A fluid has density 810 kg/m3 and flows with velocity v = z i + y2 j + x2 k, where x, y, and z are measured in meters and the components of v in meters per second. Find the rate of flow outward through the cylinder x2 + y2 = 9, 0 ≤ z ≤ 4.

Answer:

The center require to have a 4.6 inspectors, but this on you decide if you have 4 or 5 inspectors, it may not have 4.6 people working.

So then if you decide to have 5 people their utilization percentage is 83.3%, but if you decide to have 4 people their utilization rate will be 104.17% at the risk of defaulting on demand.

Step-by-step explanation:

1. Define your variables

Demand rate= 50 vehicles per hour

Service rate = 12 vehicles per hour per inspector

Inspectors= a

2. Use the formule of the center´s utilization

U= [tex]\frac{Demand}{(Service) X a}[/tex]

0,9= [tex]\frac{50}{(12) X a }[/tex]

0,9a=[tex]\frac{50}{12}[/tex]

a=[tex]\frac{50}{12X0,9}[/tex]

a= 4.6

3. According to the center´s utilization formula the center require 4,6 inspectors, but approaches 5 because people cannot be divided. With this numbers of inspectors the utilization is:

U= [tex]\frac{Demand}{(Service) X (No. inspectors)}[/tex]

U= [tex]\frac{50}{(12) X (5)}[/tex]

U= 83,3%

4. Other option that you will be used is that the center require 4 inspectors. With this numbers of inspectors the utilization is:

U= [tex]\frac{Demand}{(Service) X (No. inspectors)}[/tex]

U= [tex]\frac{50}{(12) X (4)}[/tex]

U= 104,17%

The center would require 6 inspectors to have a target utilization of 90 percent.

To determine the number of inspectors required to have a target utilization of 90 percent, we can use the formula:

Number of inspectors = Total vehicles per hour / Vehicles inspected per inspector per hour / Target utilization

Given that the center tests 50 vehicles per hour, and each inspector can inspect 12 vehicles per hour, the formula becomes:

Number of inspectors = 50 / 12 / 0.9 = 5.56

Since we cannot have a fraction of an inspector, we round up to the nearest whole number. Therefore, the center would require 6 inspectors to have a target utilization of 90 percent.

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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:

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: 465

Step-by-step explanation:

251 (days) x 24 (hours) = 6,024 hours

6,024+21 hours= 6,045

6045/13=465

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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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:

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:

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.

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: u (sub d) is inequal to zero

Step-by-step explanation: Because this is a paired t-test, our alternative hypothesis would be u(sub d) is inequal to zero.

Final answer:

The alternative hypothesis for a study on the effectiveness of a diet program would express the expectation of a statistically significant change in fitness level scores, likely a decrease if lower scores indicate better fitness, after participating in the program compared to before.

Explanation:

The correct alternative hypothesis for a study that aims to measure the effectiveness of a diet program in managing weight would look at whether there is a statistically significant difference in the fitness level scores of the participants before and after following the program. As the question suggests evaluating the effectiveness of a diet program, we are particularly interested in seeing an improvement, which would mean expecting a lower score after the program if the score represents a measure where lower is better.

Thus, the alternative hypothesis (H1) should reflect the expectation of improvement. If the fitness score is such that a lower score indicates better fitness, the alternative hypothesis would be:

H1: The mean fitness level score after the program is lower than the mean fitness level score before the program.

This implies that the diet program is effective in improving fitness levels. On the contrary, if a higher fitness score indicates better fitness, the alternative hypothesis would be framed to reflect an expected increase in the score after the program.

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:

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:

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:

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:

A. 15 and 0. B. 15 and 17

Step-by-step explanation:

A. a= 13 and b = 195.

b div a = 195 div 13. The result is the integer part of the division, so

195 div 13 = 15.

b mod a = 195 mod 13. The result is the residue of the division. In this case the division is exact, so

195 mod 13 = 0.

B. a = 24 and b=377.

b div a = 377 div 24. The result is the integer part of the division, so

377 div 24 = 15.

b mod a = 377 mod 24. The result is the residue of the division. In this case the division is not exact, so

377 = 24*15+17, then

377 mod 24 = 17.

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: The explanation is as follows:

Step-by-step explanation:

(a) [8: 9, 3, 2, 1]

q = 8

Here, coalition is as follows:

[P1, P2, P3, P4] = [9, 3, 2, 1]

for the above coalition, the combined weight is

[P1, P2, P3, P4] = 9+3+2+1 = 15 ⇒ combined weight

For simplified weighted voting system;

q = combined weight ⇒ both the terms have to be equal for a simplified weighted voting system.

But, here 8 ≠ 15

∴ It is not a simplified weighted voting system.

(b) [20: 8, 6, 3, 2, 1]

q = 20

Here, coalition is as follows:

[P1, P2, P3, P4, P5] = [8, 6, 3, 2, 1]

for the above coalition, the combined weight is

[P1, P2, P3, P4, P5] = 8+6+3+2+1 = 20 ⇒ combined weight

For simplified weighted voting system;

q = combined weight

Since,  20 = 20

∴ It is a simplified weighted voting system.

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

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:

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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Check the picture below.

Answer: SECOND OPTION.

Step-by-step explanation:

It is important to remember that a tangent to a circle is a line that touches it at one point. This point is called "Point of tangency". By definition, the angle between the tangent and the radius is 90 degrees.

In this case you can observe that EF is the radius of this circle, therefore, the angle between EF and FH measures 90 degrees.

Based on this, you can say the following:

[tex]\angle EFH=90\°[/tex]

This matches with the second option.

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

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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:

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:

(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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To find the point on the plane that is nearest to (2,0,1), we minimize the squared distance between the two points using partial derivatives and set them equal to 0. The values of x, y, and z for the point are x = 28/13, y = 3/26, and z = 27/26.

To find the point on the plane that is nearest to (2,0,1), we need to find the coordinates that satisfy the equation 4x+3y+z=10 and minimize the distance between the point and (2,0,1).

This can be done by minimizing the squared distance between the two points. Using the formula for distance, we get the squared distance as:

d^2 = (x-2)^2 + y^2 + (z-1)^2

To minimize the squared distance, we can find the partial derivatives with respect to x, y, and z and set them equal to 0.

Solving these equations, we find that x = 28/13, y = 3/26, and z = 27/26.

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Answer: 0.2195

Step-by-step explanation:

Binomial distribution formula :-

[tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex], where P(x) is the probability of x successes in the n independent trials of the experiment and p is the probability of success.

Given : The probability of that the general population has blood type A = [tex]\dfrac{1}{3}[/tex]

Sample size : n=6

Now, the probability that exactly three of them have blood type A is given by :-

[tex]P(3)=^6C_3(\dfrac{1}{3})^3(1-\dfrac{1}{3})^{6-3}\\\\=\dfrac{6!}{3!3!}(\dfrac{1}{3})^3(\dfrac{2}{3})^{3}\\\\=0.219478737997\approx0.2195[/tex]

Therefore, the probability that exactly three of them have blood type A = 0.2195

Answer:

"Decreasing the sample size."

Step-by-step explanation:

The margin of error is :

[tex]ME=z\times \frac{\sigma }{\sqrt{n}}[/tex]

Therefore, as n decreases, margin of error increases.

In the question we are asked " Which action will not reduce the margin of error."

Therefore, the correct option for which action will not reduce the margin of error is "Decreasing the sample size."

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