Solve the inhomogeneius linear ode by undetermined coefficients
Y"+4y=3sin2x

The probability that exactly two telephones will end up being replaced under warranty is found to be 0.147.

The binomial distribution is based on the binomial theorem which can be written as (a + b)ⁿ = ⁿC₀aⁿb⁰ + ⁿC₁aⁿ⁻¹b¹ + ⁿC₂aⁿ⁻²b²+ ....+ ⁿCₙa₀bⁿ.

In order to use in the probability, there should be events independent of each other and the sum of there probabilities is 1.

Suppose the total number of telephones be x.

Then, the telephones submitted under warranty is 20% × x = 0.2x.

The number of telephones to be repaired is 60% × 0.2x = 0.12x

And, those to be replaced  are given as 40% × 0.2x = 0.08x.

Now, the probability of exactly two units being replaced out of 10 is given by binomial distribution as,

P(Exactly two units being replaced) = ¹⁰C₂(0.08)²(0.92)⁸

⇒ P(Exactly two units being replaced) = 0.147

Hence, the probability for the given case is obtained as 0.147.

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To solve this problem, we can use the concept of conditional probability. Given that 20% of telephones are submitted for service while under warranty and of these, 60% can be repaired and 40% must be replaced with new units, we can calculate the probability that exactly two telephones will be replaced under warranty.

To solve this problem, we can use the concept of conditional probability. Let's break down the information given:

Now, let's calculate the probability that exactly two telephones will be replaced:

Therefore, the probability that exactly two telephones will end up being replaced under warranty is approximately 0.0572, or 5.72%.

Answer:

Domain: amount of fuel in the airplane's tank (in gallons)

The set of all real numbers from 0 to 200

Range: weight of airplane (In  pounds)

The set of all real numbers from 3000 to 4400

Step-by-step explanation:

We have the following function

[tex]W=7F+3000[/tex]

Where W represents the weight of the plane in pounds and F represents the amount of fuel in gallons.

The domain of a function is the set of values ​​"F" that can be entered in a function W(F) to obtain an output value of W.

In this case the range of the function W(F) is the whole set of values [tex]W_1, W_2, W_3, ..., W_n[/tex] that are obtained for [tex]F_1, F_2, F_3, ..., F_n[/tex]

Note that, in this case, equation W(F) is used to obtain the weight of the airplane from the amount of fuel F.

Then the domain of the function is the amount of fuel in the airplane tank (in gallons). Since the tank can only hold up to 200 gallons, and there are no negative volume units, then the domain is all real numbers between 0 and 200.

The range of the function is the weight of the plane (in pounds). Note that the minimum weight of the airplane with 0 gallons of fuel is 3000 pounds and the maximum weight with the full tank is 4400 pounds.

Then the range is all real numbers between 3000 and 4400

Answer: 0.04395

Explanation:

Given: 10 true-false questions.

So, we will have 50% chances (probability = 0.5) of being correct.

Prob( Exactly 80% score) = Prob (exactly 8 answers correct)

As we observe, if X= number of correct answers, then X~ Binomial (n=10, p=0.5)

So, Prob( Exactly 80% score) = Prob (exactly 8 answers correct)

=[tex]\binom{10}{8}\times(1/2)^{8}}\times(1/2)^{2}[/tex]

= 0.0439453125

= 0.04395

Answer:

The probability that mike will guess​ correctly is 0.0027397 or [tex]\frac{1}{365}[/tex].

Step-by-step explanation:

Consider the provided information.

The number of days in a year is 365 (Ignore leap​ years).

[tex]Probability = \frac{favorable\ outcomes}{possible\ outcomes}[/tex]

Here, favorable outcomes is 1 and total number of outcomes are 365.

Substitute these value in above formula.

[tex]Probability = \frac{1}{365}[/tex]

[tex]Probability = 0.0027397[/tex]

Thus, the probability that mike will guess​ correctly is 0.0027397 or [tex]\frac{1}{365}[/tex].

Probability of an event represents the chances of occurrence of that event.

The probability that Mike will guess Kelly's birth date correctly (ignoring leap years) is [tex]\dfrac{1}{365} \approx 0.00027[/tex]

Suppose that there are finite elementary events in the sample space of the considered experiment, and all are equally likely.

Then, suppose we want to find the probability of an event E.

Then, its probability is given as

[tex]P(E) = \dfrac{\text{Number of favorable cases}}{\text{Number of total cases}}[/tex]

Where favorable cases are those elementary events who belong to E, and total cases are the size of the sample space.

Since we know that in a year (non leap year), there are 365 days, and Kelly's birthday can be on any one of those date, so the total number of days to chose from is 365 and since birthday of Kelly is going to be on single day of whole year, so favorable case is only single.

Thus,

if E = Selecting Kelly's birth date correctly,

Then

[tex]P(E) = \dfrac{1}{365} \approx 0.00027[/tex]

Thus,

The probability that Mike will guess Kelly's birth date correctly (ignoring leap years) is [tex]\dfrac{1}{365} \approx 0.00027[/tex]

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Answer:

The inverse of h(x) is [tex]\frac{5x-6}{2}[/tex]

Step-by-step explanation:

* Lets explain how to make the inverse of a function

- To find the inverse of a function we switch x and y and then solve

  for new y

- You can make it with these steps

# write g(x) = y

# switch x and y

# solve for y

# write y as [tex]g^{-1}(x)[/tex]

* Lets solve the problem

∵ [tex]h(x)=\frac{2x+6}{5}[/tex]

# Step 1

∴ [tex]y=\frac{2x+6}{5}[/tex]

# Step 2

∴ [tex]x=\frac{2y+6}{5}[/tex]

# Step 3

∵ [tex]x=\frac{2y+6}{5}[/tex]

- Multiply each side by 5

∴ 5x = 2y + 6

- Subtract 6 from both sides

∴ 5x - 6 = 2y

- Divide both sides by 2

∴ [tex]y=\frac{5x-6}{2}[/tex]

# Step 4

∴ [tex]h^{-1}(x)=\frac{5x-6}{2}[/tex]

Answer:

g(-4) = -4

g(-2) = 1

g(1) = -4

Step-by-step explanation:

The value of the given function is -4 for all values of x other than -2 and 1 if x=-2

So,

For x=-4 the value of function will be -4.

g(-4) = -4

For x=-2

The value of function is -2.

g(-2) = 1

And for x=1, the value will be -4.

g(1) = -4 ..

Answer:

f(9) =5

Step-by-step explanation:

We have an input x and an output f(x)

The input in the last row is 9 and the output is 5

The input x=9 and the output f(9) =5

f(9) =5

The correct option is (a) because [tex]f(9)=5[/tex].

Important information:

The function notation of a point [tex](x,f(x))[/tex] or [tex](x,y)[/tex] is [tex]f(x)=y[/tex].

Using the definition of function notation. The function notation of given ordered pairs [tex](2,6),(7,3)[/tex] and [tex](9,5)[/tex] are:

[tex]f(2)=6[/tex]

[tex]f(7)=3[/tex]

[tex]f(9)=5[/tex]

Thus, option (a) is correct and the other options are incorrect.

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Answer with explanation:

Suppose, A={(a,b),(b,a), (c,d),(d,c),(p,q),(q,p),(a,a),(b,b)}

A Relation M is Symmetric , if (p,q)∈M , then (q,p)∈M.

⇒It is given that, R and S are symmetric Relation  on a Set A.

⇒If R is symmetric, then if (a,b)∈R, means,(b,a)∈R.So, R={(a,b),(b,a)}.

⇒If S is Symmetric, then if (c,d)∈S, means,(d,c)∈S.So, S={(c,d),(d,c)}.

⇒R ∩ S ={(a,b),(b,a),(c,d),(d,a)}

⇒If you will look at the elements of Set , R∩S, there is (a,b)∈ R∩S,so as (b,a)∈ R∩S.Also, (c,d)∈ R∩S,so as (d,a)∈ R∩S.

Which shows Relation in the set ,  R∩S is symmetric.

Answer:

To answer you question, we need the confidence internals formula, μ±Zc*(σ/[tex]\sqrt{N}[/tex]); N is the 20 psychology students, 0.5 the desviation standar and 4.3 is the medium based on national data, every fact in years.

Step-by-step explanation:

You need to consult the Zc value, you can find this table attached in this question; for 95%, we have a Zc value of 1.96.

Next step replace values: 4.3±1.96*(0.5/[tex]\sqrt{20}[/tex]) = 4.3±0.22 (rounding the result)

So, the confidence interval are:

4.08cm≤X≤4.52cm

Answer:

0.7564

Step-by-step explanation:

Let X be the event of watching a rented video at least once during a week,

Given,

The probability of watching a rented video at least once during a week was, p = 0.75,

So, the probability of not watching a rented video at least once during a week was, q = 1 - p = 0.25,

Binomial distributive formula,

[tex]P(x)=^nC_x p^x q^{n-x}[/tex]

Where,

[tex]^nC_x=\frac{n!}{x!(n-x)!}[/tex]

Hence, the probability that at least 5 teenagers in a group of 7 watched a rented movie at least once last week,

P(X ≥ 5) = P(X=5) + P(X=6 )+ P(X=7)

[tex]=^7C_5 0.75^5 0.25^{7-5}+^7C_6 0.75^6 0.25^{7-6}+^7C_7 0.75^7 0.25^{7-7}[/tex]

[tex]=21 (0.75)^5 (0.25)^2 + 7 (0.75)^6 0.25 + 0.75^7[/tex]

[tex]=0.756408691406[/tex]

[tex]\approx 0.7564[/tex]

The probability that at least 5 teenagers in a group of 7 watched a rented movie at least once last week is 0.3015.

The probability that at least 5 teenagers in a group of 7 watched a rented movie at least once last week can be calculated using the binomial probability distribution formula:

P(X ≥ k) = 1 - P(X < k)

where X is the number of teenagers who watched a rented movie at least once, k is the minimum number of teenagers (5 in this case), and P(X < k) is the probability that less than k teenagers watched a rented movie at least once.

In this case, the probability that a randomly selected teenager watched a rented video at least once during a week is 0.75. Therefore, the probability that a randomly selected teenager did not watch a rented video at least once is 1 - 0.75 = 0.25.

Using the binomial probability distribution formula, we can calculate the probability that less than 5 teenagers watched a rented movie at least once:

P(X < 5) = C(7, 0) * (0.25)^0 * (0.75)^7 + C(7, 1) * (0.25)^1 * (0.75)^6 + C(7, 2) * (0.25)^2 * (0.75)^5 + C(7, 3) * (0.25)^3 * (0.75)^4 + C(7, 4) * (0.25)^4 * (0.75)^3

where C(n, r) is the number of combinations of n items taken r at a time:

C(n, r) = n! / (r! * (n-r)!)

Substituting the values and evaluating the expression, we get:

P(X < 5) = 0.698486328125

Therefore, the probability that at least 5 teenagers in a group of 7 watched a rented movie at least once last week is:

P(X ≥ 5) = 1 - P(X < 5) = 1 - 0.698486328125 = 0.301513671875

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Answer:

2^64. I know 2^20 is 1048576. Cube that and multiply by 16 or grab a calculator. I'm too lazy to solve this.

The time taken to count all the grain due to him is [tex]2^{64}-1[/tex] or 18,446,744,073,709,551,615 sec .

Geometric Progression (GP) is a type of sequence where each succeeding term is produced by multiplying each preceding term by a fixed number, which is called a common ratio. This progression is also known as a geometric sequence of numbers that follow a pattern.

. The sum of infinite, i.e. the sum of a GP with infinite terms is [tex]S_{∞} = \frac{a}{(1 - r) }[/tex]such that 0 < r < 1.

The formula used for calculating the sum of a geometric series with n terms is Sn = [tex]\frac{a( r^{n} -1 )}{(r - 1)} ,[/tex] where r ≠ 1.  

According to the question

A chess board has 64 squares and all had been filled with grain

1st squares of chess board has grain = 1 = [tex]2^{0}[/tex]

2nd squares of chess board has grain = 2 = [tex]2^{1}[/tex]

3nd squares of chess board has grain = 4 = [tex]2^{2}[/tex]

4nd squares of chess board has grain  = 8 = [tex]2^{3}[/tex]

so on ..

As this is an Geometric Progression

Where

First term (a) = 1

common ratio (r) = 2

Number of terms = n = 64

Now,  

it took just 1 second to count each grain ,

Time taken to count all  the grains  

By using formula of sum of Geometric Progression  

Sn = [tex]\frac{a( r^{n} -1 )}{(r - 1)} ,[/tex] where r ≠ 1.  

substituting the values in formula

S₆₄ = [tex]\frac{1( 2^{64}-1 )}{(2-1)} ,[/tex]

S₆₄ = [tex]2^{64}-1[/tex]

S₆₄ = 18,446,744,073,709,551,615

Hence, The time taken to count all the grain due to him is [tex]2^{64}-1[/tex] or 18,446,744,073,709,551,615 sec .

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Answer:

[tex]\frac{3}{28}[/tex]

Step-by-step explanation:

Te meaning of without replacement is once the ball is picked from the stock it cannot be put back

Hre there are total 8 balls and 3 white balls

probability of picking 1st white is [tex]\frac{3}{8}[/tex]

now only 2 white and total is 7( one ball has already been picked)

therefore probability of picking 2nd white ball is [tex]\frac{2}{7}[/tex]

both the action are independent events

therefore, probability of picking 2 white balls is  [tex]\frac{3}{8}[/tex] × [tex]\frac{2}{7}[/tex]

= [tex]\frac{3}{28}[/tex]

Answer:[tex]\frac{3}{28}[/tex]

Step-by-step explanation:

Given a box contains 3 White ,2 green,2 red & 1 Blue marble

We have to draw two marbles without replacement

therefore for first draw we have 3 white marble to choose among 8 marbles

i.e.

[tex]_{1}^{3}\textrm{C}[/tex]  choices among the total of [tex]_{1}^{8}\textrm{C}[/tex] options

For second draw we 2 white marbles left therefore no of ways in which a white marble can be choosen is

[tex]_{1}^{2}\textrm{C}[/tex]

Therefore required probability is =[tex]\frac{favourable\ outcome}{Total\ outcome}[/tex]

P[tex]\left ( required\right )[/tex]=[tex]\frac{3\times2}{8\times7}[/tex]

P[tex]\left ( required\right )[/tex]=[tex]\frac{3}{28}[/tex]

Answer:

The probability is 0.66.

Step-by-step explanation:

Given,

The chance of encountering a car crash on the road is stated as 66​%,

That is, out of 100% cases the percentage of the number of car crash cases is 66%,

⇒ Total outcomes = 100%, favourable outcome = 66 %

So, the probability of occurrence a car crash = [tex]\frac{66\%}{100\%}[/tex]

[tex]=\frac{66/100}{100/100}[/tex]

[tex]=\frac{66}{100}[/tex]

[tex]=0.66[/tex]

Where, 0 < 0.66 < 1.

Hence, the indicated degree of likelihood as a probability value between 0 and 1 inclusive is 0.66.

Answer: a) 0.0088

b) 0.2997

c)  5

Step-by-step explanation:

Given : Mean : [tex]\mu = 79[/tex] minutes

Standard deviation : [tex]\sigma = 8[/tex] minutes

The formula for z-score :

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

a) For x = 60 minutes

[tex]z=\dfrac{60-79}{8}=-2.375[/tex]

The p-value =[tex]P(z\leq-2.375)=0.0087745\approx0.0088[/tex]

b) For x = 75 minutes

[tex]z=\dfrac{75-79}{8}=-0.5[/tex]

The p-value =[tex]P(60<x<75)=P(-2.375<z<-0.5)[/tex]

[tex]=P(-0.5)-P(-2.375)=0.3085-0.0088=0.2997[/tex]

c) For x = 90 minutes

[tex]z=\dfrac{90-79}{8}=1.375[/tex]

The p-value =[tex]P(z>1.375)=1-P(z<1.375)[/tex]

[tex]=1-0.9154342=0.0845658[/tex]

If the number of students in the class = 60 .

Then , the number of students will be unable to complete the exam in the allotted time =[tex]0.0845658\times60=5.073948\approx5[/tex]

The probability of completing the exam in one hour or less is 0.0087. The probability that the exam is completed in more than 60 minutes but less than 75 minutes is 0.2998. We expect about 5 students to not finish the exam in the given 90 minutes.

In statistics, when a data set is normally distributed, we use a z-score to describe the position of a raw score in terms of its distance from the mean, when measured in standard deviation units. The formula to calculate a z-score is Z = (X - μ) / σ, where X is the raw score, μ is the mean, and σ is the standard deviation.

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Answer:

2 h 16 min 23 sec

Step-by-step explanation:

Hello

the time is expressed in the sexadecimal system, which uses the number 60 as an arithmetic base,hence

1 min=60 sec

1 hora =60  min

Now, we have

[tex](8 min + 31 sec)*17=136 min +527 sec\\\\\\we\ need\ to\ convert\ this\ in\ our\ base\ 60\, using\ a\ rule\ o\ three\\\\ 60 min=1\ hour\\136 min=x ?\\\\x=\frac{ 136 h}{60}\\ x=2.26 hours\\\\[/tex]

we take the whole part as an hour, and the decimal part is multiplied by 60 to get minutes

Step 1

[tex]8min*17=136 min =2.26 h\\\\2.26h = 2\ h + 0.26h(\frac{60 min}{1 h}) \\2.26h =2h+15.6 min\\\\[/tex]

we repeat the procedure to leave the minutes as a whole part

[tex]2.26\ h =2\ h+15\ min + 0.6\ min*(\frac{60 \sec}{1 m} )\\2.26\ h =2\ h\ 15\ min\ 36\ sec[/tex]

Step 2

[tex]\\527\s*(\frac{1 min}{ 60\ sec})=8.78\ min\\ \\8.78\ min= 8\min\0.78\ min\\8.78\ min=8\ min\ 0.78\min(\frac{60\ seg}{min})\\8.78\min=8\ min \ 47\ sec\\\\now, add\\\\8 min *17 =2\ h\ 15\ min\ 36\ sec\\31 sec *17 =8\ min \ 47\ sec\\(8\ min\ 31\ sec)*17=2\ h\ 15\ min\ 83\ sec\\83 s(\frac{1 min}{60 sec})=1.38 min\\1.38\ min\ =1\ min\ 0.38\ min*(\frac{60 sec}{1\ min})\\1.38\ min=1\min\ 23 s.\\( 8min 31 sec)*17=2 h 16 min 23 sec[/tex]

Have a great day

Assume a solution of the form [tex]\Psi(x,y)=C[/tex]. Differentiating both sides gives

[tex]\Psi_x\,\mathrm dx+\Psi_y\,\mathrm dy=0[/tex]

with [tex]\Psi_x=y[/tex] and [tex]\Psi_y=x(\ln x-\ln y-1)[/tex].

Divide both sides by [tex]x[/tex] and we have

[tex]\dfrac yx\,\mathrm dx+(\ln x-\ln y-1)\,\mathrm dy=0[/tex]

Notice that

[tex]\left(\dfrac yx\right)_y=\dfrac1x[/tex]

[tex]\left(\ln x-\ln y-1\right)_x=\dfrac1x[/tex]

so the ODE is exact. Now we can look for a solution [tex]\Psi[/tex] with

[tex]\Psi_x=\dfrac yx[/tex]

[tex]\Psi_y=\ln x-\ln y-1[/tex]

Integrating the first PDE with respect to [tex]x[/tex] gives

[tex]\Psi(x,y)=y\ln x+f(y)[/tex]

and differentiating this with respect to [tex]y[/tex] gives

[tex]\Psi_y=\ln x+f'(y)=\ln x-\ln y-1\implies f'(y)=-\ln y-1\implies f(y)=-y\ln y+C[/tex]

So this ODE has general solution

[tex]y\ln x-y\ln y=C[/tex]

Given that [tex]y(1)=e[/tex], we have

[tex]e\ln1-e\ln e=C\implies C=-e[/tex]

so the particular solution is

[tex]y(\ln x-\ln y)=-e[/tex]

[tex]y\ln\dfrac xy=-e[/tex]

[tex]\boxed{y\ln\dfrac yx=e}[/tex]

Answer: 0.0427

Step-by-step explanation:

Given : The area under the standard normal curve to the left of z = -1.72 is​ 0.0427

We know that the normal curve is a bell shaped curve that is symmetric such that half of the data falls to the left of the mean (Mean lies at the middle of the curve) and half of data falls to the right.

Now, z=-1.72 lies on the left side and z=1.72 lies right side.

Since, normal curve is symmetric and the magnitude of the values if same , then the area under the standard normal curve to the left of z = -1.72 is​  equals to the area to the right of z = 1.72 is 0.0427

Therefore, the area under the standard normal curve to the right of z = 1.72  is 0.0427

The area under the standard normal curve to the right of z=1.72 is 0.9573 or 95.73%, which can be calculated by subtracting the area to the left of z from 1 (1 - 0.0427 = 0.9573).

The area under the standard normal curve to the left of z=-1.72 is 0.0427. This essentially means that 4.27% of all observations fall under this score. Conversely, the area to the right of the z-score represents the proportion of observations that are greater than z. In a normal distribution, the sum of the areas to the left and to the right of a z-score must always equal to 1 (0.0427 + x = 1), because all possible outcomes are accounted for by a normal distribution. Hence, we can find out the area to the right by simply subtracting the area to the left from 1. So, the area under the standard normal curve to the right of z=1.72 is 1 - 0.0427 = 0.9573 or 95.73%.

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Answer:

The amount would be $ 3600.

Step-by-step explanation:

Given,

The invested amount, P = $ 2000,

Annual rate of interest, r = 4 %,

Time, t = 20 years,

So, the simple interest would be,

[tex]I=\frac{P\times r\times t}{100}[/tex]

[tex]=\frac{2000\times 4\times 20}{100}[/tex]

[tex]=\frac{160000}{100}[/tex]

[tex]=\$1600[/tex]

Hence, the amount of money after 20 years,

[tex]A=P+I[/tex]

[tex]=2000+1600[/tex]

[tex]=\$ 3600[/tex]

Answer:

Step-by-step explanation:

In order to solve the inequality, follow the simple steps:

–3(5y – 4) ≥ 17 .

Dividing both sides with -3:

5y - 4 ≤ -17/3 (the sign of the inequality becomes opposite whenever a negative number is either multiplied or divided on both the sides of the inequality).

Adding 4 on both sides:

5y ≤ -5/3

Dividing 5 on both sides:

y ≤ -1/3.

This shows that all the values of y less than and equal to -1/3 satisfy the inequality.

The number line has been attached. Since it involves ≤ sign, therefore, a filled circle will be used to plot the inequality. Less than means that all the values on the left hand side of the number line will be included. This is denoted with an arrow (see the diagram)!!!

Answer:

80 degrees

Step-by-step explanation:

The entire circle represents 72 and the "slice of pie" is represented by a portion:

16/72 * 100

= 0.22 * 100

= 22.222%

.

22.222% of 360 degrees

= .22222 * 360

= 80 degrees

Each portion can be represented by 16/72

16/72 = 0.22

0.22... * 100% = 22.22...%

0.2222 * 360 = 80

Therefore, the answer is 80 degrees.

Best of Luck!

Answer: y(x) = [tex]C_{1} e^{x} + C_{2} e^{\frac{-x}{2} }[/tex]

Step-by-step explanation:

2y ′′ − y ′ − y = 0

The characteristic equation is:

[tex]2r^{2} - r - 1 = 0[/tex]

[tex]2r^{2} - 2r + r - 1 = 0[/tex]

2r(r-1) + 1(r-1) = 0

(r-1)(2r+1) = 0

[tex]r_{1} = 1 , r_{2} = \frac{-1}{2}[/tex]

∴ there are two distinct roots

so the general equation is as follows:

y(x) = [tex]C_{1} e^{r_{1}x } + C_{2} e^{r_{2}x }[/tex]

y(x) = [tex]C_{1} e^{x} + C_{2} e^{\frac{-x}{2} }[/tex]

Answer:

The P(x=45) is more that the P(x=20). Therefore x=45 successes is more likely to get.

Step-by-step explanation:

Given information: n=100 and p=1/3.

According to the binomial distribution, the probability of getting r success in n trials is

[tex]P(x=r)=^nC_rp^rq^{n-r}[/tex]

where, n is total trials, p is probability of success and q is probability of failure.

Total trials, n = 100

Probability of success, p = [tex]\frac{1}{3}[/tex]

Probability of failure, q = [tex]1-\frac{1}{3}=\frac{2}{3}[/tex]

The probability of 20 successes is

[tex]P(x=20)=^{100}C_{20}\times (\frac{1}{3})^{20}\times (\frac{2}{3})^{100-20}[/tex]

[tex]P(x=20)=\frac{100!}{20!(100-20)!}\times (\frac{1}{3})^{20}\times (\frac{2}{3})^{80}\approx 0.001257[/tex]

The probability of 45 successes is

[tex]P(x=45)=^{100}C_{45}\times (\frac{1}{3})^{45}\times (\frac{2}{3})^{100-45}[/tex]

[tex]P(x=45)=\frac{100!}{45!(100-45)!}\times (\frac{1}{3})^{45}\times (\frac{2}{3})^{55}\approx 0.004296[/tex]

The P(x=45) is more that the P(x=20). Therefore x=45 successes is more likely to get.

To find the x for a given function f(x) =5, use the graph of the function and search for places where the y-coordinate is 5. The x-coordinates of these points are the solutions.

The problem is asking for the value of x when f(x) = 5. To solve this problem, you would examine the graph of the function and look for the point(s) where the y-coordinate (the function value) is 5; the corresponding x-coordinate(s) would be your answer. For example, if seen directly above the number five on the y-axis, the line crosses at x=3, then x=3 is your solution. If it crosses again at x=-2, then x=-2 is another solution. Always remember that some problems may indeed have more than one answer, especially with functions that are not linear.

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Answer:

[tex](x+1)^{2} +(y-4)^{2} =25[/tex]

Step-by-step explanation:

In order to find the equation of the circle, first we need to know the circle's general equation, which is:

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

(h,k) is the center of the circle and r the radius of the circle.

Because the problem has given the center (-1,4) then h=-1 and k=4.

We need to find now the radius:

Using the distance equation: [tex]distance=\sqrt{(x2-x1)^{2}+(y2-y1)^{2}}[/tex] and because we have the center coordinates and an extra point (3,7) we can find the radius as:

[tex]distance=\sqrt{(3-(-1))^{2}+(7-4)^{2}}[/tex]

[tex]distance=\sqrt{4^{2}+3^{2}}[/tex]

[tex]distance=\sqrt{16+9}[/tex]

[tex]distance=\sqrt{25}[/tex]

[tex]distance=5[/tex] which means r=5

In conclusion, the equation for the given circle is [tex](x+1)^{2} +(y-4)^{2} =5^{2}[/tex] which also, can be written as [tex](x+1)^{2} +(y-4)^{2} =25[/tex]

Answer:

Option A

Step-by-step explanation:

Given that in a study of the relationship between geographical mobility (number of times a person has changed residences) and number of friends, Pearson's r^2 is reported as .40.

r square, being the coefficient of determination explains the variablity of one variable due to the variability of the other.  Here Mobility explains 40% variation in the number of friends is right answer.

Option B is wrong because r = ±6324, so cannot say positive or negative.

Similarly option c is wrong because we are unsure whether negative correlation.  Option d is wrong since 16% is not right .

Answer:

see below

Step-by-step explanation:

a

y = 2x -3

Standard form is Ax + By =C  where A is a positive integer and B and C are integers

Subtract 2x from each side

y-2x = 2x-2x -3

-2x+y = -3

We want A to be positive

Multiply each side by -1

2x -y = 3

This is in standard form

b

y = 2/3 x -7

Subtract 2/3x from each side

y-2/3x = 2/3x-2/3x -7

-2/3x+y = -7

We want A to be positive integer

Multiply each side by -3

-3*(-2/3x+y) = -7*-3

2x -3y = 21

c

y = -3x +1/2

Add 3x to each side

3x +y = -3x+3x +1/2

3x+y = 1/2

Multiply each side by 2

2(3x+y) = 1/2*2

6x+2y = 1

Answer:

(a) [tex]P_{t}=3(2)^{6t}[/tex]

(b) [tex]3(2)^{42}[/tex]

(c) 28.07 minutes

Step-by-step explanation:

A cell of some bacteria divides itself into 2 cells in every 10 minutes and initial population of the bacteria was 3.

That means sequence formed will be 3, 6, 12, 24............

We can easily say that this sequence is a geometric sequence having common ratio (r) = [tex]\frac{T_{2}}{T_{1}}=\frac{6}{3}[/tex]

r = 2

Now we know the explicit formula of a geometric sequence is given by

[tex]P_{t}=P_{0}(r)^{\frac{60t}{10}}=P_{0}(r)^{6t}[/tex]

Where a = Initial population = 3 bacteria

r = common ratio = 2

and t = time in hours

So explicit formula will be [tex]P_{t}=3(2)^{6t}[/tex]

(a) Now we have to calculate the size of population after t hours

[tex]P_{t}=3(2)^{6t}[/tex]

(b) We have to find the size of population after 7 hours or 420 minutes

[tex]P_{t}=3(2)^{6\times7}[/tex]

= [tex]3(2)^{42}[/tex]

After 7 hours bacteria population will be [tex]3(2)^{42}[/tex]

(c) Time to reach population as 21

By the explicit formula

[tex]21=3(2)^{6t}[/tex]

[tex]2^{6t}=\frac{21}{3}=7[/tex]

Now we take log on both the sides of the equation

[tex]log(2^{6t})=log(7)[/tex]

6t log2 = log 7

6t(0.301) = 0.845

t(1.806) = 0.845

t = [tex]\frac{0.845}{1.806}=0.468[/tex] hours

Or t = 0.468×60 = 28.07 minutes

Therefore, after 28.07 minutes bacterial population will be 21

The bacteria population grows following an exponential pattern, therefore the population after t hours can be calculated using the exponential growth formula with the initial population as 3 and each cell dividing every 10 minutes. To calculate the time when the population reaches a certain size, solve the exponential growth equation for t.

The growth of bacteria population can be described as exponential growth, with each cell dividing into two every 10 minutes. Given the initial population as 3 bacteria, we would need to calculate the number of divisions that occur within the specified time frame to calculate the population after t hours.

(a) To find the population after t hours, we convert the hours to minutes (since each division occurs every 10 minutes) and then calculate the number of divisions. Each bacterial division results in a doubling of the population, so we use the formula for exponential growth: N = N0 * 2^n, where N0 is the initial population (3), and n is the number of divisions (6t, because t hours is 60t minutes and each division occurs every 10 minutes, making a total of 6t divisions per hour). So the population after t hours is N = 3 * 2^(6t).

(b) To find the size of the population after 7 hours, we substitute t = 7 into the formula, to get N = 3 * 2^(6*7) = 3 * 2^42.

(c) To find out when the population reaches 21, we equate N to 21 in the formula and solve for t. So, 21 = 3 * 2^(6t). Solving this equation gives the time t in hours when the population will reach 21.

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Answer:

42 in

Step-by-step explanation:

108 in = 2l + 2w

l = 12 in

108 = 24 + 2w

84 = 2w

w = 42

The width is 21 inches, leading to a length of 33 inches, which corresponds to option c.

To find the length of the rectangular poster, we will use the formula for the perimeter of a rectangle, which is P = 2l + 2w, where P represents the perimeter, l represents the length, and w represents the width. According to the question, we have a perimeter P of 108 inches and the length is 12 inches greater than the width. If we let w represent the width, then the length can be represented as w + 12. Plugging into the perimeter formula we get: 108 = 2(w + 12) + 2w.

Now, let's solve for w:

108 = 2w + 24 + 2w

108 = 4w + 24

108 - 24 = 4w

84 = 4w

w = 84 / 4

w = 21 inches

Now that we have the width, we can find the length by adding 12 inches:

Length = w + 12

Length = 21 + 12

Length = 33 inches

Therefore, the length of the poster is 33 inches, which corresponds to option c.

Answer:

a. 50%

b. 33%  

c. 17% (I'm assuming the exercise is wrong and it has to say "white" instead of "red", because if not is the same as b.)

d. 67%

Step-by-step explanation:

a. We have a total of 18 balls, 13 are red and 5 are white. They are numbered from 1 to 18. In this case, we don't care about the color of the ball, we just need it to be not even. We have to count how many not even numbers are between 1 and 18, that is 9. So, the chances of getting a ball not even numbered are 9 in 18, that's

[tex]\frac{9}{18}*100=50\%[/tex]

b. Now we do care about the color of the ball. The red balls are numbered from 1 to 13, so we have 6 balls even numbered. That makes the chances 6 in 18 (we still have 18 in total), that's

[tex]\frac{6}{18}*100=33\%[/tex]

c. (I'm assuming the exercise is wrong and it has to say "white" instead of "red", because if not is the same as b.)

The white balls are numbered from 14 to 18, so we have 3 balls even numbered. That makes the chances 3 in 18,

[tex]\frac{3}{18}*100=17\%[/tex]

d. Let's notice that "the ball is neither red or even numbered" is the complement (exactly the opposite) of "the ball is red and even numbered", that means  

100% = Probability (ball red and even numbered) + Probability (ball neither red or even numbered)

So, Probability (ball neither red or even numbered) = 100% - Probability (ball red and even numbered) = 100% - 33% = 67%

Answer:

(0.062, 0.130)

Step-by-step explanation:

Sample size = n = 500

Number of heads that were unemployed = x = 48

Proportion of heads that were unemployed = p = [tex]\frac{x}{n}=\frac{48}{500}=0.096[/tex]

Proportion of heads that were not unemployed = q = 1 - p = 1 - 0.096 = 0.904

Confidence Level = 99%

z-value for 99% confidence level = z = 2.58

The confidence interval about a population proportion is calculated as:

[tex](p-z\sqrt{\frac{pq}{n}} , p+z\sqrt{\frac{pq}{n}})[/tex]

Using the values, we get:

[tex](0.096-2.58\sqrt{\frac{0.096 \times 0.904}{500}},0.096+2.58\sqrt{\frac{0.096 \times 0.904}{500}})\\\\ = (0.062,0.130)[/tex]

Thus, 99% confidence interval for the proportion of unemployed heads of households in the population is (0.062, 0.130)

The 99% confidence interval for the proportion of unemployed heads of households in the population is approximately 0.096 ± 0.029.

To compute the 99% confidence interval for the proportion of unemployed heads of households, we can use the formula:

Confidence interval = sample proportion ± margin of error

1. Find the sample proportion:

Divide the number of unemployed heads of households (48) by the total number of heads of households surveyed (500).

Sample proportion = 48 / 500 = 0.096

2. Calculate the margin of error:

The margin of error depends on the level of confidence and the sample size. For a 99% confidence level, we need to find the critical value, which corresponds to 99% confidence and 500 as the sample size.

The critical value for a 99% confidence level and 500 as the sample size is approximately 2.576.

Margin of error = critical value * sqrt((sample proportion * (1 - sample proportion)) / sample size)

Margin of error = 2.576 * sqrt((0.096 * (1 - 0.096)) / 500) ≈ 0.029

3. Calculate the confidence interval:

Confidence interval = sample proportion ± margin of error

Confidence interval = 0.096 ± 0.029

For more such information on: confidence interval

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