Question 2.4. This shows that the percentage in a normal distribution that is at most 1.65 SDs above average is about 95%. Explain why 1.65 is the right number of SDs to use when constructing a 90% confidence interval. (6 Points)

Thus, there is a very high probability (99.93%) that the sample mean would differ from the true mean by less than 3.273 liters, given a sample size of 109 people and a population mean of 152 liters with a variance of 484.

To answer this question, we need to use the Central Limit Theorem, which states that the sampling distribution of the sample means will be approximately normal, regardless of the distribution of the population, as long as the sample size is large enough.

First, we need to calculate the standard error of the mean, which is the standard deviation of the sampling distribution of the sample means. We can use the formula:

standard error of the mean = standard deviation / square root of sample size

Plugging in the values given, we get:

standard error of the mean = √484 / √109
standard error of the mean = 2 / 109
standard error of the mean = 0.018

Next, we need to calculate the z-score, which tells us how many standard errors the sample mean is from the true mean. We can use the formula:

z-score = (sample mean - true mean) / standard error of the mean

Plugging in the values given, we get:

z-score = (sample mean - true mean) / 0.018
3.273 = (sample mean - 152) / 0.018
(sample mean - 152) = 0.018 * 3.273
sample mean = 152 + 0.059
sample mean = 152.059

So the sample mean is 152.059 liters.

Now we can use the standard normal distribution table to find the probability that the z-score is less than 3.273. Looking up the value in the table, we get:

P(z < 3.273) = 0.9993

Therefore, the probability that the sample mean would differ from the true mean by less than 3.273 liters is approximately 0.9993, or 99.93%.

In conclusion, there is a very high probability (99.93%) that the sample mean would differ from the true mean by less than 3.273 liters, given a sample size of 109 people and a population mean of 152 liters with a variance of 484. This probability is based on the Central Limit Theorem and the standard normal distribution table.

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The probability that kid 1 gets at least 3 chocolates is 4/9 or approximately 0.444.

There are a total of [tex]{8+3-1 \choose 3-1} = {10 \choose 2} = 45[/tex] possible ways to divide the chocolates among the 3 kids, using stars and bars method.

Let's calculate the number of ways that kid 1 can get at least 3 chocolates.

If kid 1 gets 3 chocolates, there are [tex]{5+2-1 \choose 2-1} = {6 \choose 1} = 6[/tex]

ways to divide the remaining 5 chocolates between the other 2 kids.

If kid 1 gets 4 chocolates, there are[tex]{4+2-1 \choose 2-1} = {5 \choose 1} = 5[/tex]

ways to divide the remaining 4 chocolates between the other 2 kids.

If kid 1 gets 5 chocolates, there are [tex]{3+2-1 \choose 2-1} = {4 \choose 1} = 4[/tex]

ways to divide the remaining 3 chocolates between the other 2 kids.

If kid 1 gets 6, 7, or 8 chocolates, there are 3, 1, and 1 ways to divide the remaining chocolates, respectively.

Therefore, the total number of ways that kid 1 gets at least 3 chocolates is 6 + 5 + 4 + 3 + 1 + 1 = 20.

The probability that kid 1 gets at least 3 chocolates is the ratio of the number of favorable outcomes (20) to the total number of possible outcomes (45), which is 20/45 = 4/9.

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The method of sampling that is based on dividing a population into subgroups, sampling a set of subgroups, and conducting a complete census within the subgroups sampled is known as cluster sampling.

This method of sampling is often used when the population is too large to be sampled as a whole or when it is difficult to obtain a comprehensive list of the population.

In cluster sampling, the population is divided into smaller subgroups, or clusters, that are more manageable in size. These clusters are then randomly selected, and a complete census is conducted within each of the selected clusters. This means that every individual within the selected clusters is included in the sample.

Cluster sampling can be more efficient and cost-effective than other sampling methods because it reduces the amount of resources required to sample large populations. It is important to note, however, that cluster sampling can lead to increased sampling error if the selected clusters are not representative of the population as a whole.

In summary, cluster sampling is a method of sampling that involves dividing a population into subgroups, sampling a set of subgroups, and conducting a complete census within the selected subgroups. This method can be an effective way to sample large populations, but it is important to ensure that the selected clusters are representative of the population to minimize sampling error.

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Step-by-step explanation:

Divide the 12 by the 3 and you get 4 so multiply the 2 by 4 and you got your answer. Hope this helps.

The number of blue sweets in the bowl are B = 108 sweets

Given data ,

The total number of sweets in the bowl = 675 sweets

Now , ratio of red to blue sweets is 3:7

And , ratio of blue to green sweets is 4:5

So , the ratio of red : blue : green is = 12 : 28 : 35

R      B     G

3       7

        4      5

12      28      35

On simplifying the proportion , we get

75x = 675

Divide by 75 on both sides , we get

x = 9

So , the number of blue sweets is 12x = 12 x 9

B = 108 sweets

Hence , the number of blue sweets is 108

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When selecting a random sample of people to survey, Julio should consider a few important elements. Firstly, he should ensure that his sample is representative of the population he is trying to study. This means that he should try to include a diverse range of people from different ages, genders, socioeconomic backgrounds, and so on.

Additionally, he should consider the size of his sample, as larger samples generally provide more accurate results. Finally, he should strive for a random sample, where each person in the population has an equal chance of being selected, to avoid bias and ensure that his results are generalizable to the larger population. In summary, when selecting a random sample for his survey, Julio should consider factors such as representativeness, sample size, and randomness to ensure accurate and reliable results.


1. Representativeness: Ensure the sample accurately represents the population he's studying, covering various demographics such as age, gender, and socioeconomic status.

2. Sample Size: Choose an appropriate sample size (e.g., 100 people) to ensure the survey results are statistically significant and minimize sampling error.

3. Randomization: Use a random selection method, like a random number generator or drawing names from a hat, to ensure each individual has an equal chance of being chosen.

4. Avoid Bias: Make sure the selection process is free from personal or external influence, so the sample remains truly random and unbiased.

By considering these elements, Julio can ensure his random sample provides accurate and reliable data for his newspaper article.

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The GCD of 25.73 and 54.132 is 3.137, which can be written as option a, 2.5.7.13.

In mathematics, the greatest common divisor (GCD) of two or more integers, which are not all zero, is the largest positive integer that divides each of the integers.

For two integers x, y, the greatest common divisor of x and y is denoted .The greatest common divisor (GCD) of 25.73 and 54.132 can be found by using prime factorization.

25.73 can be written as 5^2.73 and 54.132 can be written as 2^2.3.11.137.
To find the GCD, we need to find the common prime factors of both numbers, which are only 3 and 137.

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Based on sample size, bottling plant B is less likely to record a mean volume of 21 fluid ounces for the day compared to bottling plant A.

To determine which bottling plant is less likely to record a mean volume of 21 fluid ounces for the day, we can use the concept of sample size and standard deviation. As the sample size increases, the standard deviation decreases, which means that the mean is more likely to be close to the population mean of 20 fluid ounces.

In this case, bottling plant B has a larger sample size (177430 sports drinks) than bottling plant A (44688 sports drinks), which means that it is less likely to record a mean volume of 21 fluid ounces for the day.

This is because the larger sample size of bottling plant B will lead to a smaller standard deviation and a more accurate estimation of the population mean.

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the given functions represent transfer functions of two systems in a control system, and they can be analyzed using various tools in the Laplace domain to determine their behavior and characteristics.

The given functions represent the transfer functions of two systems in a control system. The first transfer function, g(s), has two poles at s=1 and one pole at s=10. The poles at s=1 are repeated twice, indicating that this is a second-order system. The second transfer function, h(s), is a constant function with a value of 1.

To analyze the behavior of these systems, we can use tools such as the Laplace transform, which allows us to convert differential equations into algebraic equations that are easier to solve. The Laplace transform of g(s) can be written as G(s) = 1/(s+1)^2(s+10), and the Laplace transform of h(s) is H(s) = 1.

Once we have the transfer functions in the Laplace domain, we can use them to compute various system parameters such as the frequency response, step response, and stability. For example, the frequency response of a system is given by the magnitude and phase of the transfer function evaluated at different frequencies. The step response of a system is the output of the system when a unit step input is applied, and it can be computed using the inverse Laplace transform.

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A system is shown below where g(s) = 1/(s 1)^2(s 10) and h(s) = 1. Find the closed loop transfer function.

Answer:

The mean is the best measure of center because there are no outliers present.

The sample mean is significantly different from 10 at a 0.1 significance level.

The null hypothesis is a statement about the population parameter that we are testing, and the alternative hypothesis is the complement of the null hypothesis. The significance level is the probability of rejecting the null hypothesis when it is true.

In this case, we can use the following null and alternative hypotheses:

Null hypothesis: The population mean is equal to 10.

Alternative hypothesis: The population mean is not equal to 10.

We will use a two-tailed test because the alternative hypothesis is not directional.

We can use the t-test to test the null hypothesis. The t-test statistic is calculated as follows:

t = (sample mean - hypothesized mean) / (sample standard deviation / sqrt(sample size))

In this case:

[tex]t = (11 - 10) / (3.2 / \sqrt{10} ) = 1.58[/tex]

We need to find the critical value of t at the 0.1 significance level with 9 degrees of freedom (10 - 1 = 9). We can use a t-table or a statistical software to find this value. The critical value is ±1.833.

Since the calculated t-value of 1.58 is not greater than the critical value of ±1.833, we cannot reject the null hypothesis. We do not have sufficient evidence to conclude that the population mean is different from 10 at the 0.1 significance level.

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The rate of the plane in still air is 550 mph, and the rate of the wind is 50 mph.

Let's denote the rate of the plane in still air as "p" and the rate of the wind as "w".

When the plane flies into the wind, its effective speed is reduced by the speed of the wind.

So, the speed of the plane relative to the ground is:

p - w

Similarly, when the plane flies with the wind, its effective speed is increased by the speed of the wind, so its speed relative to the ground is:

p + w

We know that the distance traveled by the plane is 3000 miles in both cases, so we can set up two equations based on the formula:

distance = rate x time

When the plane flies into the wind:

3000 = (p - w) x 6

And when the plane flies with the wind:

3000 = (p + w) x 5

Now we have two equations with two unknowns, which we can solve for "p" and "w".

Let's start by simplifying both equations:

Equation 1: 3000 = 6p - 6w

Equation 2: 3000 = 5p + 5w

We can then solve for one of the variables in terms of the other. For example, we can solve for "p" in terms of "w" by rearranging Equation 2:

5p = 3000 - 5w

p = (3000 - 5w) / 5

We can then substitute this expression for "p" into Equation 1 and solve for "w":

3000 = 6[(3000 - 5w) / 5] - 6w

Multiplying both sides by 5:

15000 = 6(3000 - 5w) - 30w

Distributing the 6:

15000 = 18000 - 30w - 30w

Combining like terms:

15000 = 18000 - 60w

Subtracting 18000 from both sides:

-3000 = -60w

Dividing both sides by -60:

w = 50

Now that we know the rate of the wind is 50 mph, we can substitute this value into either Equation 1 or Equation 2 to solve for "p".

Let's use Equation 2:

3000 = 5p + 5(50)

3000 = 5p + 250

2750 = 5p

p = 550.

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The level curves of F(x,y) = C give the implicit general solutions to the differential equation.To find My and Nx, we need to rearrange the differential equation in the form of Mdx + Ndy = 0, where M and N are functions of x and y.

Starting with the given differential equation:

mdx + ndy = 0

We can plug in the values of m and n from the equation given:

(4x + 5y)dx + (5x + 5y)dy = 0

Now we can identify M and N:

M = 4x + 5y

N = 5x + 5y

Therefore:

My = dN/dy = 5

Nx = dM/dx = 4

Since My is not equal to Nx, the equation is not exact. To find the function F(x,y), we can use an integrating factor.

First, we find the integrating factor, u:

u(x) = e^(∫(My - Nx)/Nx dy) = e^(∫(5-4)/4 dy) = e^(y/4)

Multiplying both sides of the differential equation by u(x):

e^(y/4)(4x + 5y)dx + e^(y/4)(5x + 5y)dy = 0

We can rewrite this as:

d(e^(y/4)(4x + 5y)) = 0

Integrating both sides with respect to x:

e^(y/4)(4x + 5y) = F(y)

Therefore:

F(x,y) = e^(y/4)(4x + 5y)

Therefore, The level curves of F(x,y) = C give the implicit general solutions to the differential equation.

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A random sample of the heights (in inches) of seventh grade boys was taken the five number summary for the sample is (59, 61, 64, 67, 72) then the IQR is 6.

The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1).

To find Q1 and Q3, we need to use the five-number summary given in the problem:

Min = 59

Q1 = 61

Median = 64

Q3 = 67

Max = 72

Therefore, the IQR is:

IQR = Q3 - Q1

= 67 - 61

= 6

Hence, a random sample of the heights (in inches) of seventh grade boys was taken the five number summary for the sample is (59, 61, 64, 67, 72) then the IQR is 6.

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The probability of Carlos randomly selecting an orange block from the box is 2/53 or approximately 0.038

The probability of Carlos randomly selecting an orange block from the box can be calculated as follows:

First, we need to determine the total number of blocks in the box:

Total number of blocks = 48 brown + 3 yellow + 2 orange = 53 blocks

Next, we can calculate the probability of selecting an orange block:

Probability of selecting an orange block = Number of orange blocks / Total number of blocks

Probability of selecting an orange block = 2 / 53

Therefore, the probability of Carlos randomly selecting an orange block from the box is 2/53 or approximately 0.038 (rounded to three decimal places).

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Answer:

Graph C corresponds to x<10.

The probability of selecting three cans of apple juice is 286/10000.

The probability of selecting three cans of apple juice is calculated as follows:

Data given:

an ice chest contains 8 cans of apple juice, 6 cans of grape juice, 4 cans of orange juice, and 5 cans of mango juice.

The probability of selecting an apple juice can on the first draw = 8/23

Without replacement, the probability of selecting another apple juice can on the second draw = 7/22

Without replacement, the probability of selecting a third apple juice can on the third draw = 6/21

The probability of selecting three cans of apple juice will be:

(8/23) * (7/22) * (6/21) = 0.0286 or 286/10000

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Answer:

Step-by-step explanation:

16.5

Hypertension (high blood pressure) you first sit down but it drops to normal levels within three minutes, this may be an indication of orthostatic hypertension or postural hypotension.

Orthostatic hypertension is a condition in which blood pressure increases when a person assumes an upright posture, such as standing up from a seated position.

In some people, this can lead to a temporary increase in blood pressure that then drops back down to normal levels within a few minutes.

Postural hypotension is a related condition in which blood pressure drops significantly when a person assumes an upright posture, leading to symptoms such as dizziness, lightheadedness, and fainting.

This can be caused by a variety of factors, including dehydration, medications, and underlying health conditions.

It is important to consult with a healthcare provider if you are experiencing episodes of orthostatic hypertension or postural hypotension, as these conditions may be indicative of an underlying health problem that requires treatment.

Your healthcare provider may recommend lifestyle changes, medications, or other interventions to help manage your blood pressure and prevent complications.

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The lowest term in the expression  is 1/2

Option C is the correct answer.

We have,

Expression 2/3 - 1/6.

= 4/6 - 1/6 = 3/6

We can simplify 3/6 by dividing both the numerator and denominator by 3 to get:

3/6 = 1/2

Therefore,

The lowest term in the expression  is 1/2

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In the book "The Alchemist" by Paulo Coelho, when the owner of the crystal shop tells Santiago how expensive it is to get to Egypt, Santiago responds by saying that he wants money to buy some sheep.

Santiago explains that he was a shepherd before he started his journey to find his Personal Legend, and that he wants to buy some sheep so that he can continue pursuing his dream. He believes that he can eventually save enough money from selling the wool to fund his journey to Egypt.

The owner of the crystal shop is initially surprised that Santiago would abandon his quest for treasure to buy sheep, but ultimately gives him the money he needs to buy them.

After the owner of the crystal shop tells Santiago how expensive it is to get to Egypt, Santiago says he wants money for a ticket to Africa.

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To draw a normal curve with a mean (µ) of 61 and a standard deviation (σ) of 14, you can follow these steps:

1. Begin by plotting the mean (µ = 61) on the horizontal axis.
2. Mark one standard deviation above and below the mean (µ + σ = 61 + 14 = 75 and µ - σ = 61 - 14 = 47).
3. Draw a symmetric bell-shaped curve around the mean, with the curve extending toward the points marked in step 2.

The normal distribution curve is centered at the mean (µ = 61) and has two points of inflection representing µ - σ (47) and µ + σ (75). The curve is symmetric, with its highest point at the mean. As we move further away from the mean in either direction, the curve tapers off and approaches the horizontal axis without actually touching it. Approximately 68% of the data will fall within one standard deviation (between 47 and 75), 95% within two standard deviations, and 99.7% within three standard deviations of the mean.

To choose the correct graph of the normal curve, look for a bell-shaped curve that is symmetric around the mean (61), with points of inflection at 47 and 75. The graph should also reflect the decreasing probability density as you move further from the mean.

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After an hour, the percentage of alcohol in the vat is approximately 4.92%.

After an hour of pumping beer with 6% alcohol into the vat at a rate of 5 gallons per minute and simultaneously pumping the mixture out at the same rate, the percentage of alcohol in the vat can be calculated as follows:

Initially, the vat has 500 gallons of beer with 4% alcohol content. This means there are 20 gallons of alcohol in the vat (500 * 0.04).

Now, let's consider the beer being pumped in at a rate of 5 gallons per minute with a 6% alcohol content. Over the course of an hour (60 minutes), this amounts to 300 gallons of beer, containing 18 gallons of alcohol (300 * 0.06).

As the mixture is being pumped out at the same rate, only 500 gallons of beer remain in the vat after an hour. The total alcohol in the vat is the sum of the alcohol from the initial beer and the pumped-in beer, minus the amount of alcohol pumped out. The mixture pumped out contains the same proportion of alcohol as the mixture in the vat, so it amounts to (5 gallons/minute * 60 minutes) * X%, where X% is the percentage of alcohol in the mixture at the end of an hour.

To find X%, we can set up the following equation:

20 (initial alcohol) + 18 (alcohol pumped in) - 5 * 60 * X% = 500 * X%

Solving for X, we get:

38 - 300 * X% = 500 * X%

Rearranging and solving for X, we find that:

X% ≈ 4.92%

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Calculate Peter's speed in meters per second: 1256.64 meters / 480 seconds ≈ 2.618 m/s

To determine Peter's speed, first find the circumference of the circular track using the formula: C = πd. In this case, the diameter (d) is 100 meters.

C = π(100) ≈ 314.16 meters

Peter runs 4 laps in 8 minutes, which means he runs 4 * 314.16 = 1256.64 meters in 8 minutes.

Now convert 8 minutes to seconds: 8 minutes * 60 seconds/minute = 480 seconds

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The approximate probability of getting a z-score between +1 standard deviation and +2.5 standard deviations in a standard normal distribution is 0.1525.

To find the area under the standard normal curve between +1 standard deviation and +2.5 standard deviations, we can use a standard normal distribution table or calculator. Here are the steps:

Therefore, the area under the standard normal curve between +1 standard deviation and +2.5 standard deviations is approximately 0.1525.

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Thus, total, we have 6 + 4 + 2 = 12 four-digit integers that  that are not multiples of 1111 do the digits form an arithmetic sequence.

Let's first understand the criteria for the four-digit integers:

1. They are not multiples of 1111.
2. Their digits form an arithmetic sequence.

Now, let's analyze the four-digit integers. We can represent them as ABCD, where A, B, C, and D are the individual digits. Since the digits form an arithmetic sequence, we can represent this sequence as A, A+d, A+2d, and A+3d. Here, "A" is the first digit, "d" is the common difference, and A ≠ 0.

Since A+3d is the last digit, it must be less than 10. So, A+3d < 10. Given that A ≥ 1, we can deduce that d can take the values 1, 2, or 3.

Now let's consider each value of "d":

1. If d = 1, A can range from 1 to 6. This results in 6 possible four-digit integers.
2. If d = 2, A can range from 1 to 4. This results in 4 possible four-digit integers.
3. If d = 3, A can range from 1 to 2. This results in 2 possible four-digit integers.

In total, we have 6 + 4 + 2 = 12 four-digit integers that meet the given criteria.

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Thus, the sides of the rectangle are 10 and 12, and the area is indeed 120.

To solve this problem, we can use the formula for the area of a rectangle, which is A = L x W, where A is the area, L is the length, and W is the width.

In this case, we know that the length is x+7 and the width is x+5, and the area is 120. So we can write:

120 = (x+7)(x+5)

Expanding the brackets, we get:

120 = x^2 + 12x + 35

Rearranging, we get:

x^2 + 12x - 85 = 0

Now we can solve for x using the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = 12, and c = -85, so we get:

x = (-12 ± √(12^2 - 4(1)(-85))) / 2(1)
x = (-12 ± √(144 + 340)) / 2
x = (-12 ± √484) / 2
x = (-12 ± 22) / 2

So the two possible values of x are -17 and 5. However, we can see that the length and width of the rectangle must be positive, so we can reject the negative value and conclude that:

x = 5
Therefore, the sides of the rectangle are 10 and 12, and the area is indeed 120.

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The  area enclosed by the parabolas y² = 32(x + 8) and y² = -32(x - 6) is (64√3)/9.

We can start by simplifying the equations of the parabolas:

y² = 4(7 + 1)(x + 7 + 1)
y² = 4(7 + 1)(7 + 1 - x)

y² = 32(x + 8)
y² = -32(x - 6)

We can see that the parabolas are symmetrical with respect to the y-axis, so we only need to consider the region to the right of the y-axis. To find the points of intersection between the parabolas, we can set the right-hand sides of the equations equal to each other:

32(x + 8) = -32(x - 6)

Expanding and simplifying, we get:

64x + 256 = 192 - 32x
96x = -64
x = -2/3

So the parabolas intersect at x = -2/3. We can find the corresponding y-values by plugging this value of x into either equation:

y² = 32(x + 8)
y² = 32(-2/3 + 8)
y² = 256/3
y = ±(16√3)/3

So the points of intersection are (-2/3, (16√3)/3) and (-2/3, -(16√3)/3).

To find the area between the parabolas, we can integrate the difference between their y-values with respect to x:

A = ∫[0, -2/3] (16√3)/3 - (-16√3)/3 dx + ∫[-2/3, 0] (-16√3)/3 - (16√3)/3 dx
A = ∫[-2/3, 0] 32√3/3 dx
A = (32√3)/3 ∫[-2/3, 0] dx
A = (32√3)/3 [(0) - (-2/3)]
A = (64√3)/9

Therefore, the area enclosed by the parabolas y² = 32(x + 8) and y² = -32(x - 6) is (64√3)/9.

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For a rectangular granite slab purchased by Bob with area 18 sq. feet who replacing the laminate countertop, the width of slab is three feet.

We have a homeowner is replacing the laminate countertop with granite. For this he purchased a slab of granite that is 18 square feet. That is area of slab = 18 sq. feet

Length of slab, l = 6 feet

We have to determine the wide of slab. The shape of slab is rectangular. So, we have calculate the width of rectangular slab. Let the width be equal to ' x feet' . As we know, area of rectangle is written as product of length and width of rectangle. Thus, Area = l × x

=> 18 feet² = 6 feet × x

dividing both sides by 6

=> x = 3 feet

Hence, required width is 3 feet.

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a. The experiment is observing the number of computers sold tomorrow. The sample space consists of all possible outcomes, which in this case are: {0, 1, 2, 3, 4, 5}.
b. To define the event of selling more than three computers using set notation, we look for outcomes in the sample space where the number of computers sold is greater than 3. This would be the set: {4, 5}.
c. The probability of selling five computers is given in the question as 0.10 (or 10%).

a. The sample space would be {0, 1, 2, 3, 4, 5}, as these are the possible numbers of computers that could be sold tomorrow.

b. The event of selling more than three computers can be defined using set notation as {x | x > 3}, which means the set of all numbers (x) that are greater than 3.

c. The probability of selling five computers would be the probability of the event {5}, which is a singleton set (a set with only one element). Since each number in the sample space has an equal chance of occurring, and there are six possible outcomes in the sample space, the probability of selling five computers would be 1/6.


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