A pile of 42 coins worth $4.90, consisting of nickels and quarters. How many nickels are there?

Answer: After [tex]3.5\ seconds[/tex]

Step-by-step explanation:

Knowing that after "t" seconds, its height "h" in feet is given by the function:

 [tex]h(t) = 112t -16t^2[/tex]

The maximum height is the y-coordinate of the vertex of the parabola. Then, we can use the following formula to find the corresponding value of "t" (which is the x-coordinate of the vertex):

[tex]t=\frac{-b}{2a}[/tex]

In this case:

[tex]a=-16\\b=112[/tex]

Substituting values, we get that the ball will reach the maximum height after:

[tex]t=\frac{-112}{2(-16)}\\\\t=3.5\ seconds[/tex]

A ball thrown vertically upwards in a parabolic path reaches its maximum height at the vertex of the parabolic path represented by the function of its height. The time it takes to reach this maximum height can be calculated with the formula -b/(2a), yielding a result of 3.5 seconds in this case.

The height h of a ball thrown vertically upward is given by the function h(t) = 112t -16t^2. The maximum height of the ball can be determined by finding the maximum point of the parabola represented by the equation. The maximum point occurs at the vertex of the parabola which is determined by the formula -b/(2a), where a and b are coefficients in the quadratic equation at^2 + bt + c.

In this case, a = -16 and b = 112. So, to find the time t when the ball will reach the maximum height, we substitute these into the formula to get t = -112/(2*(-16)) = 3.5 seconds. So the ball reaches its maximum height after 3.5 seconds.

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Answer:

  B.  AAS Congruence Theorem

Step-by-step explanation:

Previous steps showed congruence of a side of the designated triangle, and two angles that do not bracket that side. In short form, you have shown congruence of ...

  Angle - Angle - Side

so the AAS congruence theorem applies.

Step-by-step explanation:

Consider the provided information.

Jason received an invoice for $5,000, and the invoice was dated on August 26.

Part 1)

We need to find the bill Jason needs to pay by September 26.

n/45 means invoice amount should be paid within 45 days after the august 26.

From August 26 to September 26 Jason complete a month.

Thus, Jason just need to pay only $5,000. Which is the invoice amount.

Part 2)

The last day of discount can be calculated by 2/10.

Here, the 2/10 represents that discount of 2% within 10 days.

Now, calculate the 10 days form August 26.

The 10 days from August 26 is September 5.

Hence, September 5 is the last day that Jason can receive the cash discount.

Part 3)

Now, we need to calculate the last day that Jason has to pay for the bill.

n/45 means invoice amount should be paid within 45 days after the august 26.

Hence, the sum of the invoice should be paid within 45 days of August 26, whatever the date.

Answer: 0.9890

Step-by-step explanation:

Given : Mean : [tex]\mu=50[/tex]

Standard deviation : [tex]\sigma =7[/tex]

We assume the random variable X is normally distributed

The formula to calculate the z-score :-

[tex]z=\dfrac{x-\mu}{\sigma}[/tex]

For x=34.

[tex]z=\dfrac{34-50}{7}=-2.2857142\approx-2.29[/tex]

The p-value =[tex]P(z>-2.29)=1-P(z<-2.29)[/tex]

[tex]=1-0.0110107=0.9889893\approx0.9890[/tex]

Hence, [tex]P(X>34)=0.9890[/tex]

Answer:

The value of [tex]\frac{dx}{dt}[/tex] is [tex]\frac{160}{x^3}[/tex].

Step-by-step explanation:

The given equation is

[tex]x^4=8y^5-240[/tex]

We need to find the value of [tex]\frac{dx}{dt}[/tex].

Differentiate with respect to t.

[tex]4x^3\frac{dx}{dt}=8(5y^4)\frac{dy}{dt}-0[/tex]              [tex][\because \frac{d}{dx}x^n=nx^{n-1},\frac{d}{dx}C=0][/tex]

[tex]4x^3\frac{dx}{dt}=40y^4\frac{dy}{dt}[/tex]

It is given that y=2 and dy/dt=1, substitute these values in the above equation.

[tex]4x^3\frac{dx}{dt}=40(2)^4(1)[/tex]

[tex]4x^3\frac{dx}{dt}=40(16)(1)[/tex]

[tex]4x^3\frac{dx}{dt}=640[/tex]

Divide both sides by 4x³.

[tex]\frac{dx}{dt}=\frac{640}{4x^3}[/tex]

[tex]\frac{dx}{dt}=\frac{160}{x^3}[/tex]

Therefore the value of [tex]\frac{dx}{dt}[/tex] is [tex]\frac{160}{x^3}[/tex].

Answer:

3) 28.28.

Step-by-step explanation:

In order to find the answer we need to establish the EOQ equation which is:

[tex]EOQ=\sqrt{2*s*d/h}[/tex] where:

s=the cost of the setup

d=demand rate

h=cost of holding

Because demand is 200/month so d=200,

the ordering cost is $20/month  so s=20, and

the carrying cost in $10/month so h=10.

Using the equation we have:

[tex]EOQ=\sqrt{2*20*200/10}[/tex]

[tex]EOQ=\sqrt{800}[/tex]

[tex]EOQ=28.28[/tex]

So, answer to 'the order quantity for product' is 3) 28.28.

a. Expected value is defined by

[tex]E[X]=\displaystyle\sum_xx\,p(x)[/tex]

so we get

[tex]E[X]=1\cdot0.05+2\cdot0.10+4\cdot0.30+8\cdot0.45+16\cdot0.10[/tex]

[tex]\boxed{E[X]=6.65}[/tex]

b. Variance is defined by

[tex]V[X]=E[(X-E[X])^2][/tex]

so with the expectation found above, we have

[tex]V[X]=E[(X-6.65)^2][/tex]

[tex]V[X]=\displaystyle\sum_x(x-6.65)^2\,p(x)[/tex]

(by definition of expectation)

[tex]V[X]=(1-6.65)^2\cdot0.05+(2-6.65)^2\cdot0.10+(4-6.65)^2\cdot0.30+(8-6.65)^2\cdot0.45+(16-6.65)^2\cdot0.10[/tex]

[tex]\boxed{V[X]=15.4275}[/tex]

c. Standard deviation is the square root of variance:

[tex]\boxed{\sqrt{V[X]}\approx3.928}[/tex]

d. I assume "shortcut formula" refers to

[tex]V[X]=E[X^2]-E[X]^2[/tex]

which is easily derived from the definition of variance. We have (by def. of expectation)

[tex]E[X^2]=\displaystyle\sum_xx^2\,p(x)[/tex]

[tex]E[X^2]=1^2\cdot0.05+2^2\cdot0.10+4^2\cdot0.30+8^2\cdot0.45+16^2\cdot0.10[/tex]

[tex]E[X^2]=59.65[/tex]

and so the variance is again

[tex]V[X]=59.65-6.65^2[/tex]

[tex]\boxed{V[X]=15.4275}[/tex]

as expected.

To compute E(X), multiply each outcome by its probability and sum them up. Compute V(X) directly from the definition and also using the shortcut formula. Compute the standard deviation of X.

To compute E(X), we need to multiply each outcome x by its corresponding probability p(x) and sum them up. So, E(X) = 1(0.05) + 2(0.10) + 4(0.30) + 8(0.45) + 16(0.10) = 7.6 GB.

To compute V(X) directly from the definition, we need to first compute the squared deviations of each outcome from the expected value, which is 7.6 GB. Then, multiply each squared deviation by its corresponding probability, and sum them up. So, V(X) = (1 - 7.6)^2(0.05) + (2 - 7.6)^2(0.10) + (4 - 7.6)^2(0.30) + (8 - 7.6)^2(0.45) + (16 - 7.6)^2(0.10) ≈ 51.64 GB^2.

The standard deviation of X is the square root of the variance, which is SD(X) ≈ √(51.64) ≈ 7.19 GB.

To compute V(X) using the shortcut formula, we can use the formula: V(X) = E(X^2) - [E(X)]^2. First, we compute E(X^2) by multiplying each outcome squared by its corresponding probability and summing them up. Then, we subtract the square of E(X) to find V(X). This gives us V(X) = (1^2)(0.05) + (2^2)(0.10) + (4^2)(0.30) + (8^2)(0.45) + (16^2)(0.10) - [7.6]^2 ≈ 51.64 GB^2.

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If [tex]H_n[/tex] denotes the [tex]n[/tex]-th hexagonal number, then this is indeed false because [tex]H_2=6[/tex].

Answer:

Part (a) The value of Z is 0.10396. Part (b) The value of Z is 0.410008.

Step-by-step explanation:

Consider the provided information.

Part (a)

In order to find the number z such that the proportion of observations that are less than z in a standard Normal distribution is 0.5414, simply find 0.5414 in the table and search for the appropriate Z-value.

Now, observing the table it can be concluded that the value of Z is 0.10396.

Part (b)

Consider the number 65.91%

The above number can be written as 0.6591.

Now, find 0.6591 in the table and search for the appropriate Z-value.

By, observing the table it can be concluded that the value of Z is 0.410008.

Using a Z-table, we find that a z-score of approximately 0.1 will give us 0.5414 of observations less than z in a standard normal distribution. Similarly, for 65.91% of observations being greater than z, we subtract this from 1 and find z to be approximately 1.0.

To find a number, z, such that a certain proportion of observations are less than z in a standard normal distribution, we use a Z-table. In the case where observations less than z comprise 0.5414 of the total, we cross reference this probability in the Z-table to find that z is approximately 0.1.

Similarly, when we need to find the number z where 65.91% of all observations from a standard normal distribution are greater than z, we subtract this percentage from 1, as we are interested in the observations to the left of z. Doing this, we get 0.3409. Checking the Z-table shows that the z-score that corresponds with this area under the curve (or probability) is approximately 1.0.

Remember that a standard normal distribution is denoted Z ~ N(0, 1), meaning it has a mean of 0 and a standard deviation of 1. When calculating z-scores, this allows us to see how many standard deviations a certain point is from the mean (µ).

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Answer: [tex]H_0:\mu\geq431\\\\H_a:\mu<431[/tex]

Step-by-step explanation:

Given : A manufacturer of banana chips would like to know whether its bag filling machine works correctly at the 431 gram setting. t is believed that the machine is underfilling the bags.

Claim : [tex]\mu<431[/tex]

We know that the null hypothesis always have equals sign , then the required set of hypothesis for the situation :-

[tex]H_0:\mu\geq431\\\\H_a:\mu<431[/tex]

Answer:

even

Step-by-step explanation:

f(-x)=f(x) means f is even

f(-x)=-f(x) means f is odd

If you get neither f(x) or -f(x), you just say it is neither.

f(x)=x^4+7x^2-30

f(-x)=(-x)^4+7(-x)^2-30

f(-x)=x^4+7x^2-30

f(-x)=f(x)

so f is even.

Notes:

(-x)^even=x^even

(-x)^odd=-(x^odd)

Examples (-x)^88=x^88   and    (-x)^85=-(x^85)

Answer: even

Step-by-step explanation:

By definition a function is even if and only if it is fulfilled that:

[tex]f(-x) = f(x)[/tex]

By definition, a function is odd if and only if it is true that:

[tex]f (-x) = -f(x)[/tex]

Then we must prove the parity for the function: [tex]f(x) = x^4 + 7x^2 - 30[/tex]

[tex]f(-x) = (-x)^4 + 7(-x)^2 - 30[/tex]

[tex]f(-x) = x^4 + 7x^2 - 30=f(x)[/tex]

Note that for this case it is true that: [tex]f(-x) = f(x)[/tex]

Finally the function is even

Answer with explanation:

Equation of Plane having Direction cosines A, B and C passing through points, p, q and r is

⇒A (x-p)+B(y-q)+C(z-r)=0

The plane passes through the points A(-2, 3,-1), B(0, 5, 2) and C(-1, -2, 1).

→A(x+2)+B(y-3)+C(z+1)=0----------(1)

→A(0+2)+B(5-3)+C(2+1)=0

2 A +2 B+3 C=0

→A(-1+2)+B(-2-3)+C(1+1)=0

A -5 B+2 C=0

[tex]\Rightarrow \frac{A}{4+15}=\frac{B}{3-4}=\frac{C}{-10-2}\\\\\Rightarrow \frac{A}{19}=\frac{B}{-1}=\frac{C}{-12}=k\\\\A=19 K,B=-K, C=-12K[/tex]

Substituting the value of A , B and C in equation (1)

⇒19 K(x+2)-K(y-3)-12 K(z+1)=0

⇒19 x +38 -y +3-12 z-12=0

⇒19 x -y -12 z +29=0, is the required equation of Plane in Cartesian form.

⇒(19 i -j -12 k)(xi +y j+z k)+29=0 ,is the required  vector equation of the plane.

Answer:

Step-by-step explanation:

For this problem you have to use combinations. from 10 choices you are choosing 3. This means you are doing 10 choose 3. If you don't know what choose is I can explain.

Any number x choose y is the same as (x factorial)/(y factorial)(x-y factorial).

In this case that is 10 factorial/3 factorial times 7 factorial. ten factorial is the same as 10*9*8*7 factorial. So in the original equation you can factor away the seven factorials to get 10*9*8/3*2*1 factoring again you get 10*3*4  which is 120.

There are 120 different ways the chef can pick 3 brands of hot sauce to mix into a gumbo.

To find the number of ways the chef can pick 3 brands of hot sauce out of 10, we can use the combination formula:

[tex]nCr = n! / (r! * (n-r)!)[/tex]

where n is the total number of items (brands of hot sauce), and r is the number of items to be chosen (3 in this case).

In this problem, n = 10 and r = 3:

10C3 = 10! / (3! * (10-3)!)

Calculating the factorials:

10! = 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1 = 3,628,800

3! = 3 × 2 × 1 = 6

(10-3)! = 7! = 7 × 6 × 5 × 4 × 3 × 2 × 1 = 5,040

Now, substitute the values:

10C3 = 3,628,800 / (6 * 5,040)

10C3 = 3,628,800 / 30,240

10C3 = 120

So, there are 120 different ways the chef can pick 3 brands of hot sauce to mix into a gumbo.

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Answer:

Choice A.

Step-by-step explanation:

Let x = price of a helmet, y = price of a bat, z = price of a ball.

Company A can provide 9 helmets, 6 bats, and 12 balls for $525.

9x + 6y + 12z = 525

Company B can provide 10 helmets, 8 bats, and 10 balls for $600.

10x + 8y + 10z = 600

Company C can provide 8 helmets, 5 bats, and 15 balls for $500.

8x + 5y + 15z = 500

Answer: Choice A.

Answer:

The correct option is A.

Step-by-step explanation:

Let the price of a helmet is x, the price of a bat is y and the price of a ball is z.

It is given that Company A can provide 9 helmets, 6 bats, and 12 balls for $525. The equation for Company A is

[tex]9x+6y+12x=525[/tex]

It is given that Company B can provide 10 helmets, 8 bats, and 10 balls for $600. The equation for Company B is

[tex]10x+8y+10x=600[/tex]

It is given that Company C can provide 8 helmets, 5 bats, and 15 balls for $500. The equation for Company C is

[tex]8x+5y+15x=500[/tex]

The system of equations is

[tex]9x+6y+12x=525[/tex]

[tex]10x+8y+10x=600[/tex]

[tex]8x+5y+15x=500[/tex]

Therefore the correct option is A.

Answer:

A. Nonnegative

Step-by-step explanation:

By definition, the absolute value of any number must be positive (i.e non-negative).Hence A is the answer.

Answer:

The product of the y-coordinates of the solutions is equal to zero

Step-by-step explanation:

we have

[tex]x^{2}+y^{2}=9[/tex] -----> equation A

[tex]x+y=3[/tex] ------> equation B

Solve by graphing

Remember that the solutions of the system of equations are the intersection point both graphs

using a graphing tool

The solutions are the points (0,3) and (3,0)

see the attached figure

The y-coordinates of the solutions are 3 and 0

therefore

The product of the y-coordinates of the solutions is equal to

(3)(0)=0

The steps to solve the system of equations involve isolating x in one equation and substituting into the other. Solving yields two solutions for y, y = 0 and y = 3. Their product is 0.

The system of equations given are [tex]x^2+y^2=9[/tex] and x+y=3. From the second equation, we can isolate x as x = 3 - y and substitute into the first equation, yielding: [tex](3 - y)^2 + y^2 = 9[/tex]. This simplifies to [tex]2y^2 - 6y + 9 = 9,[/tex]and then to [tex]2y^2 - 6y = 0[/tex]. If we factor y from this equation, we get y(2y - 6) = 0, giving two possible solutions for y: y = 0, and y = 3. As asked, the product of these y-coordinates is 0 * 3 = 0.

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Answer:

Yes.

Step-by-step explanation:

If a graph G doesn't have a circuit, we must have that

[tex]|E(G)|=|V(G)|-1[/tex]

where [tex]|E(G)|[/tex] is the number of edges of the graph and [tex]|V(G)|[/tex] the number of vertices. However, in this case it holds that

[tex]|E(G)|=72>68=|V(G)|.[/tex]

Answer: There is a probability of no correct answers is 0.4096.

Step-by-step explanation:

Since we have given that

Number of trials = 4

Probability of success i.e. getting correct answer = 0.20

We need to find the probability of no correct answers.

We would use "Binomial distribution".

Let X be the number of correct answers.

So, it becomes,

[tex]P(X=0)=(1-0.20)^4=(0.80)^4=0.4096[/tex]

Hence, there is a probability of no correct answers is 0.4096.

The probability of guessing all answers incorrectly in a multiple-choice test with 4 questions, each with 5 options, is approximately 0.41 or 41% when answers are randomly guessed, according to the binomial probability distribution.

The question you're asking pertains to the concept of binomial probability, which is a type of probability that applies when there are exactly two mutually exclusive outcomes of a trial, often referred to as 'success' and 'failure'. In this case, a 'success' refers to correctly guessing an answer, which has a probability of p = 0.20. Conversely, a 'failure' refers to incorrectly guessing an answer, and this has a probability of q = 1 - p = 0.80.

To find the probability of no correct answers from 4 trials, we employ the formula for binomial probability: P(x) = (n C x)×(p×x)*(q×(n-x)). Here, 'n' represents the number of trials (4), 'x' represents the number of successes (0 for our case), and 'n C x' denotes the number of combinations of n items taken x at a time.

By plugging in the relevant values, the binomial probability distribution gives us P(0)= (4 C 0)×(0.20×0)×(0.80×4) = 1 × 1 × 0.4096 = 0.4096. So, the probability of guessing all answers incorrectly is approximately 0.41 or 41% when answers are randomly guessed.

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Answer:

The total cost to install the 1,200 ft of the pipe is $21000.

Step-by-step explanation:

Consider the provided information.

You buy 900 ft pipe with a cost of $15 per foot and the remaining 300 ft pipe with a cost of $25 per foot.

Therefore, the total cost is:

[tex]900 \times 15+300 \times 25[/tex]

[tex]13500+7500[/tex]

[tex]13500+7500[/tex]

[tex]21000[/tex]

Hence, the total cost to install the 1,200 ft of the pipe is $21000.

Answer:

Length = 20.49 yards and Width = 12.49 yards.

Step-by-step explanation:

The area of the rectangular playground is given by 256 yards square. It is also known that one of the sides of the playground is 8 yards longer than the other side. Therefore, let the smaller side by x yards. Then the longer side will be (x+8) yards. The area of the rectangle is given by:

Area of the rectangle = length * width.

256 = x*(x+8)

x^2 + 8x = 256. Applying the completing the square method gives:

(x)^2 + 2(x)(4) + (4)^2 = 256 + 16

(x+4)^2 = 272. Taking square root on both sides gives:

x+4 = 16.49 or x+4 = -16.49 (to the nearest 2 decimal places).

x = 12.49 or x = -20.49.

Since length cannot be negative, therefore x = 12.49 yards.

Since smaller side = x yards, thus smaller side = 12.49 yards.

Since larger side = (x+8) yards, thus larger side = 12.49+8 = 20.49 yards.

Thus, the length and the width to minimize the perimeter of fencing is 20.49 yards and 12.49 yards respectively!!!

Answer:

  (B) 200 km

Step-by-step explanation:

Let A represent the distance Anna goes before transferring fuel. Let C represent the distance to Connor's house. All distances are in km. Here, we will measure fuel quantity in terms of the distance it enables.

The total distance that can be driven by the two motorbikes is ...

  2A +2C = 600

Anna can transfer to Brian an amount of fuel that is 300-2A, since she needs to get back to the depot from the stopping point. When they stop, the amount of fuel in Brian's tank is 300-A. After that transfer, the most fuel Brian can have is a full tank (300). Then ...

  (300 -A) +(300 -2A) = 300 . . . . fuel in Brian's tank after the transfer

This second equation simplifies to ...

  600 -3A = 300

  300 = 3A . . . . . . add 3A-300

  100 = A . . . . . . . . divide by 3

Using this in the first equation, we get ...

  2·100 +2C = 600

  2C = 400 . . . . . . . . subtract 200

  C = 200 . . . . . . . . . .divide by 2

The distance from the depot to Connor's house can be at most 200 km.

Answer:

78

Step-by-step explanation:

50 questions worth 2 a piece and you miss 11 questions so we are going to take 2(11) off of 50.

We are doing 100-2(11).

100-22=78.

Or if you miss 11 questions you get 39 right. So 39(2)=78.

ANSWER :

There are 50 questions, total marks are 100.

EXPLANATION:

Since each question is 2 marks, missing 11 questions is equal to missing 22 marks.

therefore, your marks will be 100-22=78 marks.

Answer:

Step-by-step explanation:

Angles are measured from the direction of motion, so the "force made good" is the force in the rope multiplied by the cosine of the angle. If the forces in the ropes (in Newtons) are represented by x and y, then we have ...

  x·cos(40°) +y·cos(55°) = 700

In order for the resultant to be in the direction of motion, the forces perpendicular to the direction of motion must cancel.

  x·sin(40°) - y·sin(55°) = 0

Here, we have assumed that the positive direction for measuring 40° is the negative direction for measuring 55°. That is, the angles are measured in opposite directions from the direction of motion.

Any of the usual methods for solving systems of linear equations can be used to solve this set. My preference is to use a graphing calculator. It gives the answers summarized above.

I'm guessing you intended to say [tex]X[/tex] has mean [tex]\mu_X=E[X]=25[/tex] and standard deviation [tex]\sigma_x=\sqrt{\mathrm{Var}[X]}=6[/tex], and [tex]Y[/tex] has means [tex]\mu_Y=E[Y]=30[/tex] and standard deviation [tex]\sigma_Y=\sqrt{\mathrm{Var}[Y]}=4[/tex].

If [tex]W=X+Y[/tex], then [tex]W[/tex] has mean

[tex]E[W]=E[X+Y]=E[X]+E[Y]=55[/tex]

and variance

[tex]\mathrm{Var}[W]=E[(W-E[W])^2]=E[W^2]-E[W]^2[/tex]

Given that [tex]\mathrm{Var}[X]=36[/tex] and [tex]\mathrm{Var}[Y]=16[/tex], we have

[tex]\mathrm{Var}[X]=E[X^2]-E[X]^2\implies E[X^2]=36+25^2=661[/tex]

[tex]\mathrm{Var}[Y]=E[Y^2]-E[Y]^2\implies E[Y^2]=16+30^2=916[/tex]

Then

[tex]E[W^2]=E[(X+Y)^2]=E[X^2]+2E[XY]+E[Y^2][/tex]

[tex]X[/tex] and [tex]Y[/tex] are independent, so [tex]E[XY]=E[X]E[Y][/tex], and

[tex]E[W^2]=E[X^2]+2E[X]E[Y]+E[Y^2]=661+2\cdot25\cdot30+916=3077[/tex]

so that the variance, and hence standard deviation, are

[tex]\mathrm{Var}[W]=3077-55^2=52[/tex]

[tex]\implies\sqrt{\mathrm{Var}[W]}=\sqrt{52}=\boxed{2\sqrt{13}}[/tex]

# # #

Alternatively, if you've already learned about the variance of linear combinations of random variables, that is

[tex]\mathrm{Var}[aX+bY]=a^2\mathrm{Var}[X]+b^2\mathrm{Var}[Y][/tex]

then the variance of [tex]W[/tex] is simply the sum of the variances of [tex]X[/tex] and [tex]Y[/tex], [tex]\mathrm{Var}[W]=36+16=52[/tex], and so the standard deviation is again [tex]\sqrt{52}[/tex].

The value is W = sqrt(52) = 7.211.

To find the standard deviation of the sum of two independent random variables X and Y, use the formula W = sqrt(X² + Y²), plugging in the given values to calculate the standard deviation of W as 7.211.

The standard deviation of the sum of two independent random variables X and Y is the square root of the sum of their variances:

W = sqrt(X² + Y²)

Substitute the given values to find the standard deviation of W:

Answer:

[tex]y=c_1x+c_2x^5[/tex]

Step-by-step explanation:

The given second order homogeneous Cauchy-Euler ordinary differential equation is

[tex]x^2y''-5xy'+5y=0[/tex]

The corresponding auxiliary equation is given by:

[tex]a {m}^{2} + (b - a)m + c = 0[/tex]

where a=1, b=-5, c=5

We substitute the coefficients into the auxiliary equation to obtain:

[tex] {m}^{2} + ( - 5 - 1)m + 5= 0[/tex]

[tex] {m}^{2} - 6m + 5= 0[/tex]

[tex](m - 1)(m - 5) = 0[/tex]

[tex] \implies \: m = 1 \: or \: m = 5[/tex]

The auxiliary equation has two distinct real roots. The general solution to the corresponding differential equation is of the form:

[tex]y=c_1x ^{m1} +c_2 {x}^{m2} [/tex]

We substitute the values to get:

[tex]y=c_1x+c_2x^5[/tex]

The provided differential equation [tex]x^2y'' - 5xy' + 5y = 0[/tex] can be solved using the method for Cauchy-Euler equations, resulting in the general solution [tex]y = c_1 x + c_2 x^5.[/tex] Therefore, the correct answer is (d) [tex]y = c_1 x^{5 + \frac{\sqrt{5}}{2}} + c_2 x^{5 - \frac{\sqrt{5}}{2}}[/tex]

To solve the given differential equation [tex]x^2y'' - 5xy' + 5y = 0[/tex], we use the method for Cauchy-Euler equations:

1. Assume a solution of the form[tex]y = x^m.[/tex]

2. Substitute [tex]y = x^m[/tex] into the differential equation:

3. Plug these into the equation:
[tex]x^2 [m(m-1)x^(m-2)] - 5x[mx^(m-1)] + 5[x^m] = 0[/tex]

4. Simplify to get the characteristic equation:
[tex]m(m-1) - 5m + 5 = 0[/tex]
[tex]m^2 - 6m + 5 = 0[/tex]

5. Solve the quadratic equation:
[tex]m = \frac{{6 \pm \sqrt{6^2 - 4 \times 5}}}{{2}}[/tex]
[tex]m = \frac{{6 \pm \sqrt{16}}}{{2}}[/tex]
[tex]m = \frac{{6 \pm 4}}{2}[/tex]
[tex]m = 5 \quad \text{or} \quad m = 1[/tex]

6. The general solution is then a linear combination of these solutions:
[tex]y = c_1 x^5 + c_2 x^1[/tex]
[tex]y = c_1 x + c_2 x^5[/tex]

[tex]y = c_1 x^{5 + \frac{\sqrt{5}}{2}} + c_2 x^{5 - \frac{\sqrt{5}}{2}}[/tex]

Answer:

[tex]y^2=2\ln (1+t^2)+4[/tex]

Step-by-step explanation:

Given that

[tex]\dfrac{dy}{dt}=\dfrac{2t}{y+yt^2}[/tex]

This is a differential equation.

Now by separating variables

[tex]y dy= \dfrac{2t}{1+t^2}dt[/tex]

Now by integrating both side

[tex]\int y dy=\int \frac{2t}{1+t^2}dt[/tex]

Now by soling above integration

We know that  integration of dx/x is lnx.

[tex]\dfrac{y^2}{2}=\ln (1+t^2)+C[/tex]

Where C is the constant.

[tex]y^2=2\ln (1+t^2)+C[/tex]

Given that when t=0 then y= -2

So by putting the above values of t and y we will find C

4=2 ln(1)+C     (we know that ln(1)=0)

So C=4

⇒[tex]y^2=2\ln (1+t^2)+4[/tex]

So solution of above equation is  [tex]y^2=2\ln (1+t^2)+4[/tex]

Answer:  There is probability that he will be asked a question in class during Week 7 is 64%.

Step-by-step explanation:

Since we have given that  

Probability that question in class being asked during Week 1 = 1%  

Probability that question in class being asked during Week 2 = 2%  

Probability that question in class being asked during Week 3 = 4%

and so on.

So, we need to find the probability that question being asked in class during week 7.

Since it forms geometric series:

1%, 2%, 4%, ........

So, we need to find the 7 th term:  

[tex]a_n=ar^{n-1}\\\\a_7=ar^{7-1}\\\\a_7=1(2)^6\\\\a_7=64\%[/tex]

Hence, there is probability that he will be asked a question in class during Week 7 is 64%.

The generating function for this sequence is

[tex]f(x)=4+4x+4x^2+4x^3+x^4+x^6+x^8+\cdots[/tex]

assuming the sequence itself is {4, 4, 4, 4, 1, 0, 1, 0, ...} and the 1-0 pattern repeats forever (as opposes to, say four 4s appearing after every four 1-0 pairs). We can make this simpler by "displacing" the odd-degree terms and considering instead the generating function,

[tex]f(x)=3+4x+3x^2+4x^3+\underbrace{(1+x^2+x^4+x^6+x^8+\cdots)}_{g(x)}[/tex]

where the coefficients of [tex]g(x)[/tex] follow a much more obvious pattern of alternating 1s and 0s. Let

[tex]g(x)=\displaystyle\sum_{n=0}^\infty a_nx^n[/tex]

where [tex]a_n[/tex] is recursively given by

[tex]\begin{cases}a_0=1\\a_1=0\\a_{n+2}=a_n&\text{for }n\ge0\end{cases}[/tex]

and explicitly by

[tex]a_n=\dfrac{1+(-1)^n}2[/tex]

so that

[tex]g(x)=\displaystyle\sum_{n=0}^\infty\frac{1+(-1)^n}2x^n[/tex]

and so

[tex]\boxed{f(x)=3+4x+3x^2+4x^3+\displaystyle\sum_{n=0}^\infty\frac{1+(-1)^n}2x^n}[/tex]

Final answer:

The generating function for the sequence is found by splitting it into two parts and expressing each as a series. The constant part can be expressed as a finite series, while the alternating sequence is a geometric series that can be simplified. Their sum yields the generating function.

Explanation:

The student has asked for a compact form for the generating function of the sequence 4, 4, 4, 4, 1, 0, 1, 0, 1, 0, 1, 0, ... .

To find the generating function for the given sequence, we can split it into two parts: The constant part (4, 4, 4, 4) and the alternating sequence (1, 0, 1, 0, ...).

The constant part can be represented as:
4 + 4x + 4x2 + 4x3 = 4(1 + x + x2 + x3)

The alternating sequence can be represented as a geometric series:
1 - x2 + x4 - x6 + ... = 1 / (1+x2)

The generating function G(x) would then be the sum of these two parts, simplifying by multiplication of the series and a fraction:

G(x) = 4(1 + x + x2 + x3) + x4 / (1 + x2)

[tex]\bf ~~~~~~~~~~~~\textit{distance between 2 points} \\\\ (\stackrel{x_1}{1}~,~\stackrel{y_1}{4})\qquad (\stackrel{x_2}{5}~,~\stackrel{y_2}{1})\qquad \qquad d = \sqrt{( x_2- x_1)^2 + ( y_2- y_1)^2} \\\\\\ d=\sqrt{(5-1)^2+(1-4)^2}\implies d=\sqrt{4^2+(-3)^2} \\\\\\ d=\sqrt{16+9}\implies d=\sqrt{25}\implies d=5[/tex]

Answer: second option.

Step-by-step explanation:

You need to use the formula for calculate the distance between two points. This is:

[tex]d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]

Given the point (1, 4) and the point (5, 1), you can say that:

[tex]x_2=5\\x_1=1\\y_2=1\\y_1=4[/tex]

Now you must substitute these values into the formula.

The distance between the points (1, 4) and (5, 1) is the following:

[tex]d=\sqrt{(5-1)^2+(1-4)^2}\\\\d=5[/tex]

This matches with the second option.