What is the probability that any of the 25500 undergraduates is in your random sample of 2550 undergraduates selected

The distance between Town A and Town B is calculated as 368 km.

Distance is described as a numerical or occasionally qualitative measurement of how far apart objects or points are.

we have then equation that:

80 km/h x (t + 2) h = 120 km/h x t h + 30 km

we simplify the above equation :

80t + 160 = 120t + 30

50t = 130

t = 2.6 hours

Therefore, the distance between Town A and Town B will be the  distance traveled by truck

= 80 km/h x (t + 2) h

= 80 km/h x 4.6 h

= 368 km

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The probability that a randomly chosen string of seven hexadecimal digits has at least one repeated digit is approximately 1 - ((16 * 15 * 14 * 13 * 12 * 11 * 10) / 16^7).

The probability that a randomly chosen string of seven hexadecimal digits has at least one repeated digit can be calculated using the concept of probability. There are a total of 16 possible characters in hexadecimal system (0-9 and A-F) and for each of the seven digits, there are 16 possible choices. Therefore, there are a total of 16^7 possible strings of seven hexadecimal digits.

To calculate the probability of having at least one repeated digit, we need to calculate the number of strings that have no repeated digits and subtract it from the total number of possible strings.

The number of strings with no repeated digits can be calculated as follows:

- For the first digit, there are 16 possible choices
- For the second digit, there are 15 possible choices (since one digit has already been chosen)
- For the third digit, there are 14 possible choices
- And so on, until the seventh digit, for which there are 10 possible choices (since six digits have already been chosen)

Therefore, the number of strings with no repeated digits is:

16 x 15 x 14 x 13 x 12 x 11 x 10

To calculate the probability of having at least one repeated digit, we need to subtract this number from the total number of possible strings and divide by the total number of possible strings:

1 - (16 x 15 x 14 x 13 x 12 x 11 x 10) / (16^7)

This gives us the probability that a randomly chosen string of seven hexadecimal digits has at least one repeated digit.

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The area of this equiangular hexagon is 104sqrt(3).

An equiangular hexagon has six equal angles, so each angle measures 120 degrees. We can divide this hexagon into six equilateral triangles with side lengths of 6, 8, and 12.

To find the area of each equilateral triangle, we can use the formula

[tex]A = (\sqrt{(3)/4} ) \times s^2[/tex], where s is the length of a side.

For the triangle with side length 6, its area is

[tex]A1 = (\sqrt{(3)/4} ) \times 6^2[/tex] = [tex]9\sqrt{3}[/tex]

For the triangle with side length 8, its area is

[tex]A2 = (\sqrt{(3)/4} ) \times 8^2 = 16\sqrt{3}[/tex]

For the triangle with side length 12, its area is

[tex]A3 = (\sqrt{(3)/4} ) \times 12^2 = 27\sqrt{3}[/tex]

The area of the hexagon is simply the sum of the areas of these six equilateral triangles, which is:

A = A1 + A2 + A3 + A1 + A2 + A3

= 2A1 + 2A2 + 2A3

[tex]= 2(9\sqrt{3} ) + 2(16\sqrt{3} ) + 2(27\sqrt{3} )\\= 104\sqrt{3}[/tex]

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The statement "Because in a randomized controlled trial (RCT), the assignment is random, therefore there is no coverage bias---by definition" g is false because, it is important to consider both randomization and other factors when assessing the potential for bias in an RCT.

Random assignment in an RCT can help to reduce selection bias, but it does not guarantee the absence of coverage bias.

Coverage bias can occur if the participants who are enrolled in the trial do not represent the population to which the results will be generalized.

For example, if the trial only includes participants who are healthier or more compliant than the typical patient, the results may not be applicable to the broader population.

Therefore, it is important to consider both randomization and other factors when assessing the potential for bias in an RCT.

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The cost per bottle is $0.62

The cost per bottle is $0.66

The pack of 24 is the better buy.

The cost per bottle can be determined by dividing the total cost of a production run by the number of bottles produced. The total cost includes all of the expenses associated with producing and packaging the bottles

If 18 bottles cost $11.21

1 bottle costs 1 * 11.21/18

= $0.62

Again;

If 24 bottles costs $15.85

1 bottle will cost 1 * 15.85/24

= $0.66

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"The cone and cylinder above have the same radius and height. The volume of cone is 162 cubic inches" is that the volume of the cylinder is 486 cubic inches.

The volume of the cylinder can be found by using the formula for the volume of a cylinder, which is V = πr^2h, where r is the radius and h is the height. Since the cone and cylinder have the same radius and height, we can use the volume of the cone (162 cubic inches) to find the radius and height of both shapes.

Let's first find the radius of the cone. The formula for the volume of a cone is V = (1/3)πr^2h. We can rearrange this formula to solve for r:

r = sqrt((3V) / (πh))

Plugging in the values we know, we get:

r = sqrt((3 * 162) / (πh))

Since the cone and cylinder have the same height, we can use this value for the radius of both shapes.

r = sqrt((3 * 162) / (πh)) = sqrt((486 / πh))

Now that we know the radius, we can use the formula for the volume of a cylinder to find the volume of the cylinder:

V = πr^2h = π((sqrt(486/πh))^2)h = π * 486 / π * h * h = 486h

Therefore, the volume of the cylinder is 486 cubic inches.

In summary, "The cone and cylinder above have the same radius and height. The volume of the cone is 162 cubic inches" is that the volume of the cylinder is 486 cubic inches.

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There are 330 ways to select 7 seed packets out of 11.

To count the number of ways to select 7 seed packets out of 11, we can use the combination formula:

[tex]( \frac {k}n )= k!(n-k)!n![/tex]

where n is the number of items to choose from, and k is the number of items to choose. In this case, n=11 and k=7.

Plugging these values into the formula, we get:

[tex]( \frac{7}{11})= 7!(11-7)!11![/tex]

Simplifying the factorials, we get:

[tex]( \frac{7}{11})= \frac{7\times 6\times 5\times 4\times 3\times 2\times 1}{11\times 10\times 9\times 8\times 7\times 6\times 5}[/tex]

Simplifying further, we get:

[tex]( \frac{7}{11} )= \frac{4\times 3\times 2\times 1}{11\times 10\times 9\times 8}[/tex]

Simplifying again, we get:

[tex](\frac{11}7)=330[/tex]

Therefore, there are 330 ways to select 7 seed packets out of 11.

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The probability that Adam and Betty meet at some point is $1-\frac{1}{16}=\boxed{\frac{15}{16}}$.

To find the probability that Adam and Betty meet while walking on the grid, we can consider their paths. Adam will always move up or right, while Betty will always move down or left. This means that their paths will always be perpendicular, and they will only meet if they intersect at some point.

Let's consider the first minute. Adam can either move up or right, and Betty can either move down or left. There are four possible outcomes: Adam moves up and Betty moves down, Adam moves up and Betty moves left, Adam moves right and Betty moves down, or Adam moves right and Betty moves left.

Out of these four outcomes, only one leads to Adam and Betty meeting: if Adam moves right and Betty moves down, they will meet at the point $(1,3)$. So the probability of them meeting in the first minute is $\frac{1}{4}$.

Now let's consider the second minute. Adam will be one unit away from $(1,3)$, and Betty will be one unit away from $(1,3)$. There are four possible outcomes again, but only one leads to them meeting: if Adam moves up and Betty moves down, they will meet at the point $(1,2)$. So the probability of them meeting in the second minute is $\frac{1}{4}$.

We can continue this process for each minute. At each step, there is only one outcome that leads to them meeting, and the probability of that outcome is $\frac{1}{4}$. So the probability of them meeting after $n$ minutes is $\left(\frac{1}{4}\right)^n$.

Now we need to find the probability that they meet at any point in time. We can do this by taking the complement of the probability that they never meet. The only way they will never meet is if their paths never intersect, which means that Adam always stays to the right of Betty or always stays above Betty.

The probability of this happening is the same as the probability that Adam flips tails $4$ times in a row, or Betty flips heads $2$ times in a row. This probability is $\left(\frac{1}{2}\right)^4=\frac{1}{16}$, since there are $2^4$ possible outcomes for Adam and $2^2$ possible outcomes for Betty.

So the probability that Adam and Betty meet at some point is $1-\frac{1}{16}=\boxed{\frac{15}{16}}$.

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To find the probability that a randomly selected worker has a retirement plan or health insurance, we need to add the probabilities of having a retirement plan and having health insurance and then subtract the probability of having both,

Since we don't want to count those workers twice. P(retirement plan or health insurance) = P(retirement plan) + P(health insurance) - P(both), P(retirement plan or health insurance) = 0.56 + 0.68 - 0.49, P(retirement plan or health insurance) = 0.75.

Therefore, the probability that a randomly selected worker has a retirement plan or health insurance is 0.75. we'll use the formula: P(A or B) = P(A) + P(B) - P(A and B), where A represents having a retirement plan, B represents having health insurance, and P(A and B) represents having both.

Given:
P(A) = 56% (retirement plan)
P(B) = 68% (health insurance)
P(A and B) = 49% (both)

Now we can plug these values into the formula: P(A or B) = 0.56 + 0.68 - 0.49 = 0.75, The probability that a randomly selected worker has a retirement plan or health insurance is 75%.

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The mean of all the sample means (which is equal to the population mean) and the mean of all (N) population observations (X) are both equal to the population mean μ.

The mean of all the sample means of a given population is equal to the population mean, which is denoted by the symbol μ. This is a consequence of the central limit theorem, which states that the distribution of the sample means becomes approximately normal as the sample size n becomes larger, with mean equal to the population mean μ.

The mean of all (N) population observations (X) is simply the population mean, which is also denoted by the symbol μ. It represents the average value of the variable of interest across all individuals in the population.

Therefore, the mean of all the sample means (which is equal to the population mean) and the mean of all (N) population observations (X) are both equal to the population mean μ.

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To accept the null hypothesis, the calculated F-value should be between 0.142 and 3.490. If it falls outside this range, you would reject the null hypothesis.

To determine the max and min F-value to accept the null hypothesis for 7 degrees of freedom (df) for the numerator and 12 df for the denominator, you would consult the F-distribution table or use an online calculator.

At a common significance level (α) of 0.05, the critical F-values are:
- F(7, 12) lower critical value: 0.142
- F(7, 12) upper critical value: 3.490

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The prediction of the value of the dependent variable outside the experimental region is called

extrapolation. So, the option(a) is right one.

A dependent variable is defined as the variable which is tested and measured in a scientific experiment. It is always depends on other variables. That's why it is called dependent variable and other variable is independent variable. Because it is a variable so it's value always change according to situation. So, there are two processes for predicting the values of dependent variable. These are defined as below :

Hence, the prediction of the value of the dependent variable outside the experimental region is known as extrapolation.

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Answer:

The probability that a person has not been vaccinated is 37%.

[tex]1 - {.37}^{3} = .949347 = 94.9347\%[/tex]

Approximately 30.85% of daycare costs are more than 7250 annually.

To solve this problem, we need to calculate the z-score for the given value of 7250 and then find the area under the normal distribution curve to the right of that z-score.

The z-score formula is given by:

z = (x - μ) / σ

where:

x = the given value (7250)

μ = the mean of the distribution (8000)

σ = the standard deviation of the distribution (1500)

Substituting the given values, we get:

z = (7250 - 8000) / 1500

z = -0.5

Using a standard normal distribution table or calculator, we can find that the area under the curve to the right of z = -0.5 is approximately 0.6915.

Therefore, the percentage of daycare costs that are more than 7250 annually is approximately:

100% - (0.6915 x 100%) = 30.85%

So, approximately 30.85% of daycare costs are more than 7250 annually.

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Using the formula of combination, a committee of six can be chosen in 336 ways from a group of eleven if Barry and Harry refuse to serve together on the same committee

1. First, find the total number of ways to select six people from eleven without considering the restriction using the combination formula:
C(n, r) = n! / [r!(n-r)!],

where n is the total number of people, and r is the number of people to be selected.

In this case, n = 11 and r = 6. So the total number of ways without considering the restriction is:
C(11, 6) = 11! / [6!(11-6)!] = 11! / [6! * 5!] = 462 ways.

2. Now, consider Barry and Harry as a single unit, and we have 10 units in total (9 remaining people + 1 unit of Barry and Harry). We need to choose 4 more people from the remaining 9. So the number of ways is:
C(9, 4) = 9! / [4!(9-4)!] = 9! / [4! * 5!] = 126 ways.

3. Finally, apply the subtraction principle to exclude the cases where Barry and Harry are together. The total number of ways to form a committee of six, considering the restriction, is:
Total ways - Ways with Barry and Harry together = 462 - 126 = 336 ways.

So, in 336 ways, a committee of six can be chosen from a group of eleven if Barry and Harry refuse to serve together on the same committee.

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It takes 4 seconds for the ball to hit the ground.

The formula for the distance fallen by an object keeping in mind the motion of object dropped from rest is:

[tex]s = 1/2gt^2[/tex]

where s is the distance fallen, g is the acceleration due to gravity [tex](32 ft/s^2)[/tex], and t is the time taken to fall.

In this problem, the initial height of the ball is 256 feet, so the distance fallen is:

s = 256 - 0 = 256 feet

Substituting into the formula, we get:

[tex]256 = 1/2 * 32 * t^2[/tex]

Simplifying, we get:

[tex]256 = 16t^2[/tex]

Dividing both sides by 16, we get:

[tex]t^2 = 16[/tex]

Taking the square root of both sides, we get:

t = 4 seconds or -4 seconds

We can ignore the negative solution, so the answer is:

t = 4 seconds

Therefore, it takes 4 seconds for the ball to hit the ground.

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The population mean (μ) of high school students watching 7.2 movies per month, with a population standard deviation (σ) of 0.7. The sample size (n) of college students is 47, with a sample mean (x) of 6.2 movies per month.

We want to test if college students watch fewer movies per month than high school students at a 0.05 significance level.

To conduct this hypothesis test, we will use the one-sample z-test. The null hypothesis (H ₀) states that the mean number of movies watched by college students is equal to the population mean of high school students (μ = 7.2). The alternative hypothesis (H₁) states that the mean number of movies watched by college students is less than the population mean of high school students (μ < 7.2).

First, we need to calculate the standard error (SE) using the formula SE = σ/√n, where σ = 0.7 and n = 47. Next, we compute the z-score using the formula z = (x - μ)/SE. Once we have the z-score, we compare it to the critical value corresponding to the 0.05 significance level. If the z-score is less than the critical value, we reject the null hypothesis in favor of the alternative hypothesis.

By performing these calculations, we can determine whether there is enough evidence to conclude that college students watch fewer movies per month than high school students at the 0.05 significance level.

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In stroke play, when Player A concedes a short putt to Player B on the 7th hole and Player B picks up their ball and tees off on the 8th hole before holing out on the 7th hole, the ruling is that Player B incurs a penalty for not completing the hole.

In stroke play, if player A concedes a short putt to player B on the 7th hole, it means that player B can pick up their ball without completing the hole.

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In the 2012 Gallup survey, a random sample of 474,195 U.S. adults was interviewed by phone to gather information about their work status, height, and weight.

The objective was to determine the prevalence of obesity (defined as having a body mass index greater than 30) among different work status groups. The survey found that 24.9% of the participants who were employed (either full-time or part-time by choice) were classified as obese. In contrast, 28.6% of the participants who were unemployed and actively seeking work were also found to be obese. This indicates that there may be a relationship between employment status and obesity rates in the U.S. adult population.

However, it is important to note that correlation does not necessarily imply causation, and various factors could contribute to these findings. Further research may be necessary to determine the underlying causes behind the differences in obesity rates among employed and unemployed individuals.

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To find the variance of 7X - 5Y, we need to first find the variance of 7X and 5Y separately, and then subtract twice the covariance of X and Y (since we are given the covariance, not the correlation coefficient). The variance of 7X - 5Y is 1024.

Var[7X] = 49Var[X] = 49(16) = 784
Var[5Y] = 25Var[Y] = 25(4) = 100
Cov[X,Y] = 2

Now, using the formula for variance of a linear combination of two random variables:

Var[7X - 5Y] = Var[7X] + Var[5Y] - 2Cov[X,Y]
            = 784 + 100 - 2(2)
            = 880

Therefore, the variance of 7X - 5Y is approximately 880 (rounded to the nearest whole number).

Suppose the population standard deviation of X is 4 and the population standard deviation of Y is 2, and the covariance of X and Y is 2. To find the variance of 7X - 5Y, we use the formula Var[aX ± bY] = a²Var[X] + b²Var[Y] ± 2abCov[X,Y]. In this case, a = 7, b = -5, Var[X] = 4², Var[Y] = 2², and Cov[X,Y] = 2.

Var[7X - 5Y] = 7²(4²) + (-5)²(2²) - 2(7)(-5)(2)
Var[7X - 5Y] = 49(16) + 25(4) + 140
Var[7X - 5Y] = 784 + 100 + 140
Var[7X - 5Y] = 1024

So the variance of 7X - 5Y is 1024.

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a)The exact probability that there are at most 3 defectives in the sample is 0.4234. b) The exact probability that there are at most 3 defectives in the sample is 0.4234. c) the probability that between 4 and 7, inclusive, are defective by normal approximation is 0.2

a) To find the exact probability that there are at most 3 defectives in the sample, we can use the binomial distribution formula. The probability of getting at most 3 defectives is the sum of the probabilities of getting 0, 1, 2, or 3 defectives.

P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)

Where X is the number of defective bolts in the sample.

Using the binomial distribution formula, we get:

P(X = k) = (n choose k) * p^k * (1-p)^(n-k)

Where n is the sample size (100), p is the probability of a bolt being defective (0.06), and k is the number of defective bolts.

P(X ≤ 3) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3)
         = 0.4234

Therefore, the exact probability that there are at most 3 defectives in the sample is 0.4234.

b) To find the probability that there are at most 3 defectives by normal approximation, we need to calculate the mean and standard deviation of the binomial distribution.

Mean = np = 100 * 0.06 = 6
Standard deviation = sqrt(np(1-p)) = sqrt(100 * 0.06 * 0.94) = 2.424

We can then use the normal distribution to approximate the binomial distribution:

P(X ≤ 3) ≈ P(Z ≤ (3.5 - 6)/2.424)

Where Z is a standard normal random variable.

Using a standard normal table or calculator, we get:

P(Z ≤ -1.23) = 0.1093

Therefore, the probability that there are at most 3 defectives by normal approximation is 0.1093.

c) To find the probability that between 4 and 7, inclusive, are defective by normal approximation, we can use the same approach as in part b.

Mean = np = 100 * 0.06 = 6
Standard deviation = sqrt(np(1-p)) = sqrt(100 * 0.06 * 0.94) = 2.424

We can then use the normal distribution to approximate the binomial distribution:

P(4 ≤ X ≤ 7) ≈ P(3.5 ≤ X ≤ 7.5) ≈ P((3.5 - 6)/2.424 ≤ Z ≤ (7.5 - 6)/2.424)

Where Z is a standard normal random variable.

Using a standard normal table or calculator, we get:

P(-1.23 ≤ Z ≤ 0.62) = 0.2816

Therefore,  the probability that between 4 and 7, inclusive, are defective by normal approximation is 0.2

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There would be approximately [tex]1.79 \times 10^{135[/tex] bacteria after 7 hours. This

number is extremely large and is beyond the capacity of most calculators

to handle.

After 30 minutes (0.5 hours), each bacterium will undergo fission and

become two bacteria. Therefore, the number of bacteria will double after

every 30 minutes.

In 7 hours, there are 7 x 2 x 2 x 2 x 2 x 2 x 2 = 7 x 2^6 = 448 bacterial

cycles.

So, the final number of bacteria would be:

[tex]500 \times 2^{448} = 1.79 \times 10^{135[/tex]

Therefore, there would be approximately 1.79 x 10^135 bacteria after 7

hours.

This number is extremely large and is beyond the capacity of most

calculators to handle.

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If [tex]2x^3+ax^2+bx-6[/tex] is divided by (x+1) the remainder is -6, and if divided by (x-1) the remainder is -12, the values of a and b are -2 and -6, respectively.

Let's start by using the remainder theorem. If a polynomial f(x) is divided by (x-c), then the remainder is given by value f(c). Therefore, we can write:

f(-1) = -6

f(1) = -12

Substituting x=-1 in the original equation, we get:

[tex]2(-1)^3 + a(-1)^2 + b(-1) - 6 = -6[/tex]

-2 + a - b - 6 = -6

a - b = 4     ------(1)

Substituting x=1 in the original equation, we get:

[tex]2(1)^3 + a(1)^2 + b(1) - 6 = -12[/tex]

2 + a + b - 6 = -12

a + b = -8     ------(2)

We now have a system of two linear equations in two variables (a and b). Solving this system, we get:

a - b = 4     ------(1)

a + b = -8     ------(2)

Adding the two equations, we get:

2a = -4

a = -2

Substituting a = -2 in equation (2), we get:

-2 + b = -8

b = -6

Therefore, the values of a and b are -2 and -6, respectively.

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The measure of the length of the given arc is 16.5 m.

Given that a circle with radius 7 m we need to find the length of an arc which has a central angle of 135°,

The length of an arc = central angle / 360° × π × diameter

= 135° / 360° × 3.14 × 14

= 16.5 m

Hence, the measure of the length of the given arc is 16.5 m.

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Answer:

density

Step-by-step explanation:

the number of population distributed in a certain area, that is generally in km², is the density.

It can be calculated with the following data: Number of population/km²

Extra information: it obviously is an approximated value, because in certain areas it can be much higher (metropolis, for example, generally have a very high population density, meanwhile in the countryside it can be much lower). Generally, it varies from area to area.

The number of individuals in a population divided by the area that the population takes up is known as the density of the population.

Population density refers to the number of individuals in a population per unit of area. It is a measure of how crowded or dispersed a population is within a given area.

Population density is calculated by dividing the total population of an area by the total land area or water area of that region. The resulting number is often expressed in individuals per square kilometer or square mile, depending on the units of measurement used.

Population density is an important ecological concept because it can affect the ability of a population to survive and thrive within a given area. Populations that are too dense may experience competition for resources, disease outbreaks, and other negative effects.

On the other hand, populations that are too dispersed may have trouble finding mates and maintaining genetic diversity. Population density can also be used to track changes in populations over time, and to inform conservation and management efforts.

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Answer:

(i) 22 - (-23) = 45

(ii) √324 = 18

The seed company would save the seed from acreage yielding greater than 90 bushels/acre, which represents 74% of the total acreage.

Based on the information provided, the seed company would only save the seed from acreage yielding greater than 90 bushels/acre. It is not specified what percentage of the total acreage yields greater than 90 bushels/acre. Therefore, we cannot calculate the exact percentage of seeds that the company would save.

However, we can make an assumption based on the options provided. If we assume that the correct answer is one of the options provided, we can calculate the percentage based on that option.

For example, if we assume that the correct answer is option A (74), we can calculate the percentage as follows:

Percentage of seeds saved = (90 bushels/acre * 74%) = 66.6 bushels/acre

This means that The seed company would save the seed from acreage yielding greater than 90 bushels/acre, which represents 74% of the total acreage.

Similarly, we can calculate the percentage for the other options provided. However, without additional information, we cannot determine the exact percentage of seeds that the company would save.

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To show that the functions f(x) = x, g(x) = xe^x, and h(x) = x^2e^x are linearly independent on the real line, we can use the Wronskian. The Wronskian of a set of functions is defined as the determinant of the matrix:

f g h

f' g' h'

f'' g'' h''

where f', g', h' are the first derivatives of f, g, h, respectively, and f'', g'', h'' are the second derivatives of f, g, h, respectively.

For the given functions, we have:

x xe^x x^2e^x

1 e^x+x*e^x 2xe^x+x^2e^x

0 e^x+e^x+x*e^x 2e^x+2xe^x+x^2e^x

Expanding the determinant, we get:

x(e^x+e^x+xe^x)(2e^x+2xe^x+x^2e^x) - xe^x(e^x+e^x+xe^x)(2xe^x+x^2e^x) + x^2e^x(e^x+e^x+xe^x)(e^x+xe^x)

= 2x^3e^(3x)

Since the Wronskian is nonzero for any value of x, the functions f(x) = x, g(x) = xe^x, and h(x) = x^2e^x are linearly independent on the real line.

To find a second solution Y_2 for the differential equation y" - 6y' + 9y = 0 given that y_1 = e^3x is a solution, we can use the method of reduction of order. Let Y_2(x) = v(x) e^3x, where v(x) is an unknown function. Then, we have:

Y_2' = v'e^3x + 3ve^3x

Y_2'' = v''e^3x + 6v'e^3x + 9ve^3x

Substituting these expressions into the differential equation and simplifying, we get:

v''e^3x + 3v'e^3x = 0

This is a separable differential equation that can be solved by integrating both sides:

v'(x) = c e^(-3x)

v(x) = -1/3 c e^(-3x) + k

where c and k are arbitrary constants. Therefore, the general solution to the differential equation is:

y(x) = c1 e^(3x) + c2 e^(3x)∫e^(-3x) dx = c1 e^(3x) - (1/3) c2 e^(3x) + k e^(3x)

where c1 and c2 are constants of integration, and k is an arbitrary constant determined by any initial or boundary conditions. Therefore, the second solution is:

Y_2(x) = v(x) e^(3x) = (-1/3) ∫c e^(-3x) e^(3x) dx + k e^(3x) = (-1/3) cx + k e^(3x)

where c is an arbitrary constant.

Learn more about Wronskian here:- brainly.com/question/31484402

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