HELP!!!!! Please. I really need it.

Answer:

9 and 40investment is costing 542

Volume is a three-dimensional scalar quantity. The correct option is B, (4/3)π(10)³ - (4/3)π(4)³.

We have been given that a large sphere has a diameter of 20 feet. A smaller sphere with a radius of 4 feet is cut out of the center of the larger sphere. We are asked to find the volume outside smaller sphere and inside larger sphere.

A volume is a scalar number that expresses the amount of three-dimensional space enclosed by a closed surface.

The volume, in cubic units, of the shaded part of the sphere is the difference between the volume of the larger sphere and the smaller sphere. Therefore, the volume can be written as,

The volume of the sphere = (4/3)π(10)³ - (4/3)π(4)³

Hence, the correct option is B, (4/3)π(10)³ - (4/3)π(4)³.

Complete Question:

The large sphere has a diameter of 20 feet. A large sphere has a diameter of 20 feet. A smaller sphere with a radius of 4 feet is cut out of the center of the larger sphere. Which expression represents the volume, in cubic units, of the shaded part of the sphere? Four-thirdsπ(103) + Four-thirdsπ(43) Four-thirdsπ(103) – Four-thirdsπ(43) Four-thirdsπ(203) + Four-thirdsπ(43) Four-thirdsπ(203) – Four-thirdsπ(43)

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Answer:

and H(3,7), I(2,2), J(5,2).

To solve for the perimeter, we need to find the distance between each pair of consecutive points and add them up.

The distance between E and F is 3 units, between F and G is approximately 5.83 units, between G and H is 5 units, between H and I is approximately 5.83 units, and between I and J is 3 units.

So the perimeter of EFG is approximately 22.66 units.

Now, we need to check if HIJ has the same side lengths.

The distance between H and I is 5 units, between I and J is 3 units, and between J and H is approximately 5.83 units.

Therefore, HIJ does not have the same side lengths as EFG since their perimeters are different.

(a) The experimental probability of rolling a 3 is approximately 8%.

(b) The experimental probability of rolling a 6 is 25%.

(c) The experimental probability of rolling a number less than 4 is 50%.

The experimental probability of rolling a 3 can be determined from the result presented in the table as shown below.

from the result presented, there a total outcome of 12

number of rolling a 3 in the result = 1

P(3) = 1/12 = 0.083

P(3) ≈ 8%

The experimental probability of a rolling a 6 is calculated as;

number of 6 obtained in the result = 3

P(6) = 3/12

P(6) = 0.25

P(6) = 25%

The experimental probability of rolling a number less than 4:

P ( less than 4) = P(1) + P(2) + P(3)

P ( less than 4) = 17% + 25% + 8%

P ( less than 4) = 50%

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The probability that the water in Willow Pond will have a pH of 7 in at least 4 out of 5 tests is approximately 0.0278 or 2.78%

This means that it is not very likely that the pond water will have a pH of 7 in at least 4 out of Laura's 5 tests.

Laura's simulation involves picking a shell from a bag with 3 green shells and 7 white shells.

Each pick represents one of Laura's tests, and picking a green shell represents a pH of 7.

Based on the results of 300 trials, we can calculate the probability of getting at least 4 green shells (pH of 7) out of 5 tests using the binomial distribution formula:

P(X >= 4) = 1 - P(X < 4)

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

where X is the number of green shells (pH of 7) in 5 tests.

Using the table, we can see that P(X=0) = 50/300, P(X=1) = 109/300, P(X=2) = 92/300, and P(X=3) = 40/300.

Therefore,

P(X >= 4) = 1 - (50/300 + 109/300 + 92/300 + 40/300) = 0.0278.

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Answer: The median changed the most.

Old median = 79.5

new median = 86.5

===============================================

Explanation:

To find the mean, we add up the values and divide by 12 since there are 12 numbers in this list.

mean = (add up the values)/(number of values)

mean = (58+61+71+77+91+100+105+102+95+82+66+57)/12

mean = 80.41667 approximately

--------

To get the median, we need to sort the numbers from smallest to largest

57, 58, 61, 66, 71, 77, 82, 91, 95, 100, 102, 105

There are n = 12 items in this set.

Because n = 12 is an even number, the median is between slots n/2 = 12/2 = 6 and 7

The midpoint of those values is (77+82)/2 = 79.5 which is the median.

--------

The range is the difference between the min and max

range = max - min = 105 - 57 = 48

The IQR will involve splitting the sorted set into two halves

L = lower half = stuff below the median

L =  {57, 58, 61, 66, 71, 77}

U = upper half = stuff above the median

U = {82, 91, 95, 100, 102, 105}

The median of set L is (61+66)/2 = 63.5 which is the value of Q1.

The median of set U is (95+100)/2 = 97.5 which is the value of Q3

IQR = interquartile range

IQR = Q3 - Q1

IQR = 97.5 - 63.5

IQR = 34

--------

Here is a summary of what we calculated

If we were to replace the "71" with "93", and redo the calculations, then we'll get these results:

The range and IQR stay the same, but the mean and median values are different.

Let's see which of those two values changed the most.

The median has changed the most because the +7 is larger than +1.83333

The required probability of 911 calls are as follow,

For all six calls are emergency is equals to 0.000001.

For two or more calls are not emergencies is 0.999999.

Smallest number of calls for at least one emergency call is 20.

Probability that all six calls are emergencies is,

P(all six are emergencies) = (0.15)^6

                                           ≈ 0.000001

Probability that all six calls are emergencies is approximately 0.000001.

Probability that two or more calls are not emergencies,

Calculate it by the complement of the probability that all six calls are emergencies, and then subtracting this from 1,

P(at least two are not emergencies)

= 1 - P(all six are emergencies)

= 1 - (0.15)^6

≈ 0.999999

Probability that two or more calls are not emergencies is approximately 0.999999.

Let X be the number of calls that need to be answered to be at least 96%.

At least one call is an emergency.

Calculate the smallest value of X ,

P(at least one call is an emergency) ≥ 0.96

Using the complement rule,

P(no call is an emergency) ≤ 1 - 0.96 = 0.04

Each call is independent,

Probability that no call is an emergency can be calculated as,

P(no call is an emergency) = 0.85^X

Solve the inequality,

0.85^X ≤ 0.04

Taking logarithms of both sides, we get,

X log(0.85) ≤ log(0.04)

X ≥ log(0.04) / log(0.85)

X ≥ 19.83

Smallest number of calls that the 911 operators need to answer to be at least 96% sure that at least one call is an emergency is 20.

Therefore, the required probability for the 911 calls with 85% not emergency is given by ,

probability of all six calls are emergency is 0.000001.

probability of two or more calls are not emergencies is 0.999999.

required smallest number of calls for at least one emergency call is 20.

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The distance between the van and Smithville is changing at a rate of  14.50 miles per hour

1. Calculate the total distance between the van and Smithville by subtracting the distance the van has travelled (121 miles) from the total distance the van is from Smithville (186 miles): 186- 121 = 65 miles.

2. Calculate the rate of change by dividing the total distance (65 miles) by the amount of time it has taken the van to travel that distance (121 miles/27 miles per hour = 4.48 hours): 65/4.48= 14.50 miles per hour.

3. Round the rate of change to the nearest hundredth: 14.50 rounded is 14.5.

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The statements that are True about the Confounding variables are :

(a) Variables can make the relationship look different than it really is,

(b) Variables that are associated with both exposure and outcome.

The Confounding-Variables are defined as variables that are associated with both the exposure and outcome variables in a study.

These variables can make the relationship between the exposure and outcome variables look-different than it really is, which can lead to biased results.

The confounding variables are associated with both the exposure and outcome variables it means that if the relationship between the exposure and outcome variables is not properly adjusted for the confounding variable, the effect of the exposure on the outcome may be overestimated or underestimated.

Therefore, the correct options are (a) and (b).

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The given question is incomplete, the complete question is

Which of the following is/are true about confounding variables?

(a) Variables can make the relationship look different than it really is

(b) Variables that are associated with both exposure and outcome

(c) Variable that decreases the independent variable's impact on the dependent variable

(d) Variables that can increase the chance of a type ii error while eliminating the chance of a type i error

a. Yes, the estimate of age based on 500 plots is influenced by sampling error.

Sampling error occurs because the sample is only a portion of the entire population, and the sample may not perfectly represent the whole population.

Therefore, the estimate of the mean age derived from the sample might deviate from the true mean age of the entire population.

b. If the investigators had used a sample of only 100 plots instead of 500, the sampling error of the estimate of mean age would likely increase. As the sample size decreases, the likelihood of the sample representing the whole population accurately decreases as well.

Consequently, the deviation between the sample means age and the true population mean age would likely be larger, leading to a higher sampling error.

Sampling error refers to the difference or discrepancy between a sample's characteristics and the corresponding characteristics of the entire population from which the sample was drawn. It arises due to the fact that a sample is only a subset of the population, and the sample may not perfectly represent the whole population. Sampling error can impact the accuracy of statistical inferences and conclusions drawn from the sample, such as estimates of mean, variance, or correlation.

As the sample size decreases, the likelihood of the sample representing the whole population accurately decreases as well, leading to higher sampling error.

Therefore, minimizing sampling error is crucial in ensuring the validity and reliability of research findings.

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The volume of the cube-shaped  = 1,331 volume of the hole cut out of the block = 125 cubic inches Subtract 1331 - 125 = 1206, the correct option is A

volume of the hole cut out of the block = 5 inches x 5 inches x 5 inches

volume of the hole cut out of the block = 125 cubic inches.

volume of the cube-shaped block = 11 inches x 11 inches x 11 inches

volume of the cube-shaped block = 1,331

To find the volume of the wood remaining after the hole is cut out, we can subtract the volume of the hole from the volume of the block:

1,331 cubic inches - 125 cubic inches = 1,206 cubic inches

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The best question to determine the evidence for the given hypotheses is "If we examine the proportion of students at their campus who still live at home with their parents, how likely is that proportion to be more than 36%?" so, the correct option is D).

This question directly relates to the alternative hypothesis which states that the proportion of millennial students living at home with their parents is more than 36%.

Therefore, we need to determine the probability of observing a proportion of 36% or higher in a sample of 300 students if the true proportion is actually 36%.

So, the correct answer is D).

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--The given question is incomplete, the complete question is given

" Question 3 Select one answer Living with parents: The Pew Research Center reported that 36% of American Mille nnials (adults ages 18-31) still live at home with their parents. 10 points A group of students wants to conduct a study to determine whether this result is true for students at their campus. They survey 300 randomly selected students at their campus and determine that 43% of them live at home with their parents. With this data, they test the following hypotheses H o: Of Mille nnial students at their campus, 36% live at home with their parents. Ha: More than 36% of Millennial students at their campus live at home with their parents In order to assess the evidence, which question best describes what we need to determine?

A. If we examine a sample of students at their campus and determine the proportion who  still live at home with their parents, how likely is that proportion to be 36%?

B. If we examine a sample of students at their campus and determine the proportion who  still live at home with their parents,  how likely is that proportion to be 43% or more?

C.  If we examine the proportion of students at their campus who  still live at home with their parents, still live at home with their how likely is that proportion to be more than 36%?

D. IF if we examine the proportion of students at their campus who still live at home with their parents,  how likely is that proportion to be 36%?

E. If we examine the proportion of students at their campus who still live at home with their still live at home with their parents, how likely is that proportion to be 43%?"--

An event where two or more things happen at the same time is called B. Compound event

The term that describes an event where two or more things happen at the same time is a compound event.

A compound event is an event that involves two or more independent events occurring at the same time. For example, tossing a coin and rolling a dice at the same time is a compound event.

In contrast, a dependent event is an event where the outcome of one event affects the outcome of another event.

An organized list is a method used to determine the total number of possible outcomes of an event, usually for small sets of outcomes. It involves listing all possible outcomes in an organized manner.

Therefore, in the context of the question, the correct answer is B. Compound event.

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The car traveled at a faster rate in the last six minutes compared to the first six minutes.

In the first six minutes, the car traveled 2 miles, which means it had a speed of (2 miles)/(6 minutes) = 0.33 miles per minute.

In the last six minutes, the car traveled 8 miles, which means it had a speed of (8 miles)/(6 minutes) = 1.33 miles per minute.

Comparing the two speeds, we can see that the car traveled at a faster rate in the last six minutes. This could be because the driver increased the speed or drove on a more favorable road surface. It is also possible that the driver had to slow down in the first six minutes due to traffic or other road conditions.

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The exact probability that a student response is from a freshman, given the student prefers to watch reality shows on television, is approximately 46.15%.

To solve this question, we'll use Bayes' theorem:

P(Freshman | Reality Show) = (P(Reality Show | Freshman) * P(Freshman)) / P(Reality Show)

We know:

We need to find P(Reality Show), which is the probability that a randomly chosen student prefers reality shows. We can find this by adding the probability of each class preferring reality shows:

P(Reality Show) = P(Reality Show & Freshman) + P(Reality Show & Sophomore) + P(Reality Show & Junior) + P(Reality Show & Senior)

We can calculate each probability by multiplying the probability of the class preferring reality shows with the probability of the class:

P(Reality Show & Freshman) = P(Reality Show | Freshman) * P(Freshman) = 0.6 * 0.3 = 0.18

P(Reality Show & Sophomore) = P(Reality Show | Sophomore) * P(Sophomore) = 0.4 * 0.25 = 0.1

P(Reality Show & Junior) = P(Reality Show | Junior) * P(Junior) = 0.3 * 0.2 = 0.06

P(Reality Show & Senior) = P(Reality Show | Senior) * P(Senior) = 0.2 * 0.25 = 0.05

P(Reality Show) = 0.18 + 0.1 + 0.06 + 0.05 = 0.39

Now we can calculate the probability:

P(Freshman | Reality Show) = (P(Reality Show | Freshman) * P(Freshman)) / P(Reality Show) = (0.6 * 0.3) / 0.39 ≈ 0.4615

The exact probability that a student response is from a freshman, given the student prefers to watch reality shows on television, is approximately 46.15%.

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There are 1000 students in total:

300 freshmen

250 sophomores

200 juniors

250 seniors

The preferences for watching reality shows are as follows:

60% of freshmen prefer reality shows

40% of sophomores prefer reality shows

30% of juniors prefer reality shows

20% of seniors prefer reality shows

What is the exact probability that a student response is from a freshman, given the student prefers to watch reality shows on television, considering the above student population and preferences?

Step-by-step explanation:

Sin(30 degrees) = 1/2

The value of sin 30° is 1/2 and it lies in 1 st quadrant.

Trigonometry is a branch of mathematics that studies relationships between side lengths and angles of triangles.

sin 30° can be calculated using the unit circle or a calculator that has a sin function.

In the unit circle, 30° is in the first quadrant, and the sine of an angle in the unit circle is defined as the y-coordinate of the point on the circle that corresponds to that angle.

For a 30° angle, the point on the unit circle is (cos 30°, sin 30°)

= (√3/2, 1/2).

Therefore, the value of sin 30° is 1/2.

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the amount of money available can be represented by the following inequality: 5a + 7b ≤ 200

What is inequality?

Mathematical expressions with inequalities on both sides are known as inequalities. In an inequality, we compare two values as opposed to equations. In between, the equal sign is changed to a less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

Since the aprons with 2 pockets cost $5 each and the aprons with 4 pockets cost $7 each, the constraint on the number of aprons that can be purchased based on

Hence, the amount of money available can be represented by the following inequality: 5a + 7b ≤ 200

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Answer:

9/8, 1.125, or 1 1/8

Answer: [tex]\frac{21}{8}[/tex] or 2.625

Step-by-step explanation:

Improper fractions

First, we want to convert 1 3/4 to an improper fraction.

We multiply the denominator(4)  by the coefficient (1) and add it to the numerator (3).

This gets us [tex]\frac{7}{4}[/tex]

Keep change flip

When dividing fractions, a helpful thing to remember is "keep change flip"

This means you keep the first number as it is, change the division sign to multiply, and flip the divisor, in this case 2/3, to be 3/2.

So, the problem becomes [tex]\frac{7}{4} *\frac{3}{2}[/tex]

Now, you jut multiply the numerators and denominators

[tex]\frac{7*3}{4*2} =\frac{21}{8}=2.625[/tex]

Answer: y = ax2 + bx + c

Step-by-step explanation:

Step 1: The equation of any parabola is given by y = ax2 + bx + c, where a, b, and c are constants.

Step 2: We can calculate the axis of symmetry (x-coordinate) by using the formula x = -b/2a.

Step 3: We can calculate the value of a by looking at the width of the parabola.

Step 4: Once we have the values of a and b, we can substitute them into the equation to get the equation of the parabola: y = ax2 + bx + c.

Answer:

(b/2)² + 1 = x² - 3x + (b/2)²

b²/4 + 1 = x² - 3x + b²/4

b²/4 - b²/4 = x² - 3x - 1

x² - 3x - 1 = 0

x² - 3x = 1

x² - 3x + 9/4 = 1 + 9/4

(x - 3/2)² = 13/4

x - 3/2 = ±(√13/4)

x = ±√13/2 + 3/2

so the number that must be added will be 3/2

Hence, the circle's basic shape is as follows: [tex]x^2 - 30x + y^2 - 14y + 199=0[/tex] as the equation for a circle from standard form to general form.

All the points on a planar that are equally spaced from a specific point known as the circle's centre make up the geometric shape known as a circle. The radius of a circle is the separation between the centre and any point along its circumference. The fact that all of a circle's radii (plural of radius) are the same length, that the inner diameter (the distance it around circle) is proportional to the diameter (the length across the circle passing through the centre), and that the area enclosed by such a circle is inversely proportional to the square of its radius are just a few of the many significant properties of circles. From geometry and mathematics via architecture, art, and science, circles are employed in a variety of disciplines.

given

We must expand the square terms and simplify the equation in order to translate the equation for a circle from standard form to general form.

Using the circle's conventional form as a starting point:

[tex](x - 15)^2 + (y - 7)^2 = 25[/tex]

Adding phrases to the squares

[tex]x^2 - 30x + 225 + y^2 - 14y + 49 = 25[/tex]

Simplifying by grouping together all the terms:

[tex]x^2 - 30x + y^2 - 14y + 199 = 0[/tex]

Hence, the circle's basic shape is as follows: [tex]x^2 - 30x + y^2 - 14y + 199=0[/tex] as the equation for a circle from standard form to general form.

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A complete graph is one in which there is an edge connecting every vertex to every other vertex. In order to determine for what values of n a complete graph with n vertices has an Euler circuit and a Hamiltonian circuit, let's discuss the definitions and requirements of each type of circuit.

1. Euler Circuit: An Euler circuit is a path that traverses each edge of a graph exactly once and returns to its starting vertex. For a graph to have an Euler circuit, all vertices must have an even degree (number of edges connected to the vertex).

2. Hamiltonian Circuit: A Hamiltonian circuit is a path that visits every vertex in a graph exactly once and returns to its starting vertex. A complete graph always has a Hamiltonian circuit, regardless of the number of vertices.

Now, let's determine for what values of n a complete graph has an Euler circuit:

In a complete graph with n vertices, each vertex is connected to every other vertex, which means the degree of each vertex is (n-1). For a complete graph to have an Euler circuit, all vertices must have an even degree. This implies that (n-1) must be even, so n must be odd.

So, for a complete graph with n vertices to have an Euler circuit, n must be an odd number.

In summary:
- A complete graph with n vertices always has a Hamiltonian circuit.
- A complete graph with n vertices has an Euler circuit if n is an odd number.

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Step-by-step explanation:

To graph the boundary lines of the linear inequalities, we need to first set up the inequalities based on the given information. Let x be the number of popcorn bags and y be the number of cotton candy bags. Then we have:

Chris needs at least 7 bags of popcorn, so the inequality for popcorn is x ≥ 7.

Chris has $56 to spend, so the total cost of popcorn and cotton candy cannot exceed $56, which gives the inequality 3x + 7y ≤ 56.

To graph the boundary lines, we need to first graph the lines corresponding to these two inequalities:

The line x = 7 is a vertical line passing through the point (7,0), because the only restriction on the popcorn is that Chris needs at least 7 bags.

The line 3x + 7y = 56 is a diagonal line with x-intercept 56/3 and y-intercept 8, because these are the points where the line intersects the x and y axes, respectively.

We can now plot these lines on a coordinate plane:

|

8 | x = 7

| o

7 | |

| | 3x + 7y = 56

6 | |

| o

5 |

| o

4 |

|

3 |

|

2 | o

|

1 | o

|____________________________

1 2 3 4 5 6 7 8 9 10

The shaded region below and to the left of the diagonal line represents the solution set to the inequalities, because it includes all the points where Chris can buy at least 7 bags of popcorn and stay within his budget.

Finally, we can plot the given points and identify which ones are part of the solution set:

(3, 7) is not part of the solution set because it is above the diagonal line.

(2, 2) is part of the solution set because it is below and to the left of the diagonal line.

(8, 1) is not part of the solution set because it is above the diagonal line.

(10, 2) is not part of the solution set because it is above the diagonal line.

(5, 5) is part of the solution set because it is below and to the left of the diagonal line.

Therefore, the solution set consists of the points (2, 2) and (5, 5).

The experimental probability of selecting a green marble on the next attempt is 1/6 or approximately 0.1667 or 16.67%.

What is probability?

Probability is a measure of the likelihood of an event occurring.

The experimental probability of an event happening is calculated by dividing the number of times the event occurs by the total number of trials or attempts.

In this case, Diane has already selected 2 green marbles and 10 other marbles from the bag. So, the total number of marbles left in the bag is 12.

Since there are 2 green marbles left in the bag, the probability of selecting a green marble on the next attempt is:

2 (number of green marbles) / 12 (total number of marbles) = 1/6

So, the experimental probability of selecting a green marble on the next attempt is 1/6 or approximately 0.1667 or 16.67%.

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The value of x in the given inequality is x is greater than or equal to 16.

Using symbols like (less than), > (greater than), (less than or equal to), or (greater than or equal to), an inequality compares two values or expressions and illustrates their connection (not equal to). When values or phrases are being compared, it is said that there exist inequalities. For instance, the inequality x + 2 7 signifies that "x plus 2 has a value less than 7."

The given inequality is x - 4 ≥ 12.

Add 4 on both sides of the inequality:

x - 4 + 4 ≥ 12 + 4

x ≥ 16

Hence, the value of x in the given inequality is x is greater than or equal to 16.

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The probability that a randomly selected lobster weighs more than 2.5 lb is 0.0026.

Let X be the weight of a randomly selected Maine lobster. We know that X is normally distributed with mean μ = 1.8 lb and standard deviation σ = 0.25 lb.

We need to find the probability that a randomly selected lobster weighs

a) less than 1.5 lb

b) between 1.6 and 2 lb

c) more than 2.5 lb

To solve these problems, we need to standardize the variable X using the standard normal distribution

Z = (X - μ) / σ

a) To find the probability that a randomly selected lobster weighs less than 1.5 lb, we need to find P(X < 1.5). Standardizing X, we have

Z = (1.5 - 1.8) / 0.25 = -1.2

Using a standard normal distribution table or calculator, we find that P(Z < -1.2) = 0.1151.

Therefore, the probability that a randomly selected lobster weighs less than 1.5 lb is 0.1151.

b) To find the probability that a randomly selected lobster weighs between 1.6 and 2 lb, we need to find P(1.6 < X < 2). Standardizing X, we have

Z1 = (1.6 - 1.8) / 0.25 = -0.8

Z2 = (2 - 1.8) / 0.25 = 0.8

Using a standard normal distribution table or calculator, we find that P(-0.8 < Z < 0.8) = 0.5328

Therefore, the probability that a randomly selected lobster weighs between 1.6 and 2 lb is 0.5328.

c) To find the probability that a randomly selected lobster weighs more than 2.5 lb, we need to find P(X > 2.5). Standardizing X, we have:

Z = (2.5 - 1.8) / 0.25 = 2.8

Using a standard normal distribution table or calculator, we find that P(Z > 2.8) = 0.0026.

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Answer:

Therefore, the volume of the cylinder is approximately 2,262.9 cubic centimeters.

Step-by-step explanation:

The formula for the volume of a cylinder is V = πr^2h, where r is the radius of the base and h is the height of the cylinder.

Substituting the given values, we get:

V = π(6 cm)^2(20 cm)

V = 720π cm^3

To find the volume to the nearest tenth, we can use the approximation π ≈ 3.14 and calculate:

V ≈ 720(3.14) cm^3

V ≈ 2,262.8 cm^3

Rounding to the nearest tenth, we get:

V ≈ 2,262.8 cm^3 ≈ 2,262.9 cm^3

Therefore, the volume of the cylinder is approximately 2,262.9 cubic centimeters.

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The shaded area under the curve is approximately 9.11, which confirms our result.

In calculus, an integral is a mathematical object that represents the area between a curve and the x-axis. It is a fundamental concept in calculus, and is used to calculate quantities such as the area under a curve, the volume of a solid, and the work done by a force. The process of finding an integral is called integration, and it involves finding an antiderivative (or indefinite integral) of a function, which is a function whose derivative is equal to the original function. The definite integral is then calculated by evaluating the antiderivative at two limits, which represent the beginning and end points of the area being calculated.

Here,

We can start by expanding the integrand using the distributive property:

(9 - t)√t = 9√t - t√t

Now we can integrate each term separately using the power rule of integration:

∫(9 - t)√t dt = ∫9√t dt - ∫t√t dt

= 18/2 * [tex]t^{(1/2)}[/tex] - 2/3 * [tex]t^{(3/2)}[/tex] + C

where C is the constant of integration.

Evaluating the definite integral from 0 to 2:

∫(9 - t)√t dt from 0 to 2 = [18/2 * [tex]2^{(1/2)}[/tex] - 2/3 * [tex]2^{(3/2)}[/tex]] - [18/2 * [tex]0^{(1/2)}[/tex] - 2/3 * [tex]0^{(3/2)}[/tex]]

= 9√2 - 4/3

≈ 9.11

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a. The two parameters of the gamma distribution are

shape parameter (α) and velocity parameter (β).

The mean of the gamma distribution = α/β and the standard deviation = sqrt(α)/β.

mean = 5 seconds

standard deviation =2.5 seconds.

α/β = 5 (equation 1)

sqrt(α)/β = 2.5 (equation 2)

Squaring equation 2 and multiplying both sides by β^2 yields:

[tex]α = 6.25β^2[/tex]

Substituting this value of α into Equation 1 yields:

[tex]6.25β^2/β = 5[/tex]

Simplified, it looks like this:

β = 0.8

Substituting this value of β into Equation 1 yields:

a = 4

Therefore, the parameters of the gamma distribution are α = 4 and β = 0.8.

b. I need to find the probability that this software will take more than 10 seconds to run.

The probability density function (PDF) of the gamma distribution is given by

[tex]f(x) = (β^α * x^(α-1) * e^(-βx)) / Γ(α)[/tex]

where Γ(α) is the gamma function.

Using the values ​​of α and β obtained in part (a), we can write the PDF as

[tex]f(x) = (0.8^4 * x^(4-1) * e^(-0.8x)) / Γ(4)[/tex]

I need to find the probability that the execution time exceeds 10 seconds. This can be written as:

P(X > 10) = ∫(10 to infinity) f(x) dx

Using software or a calculator, we can evaluate this integral to get:

P(X > 10) = 0.0559 (rounded to four decimal places)

Therefore, the probability is 0.0559.

c.Since it has already taken more than 5 seconds, we need to find the probability of this software where will take more than 10 seconds to run.

This is a conditional probability and should be calculated using Bayes' theorem.

P(X > 10 | X > 5) = P(X > 10 and X > 5) / P(X > 5)

The numerator can be simplified to

P(X > 10 and X > 5) = P(X > 10)

Using the results obtained in part (b), we can write

P(X > 10 and X > 5) = 0.0559

The denominator can be written as:

P(X > 5) = ∫(5 to infinity) f(x) dx

Using the same PDF as before, we can evaluate this integral and get:

P(X > 5) = 0.2615 (rounded to four decimal places)

Substituting these values ​​into the conditional probability formula gives:

P(X > 10 | X > 5) = 0.0559 / 0.2615

Simplified, it looks like this:

P(X > 10 | X > 5) = 0.214

Therefore, the probability that his single run of this software took longer than 5 seconds to take longer than 10 seconds is 0.214 (rounded to three decimal places).

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