply concepts from normal distribution.

From the properties of the normal distribution, we know that 67% corresponds to an area under the curve to the left of the mean. Using the standard normal distribution table (z-table), we can find the z-score that corresponds to 0.67 (67%). The z-score close to this area under the curve is approximately 0.44.

The formula to convert a z-score to an actual score is given as:

X = μ + (z × σ)

Where:

We are given X (3.21), z (0.44), and σ (0.43), and we need to find the mean μ.

Substituting the known values into the formula, we get:

3.21 = μ + (0.44 × 0.43)

Now we solve for μ:

3.21 = μ + 0.1892

μ = 3.21 - 0.1892

μ ≈ 3.02

So, the average wingspan of the albatrosses is approximately 3.02 meters rounded to two decimal places.

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Answer:

Step-by-step explanation:

Given

height of tank [tex]h=5\ ft[/tex]

Width of tank [tex]w=3\ ft[/tex]

length of tank [tex]L=2\ ft[/tex]

suppose a layer of water at height h of thickness dh from bottom needed to be pump out

So distance moved by this layer to come out of tank is [tex]\Delta h=5-h[/tex]

weight density of water [tex]\rho =50\ pounds/ft[/tex]

Force required to hold this layer up [tex]F_s=2\times 3\times \Delta h\times 50=300\Delta h[/tex]

Work done to remove the water

[tex]W=\int_{0}^{5}300\Delta hdh[/tex]

[tex]W=\int_{0}^{5}300\left ( 5-h\right )dh[/tex]

[tex]W=3750\ Pound-ft[/tex]

The work does it take to pump all of the liquid out of the top of the tank is 3750-pound feet.

Given that

A tank in the shape of a right rectangular prism has a height of 5 feet, width of 3 feet, and length of 2 feet.

It is full of a liquid weighing 50 pounds per cubic foot.

How much work does it take to pump all of the liquid out of the top of the tank?

A tank in the shape of a right rectangular prism has a height of 5 feet, width of 3 feet, and length of 2 feet.

The distance moved by this layer to come out of the tank is;

[tex]\rm \triangle h = 5-h[/tex]

The force required to hold this layer up is;

[tex]\rm Force = \triangle Height \times Width \times length \times liquid \ weighing\\ \\ Force = 2 \times 3 \times (5-h) \times 50\\ \\ Force = 300 (5-h)[/tex]

The work does it take to pump all of the liquid out of the top of the tank is calculated by;

[tex]\rm Work = \int\limits^5_0 {300(5-h)} \, dh\\ \\ Work = 300(5\int\limits^5_0 {} \, dh - \int\limits^5_0 {h} \, dh) \\ \\ Work = 300(5[h]^5_0- [\dfrac{h^2}{2}]^5_0)\\ \\ Work = 300(5(5-0)-\dfrac{5^2}{2}-\dfrac{0^2}{2})\\ \\ Work = 300(25-\dfrac{25}{2})\\ \\ Work = 300\times \dfrac{50-25}{2}\\ \\ Work = 300 \times \dfrac{25}{2}\\ \\ Work = 300 \times (12.5)\\ \\ Work = 3750 \ pounds \ feet[/tex]

Hence, The work does it take to pump all of the liquid out of the top of the tank is 3750-pound feet.

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Answer:

MARK ME BRAINIEST OR ILL BREAK MY KEYBOARD

Step-by-step explanation:

BBBBBBBBBBBBB

Answer:

[tex] y=(x-29)^2 +60[/tex]

Step-by-step explanation:

For this case we have the original function [tex] y =x^2[/tex]

So let's do the transformations one by one.

The general expression for a parabola like the formula given is:

[tex] y = (x-h)^2 +k[/tex]

If we want to do a shift on the vertical axis we need to modify the value of k, since we want 60 units upward the value of k =60, and then the formula after the first transformation would be:

[tex] y = x^2 + 60[/tex]

For the other part related to the movement on the x axis 29 units to the right we need to modify the value of h in the general expression , since is a translation to the right the value of h = 29 and if we replace we got:

[tex] y=(x-29)^2 +60[/tex]

And that would be our final expression after the transformations on the y and x axis.

On the figure attached we see the original function in red, the blue function represent the shift upward and the green one the two tranformations at the sam time to check that we did the procedure right.

Answer:

The volume of remaining sphere is 489.84 cubic inches.

Step-by-step explanation:

We are given the following in the question:

A hole 2 inches in radius is drilled out of a solid sphere of radius 5 inches.

Radius of sphere = 5 inches

Radius of hole = 2 inches

Volume of sphere =

[tex]\dfrac{4}{3}\pi r^3[/tex]

where r is the radius of sphere.

Volume of sphere =

[tex]\displaystyle\frac{4}{3}\pi (5)^3\\\\=\frac{4}{3}\times 3.14\times (5)^3\\\\=523.33\text{ cubic inches}[/tex]

Volume of hole =

[tex]\displaystyle\frac{4}{3}\pi (2)^3\\\\=\frac{4}{3}\times 3.14\times (2)^3\\\\=33.49\text{ cubic inches}[/tex]

Volume of remaining solid =

Volume of sphere - Volume of hole

[tex]=523.33 - 33.49\\=489.84\text{ cubic inches}[/tex]

The volume of remaining sphere is 489.84 cubic inches

Answer:

720 different choices of officers are possible if there are no restrictions.

The number of ways in which Worf and Troi will not serve together is 672.

Step-by-step explanation:

Consider the provided information.

We have 3 post admiral, captain, and commander, and 10 Starfleet officers.

Part (A) there are no restrictions?  

We need to select 3 people out of 10. And these 3 people can again rearranged into different rank.

Thus, the number of ways are: [tex](^{10}C_3)3!=\left(\dfrac{10!}{3!7!}\right)3!=720[/tex]

Hence, 720 different choices of officers are possible if there are no restrictions.

Part (B) Worf and Troi will not serve together?

Subtract those cases in which both of them are selected from total number of ways.  

If both of them selected then we need to select only 1 person out of 8. And further they can rearranged into 3 different rank.

Thus, the number of ways are: [tex]720-(^{8}C_1)3!=720-8\times3!=672[/tex]

Hence, the number of ways in which Worf and Troi will not serve together is 672.

Answer:

2:3

Step-by-step explanation:

the ratio of 28 hours to 42 hours is 28:42

28/42

7 divides both the numerator and denominator into small terms

4/6

2 divides both the numerator and denominator into small terms

2/3 = 2:3

Answer:

A. x = -4 + 3t, y = 3 - 4t: - 2 lessthanorequalto t lessthanorequalto 3

Step-by-step explanation:

Given that a line in two dimension passes through (-4,3) and (2,-5) oriented in the direction of increasing x

We can write the line equation in parametric form as

[tex]\frac{x-x_1}{x_2-x_1} =\frac{y-y_1}{y_2-y_1} \\\frac{x+4}{6} =\frac{y-3}{-8} =t\\x=-4+6t\\y = 3-8t[/tex]

The values of t when x=-4 is 0 and when x =2 is 1

So t varies from 0 to 1

If instead of t we give t' say which is t +2

then we have

t he parametric equations as

x =-4+3t and y = 3-4t

For  x=-4, t =2 and for x = 2 , t =3

So option A is right.

Answer: (18/5) * 11

Step-by-step explanation:

In equations of proportion, to proceed, we need to determine the constant of proportion, let's denote as "k"

Since y ~ x

Means y=kx.

To determine the value of k, we input initial values of y and x.

Initial value of y = 18

Initial value of x = 5

The equation becomes :

18 = 5k

k = 18/5.

Now, If given the value of x as 11,to determine the value of y we go back to the equation.

y= kx

y = 18/5 * 11.

The correct expression to find the value of y when x is 11 is [tex]\( y = \frac{18}{5} \times 11 \)[/tex].

Given that y varies directly as x, we can express this relationship using the formula [tex]\( y = kx \)[/tex], where k is the constant of proportionality. To find the value of k, we use the given values of y and x when [tex]\( y = 18 \)[/tex] and [tex]\( x = 5 \)[/tex]. Thus, we have:

[tex]\[ k = \frac{y}{x} = \frac{18}{5} \][/tex]

Now, we want to find the value of y when [tex]\( x = 11 \)[/tex]. Using the direct variation formula with our calculated k:

[tex]\[ y = kx = \frac{18}{5} \times 11 \][/tex]

This expression will give us the value of y when x is 11. The other options provided are incorrect because they either divide by 11 or use the inverse relationship, which does not apply in the case of direct variation.

Answer:

C. Total flow

Step-by-step explanation:

The total amount of fluid that has passed a designated point is designated as the total flow.

The flow rate is the rate at which fluid passes through a certain point, not the total amount of fluid.

Laminar and turbulent flow are different types of fluid flow patterns and don't characterize a volume of fluid.

Therefore, the correct answer is C. Total flow.

Answer:

1. Linear relation

2. Linear relation

3. Non Linear relation

4. Non Linear relation

Step-by-step explanation:

i) the number of pages a copy machine can print over time, expressed by n = 30m is a linear relation where the value of 'n' is directly proportional to 'm'.

ii) The amount of money an hourly worker makes over time, expressed by a = 15h, is a linear relation where the value of 'a' is directly proportional to 'h'.

iii) The temperature outside at noon each day for a year is a non-linear relation because the temperature will be different every day and not by the same amount.

iv) The height increase of a child over 3 years is also a non-linear relation as the height increase will vary from year to year and not by the same amount.

Answer:

E (Y) = 3

Step-by-step explanation:

If a 4-sided die is being rolled repeatedly; and the odd-numbered rolls (1st 3rd,5th, etc.)

The probability of odd number roll will be, p(T) = [tex]\frac{1}{2}[/tex]

However, on your even-numbered rolls, you are victorious if you get a 3 or 4. Also, the  probability of even number roll, p(U) = [tex]\frac{1}{2}[/tex]

In order to  calculate: E (Y); We can say Y to be the number of times you roll.

We know that;

E (Y) = E ( Y|T ) p(T) + E ( Y|U ) p(U)

Let us calculate E ( Y|T ) and E ( Y|U )

Y|T ≅ geometric = [tex]\frac{1}{4}[/tex]

Y|U ≅ geometric = [tex]\frac{1}{2}[/tex]

also; x ≅ geometric (p)

∴ E (x) = [tex]\frac{1}{p}[/tex]

⇒ [tex]\frac{Y}{T}[/tex] = 4 ; also [tex]\frac{Y}{U}[/tex] = 2

E (Y) = 4 × [tex]\frac{1}{2}[/tex] + 2 ×

        = 2+1

E (Y) = 3

Answer:

a) 68% of the investments had a return of between 10% and 30%.

b) 32% of investments had a return that was either less than 10% or morethan 30%.

Step-by-step explanation:

We can use the Empirical Rule to solve this question:

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

68% of the measures are within 1 standard deviation of the mean. This also means that 32% of the measures are more than 1 standard deviation from the mean.

95% of the measures are within 2 standard deviation of the mean. This also means that 5% of the measures are more than 2 standard deviations from the mean.

99.7% of the measures are within 3 standard deviations of the mean. This also means that 0.3% of the measures are more than 3 standard deviations from the mean.

In this problem, we have that:

Mean = 20%.

Standard deviation = 10%.

a. What proportion of the investments had a return of between 10% and 30%?

10 is the mean subtracted by 1 one standard deviation

30 is one standard deviation added to the mean.

So 10 and 30 are within 1 standard deviation of the mean. So 68% of the investments had a return of between 10% and 30%.

b. What proportion of investments had a return that was either less than 10% or morethan 30%?

This is the proportion of investments that were farther than one standard deviation of the mean.

By the Empirical Rule, 32% of investments had a return that was either less than 10% or morethan 30%.

Answer:there are 16 goats and 22 chickens.

Step-by-step explanation:

Let x represent the number of goats in the field.

Let y represent the number if chicken in the field.

After a quick count, you determine there are 38 heads and 108 feet in the field. A goat has one head. A chicken also has one head. It means that

x + y = 38

A goat has four legs. A chicken has 2 legs. It means that

4x + 2y = 108 - - - - - - - - - - - 1

Substituting x = 38 - y into equation 1, it becomes

4(38 - y) + 2y = 108

152 - 4y + 2y = 108

- 4y + 2y = 108 - 152

- 2y = - 44

y = - 44/ - 2

y = 22

x = 38 - y = 38 - 22

x = 16

By the chain rule,

[tex]\dfrac{\partial z}{\partial u}=\dfrac{\partial z}{\partial x}\dfrac{\partial x}{\partial u}+\dfrac{\partial z}{\partial y}\dfrac{\partial y}{\partial u}[/tex]

where [tex]u\in\{r,t\}[/tex].

We have component partial derivatives

[tex]\dfrac{\partial z}{\partial x}=\dfrac{2x}{x^2+y}=\dfrac{2re^t}{r^2e^{2t}+te^r}[/tex]

[tex]\dfrac{\partial z}{\partial y}=\dfrac1{x^2+y}=\dfrac1{r^2e^{2t}+te^r}[/tex]

[tex]\dfrac{\partial x}{\partial r}=e^t[/tex]

[tex]\dfrac{\partial x}{\partial t}=re^t[/tex]

[tex]\dfrac{\partial y}{\partial r}=te^r[/tex]

[tex]\dfrac{\partial y}{\partial t}=e^r[/tex]

Putting the appropriate pieces together and setting [tex](r,t)=(1,2)[/tex], we get

[tex]\dfrac{\partial z}{\partial r}(1,2)=\dfrac{2e^3+2}{e^3+2}[/tex]

[tex]\dfrac{\partial z}{\partial t}(1,2)=\dfrac{2e^3+1}{e^3+2}[/tex]

Final answer:

To find the probability of selecting one ace and one of either a ten, jack, queen, or king from a deck with replacement, calculate the probability for each draw and account for both possible orders of drawing these cards the final probability is 2 * ((4/52) * (16/52)) to account for both orders.

Explanation:

To calculate the probability that one card is an ace and the other is either a ten, jack, queen, or king when two cards are selected randomly from an ordinary deck, we approach this problem considering the combinations and probabilities of each event happening.

First, let's establish the total possibilities. Since we are replacing the card back into the deck after the first draw, the number of possibilities for each draw remains 52. So, the total number of ways to draw any two cards (with replacement) is 52 * 52.

The probability of drawing an ace (P(Ace)) on the first draw is 4/52 because there are 4 aces in a deck of 52 cards.

The probability of drawing a ten, jack, queen, or king (P(10/J/Q/K)) on the second draw is 16/52, as there are 4 of each of these cards in the deck, totaling 16 cards.

To find the probability of both events happening, you multiply the probabilities of each event: P(Ace) * P(10/J/Q/K). This gives you (4/52) * (16/52).

However, the order in which the ace and the 10/J/Q/K are drawn matters, as the ace could also be drawn second. This means we need to calculate the probability again with the ace being drawn second and the 10/J/Q/K card drawn first, which is the same calculation. Therefore, the final probability is 2 * ((4/52) * (16/52)) to account for both orders.

To calculate the probability of drawing an ace followed by a ten, jack, queen, or king, we find the probabilities of each event separately and multiply them since the events are independent, assuming the card is replaced after the draw.

The probability of selecting one ace and another card which is a ten, jack, queen, or king involves understanding the composition of a standard deck and calculating the odds of each draw.

A standard deck has 52 cards with 4 aces and 16 face cards (including the tens). The chance of drawing an ace (event A) is therefore 4/52. For the second draw (event B), with the card being replaced after the first draw, the odds of drawing either a ten, jack, queen, or king from any suit remain identical, since we're back to having a full deck.

To find the overall probability, we calculate the probability of both events occurring, which involves multiplying the probabilities of each individual event, assuming they are independent. This is represented by: P(A and B) = P(A) * P(B).

For the specific example of drawing a four (event A) then a five (event B) when replacing the card back into the deck after each draw, the probability of event A is 1/13 since there are four fours in the deck and the probability of event B is also 1/13 for the same reason. Therefore, the combined probability is (1/13)*(1/13), which simplifies to 1/169.

Answer:

The sample standard deviation for given data is 222.69        

Step-by-step explanation:

We are given the following data set:

508, 657, 214, 958, 765, 449, 338, 497

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{4386}{8} = 548.25[/tex]

Sum of squares of differences =

1620.0625 + 11826.5625 + 111723.0625 + 167895.0625 + 46980.5625 + 9850.5625 + 44205.0625 + 2626.5625 = 396727.5

[tex]\text{Sample standard Deviation} = \sqrt{\dfrac{396727.5}{7}} =222.69[/tex]

The sample standard deviation for given data is 222.69        

To calculate the sample standard deviation using a calculator, enter the data points in statistical mode, use the function labeled as 'sx' or 'σx' for the sample, not the population, then calculate and round to the nearest tenth. The symbol for sample standard deviation is typically 's' or 'sx' on most calculators.

To find the sample standard deviation of the dataset {508, 657, 214, 958, 765, 449, 338, 497}, you will need to use your calculator's statistical functions. Use the statistical mode in your calculator that corresponds to Equation 4.1.1 for calculating the standard deviation of a sample, not the population. Here are the typical steps for calculating it:

The symbol s or sx represents the sample standard deviation on most calculators, while σ and σx are typically used for a population's standard deviation (which is not needed in this question).

As an example, if your calculated variance is .5125, the standard deviation would be the square root of the variance: S = √.5125 = .715891, and when rounded to two decimal places, s = .72. Always consult your calculator's manual for specific instructions, as they can vary between models.

Answer:

See the proof below.

Step-by-step explanation:

Assuming this complete question: "For each given p, let Z have a binomial distribution with parameters p and N. Suppose that N is itself binomially distributed with parameters q and M. Formulate Z as a random sum and show that Z has a binomial distribution with parameters pq and M."

Solution to the problem

For this case we can assume that we have N independent variables [tex]X_i[/tex] with the following distribution:

[tex] X_i Bin (1,p) = Be(p)[/tex] bernoulli on this case with probability of success p, and all the N variables are independent distributed. We can define the random variable Z like this:

[tex] Z = \sum_{i=1}^N X_i[/tex]

From the info given we know that [tex] N \sim Bin (M,q) [/tex]

We need to proof that [tex] Z \sim Bin (M, pq)[/tex] by the definition of binomial random variable then we need to show that:

[tex] E(Z) = Mpq[/tex]

[tex] Var (Z) = Mpq(1-pq)[/tex]

The deduction is based on the definition of independent random variables, we can do this:

[tex] E(Z) = E(N) E(X) = Mq (p)= Mpq[/tex]

And for the variance of Z we can do this:

[tex] Var(Z)_ = E(N) Var(X) + Var (N) [E(X)]^2 [/tex]

[tex] Var(Z) =Mpq [p(1-p)] + Mq(1-q) p^2[/tex]

And if we take common factor [tex]Mpq [/tex] we got:

[tex] Var(Z) =Mpq [(1-p) + (1-q)p]= Mpq[1-p +p-pq]= Mpq[1-pq][/tex]

And as we can see then we can conclude that   [tex] Z \sim Bin (M, pq)[/tex]

To show that X has a binomial distribution with parameters p*n and q*m, we can formulate X as a random sum of indicator random variables Zi = 1 if the i-th trial is a success and 0 otherwise.

Let X be a binomial random variable with parameters p and n, and let Y be a binomial random variable with parameters q and m. To show that X has a binomial distribution with parameters p*n and q*m, we can formulate X as a random sum of indicator random variables Z1, Z2, ..., Zn, where Zi = 1 if the i-th trial is a success and 0 otherwise. Then X can be expressed as X = Z1 + Z2 + ... + Zn. Since each Zi is independent and follows a Bernoulli distribution with parameter p, the sum of n independent Bernoulli random variables is a binomial random variable with parameters p and n. Therefore, X has a binomial distribution with parameters p*n and q*m.

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Answer:

You didn't give the differential equations, but I'll explain how to identify the independent variable, dependent variable, how to know the order, linearity, and nonlinearity of a differential equation.

Step-by-step explanation:

DIFFERENTIAL EQUATION

This is any equation that involves differential coefficients. It is a relationship between an independent variable, x, a dependent variable, y, and one or more derivatives of y with respect to x.

Examples

(1) xd²y/dx² + 7dy/dx = 0

(2) y²dy/dx + 2x = 0

(3) xd³y/dx³ = y½ + 1

(4) 2xy'' - 3y' + 5y = 0

(5) (y''')² + 30xy = 0

Note that the dependent variable is always the numeratior, and the independent, denominator, in a different coefficient. In the case of our examples, y is the dependent variable, and x is the independent.

Example (4) is another way of writing a differential coefficient, y' (read as y-prime) is the same as dy/dx (read as dee-y dee-x). In some cases when the independent variable is time t, it is written as ÿ, which is the same as d²y/dt² (read as dee-two-y dee-t-squared)

ORDER

This is the order of the highest derivative in a differential equation. You need not consider other derivatives, just the highest.

In the examples, the orders are

(1) two

(2) one

(3) three

(4) two

(5) three

LINEAR DIFFERENTIAL EQUATION

This is the kind of differential equation in which the functions of the dependent variable are linear. There are no powers of the dependent variable and/or its derivatives, there are no products of the dependent variable and its derivative, there are no functions of the dependent variable like cos, sin, exp, etc.

NONLINEAR DIFFERENTIAL EQUATION

If any condition for linearity is not met, then it is nonlinear.

(1) Linear

(2) Nonlinear because y is the dependent variable, and y² is nonlinear, and even still, it multiplies a derivative.

(3) Nonlinear because y½ is nonlinear

(4) Linear

(5) Nonlinear because (y''')² in nonlinear.

Understanding this, you can determine the order, linearity or nonlinearity of any differential equation. Cheers!

Equations 1 and 3 are nonlinear due to their respective terms, while Equation 2 is linear. The orders are first for Equations 1 and 2, and second for Equation 3.

Let's analyse each of the given differential equations step by step.

Equation 1: y' = y - x²

Equation 2: xy' = 2y

Equation 3: x'' + 5x = e-x

Thus,

1. [tex]\( y' = y - x^2 \)[/tex]

  (a) Independent variable: x, Dependent variable: y

  (b) First order

  (c) Nonlinear because of the term -x².

2. xy' = 2y

  (a) Independent variable: x, Dependent variable: y

  (b) First order

  (c) Nonlinear because of the product xy.

3. [tex]\( x'' + 5x = e^{-x} \)[/tex]

  (a) Independent variable: t (not explicitly given), Dependent variable: x

  (b) Second order

  (c) Linear because it has no products or powers of x other than x and x''.

Complete question: In problems below

(a) identify the independent variable and the dependent variable of each equation (use 't' for the independent variable if an independent variable is not given explicitly): (b) give the order of each differential equation (enter '1' for first order. '2' for second order and so on: do not include the quotes); and (C) state whether the equation is linear or nonlinear If your answer to (C) is nonlinear, make sure that you can explain why this is true

Equation:

1. y' = y-x²

2. xy' = 2y

3. [tex]\( x'' + 5x = e^{-x} \)[/tex]

Answer:

The values of x that makes these vectors orthogonal are x = 2 and x = 4.

Step-by-step explanation:

Orthogonal vectors

Suppose we have two vectors:

[tex]v_{1} = (a,b,c)[/tex]

[tex]v_{2} = (d,e,f)[/tex]

Their dot product is:

[tex](a,b,c).(d,e,f) = ad + be + cf[/tex]

They are ortogonal is their dot product is 0.

Solving quadratic equations:

To solve this problem, we are going to need tosolve a quadratic equation.

Given a second order polynomial expressed by the following equation:

[tex]ax^{2} + bx + c, a\neq0[/tex].

This polynomial has roots [tex]x_{1}, x_{2}[/tex] such that [tex]ax^{2} + bx + c = (x - x_{1})*(x - x_{2})[/tex], given by the following formulas:

[tex]x_{1} = \frac{-b + \sqrt{\bigtriangleup}}{2*a}[/tex]

[tex]x_{2} = \frac{-b - \sqrt{\bigtriangleup}}{2*a}[/tex]

[tex]\bigtriangleup = b^{2} - 4a[/tex]

Find all values of x such that (4, x, −6) and (2, x, x) are orthogonal.

[tex](4,x,-6)(2,x,x) = 8 + x^{2} - 6x[/tex]

These vectors are going to be orthogonal if:

[tex]x^{2} -6x + 8 = 0[/tex]

This is a quadratic equation, in which [tex]a = 1, b = -6, c = 8[/tex]. So

[tex]\bigtriangleup = 6^{2} - 4*1*8 = 4[/tex]

[tex]x_{1} = \frac{-(-6) + \sqrt{4}}{2} = 4[/tex]

[tex]x_{2} = \frac{-(-6) - \sqrt{4}}{2} = 2[/tex]

The values of x that makes these vectors orthogonal are x = 2 and x = 4.

The values of x that make the vectors (4, x, −6) and (2, x, x) orthogonal are x = 2 and x = 4, determined by setting their dot product to zero and factoring the resulting quadratic equation.

To find all values of x such that the vectors (4, x, −6) and (2, x, x) are orthogonal, we need to perform the dot product of the vectors and set it equal to zero. Two vectors are orthogonal if their dot product is zero.

The dot product is calculated as follows:

Next, we factor the quadratic equation:

Hence, the two values of x that make the vectors orthogonal are x = 2 and x = 4.

Answer:

Option B) No. The standard deviation cannot be negative.

Step-by-step explanation:

We are given the following in the question:

[tex]\mu = 2.0\\\sigma = -3.5[/tex]

The above results are not possible.

Option B) No. The standard deviation cannot be negative.

Final answer:

The results from Excel indicating that the standard deviation of a discrete probability distribution is negative are incorrect, as B. standard deviation cannot be negative.

Explanation:

When analyzing a discrete probability distribution in Excel, we expect certain characteristics from the mean (mu) and standard deviation (sigma). The mean represents the expected value or average of the distribution, while the standard deviation measures how much the values in the distribution vary from the mean. In the context of a probability distribution, a standard deviation cannot be negative as it represents the square root of variance, which is a squared value and therefore nonnegative.

Given the Excel results showing mu equals 2.0 and sigma equals negative 3.5, we can immediately identify an error. A negative standard deviation does not make sense in the real world since it is a measure of dispersion. Therefore, the correct answer to whether these results can be correct is: B. No. The standard deviation cannot be negative.

Answer:

78,79,80

Step-by-step explanation:

use algebraic values. in this case, x, x+1, x+2

Final answer:

The three consecutive integers that sum up to 237 are 78, 79, and 80. To find these, an algebraic equation is set up and solved step by step to identify the smallest integer followed by the next two consecutive ones.

Explanation:

The sum of three consecutive integers is 237. To find these integers, you can set up an algebraic expression for the problem. Let the smallest integer be x. Then the next two consecutive integers would be x + 1 and x + 2. The sum can be written as:

x + (x + 1) + (x + 2) = 237

Combine like terms to solve for x:

3x + 3 = 237

3x = 234

x = 78

Now we have the smallest integer. The next two consecutive integers are:

x + 1 = 79

x + 2 = 80

Therefore, the three consecutive integers are 78, 79, and 80.

Answer:

x1 =1

x2 =2

x3 =4

Step-by-step explanation:

Given is a systems of equations in 3 variables.

No of equations given = 3

[tex]x1 + x2 + x3 = 7 ... I\\x1 - x2 + 2x3 = 7 ... II\\5x1 + x2 + x3 = 11 ... III[/tex]

subtract equation 1 form equation 3

We get

[tex]4x1=4\\x1=1[/tex]

Substitute this value in 2 and 3

[tex]x2-2x3 = -6 ... iv\\x2+x3 =6 ... v[/tex]

subtract  iv from v

3x3 = 12

x3=4

Substitute in v

x2 =2

solution is

x1 =1

x2 =2

x3 =4