Two new wind-farm tower projects are proposed for a small company that installs them in south western Wisconsin. Project A will cost $250,000 to complete and is expected to have an annual net cash flow of $75,000. Project B will cost $150,000 to complete and should generate annual net cash flows of $52,000. As a small company, the owner and senior management team are very concerned about their cash flow. Use the payback period method and determine which project is better from a cash flow standpoint. Can someone show me how I can help my son solve this problem so I can guide him through it?

Answer:

a) Parameter

b) Statistic

c) Statistic

d) Parameter

e) Statistic

Step-by-step explanation:

For this case we need to remmber that a parameter describe a population of interest is fixed and not changes , and a statistic is a value that describe the sample size selected and can change between samples.

a. There are 100 senators in the 114th Congress, and 54% of them are Republicans.

The 54% here is a parameter since represent the proportion for all the population of interest on this case.

b. In a 2011 Gallup poll of 1008 adults living in the United States, 11% said they are satisfied with the condition of the national economy.

The 11% here is a statistic since we have a random sample and from this sample we calculate the proportion of interest for this case.

c. A survey of hospital records in 120 hospitals throughout the world shows the mean height of 180 cm for adult males.

The mean height of 180 cm is a statistic since we have a survey not all the population of interest

d. The 59 players on the roster of a championship football team have a mean weight of 248.6 pounds with a standard deviation of 44.6 pounds.

The 44.6 pounds is a parameter since we are interested on all the possible players and we have the info for all of them

e. In a random sample of households in the United States, it is found that 51% of the sampled households have at least one high‑definition television.

The 51% here is a statistic since we have a result from a sample not from the population

Answer: In my opinion, I would say b is the correct answer

Step-by-step explanation:

Answer:

a) [tex]Q(t) = 112.8*(0.766)^{t}[/tex]

b) When t = 10, Q = 7.845.

Step-by-step explanation:

The value of a quantity after t years is given by the following formula:

[tex]Q(t) = Q_{0}(1 + r)^{t}[/tex]

In which [tex]Q_{0}[/tex] is the initial quantity and r is the rate that it changes. If it increases, r is positive. If it decreases, r is negative.

a) Write a formula for Q as a function of t.

The initial value of a quantity Q (at year t = 0) is 112.8.

This means that [tex]Q_{0} = 112.8[/tex].

The quantity is decreasing by 23.4% per year.

This means that [tex]r = -0.234[/tex]

So

[tex]Q(t) = 112.8*(1 - 0.234)^{t}[/tex]

[tex]Q(t) = 112.8*(0.766)^{t}[/tex]

b) What is the value of Q when t = 10?

This is Q(10).

[tex]Q(t) = 112.8*(0.766)^{t}[/tex]

[tex]Q(t) = 112.8*(0.766)^{10} = 7.845[/tex]

When t = 10, Q = 7.845.

Answer:

[tex]^{13}C_3 \times ^{12}C_2 = 22308[/tex]

Step-by-step explanation:

We have 5 spaces. In our hand.

_ _ _ _ _

a standard deck of 52 cards contains 13 cards for each suit.

so we have 13 spades and 13 diamonds in total. There arrangements don't matter (it doesn't matter if the first card is 9 of spades or the second is 9 of spades, all of these will be counted as one arrangement)

[tex]^{13}C_3[/tex]: to choose the 3 spade card from a total of 13 spades.

[tex]^{12}C_2[/tex]: to choose the 2 diamond cards from a total of 13 diamonds.

and that is it!

we're gonna multiply the two values.

[tex]^{13}C_3 \times ^{12}C_2[/tex]

Answer:

116.45 is the minimum score needed to be stronger than all but 5% of the population.

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 100

Standard Deviation, σ = 10

We are given that the distribution of score is a bell shaped distribution that is a normal distribution.

Formula:

[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]

We have to find the value of x such that the probability is 0.05

P(X > x)  

[tex]P( X > x) = P( z > \displaystyle\frac{x - 100}{10})=0.05[/tex]  

[tex]= 1 -P( z \leq \displaystyle\frac{x - 100}{10})=0.05[/tex]  

[tex]=P( z \leq \displaystyle\frac{x - 100}{10})=0.95 [/tex]  

Calculation the value from standard normal z table, we have,  

[tex]P(z<1.645) = 0.95[/tex]

[tex]\displaystyle\frac{x - 100}{10} = 1.645\\x =116.45[/tex]  

Hence, 116.45 is the minimum score needed to be stronger than all but 5% of the population.

Final answer:

To solve the given initial value problem, we'll use the method of integrating factors. After rearranging the equation to standard form, we'll find the integrating factor and proceed to solve for y.

Explanation:

To find the solution of the given initial value problem, we'll use the method of integrating factors. First, we'll rearrange the equation to the standard form: y' + (6/t)y = t - 1/t + 1. Now, let's identify the integrating factor, which is e^(∫6/t dt) = e^(6ln|t|) = t^6. Multiply both sides of the equation with t^6 to get t^7y' + 6t^5y = t^7 - t^6 + t^6. Now, we can integrate both sides and solve for y.

Answer:

  (-1, -5)

Step-by-step explanation:

When you graph the points, you find that the lines intersect at the point (-1, -5). That is the solution to the system of equations.

  (x, y) = (-1, -5)

The solution to a system of linear equations is the point where the lines intersect. Given the data points, we can infer the linear relationship of each equation. However, without more information, we cannot calculate the exact point of intersection.

The solution to a system of linear equations is the point where the two lines intersect. From the provided data for Equation A and Equation B, we can infer their linear relationship by finding the slope and y-intercept.

For Equation A, the points (0, -2) and (2, 4) gives a slope of (4 - -2) / (2 - 0) = 3 and y-intercept of -2. For Equation B, the points (-3,-9) and (-1, -5) gives a slope of (-5 - -9) / (-1 - -3) = 2 and y-intercept of -3.

As the provided slopes and y-intercepts are different for these two equations, they would intersect at a point. To find this point, we set the two equations equal to each other and solve for x and y. However, with the provided data points, we cannot calculate the exact point of intersection without further information.

So, the solution to the system of equations would be the x and y values at the point of intersection of these two lines.

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Answer:  a. 242.5 pounds

b.  236 pounds

c . 208 and 278 pounds

The results are unlikely to be representative of all players in that​ sport's league because players randomly selected from championship sports team not the whole league.

Step-by-step explanation:

Given : Listed below are the weights in pounds of 11 players randomly selected from the roster of a championship sports team.

278    303    186    292    276    205    208    236    278    198    208

Mean = [tex]\dfrac{\text{Sum of weights of all players}}{\text{Number of players}}[/tex]

[tex]=\dfrac{278+303+186+292+276+205+208+236+278+198+208}{11}\\\\=\dfrac{2668}{11}\approx242.5\text{ pounds}[/tex]

For median , first arrange weights in order

186, 198 , 205 , 208, 208 , 236 , 276 , 278 ,278, 292 , 303

Since , number of data values is 11 (odd)

So Median = Middlemost value = 236 pounds

Mode = Most repeated value= 208 and 278

The results are unlikely to be representative of all players in that​ sport's league because players randomly selected from championship sports team not the whole league.

The mean, median and mode values of the distribution are :

Given the data :

Arrange the data in ascending order :

The mean = ΣX/ n

The median ;

The mode :

Therefore ;

Mean = 242.5 pounds

Median = 236 pounds

Mode = 208 and 278

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Final answer:

A linear system with three equations and five variables does not have to be consistent. The statement 'A linear system with three equations and five variables must be consistent' is false

Explanation:

A linear system with three equations and five variables does not have to be consistent. In fact, it is possible for the system to be inconsistent.

The statement that a linear system with three equations and five variables must be consistent is False. In linear algebra, the consistency of a system depends on whether there are any contradictions among the equations. For a system to be consistent, it must have at least one solution.

For example, consider the system of equations:

x + y + z = 5

2x + 3y + 4z = 10

5x + 2y + 3z = 8

Since there are more variables than equations, there will be infinitely many solutions if the system is consistent. But if the system is inconsistent, there will be no solution.

Therefore, the statement 'A linear system with three equations and five variables must be consistent' is false

Answer: The speed that Jesse travels in still water is 6 km/hr.

Step-by-step explanation:

Let the speed that Jesse travels in still water be 'x'.

Distance = 72 km

The trip to the camp area with a 2-km/hr current takes 9 hr less time than the return trip against the current.

Speed of current = 2 km/hr

According to question, we get that

[tex]\dfrac{72}{x-2}-\dfrac{72}{x+2}=9\\\\\dfrac{x+2-(x-2)}{x^2-4}=\dfrac{9}{72}\\\\\dfrac{4}{x^2-4}=\dfrac{1}{8}\\\\32=x^2-4\\\\32+4=x^2\\\\x^2=36\\\\x=\sqrt{36}\\\\x=6[/tex]

Hence, the speed that Jesse travels in still water is 6 km/hr.

Jesse's speed in still water is determined by setting up a system of equations using the distance equals rate times time formula for both downstream and upstream travel. By accounting for the time difference and the current speed, we solve for the variable representing Jesse's speed in still water.

Jesse takes a kayak trip traveling 72 km with and against a current, and we need to find Jesse's speed in still water. Let's denote the speed in still water as v (km/hr) and the current speed as 2 km/hr. The trip downstream increases Jesse's speed to (v + 2) km/hr, and upstream decreases it to (v - 2) km/hr.

Using the distance equals rate times time formula (d = rt), we can write the following equations for the time taken downstream (td) and upstream (tu):

Given that it takes 9 hours less to travel downstream, we have tu = td + 9.

By solving these linear equations, we find the system:

Combining these gives us:

72 / (v + 2) + 9 = 72 / (v - 2)

By solving this equation, we find the value of v, Jesse's speed in still water.

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Answer:

x^2+y^2+z^2-10x-6y-10z +50 =0

Step-by-step explanation:

Given that a sphere is contained in the first octant

Centre of the sphere is given as (5,3,5)

Since this is contained only in the first octant radius should be at most sufficient to touch any one of the three coordinate planes

When it touches we can get the maximum sphere

We find that y coordinate is the minimum of 3 thus radius can be atmost 3 so that then only it can touch y =0 plane i.e. zx plane without crossing to go to the other octants.

Hence radius =3

Equation of the sphere would be

[tex](x-5)^2 +(y-3)^2+(z-5)^2 = 3^2\\x^2+y^2+z^2-10x-6y-10z +50 =0[/tex]

Answer:

f(14) = 28

Step-by-step explanation:

The function that satisfy the equations is

f(x) = 3x - x

f(10) = 3*10 - 10 = 30 - 10 = 20

f'(x) = 3 - 1 = 2.

Therefore,

f(14) = 3(14) - 14 = 42 - 14 = 28.

f(14) = 28

Answer:

False. See the explanation below.

Step-by-step explanation:

We need to proof if the following statement "If a linear system has four equations and seven variables, then it must have infinitely many solutions." is false.

And the best way to proof that is false is with a counterexample.

Let's assume that we have seven random variables given by [tex]a_1, a_2, a_3, a_4, a_5, a_6, a_7[/tex] and we have the following four equations given by the following system:

[tex] a_1 +a_2 +a_3 +a_4 +a_5 +a_6 +a_7 =1[/tex]   (1)

[tex] a_1 +a_2= 0[/tex]    (2)

[tex] a_3 +a_4 +a_5 =1[/tex]   (3)

[tex] a_6 +a_7 =1[/tex]   (4)

As we can see we have system and is inconsistent since equation (1) is not satisfied by equation (2) ,(3) and (4) if we add those equations we got:

[tex] a_1 +a_2 +a_3 +a_4 +a_5 +a_6 +a_7 = 0+1+1= 2 \neq 1[/tex]

So then we can have a system of 7 variables and 4 equations inconsistent and with not infinitely solutions for this reason the statement is false.

A linear system can have more variables than equations, this is referred to as an underdetermined system. It's often believed that such systems have infinitely many solutions, but it's not necessarily the case. Such a system could also have no solutions if the system is inconsistent, signifying that not all underdetermined systems yield infinite solutions.

Contrary to the common belief, the statement that a linear system with more variables than equations must have infinitely many solutions is not always true. A system having more variables than equations is termed as underdetermined. Indeed, such systems often have infinite solutions, but not necessarily.

A system could also be inconsistent, meaning there are no solutions. As an example, consider the system of equations where:

Even though there are more variables (3) than equations (4), there are still no solutions as these equations contradict each other.

This emphasizes the point that, for a linear system to have infinitely many solutions, it must have at least one free variable and must be consistent in the first place.

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Answer:

p ∈ IR - {6}

Step-by-step explanation:

The set of all linear combination of two vectors ''u'' and ''v'' that belong to R2

is all R2 ⇔

[tex]u\neq 0_{R2}[/tex]      

[tex]v\neq 0_{R2}[/tex]

And also u and v must be linearly independent.

In order to achieve the final condition, we can make a matrix that belongs to [tex]R^{2x2}[/tex] using the vectors ''u'' and ''v'' to form its columns, and next calculate the determinant. Finally, we will need that this determinant must be different to zero.

Let's make the matrix :

[tex]A=\left[\begin{array}{cc}3&1&p&2\end{array}\right][/tex]

We used the first vector ''u'' as the first column of the matrix A

We used the  second vector ''v'' as the second column of the matrix A

The determinant of the matrix ''A'' is

[tex]Det(A)=6-p[/tex]

We need this determinant to be different to zero

[tex]6-p\neq 0[/tex]

[tex]p\neq 6[/tex]

The only restriction in order to the set of all linear combination of ''u'' and ''v'' to be R2 is that [tex]p\neq 6[/tex]

We can write : p ∈ IR - {6}

Notice that is [tex]p=6[/tex] ⇒

[tex]u=(3,6)[/tex]

[tex]v=(1,2)[/tex]

If we write [tex]3v=3(1,2)=(3,6)=u[/tex] , the vectors ''u'' and ''v'' wouldn't be linearly independent and therefore the set of all linear combination of ''u'' and ''b'' wouldn't be R2.

Answer:

95% of monsoon rainfall lies between 688 mm and 1016 mm.

Step-by-step explanation:

We are given the following information in the question:

Mean, μ = 852 mm

Standard Deviation, σ = 82 mm

We are given that the distribution of monsoon rainfall is a bell shaped distribution that is a normal distribution.

68 ‑ 95 ‑ 99.7 rule

We have to find the monsoon rains fall in the middle 95 % of all years.

By the rule 95% of data lies within two standard deviation of mean.Thus,

[tex]\mu + 2\sigma = 852 + 2(82) = 1016\\\mu - 2\sigma = 852 - 2(82) = 688[/tex]

Thus, 95% of monsoon rainfall lies between 688 mm and 1016 mm.

Using the Empirical Rule, we can determine that 95% of the time, the monsoon rains in India will fall between the amounts of 688 mm and 1016 mm. This is determined by subtracting and adding two standard deviations from the mean amount of rainfall.

The question pertains to the principle of the Normal Distribution curve in statistics, specifically using the 68 - 95 - 99.7 rule. Also known as the Empirical Rule, this principle suggests that for a Normal Distribution: approximately 68% of data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations. Given that the mean monsoon rainfall is 852 mm and standard deviation is 82 mm, we can calculate the range for the middle 95% of all years.

To find this range, we add and subtract two standard deviations from the mean. Therefore for two standard deviations (164 mm, since 82 mm x 2 = 164 mm), our range is: 852 mm - 164 mm = 688 mm and 852 mm + 164 mm = 1016 mm. Therefore, in 95% of all years, the monsoon rainfall in India falls between 688 mm and 1016 mm.

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Answer:

[tex] A = \frac{74}{3} -\frac{7}{2} +36 = \frac{127}{6} +36 = \frac{343}{6}[/tex]

Step-by-step explanation:

For this case we have these two functions:

[tex] x+y^2 = 12[/tex]   (1)

[tex] x+y=0[/tex]   (2)

And as we can see we have the figure attached.

For this case we select the x axis in order to calculate the area.

If we solve y from equation (1) and (2) we got:

[tex] y = \pm \sqrt{12-x}[/tex]

[tex] y = -x[/tex]

Now we can solve for the intersection points:

[tex] \sqrt{12-x} = -\sqrt{12-x}[/tex]

[tex] 12-x = -12+x[/tex]

[tex] 2x=24 , x=12[/tex]

[tex] \sqrt{12-x} =-x[/tex]

[tex] 12-x = x^2[/tex]

[tex] x^2 +x -12=0 [/tex]

[tex] (x+4)*(x-3) =0[/tex]

And the solutions are [tex] x =-4, x=3[/tex]

So then we have in total 3 intersection point [tex] x=12, x=-4, x=3[/tex]

And we can find the area between the two curves separating the total area like this:

[tex] \int_{-4}^3 |\sqrt{12-x} - (-x)| dx +\int_{3}^{12}|-\sqrt{12-x} -\sqrt{12-x}|dx[/tex]

[tex] \int_{-4}^3 |\sqrt{12-x} + x| dx +\int_{3}^{12}|-2\sqrt{12-x}|dx[/tex]

We can separate the integrals like this:

[tex] \int_{-4}^3 |\sqrt{12-x} dx +\int_{-4}^3 x +2\int_{3}^{12}\sqrt{12-x} dx[/tex]

For this integral [tex] \int_{-4}^3 |\sqrt{12-x} dx [/tex] we can use the u substitution with [tex]u = 12-x[/tex] and after apply and solve the integral we got:

[tex] \int_{-4}^3 |\sqrt{12-x} dx =\frac{74}{3}[/tex]

The other integral:

[tex] \int_{-4}^3 x dx = \frac{3^2 -(-4)^2}{2} =-\frac{7}{2}[/tex]

And for the other integral:

[tex]2\int_{3}^{12}\sqrt{12-x} dx[/tex]

We can use the same substitution [tex] u = 12-x[/tex] and after replace and solve the integral we got:

[tex]2\int_{3}^{12}\sqrt{12-x} dx =36[/tex]

So then the final area would be given adding the 3 results as following:

[tex] A = \frac{74}{3} -\frac{7}{2} +36 = \frac{127}{6} +36 = \frac{343}{6}[/tex]

The task is to sketch the region enclosed by the parabola [tex]x + y^2 = 12[/tex]and the line [tex]x + y = 0,[/tex] and then find the area of that region. The area can be found by integrating with respect to y, using vertical slices through the region and the points of intersection as limits.

The student has asked to sketch the region enclosed by two equations and then find the area of that region. It looks like there may be some confusion with the equations provided, as they seem to contain typos. Assuming the correct equations are [tex]x + y2 = 12[/tex] and [tex]x + y = 0[/tex], we can proceed by first graphing these two curves.

After sketching, you would see that the curve [tex]x + y2 = 12[/tex] forms a parabola that opens to the right, and the line [tex]x + y = 0[/tex]is a straight line that has a negative slope and passes through the origin. The region enclosed by these two would resemble a 'cap' shape, with the point of intersection providing the limits for integration.

To find the area of this region, you could choose to integrate with respect to y since the equations can be easily solved for x, and the structure of the parabola suggests that vertical slices would be easier to work with. The integration limits would be the y-values where the curves intersect. After setting up the integral, you would calculate the definite integral to find the area.

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Answer: 5&3/4ths

Step-by-step explanation:

[tex]5+\frac{5}{16} +\frac{7}{16} \\\\5+\frac{5+7}{16} \\\\\5\frac{12}{16}=5\frac{3}{4}[/tex]

Now you know the answer as well as the formula. Hope this helps, have a BLESSED AND WONDERFUL DAY!

- Cutiepatutie ☺❀❤

Answer: c. 8!

Step-by-step explanation:

We know , that if we line up n things , then the total number of ways to arrange n things in a line is given by :-

[tex]n![/tex] ( in words :- n factorial)

Therefore , the number of ways 8 cars can be lined up at a toll booth would be 8! .

Hence, the correct answer is c. 8! .

Alternatively , we also use multiplicative principle,

If we line up 8 cars , first we fix one car , then the number of choices for the next place will be 7 , after that we fix second car ,then the number of choices for the next place will be 6 , and so on..

So , the total number of ways to line up 8 cars = 8 x 7 x 6 x 5 x 4 x 3 x 2 x1 = 8!

Hence, the correct answer is c. 8! .

Answer:

The required equation would be [tex]\frac{dT}{dt}=k|M-T|[/tex]

Where, k is constant of proportionality.

Step-by-step explanation:

Differential equation : is an equation that contains one or more functions and their derivatives.

Where, derivatives shows the rate of change of a variable with respect to other variable.

Given,

Change in temperature T of coffee with respect to t ( time ) is proportional to difference between the fixed temperature M and temperature of coffee,

i.e. [tex]\frac{dT}{dt}\propto |M - T|[/tex]    ( if M > T, Change is M - T if M < T then change is T - M)

[tex]\frac{dT}{dt}=k|M-T|[/tex]

Where,

k = constant of proportionality.

Answer:

Check the boxplot below,plus the comments

Step-by-step explanation:

1) Completing the question:

Construct a boxplot for the following data and comment on the shape of the distribution representing the number of games pitched by major league baseball’s earned run average (ERA) leaders for the past few years.

30 34 29 30 34 29 31 33 34 27  30 27 34 32

2) Arranging the distribution orderly to find the 2nd Quartile (Median):

27  27  29  29  30  30  30  31  32  33  34  34  34  34

Since n=14, dividing the sum of the 14th and 15th element by two:

[tex]Md=Q_{2}=\frac{30+31}{2} =30.5[/tex]

3) Calculating the Quartiles, the Upper and the Lower one comes:

[tex]Q_{1}=29.25\\Q_{3}=33.75[/tex]

4) Boxplotting (Check it below)

5) Notice that since the values are very close, then the box is not that tall. The difference between the Interquartile Range is not so wide what makes it shorter. Check the values on the table below.

To construct a boxplot, organize the data, calculate the median, Q1 and Q3, and then draw the boxplot. The shape of the distribution can be assessed by observing the boxplot - if it's symmetric, positively skewed, or negatively skewed.

To construct a boxplot for a given set of data, follow these steps:

With respect to the shape of the distribution, by closely looking at the boxplot you can determine if the distribution is symmetric (box is centered about the median), positively skewed (box is shifted towards the lower end of the scale) or negatively skewed (box is shifted towards the upper end).

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Answer:

c. Asking people leaving a local election to take part in an exit poll

Step-by-step explanation:

Asking people leaving a local election to take part in an exit poll best represents the highest potential for nonresponse bias in a sampling strategy because of the importance of the local election compared to the exit polls.

It is worthy of note that nonresponse bias occurs when some respondents included in the sample do not respond to the survey. The major difference here is that the error comes from an absence of respondents not the collection of erroneous data. ...

Oftentimes, this form of bias is created by refusals to participate for one reason or another or the inability to reach some respondents.

Answer:

Please refer to the attachment below.

Step-by-step explanation:

Please refer to the attachment below for explanation.

Answer:

y=1/6 · ln |x|+c .

Step-by-step explanation:

From Exercise we have the differential equation

4xy dx= (4y6x²) dy.

We calculate the given differential equation, we get

4xy dx= (4y6x²) dy

xy dx=6yx² dy

6 dy=1/x dx

∫ 6 dy=∫ 1/x dx

6y=ln |x|+c

y=1/6 · ln |x|+c

Therefore, we get that the solution of the given differential equation is

y=1/6 · ln |x|+c .

The differential equation presented can be rewritten and solved using an integrating factor. The integrating factor in this case is 1, yielding the solution x^2 y = C.

To solve the given differential equation, we first need to rewrite it in a recognizable form namely, in the form Mdx + Ndy = 0. The given differential equation can be rewritten as 4xy dx + (4y - 6x2 ) dy = 0.

Now, we need to find an integrating factor which is e^∫(M_y - N_x) / N dx. In this case, M = 4xy, N = 4y - 6x2, M_y = 4x and N_x = -12x. Substituting these values into the equation ∫(M_y - N_x) / N dx, we find that the integrating factor is e^0=1.

With the integrating factor being 1, the given differential equation can be rewritten as d(x2y) = 0. Integrating both sides of this equation, we get x2y = C.

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Answer:

0.140625

Step-by-step explanation:

Given that multiple-choice questions each have four possible answers (a comma b comma c comma d )​, one of which is correct. Assume that you guess the answers to three such questions.

Each question is independent of the other with constant probability

p = Prob for correct guess = 1/4 = 0.25

q = prob for wrong guess = 1-p = 0.75

Hence

[tex]P(CWW)\\= P(C)*P(W)*P(W)[/tex], since each question is independent of the other

=[tex]0.25*0.75*0.75\\= 0.140625[/tex]

Answer:

Var(u​|X​=1) = 1

Var(Y|X​=1) = 1

​Var(u|X​) = 3

Var(Y|X​) = 9

Step-by-step explanation:

First, we need to identify the properties of the variance, so:

1. Var(a) = 0 , if a is constant

2. [tex]Var(aX) = a^{2} V(X)[/tex],  Where a is constant and X is a random variable

3. [tex]Var(aX+b)=a^{2} Var(X)+0[/tex], Where a and b are constants and X is a random variable.

4. [tex]Var(X + Y)=Var(X) + Var(Y)[/tex], Where X and Y are random variables and are independents.

Then, if ​Y=1​+X+u​, where X​, Y​, and ​u=v+X are random​ variables, v is independent of X​; ​E(v​)=0, ​Var(v​)=1​, ​E(X​)=1, and ​Var(X​)=2, the variance of the following cases are calculated as:

Var(u|X​=1) = Var( v + X | X = 1) = Var( v + 1 )    

                 = Var(v) + Var(1)                            

                  = 1 + 0 = 1

Var(Y|X​=1) = Var( 1 + X + u | X = 1 )

                 = Var (1 + X + v + X|X=1)

                 = Var(1 + 1 + v + 1)                      

                 = Var(3) + Var(v)                      

                 = 0 + 1 = 1

​Var(u|X​) = Var ( v + X | X)

             = Var ( v + X)

             = Var(v) + Var(X)

             =  1 + 2  = 3

Var(Y|X​) = Var ( 1 + X + u | X)

             = Var ( 1 + X + v + X | X)

             = Var ( 1 + 2X + v)

             = Var(1) + Var(2X) + Var(v)

              [tex]= Var(1)+2^{2}*Var(X)+Var(v)[/tex]

             = 0 + 4*2 + 1 = 8 + 1 = 9

Answer:

The required probability is [tex]\dfrac{5}{12}[/tex].

Step-by-step explanation:

If a fair dice is rolled then total outcomes are

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

We need to find the probability that the second die lands on a higher value than the first.

So, total favorable outcomes are

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

Formula for probability:

[tex]Probability=\dfrac{\text{Favorable outcomes}}{\text{Total outcomes}}[/tex]

[tex]Probability=\dfrac{15}{36}[/tex]

[tex]Probability=\dfrac{5}{12}[/tex]

Therefore, the required probability is [tex]\dfrac{5}{12}[/tex].

Answer:

C makes most sence

Step-by-step explanation:

Answer:

The correct answer is B=0.0262

Step-by-step explanation:

91÷3525 = 0.0262

The attached picture below gives a step by step explanation of how I arrived at my answer.

Please let's endeavor to always upload complete questions to avoid wrong answers.

Answer:

Cost function C(x) where x is the cubic feet of water used.

[tex]C(x) = 75 + 0.03x[/tex]

Step-by-step explanation:

We are given the following in the question:

The municipal water supply are billed a fixed amount monthly plus a charge for each cubic foot of water used.

Let x be the fixed amount and y be the cost for each cubic foot of water used.

A household using 1700 cubic feet was billed $126.

Thus, we can write:

[tex]x + 1700y = 126[/tex]

A household using 2500 cubic feet was billed $150.

[tex]x + 2500y = 150[/tex]

Solving the two equation by eliminating x by subtracting the equations, we get,

[tex]x+2500y-(x + 1700y) = 150-126\\800y = 24\\\\y =\dfrac{24}{800} = 0.03\\\\x+1700(0.03) = 126\\\Rightarrow x = 126-(1700)(0.03) = 75[/tex]

Thus, the fixed cost is $75 and the cost per cubic foot of water is $0.03

Cost function can be written as:

[tex]C(x) = 75 + 0.03x[/tex]

where x is the cubic feet of water used.

Final answer:

To derive the total cost equation for the water bill, a system of linear equations was solved to find the fixed charge and the variable rate per cubic foot. The cost function is C(x) = 75 + 0.03x, where C represents the total cost and x represents cubic feet of water used.

Explanation:

To find the equation that represents the total cost of a resident's water bill as a function of the cubic feet of water used, we first need to establish the fixed charge and the variable charge per cubic foot. We do this by setting up a system of linear equations based on the information provided about the two households.

Let F be the fixed monthly charge and V be the variable charge per cubic foot. We can set up two equations based on the given information:

Subtracting the first equation from the second gives us:

800V = 24

Dividing both sides by 800, we find:

V = 0.03

Now, we plug the value of V back into the first equation to find F:

F + 1700(0.03) = 126

F + 51 = 126

F = 126 - 51

F = 75

Therefore, the equation for the total cost C, in dollars, as a function of x, the cubic feet of water used, is:

C(x) = 75 + 0.03x

Answer: 1. Route ABC is a right triangle.

2. Route CDE ia not a right triangle.

3. Distance HJ= 23.32miles

4. Distance GE = 17

5. Missing length= 35

6. The sides 16, 60, 62 do NOT belong to a right triangle.

Step-by-step explanation:

Going by Pythagoras theorem, for triangle to be proven to be a right triangle, the condition below must be satisfied.

Hypotenuse² = Opposite² + Adjacent²

For question 1,

Hyp =13, opp = 5, Adj is 12

Going by Pythagoras rule.

Since 13²= 5² + 12²

Then triangle ABC is a right triangle.

For question 2,

Using the same Pythagoras theorem to prove,

In triangle CDE,

Hyp= 22, opp= 18, Adj = 14

Since 22² is not = 18² + 14²

then CDE is not a right triangle.

For question 3,

For triangle HIJ, since it is confirmed to be a right triangle, then we use the Pythagoras theorem to calculate the missing side.

Longest side if the triangle= IJ = hypotenuse = 25

HI = 9.

IJ² = HI² + HJ²

HJ²= IJ² - HI²

HJ² = 25² - 9²

HJ² = 625 - 81

HJ= √544

HJ = 23.32miles

For question 4,

FGE is also shown to be a right triangle and the missing side GE is the longest side which is also the hypotenuse.

FG= 8, FE =15

Using the Pythagoras theorem,

Hyp² = FG² + FE²

GE² = 8² + 15²

GE² = 289

GE = √289

GE = 17.

For question 5,

The hypotenuse is given as 37, one side is given as 12, let's call the missing side x

Going by Pythagoras theorem,

37² = 12²+ x²

x²= 37² - 12²

x²= 1225

x=√1225

x=35.

The missing side is 35inches.

For number 6,

The numbers given are 16, 60, 62

To know if three sides belong to a right angle, we simply put them to test using Pythagoras theorem.

It is worthy of note that the longest side is the hypotenuse.

This brings us to the equation to check below that since:

62² Is not = 60² + 16²

Then the side lengths 16, 60, 62 do not belong to a right angle.