What the area of the small triangle

(a) The probability that at least 2 students share a birthday in a classroom of 25 students is about 0.5687.

(b) The probability that at least 2 students share a birthday in a classroom of 25 students, given that no students were born in January, is about 0.9585.

(c) If your birthday is on any of the 365 days of the year, then the probability that at least one student in a classroom of 25 shares your birthday is about 0.0641.

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

(a) The probability that none of the 25 students share a birthday is (365/365) * (364/365) * (363/365) * ... * (341/365), which is the probability that the first student has a unique birthday, the second student has a unique birthday, and so on. So the probability that at least 2 students share a birthday is 1 minus this probability:

P(at least 2 students share a birthday) = 1 - (365/365) * (364/365) * (363/365) * ... * (341/365)

≈ 1 - 0.4313

≈ 0.5687

So the probability that at least 2 students share a birthday in a classroom of 25 students is about 0.5687.

(b) If no students were born in January, then there are only 11 months in which the students could have been born. The probability that none of the 25 students share a birthday is (31/334) * (31/333) * (31/332) * ... * (31/310), which is the probability that the first student has a unique birthday, the second student has a unique birthday, and so on. So the probability that at least 2 students share a birthday is 1 minus this probability:

P(at least 2 students share a birthday | no students born in January) = 1 - (31/334) * (31/333) * (31/332) * ... * (31/310)

≈ 1 - 0.0415

≈ 0.9585

So the probability that at least 2 students share a birthday in a classroom of 25 students, given that no students were born in January, is about 0.9585.

(c) The probability that at least one student shares your birthday depends on when your birthday is. If your birthday is on any of the 365 days of the year, then the probability that at least one student shares your birthday is:

P(at least one student shares your birthday) = 1 - (364/365)²⁵

≈ 0.0641

So if your birthday is on any of the 365 days of the year, then the probability that at least one student in a classroom of 25 shares your birthday is about 0.0641.

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We can be 90% confident that the true proportion of the population who pay for Company ABC's healthcare falls between 0.630 and 0.738. This means that if we were to take repeated random samples from the population and calculate the confidence intervals using the same method, 90% of these intervals would contain the true population proportion.

To calculate the 90% confidence interval for the proportion of the population who pay for Company ABC's healthcare, we can use the formula:

CI = p' ± z* SE

where:

p' = sample proportion (0.684 in this case)

q' = complement of sample proportion (0.316 in this case)

n = sample size (380 in this case)

z = z-score for the desired confidence level (1.645 for 90% confidence)

SE = standard error of the proportion, calculated as SE = √(p' * q' / n)

Substituting the given values, we get:

SE = √(0.684 * 0.316 / 380) = 0.0316

CI = 0.684 ± 1.645 * 0.0316

CI = (0.630, 0.738)

Therefore, we can be 90% confident that the true proportion of the population who pay for Company ABC's healthcare falls between 0.630 and 0.738. This means that if we were to take repeated random samples from the population and calculate the confidence intervals using the same method, 90% of these intervals would contain the true population proportion.

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To form a triangle, the set of line segments must meet the requirement that the sum of the lengths of any two sides is greater than the length of the third side. In other words, if we have three line segments, a, b, and c, to form a triangle, we need a + b > c, b + c > a, and a + c > b.

Therefore, we cannot determine the set of line segments that meet this requirement based on the information given in the question. We would need more information about the lengths of the line segments in order to determine which sets could form a triangle.

For example, if the set of line segments was {3, 4, 5}, then this set would meet the requirement to form a triangle because 3 + 4 > 5, 4 + 5 > 3, and 3 + 5 > 4. However, if the set of line segments was {1, 2, 6}, then this set would not meet the requirement to form a triangle because 1 + 2 < 6.

In summary, the set of line segments that meet the requirements to form a triangle depends on the lengths of the segments themselves, and more information is needed to answer this question.

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The given polynomial has seven terms.

It should be noted that to count the number of terms in a polynomial, we need to identify each individual term, which are separated by plus or minus signs.

In this case, we can write the polynomial as:

-10x^6 - 2x^5 + 7x^3 - 9x^2 + 4x^1 + 5.5x^1 - 1

The individual terms are:

-10x^6, -2x^5, 7x^3, -9x^2, 4x^1, 5.5x^1, -1

There are seven terms in total.

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De acuerdo con la información, podemos inferir que el valor de la hipotenusa es 40.

Para calcular el valor de la hipotenusa de este triángulo debemos utilizar el Teorema de Pitágoras. Entonces, debemos aplicar esta fórmula:

En este caso, el valor de a sería 32, el valor de b sería 24. Una vez remplazamos los valores debemos solucionar la fórmula para hallar el valor de c (hipotenusa):

Entonces podemos inferir que 40 es el valor de la hipotenusa de este triángulo.

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Yes, we can use z test.

The normal approximation to the binomial can be used for the hypothesis test of population proportion for option b but not for options a, c and d.

The Z-test is used to test the hypothesis about population parameters when the sample size is large and the population standard deviation is known or can be estimated from the sample.

For the case of a hypothesis test for a population proportion,

the Z-test can be used when the sample size is large enough to satisfy the conditions of normal approximation to the binomial distribution.

The conditions for normal approximation to the binomial distribution are:

The sample size, n, is large enough that np ≥ 10 and n(1-p) ≥ 10, where p is the population proportion.

The observations are independent.

Based on these conditions, we can determine whether or not the normal approximation to the binomial should be used for each of the following cases:

a. n=20 and H0:p=0.3.

In this case, np = 20 × 0.3 = 6 and n(1-p) = 20 × 0.7 = 14, both of which are less than 10. Therefore, the normal approximation to the binomial should not be used.

b. n=70 and H0:p=0.2.

In this case, np = 70 × 0.2 = 14 and n(1-p) = 70 × 0.8 = 56, both of which are greater than 10. Therefore, the normal approximation to the binomial can be used.

c. n=100 and H0 : p=0.08.

In this case, np = 100 × 0.08 = 8 and n(1-p) = 100 × 0.92 = 92, both of which are less than 10.

So, the normal approximation to the binomial should not be used.

d. n=150 and H0:p=0.01.

In this case, np = 150 × 0.01 = 1.5 and n(1-p) = 150 × 0.99 = 148.5, both of which are less than 10. Therefore, the normal approximation to the binomial can be used for the hypothesis test of population proportion for option b but not for options a, c and d.

In the cases where the normal approximation cannot be used, alternative methods like the exact binomial test or the chi-square test can be used.

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If we have coins = [1, 2, 5] and amount = 5, then the function should return 4, because there are four combinations of coins that make up an amount of 5: [1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 2, 2], and [5].

This problem can be solved using dynamic programming. Let's define dp[i][j] as the number of combinations of coins using the first i coins to make up an amount j. Then we can use the following recurrence relation:

dp[i][j] = dp[i-1][j] + dp[i][j-coins[i]]

The first term on the right-hand side of the equation corresponds to the case where we don't use the i-th coin, while the second term corresponds to the case where we use the i-th coin at least once. Note that we only need to consider cases where j >= coins[i], because it's impossible to make up an amount less than the value of the i-th coin using that coin.

We can initialize dp[0][0] = 1, because there is exactly one way to make up an amount of zero using no coins. Finally, the answer to the problem is dp[n][amount], where n is the total number of coins.

Here's the Python code:

def change(amount, coins):

   n = len(coins)

   dp = [[0] * (amount+1) for _ in range(n+1)]

   dp[0][0] = 1

   for i in range(1, n+1):

       dp[i][0] = 1

       for j in range(1, amount+1):

           dp[i][j] = dp[i-1][j]

           if j >= coins[i-1]:

               dp[i][j] += dp[i][j-coins[i-1]]

   return dp[n][amount]

For example, if we have coins = [1, 2, 5] and amount = 5, then the function should return 4, because there are four combinations of coins that make up an amount of 5: [1, 1, 1, 1, 1], [1, 1, 1, 2], [1, 2, 2], and [5].

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To determine if there is a significant difference between the mean distance walked for patients using a treadmill and the mean for the control group, we need to conduct a two-sample t-test.

Let us define our hypotheses:

Null hypothesis: The mean distance walked for patients using a treadmill is not significantly different from the mean for the control group.

Alternative hypothesis: The mean distance walked for patients using a treadmill is significantly different from the mean for the control group.

We will use a significance level of α = 0.05, which means we are willing to accept a 5% chance of making a type 1 error (rejecting the null hypothesis when it is true).

Using the Ti-84 plus calculator, we can perform a two-sample t-test by selecting "STAT" then "TESTS" then "2-SampTTest". We will enter the necessary data including the mean distance, standard deviation, and sample size for each group.

The calculator output will give us the t-value and the p-value. If the p-value is less than our significance level, we can reject the null hypothesis and conclude that there is a significant difference between the mean distance walked for patients using a treadmill and the mean for the control group.

In this case, we do not have the actual data for the sample mean and sample size for each group. We only have the mean distance and standard deviation for each group. Therefore, we cannot perform the t-test without additional information.

To determine whether treadmill exercise could improve the walking ability of patients suffering from claudication, we need to perform a hypothesis test using the provided data. The terms involved in this process include:

1. Sample: The group of patients who walked on a treadmill.
2. Control group: The group of patients who did not walk on a treadmill.
3. Mean distance: The average distance walked in six minutes.
4. Standard deviation: A measure of the variation in the distances walked by patients.
5. Level of significance (α): A threshold to determine if there is a significant difference between the two groups.

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The value of the function 5⋅f(1)+5⋅g(9) is -55

Here the graph of function y = f(x) and function y = g(x) is shoen in attached figure.

We need to find the value of expression 5⋅f(1)+5⋅g(9)      .......(1)

From the graph, the value of function f(x) for x = 1 would be,

y = f(1)

y = -5

From the graph, the value of function g(x) for x = 9 would be,

y = g(9)

y = -6

Substitute these values in expression (1),

5⋅f(1)+5⋅g(9)

= (5 × (-5)) + (5 × (-6))

= -25 - 30

= -55

This is the required value of expression.

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the approximation of the number of people who leave the building during the first 15 minutes of the evacuation drill using a right Riemann sum with three subintervals is 375 people.

To approximate the number of people who leave the building during the first 15 minutes of the evacuation drill using a right Riemann sum with three subintervals, we can divide the interval [0, 15] into three subintervals of equal width:

[0, 5], [5, 10], [10, 15]

The right Riemann sum is then given by:

Δt [f(5) + f(10) + f(15)]

where Δt is the width of each subinterval (Δt = 5), f(t) is the rate of people leaving the building at time t (in people per minute), and the values of f(t) are given in the table.

Plugging in the values, we get:

Δt [f(5) + f(10) + f(15)]

= 5 [20 + 25 + 30]

= 5 [75]

= 375

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If the space is two-dimensional: c. 4

If the space is one-dimensional: b. 1

If the space is four-dimensional: a. 7

In rigid-body physics, the location of a rigid body can be specified by three coordinates (x, y, z) in three-dimensional space. The orientation of a rigid body can be specified by three angles (roll, pitch, yaw) or by a rotation matrix.

If the space is two-dimensional, the location of a rigid body can be specified by two coordinates (x, y). The orientation can be specified by one angle or by a 2x2 rotation matrix. Therefore, the total number of coordinates required is 3.

If the space is one-dimensional, the location of a rigid body can be specified by one coordinate (x). Since there is only one dimension, there is no need to specify orientation. Therefore, the total number of coordinates required is 1.

If the space is four-dimensional, the location of a rigid body can be specified by three coordinates (x, y, z) as in three-dimensional space. The orientation can be specified by four parameters, such as quaternions, which require four coordinates. Therefore, the total number of coordinates required is 7.

So, the answers are:

If the space is two-dimensional: c. 4

If the space is one-dimensional: b. 1

If the space is four-dimensional: a. 7

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Answer: (-1,-1)

Step-by-step explanation: i just did this one on a math assignment lol

The value of the given expression is 3.75.

Given is an expression, 3/2·∛16, we need to solve it,

3/2·∛16
= 1.5 × ∛16

= 1.5 × ∛2×2×2×2

= 1.5 × 2∛2

= 1.5 × 2 × 1.25

= 3.75

Hence, the value of the given expression is 3.75.

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If the standard length of one kind of nail is 5 cm, and we want to test whether the nails produced on a particular day fit the standard requirement, we would set up the following hypotheses:

Null hypothesis (H0): The mean length of nails produced on the particular day is equal to the standard length of 5 cm.
Alternative hypothesis (Ha): The mean length of nails produced on the particular day is not equal to the standard length of 5 cm.

To test these hypotheses, we would take a sample of nails produced on the particular day and measure their lengths. We would then calculate the sample mean and compare it to the standard length of 5 cm using a hypothesis test.

In conclusion, If the sample mean is significantly different from the standard length, we would reject the null hypothesis and conclude that the nails produced on the particular day do not meet the standard length requirement. If the sample mean is not significantly different from the standard length, we would fail to reject the null hypothesis and conclude that the nails produced on the particular day meet the standard length requirement.

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Answer:

16 meters

Step-by-step explanation:

Factors of 64: 1, 2, 4, 8, 16, 32, 64

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

The greatest common factor of 64 and 48 is 16.

Answer:

Step-by-step explanation:

To find the length of the longest scale that can measure 64 m and 48 m exactly, we need to find the greatest common factor (GCF) of 64 and 48. The GCF is the largest number that divides both numbers without leaving a remainder. One way to find the GCF is to list the factors of both numbers and find the largest one that they have in common. For example:

Factors of 64: 1, 2, 4, 8, 16, 32, 64

Factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48

The largest factor that both numbers have in common is 16. Therefore, the GCF of 64 and 48 is 16.

This means that the longest scale that can measure both lengths exactly is 16 m. We can check this by dividing both lengths by 16 and seeing that there is no remainder:

1664​=4

1648​=3

When it comes to printing full-color pictures, it usually involves using the CMYK color model which stands for cyan, magenta, yellow, and black.

These colors are combined in different amounts to create a wide range of colors and shades. Therefore, the answer to your question is three plus black. The colors cyan, magenta, and yellow are combined to produce a wide range of colors, while the black is added to enhance contrast and to create darker tones.

This combination of colors provides a more accurate and vibrant representation of the original picture. It is important to note that there are other color models used for printing such as RGB (red, green, and blue) which is used for digital displays, but for printing purposes, CMYK is the most common.

In conclusion, full-color pictures are printed using three colors (cyan, magenta, and yellow) plus black to enhance contrast and create darker tones. This combination of colors provides a more accurate and vibrant representation of the original picture.

This model consists of four ink colors: cyan (C), magenta (M), yellow (Y), and key (black, K). These four inks are combined in various proportions to produce a wide range of colors in the final print. Therefore, the correct answer would be four colors (CMYK), including black.

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Answer: TWO THIRDS

Step-by-step explanation:

Start by determining consonants

M, T, H, S, F, N  (6)

Then find the total

M, A, T, H, I, S, F, U, N (9)

Your probability is (6/9), which simplifies to TWO THIRDS.

The probability that someone is driving slower than 20 mph on the 405 freeway is approximately 7.08%.

We are given that the average speed on the 405 freeway is 42 mph and is normally distributed with a standard deviation of 15 mph.

To solve this problem, we need to use the standard normal distribution since we are given the mean and standard deviation of the speed on the 405 freeway.

We know that

[tex]Z=\frac{X-\mu}{\sigma}[/tex]

Here, X = 20

[tex]\mu=42[/tex]

[tex]\sigma=15[/tex]

Z=(20-42)/15

Z=-22/15

Z=-1.47

Using a standard normal distribution table

P(Z<-1.47)=0.0708

or 7.08%

Therefore, the probability that someone is driving slower than 20 mph on the 405 freeway is approximately 0.0708 or 7.08%.

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The constant of proportionality in the table is 2.50.

In Mathematics and Geometry, a proportional relationship refers to a type of relationship that produces equivalent ratios and it can be modeled or represented by the following mathematical equation:

y = kx

Where:

Next, we would determine the constant of proportionality (k) by using various data points as follows:

Constant of proportionality, k = (7.50 - 5)/(3 -2)

Constant of proportionality, k = 2.50.

Therefore, the required linear equation is given by;

y = kx

y = 2.50x

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This scene is an example of dramatic irony used to create suspense since the audience knows that this joyous occasion will ultimately lead to tragedy.

In the scene, Lord Capulet is preparing for his daughter Juliet's wedding to Paris, while the audience knows that Juliet is already secretly married to Romeo. Lord Capulet's excitement and eagerness to prepare for the wedding create suspense and tension for the audience, who knows that this joyous occasion will ultimately lead to tragedy.

Furthermore, the use of music within the scene also adds to the suspense. The audience hears the music, which signifies the arrival of the wedding party, but also knows that this will lead to the revelation of Juliet's secret marriage.

The urgency in Lord Capulet's instructions to the Nurse to wake up Juliet and make haste heightens the tension for the audience, who are aware of the impending disaster.

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Complete Question:

Read the excerpt from Act IV, scene iii of Romeo and Juliet.

Capulet Good faith! this day:

The county will be here with music straight,

For so he said he would. [Music within.] I hear him near.

35

This scene is an example of dramatic irony used to create suspense since the audience knows that

Kayla ran a total of 3 and 1/3 miles last week.

Kayla ran 2/3 miles each day for 5 days. To find the total distance she ran, we need to multiply the distance she ran each day (2/3 miles) by the number of days she ran (5 days).

So, we can set up the multiplication like this:

Total distance Kayla ran = (2/3 miles) x (5 days)

To multiply fractions, we multiply the numerators (top numbers) together and the denominators (bottom numbers) together.

So, the equation becomes:

Total distance Kayla ran = (2/3) x 5 miles

Multiplying 2/3 by 5 gives us:

Total distance Kayla ran = 10/3 miles

However, we usually want our answer to be in a simplified form. To simplify a fraction, we divide the numerator and denominator by their greatest common factor.

In this case, the greatest common factor of 10 and 3 is 1. So, our simplified answer is:

Total distance Kayla ran = 10/3 miles or 3 and 1/3 miles

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Answer:

As a lawn mower, I can earn $12 per hour without incurring any direct costs. However, my opportunity cost of time varies. For the first three hours, I could have earned $9 per hour doing other activities. Thereafter, my opportunity cost increases to $15 per hour due to other commitments. As such, my total earnings from lawn mowing depends on the number of hours I work, and I should prioritize lawn mowing in the first three hours to maximize my earnings.

By the pigeonhole principle, there must be at least two people at the party.

What is pigeonhole principle?

The pigeonhole principle is a fundamental concept in mathematics that states that if there are n items and k containers, and n > k, then at least one of the containers must contain more than one item.

This is an example of the pigeonhole principle. The pigeonhole principle states that if you have n pigeons and fewer than n pigeonholes, then there must be at least one pigeonhole with more than one pigeon in it.

In this case, we have 19 people and only 18 possible pigeonholes (since you cannot put two people in the same spot).

Therefore, by the pigeonhole principle, there must be at least one pigeonhole (i.e., a spot at the party) with more than one pigeon (i.e., more than one person). In other words, there must be at least two people at the party.

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In this scenario, a computer software magazine compares the rates of malware infection for computers protected by security software A and B.

To construct a 98% confidence interval for the difference between the proportions of infection for the two types of security software packages, follow these steps:

1. Calculate the proportions of infection for each security software (pA and pB):
  pA = 80/800 = 0.1
  pB = 45/900 = 0.05

2. Calculate the combined proportion (pC) and the standard error (SE) for the difference between proportions:
  pC = (80 + 45) / (800 + 900) = 125/1700 = 0.0735
  SE = sqrt((pC * (1 - pC)) * ((1/800) + (1/900))) = 0.0204

3. Find the z-score for a 98% confidence interval (z = 2.33 for a 98% CI).

4. Calculate the margin of error (ME) using the z-score and standard error:
  ME = z * SE = 2.33 * 0.0204 = 0.0475

5. Calculate the confidence interval for the difference between proportions:
  CI = (pA - pB) ± ME = (0.1 - 0.05) ± 0.0475 = 0.05 ± 0.0475 = (0.0025, 0.0975)

The 98% confidence interval is (0.0025, 0.0975), which can be rounded to (0.002, 0.098). This interval does not contain zero, which means it contradicts the claim that the proportion of infections is the same for the two types of security software. Therefore, the answer is (0.02, 0.098); yes.

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The 98% confidence interval for the difference in proportions of patients needing pain medication between the old and new procedures is (-0.003, 0.061).

To construct a 98% confidence interval for the difference in the proportions of patients needing pain medication between the old and new procedures, we can use the formula:

p1 - p2 ± z*sqrt(p1(1-p1)/n1 + p2(1-p2)/n2)

where p1 is the proportion of patients in the old procedure group who required medication, p2 is the proportion of patients in the new procedure group who required medication, n1 is the sample size of the old procedure group, n2 is the sample size of the new procedure group, and z is the critical value for a 98% confidence interval (which is approximately 2.33).

Plugging in the given values, we get:

12/65 - 14/90 ± 2.33sqrt((12/65)(53/65)/65 + (14/90)*(76/90)/90)

Simplifying this expression, we get:

-0.003 < 0.052 < 0.061

Therefore, the 98% confidence interval for the difference in proportions is (-0.003, 0.061).

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The correct statement regarding the 95% confidence interval is given as follows:

The interval is from 57. 2% to 63. 2%. Since the value of 63% lies in the interval, Davenport's claimed acceptance rate seems accurate.

A confidence interval of proportions has the bounds given by the rule presented as follows:

[tex]\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}[/tex]

In which the variables used to calculated these bounds are listed as follows:

The confidence level is of 95%, hence the critical value z is the value of Z that has a p-value of [tex]\frac{1+0.95}{2} = 0.975[/tex], so the critical value is z = 1.96.

The parameters for this problem are given as follows:

[tex]n = 1000, \pi = \frac{602}{1000} = 0.602[/tex]

The lower bound of the interval is given as follows:

0.602 - 1.96 x sqrt(0.602 x 0.398/1000) = 0.572 = 57.2%.

The upper bound of the interval is given as follows:

0.602 + 1.96 x sqrt(0.602 x 0.398/1000) = 0.632 = 63.2%.

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the city grew by approximately 3 percent from 2010 to 2020.

To find out the percent growth from 2010 to 2020, we need to calculate the net percent change over the entire time period. We can start by finding out the population of the city in 2015.

From 2010 to 2015, the population grew by 6.6%. Using this percentage increase, we can find the population in 2015 by multiplying the 2010 population by 1.066:

179,000 x 1.066 = 190,294

Therefore, the population in 2015 was approximately 190,294.

From 2015 to 2020, the population fell by 3.4%. Using this percentage decrease, we can find the population in 2020 by multiplying the 2015 population by 0.966:

190,294 x 0.966 = 183,902

Therefore, the population in 2020 was approximately 183,902.

To find the net percent change from 2010 to 2020, we can use the formula:

[(final value - initial value) / initial value] x 100

Plugging in the numbers we found, we get:

[(183,902 - 179,000) / 179,000] x 100 = 2.7%

Therefore, to the nearest whole number, the city grew by approximately 3% from 2010 to 2020.

The city experienced a period of growth from 2010 to 2015, followed by a period of decline from 2015 to 2020. However, overall, the city still managed to grow by approximately 3% over the entire time period.


1. Calculate the population in 2015 by applying the 6.6% growth:
2015 Population = 179,000 * (1 + 6.6/100) = 179,000 * 1.066 ≈ 190,814

2. Calculate the population in 2020 by applying the 3.4% decline:
2020 Population = 190,814 * (1 - 3.4/100) = 190,814 * 0.966 ≈ 184,302

3. Calculate the overall percentage growth from 2010 to 2020:
Percentage Growth = ((2020 Population - 2010 Population) / 2010 Population) * 100
Percentage Growth = ((184,302 - 179,000) / 179,000) * 100 ≈ 2.96%


Considering the population growth and decline in the respective periods, the city's population grew by approximately 3% from 2010 to 2020, to the nearest whole number.

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The given linear programming problem can be analyzed by examining its constraints and objective function.

We are asked to minimize the objective function Z = 1X + 1Y, subject to the constraints:

1. 5X + 3Y < 30
2. 3X + 4Y > 36
3. Y < 7
4. X, Y > 0

To determine whether the problem exhibits infeasibility, unboundedness, or alternate optimal solutions, we'll analyze its feasible region.

Step 1: Plot the constraints on a graph and find the feasible region.
Step 2: Analyze the feasible region and identify its properties.

After plotting the constraints, we find that there is no common area satisfying all constraints.

This indicates that the problem exhibits infeasibility, meaning there is no solution that satisfies all the constraints simultaneously.

In this case, there are no alternate optimal solutions or unboundedness present since no feasible solution exists.

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Answer: x = 12, -2

Step-by-step explanation:

Answer:

The anser is 12,-2

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a. The mean of the sampling distribution of [tex]\bar{X}[/tex] is 75 and the standard deviation is 0.4.

b. The chance that the average score of this group of students is more than 76 is 0.0062.

c. The probability that the sample mean deviates from the population mean by less than 2 is 1.

d. The 98% confidence interval for µ is (74.07, 75.93).

(a) Mean of  [tex]\bar{X}[/tex] = μ = 75

Standard deviation of [tex]\bar{X}[/tex] = σ/√n = 4/√100 = 0.4

Therefore, the mean of the sampling distribution of [tex]\bar{X}[/tex] is 75 and the standard deviation is 0.4.

(b) z = ([tex]\bar{X}[/tex] - μ) / (σ/√n)

where [tex]\bar{X}[/tex] = 76, μ = 75, σ = 4, and n = 100.

Substituting the values, we get:

z = (76 - 75) / (4/√100) = 2.5

Using a standard normal distribution table, we can find the probability that a z-score is greater than 2.5. This probability is approximately 0.0062. As a result, the chance that the average score of this group of students is more than 76 is 0.0062.

(c) We need to calculate the z-scores for [tex]\bar{X}[/tex] = 75 + 2 = 77 and [tex]\bar{X}[/tex] = 75 - 2 = 73 using the formula:

z = ([tex]\bar{X}[/tex] - μ) / (σ/√n)

Substituting the values, we get:

z1 = (77 - 75) / (4/√100) = 5

z2 = (73 - 75) / (4/√100) = -5

Using a standard normal distribution table, we can find the probability that a z-score is between -5 and 5. This probability is approximately 1. Therefore, the probability that the sample mean deviates from the population mean by less than 2 is 1.

(d) To construct a 98% confidence interval for µ, we can use the formula:

[tex]\bar{X}[/tex] ± zα/2 (σ/√n)

where [tex]\bar{X}[/tex] = 75, σ = 4, n = 100, and zα/2 is the z-score corresponding to the 98% confidence level, which can be found using a standard normal distribution table. For a 98% confidence level, zα/2 = 2.33.

Substituting the values, we get:

75 ± 2.33 (4/√100)

Simplifying, we get:

75 ± 0.93

Therefore, the 98% confidence interval for µ is (74.07, 75.93).

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