HS HONORS GEOMETRY!!!
PLEASE HELP ME THIS IS URGENT I WILL GIVE BRAINLIEST ​

(a) The probability that both inspectors assign the same safety score is 0.80.

(b) The probability that the second inspector assigns a higher safety score than the first inspector is 0.11.

(c) The expectation of marginal probability mass function of the score assigned by the first inspector is 2.83 and variance is 0.83.

(d) The expectation of marginal probability mass function of the score assigned by the first inspector is 3.00 and variance is 0.77.

(e) The scores assigned by the two inspectors are not independent of each other.

(f) The covariance of the scores assigned by the two inspectors is -0.0148.

(a) The probability that both inspectors assign the same safety score is the sum of the joint probabilities where X and Y take the same value:

P(X = Y) = P(1,1) + P(2,2) + P(3,3) + P(4,4) = 0.09 + 0.15 + 0.24 + 0.32 = 0.80

(b) The probability that the second inspector assigns a higher safety score than the first inspector is given by:

P(Y > X) = P(2,1) + P(3,1) + P(4,1) + P(3,2) + P(4,2) + P(4,3) = 0.03 + 0.01 + 0.01 + 0.03 + 0.01 + 0.02 = 0.11

(c) The marginal probability mass function of the score assigned by the first inspector is obtained by summing the joint probabilities over all values of Y:

P(X = 1) = 0.09 + 0.02 + 0.01 + 0.00 = 0.12

P(X = 2) = 0.03 + 0.15 + 0.01 + 0.01 = 0.20

P(X = 3) = 0.01 + 0.03 + 0.24 + 0.02 = 0.30

P(X = 4) = 0.01 + 0.01 + 0.04 + 0.32 = 0.38

The expectation of X is given by:

E(X) = 10.12 + 20.20 + 30.30 + 40.38 = 2.83

The variance of X is given by:

Var(X) = E(X^2) - [E(X)]^2

= (1^20.12 + 2^20.20 + 3^20.30 + 4^20.38) - (2.83)^2

= 0.83

(d) The marginal probability mass function of the score assigned by the second inspector is obtained by summing the joint probabilities over all values of X:

P(Y = 1) = 0.09 + 0.03 + 0.01 + 0.00 = 0.13

P(Y = 2) = 0.02 + 0.15 + 0.01 + 0.01 = 0.19

P(Y = 3) = 0.01 + 0.03 + 0.24 + 0.02 = 0.30

P(Y = 4) = 0.01 + 0.01 + 0.04 + 0.32 = 0.38

The expectation of Y is given by:

E(Y) = 10.13 + 20.19 + 30.30 + 40.38 = 3.00

The variance of Y is given by:

Var(Y) = E(Y^2) - [E(Y)]^2

= (1^20.13 + 2^20.19 + 3^20.30 + 4^20.38) - (3.00)^2

= 0.77

(e) To determine if the scores assigned by the two inspectors are independent of each other, we need to check if the joint probability mass function can be expressed as the product of the marginal probability mass functions.

If the scores are independent, then P(X = i and Y = j) = P(X = i)P(Y = j) for all possible values of i and j.

Checking the probabilities in the given joint probability mass function, we can see that P(X = 1 and Y = 2) = 0.03, P(X = 1) = 0.12, and P(Y = 2) = 0.20.

Since P(X = 1 and Y = 2) is not equal to P(X = 1)P(Y = 2), the scores assigned by the two inspectors are not independent of each other.

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I hope this answer helps .

A sample size of 105 was used in the study with 99% of confidence interval.

A range of values that, with a certain degree of confidence, is likely to include the population mean is known as a confidence interval for a mean. The formula for calculating the confidence interval is:

CI = X ± Zα/2 × σ/√n

where X is the sample mean, σ is the population standard deviation, n is the sample size, and Zα/2 is the critical value of the normal distribution at α/2. Here, for a 99% confidence level, α = 0.01 and Zα/2 = 2.576).

We can use this formula to solve for n:

n = (Zα/2 × σ / E)²

where E is the margin of error, which is half of the confidence interval.

In this case, we have a 99% confidence interval with a range of 152 to 160 and a standard deviation of 15.

The margin of error (E) can be calculated as half of the range of the confidence interval:

E = (160 - 152) / 2 = 4

By adding these values to the previous formula, we obtain:

n = (2.576 × 15 / 4)² = 105

Therefore, we can conclude that a sample size of at least 105 was used in this study.

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Complete question is:

A 99% confidence interval for a population mean was reported to be 152 to 160. If standard deviation is 15, what sample size was used in this study? (Round your answer up to the next whole number.)

On solving the provided question we can say that So, the discriminant of the given quadratic equation is 109.

A quadratic equation is x ax2+bx+c=0, which is a single variable quadratic polynomial. a 0. Because this polynomial is of second order, the Basic Theorem of Algebra implies that it has at least one solution. Simple or complex solutions are possible. A quadratic equation is a quadratic equation. This means it has at least one word that must be squared. One of the most common solutions for quadratic equations is "ax2 + bx + c = 0." where a, b, and c are numerical coefficients or constants. where the variable "X" is unnamed.

The given quadratic equation is [tex]3x^2 - 5x - 7 = 0[/tex]

(a) Here, the equation is [tex]3x^2 - 5x - 7 = 0[/tex], so a = 3, b = -5, and c = -7. Therefore, the discriminant is:

[tex]b^2 - 4ac = (-5)^2 - 4(3)(-7) = 25 + 84 = 109[/tex]

So, the discriminant of the given quadratic equation is 109.

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I can't get how to type out the inequality sign, so I'm gonna use the equal to sign (=) even though it's wrong. If you get what I mean. I'll write the full answer in english tho.

x-4=12

collecting like terms, we have;

X=12+4

X=16( X is greater or equal to 16)

So for the graph, it'll be on 16 facing the the positive side( that's to the right) with the the end shaded.

Hope it helps

The triangle's hypotenuse, according to the Pythagorean equation, measures 20 inches.

Let's start by using the Pythagorean theorem to relate the hypotenuse and the legs of the right triangle. According to the theorem, we have:

(5y)² = (x+8)² + (x+3)²

25y² = 2x² + 22x + 73

Now, we can use the given information about the perimeter and the difference between the hypotenuse and the shorter leg to set up a system of equations. We know that the perimeter of the triangle is:

P = (x+8) + (x+3) + 5y = 2x + 5y + 11

And we also know that the difference between the hypotenuse and the shorter leg is:

5y - (x+3) = 10

Solving for y in the second equation gives us:

y = (x+13)/5

Substituting this value into the first equation and solving for x gives us:

2x + 25(x+13)/25 = 60

2x + x + 13 = 60

3x = 47

x = 47/3

Substituting this value of x back into the equation for y gives us:

y = (47/3 + 13)/5 = 4

So the hypotenuse of the triangle is:

5y = 20

Therefore, the hypotenuse of the triangle is 20 inches long.

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3k+5k=8k
96/8=12
k=12
as we know blue-eyed are 3k and brown-eyed 5k so we need to put 12 instead of k
5x12=60

Answer:

Step-by-step explanation:

Step 1: We know that the cosine of an angle in Quadrant III is negative.

Step 2: We convert the given angle A into radians by multiplying it by pi/180.

A = (7/√120) * (π/180) = (7π)/(12√5)

Step 3: We use the cosine formula to find the exact value of cos A.

cos A = cos((7π)/(12√5))

Step 4: We simplify the result using trigonometric identities.

cos A = (-1/3)√5

Since 3 has the highest frequency, the mode is 3.Now, the curve is positively skewed if mean is bigger than mean.

Total number is a term used to refer to the sum of two or more individual elements. It is commonly used in the context of mathematics and can refer to the result of addition, subtraction, multiplication, or division. For example, the total number of apples in a basket could be five, the total number of days in a week could be seven, and the total number of people in a family could be four. In any case, the total number is the sum of all the individual elements that are being considered.

Presented to us is a distribution:

{4.5, 3, 1, 2, 4, 3, 6, 4.5, 4, 5, 2, 1, 3, 4, 3, 2}

We must determine the mean.

Mean is equal to the sum of all observations divided by the total number of observations.

Mean = 52/16

Mean = 3.25

We now determine the mode, which is the observation with the highest frequency, or how frequently it has occurred.

observations per unit of time

4.5 2

3 4

1 2

2 3

4 3

6 1

5 1

Since 3 has the highest frequency, the mode is 3.
Now, the curve is positively skewed if mean is bigger than mean.

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Complete questions as follows-
What is the mean of the given distribution, and which type of skew does it exhibit?
{4.5, 3, 1, 2, 4, 3, 6, 4.5, 4, 5, 2, 1, 3, 4, 3, 2}

WHAT IS THE MEAN? and WHAT TYPE OF SKEW EXHIBITS? negative, positive, symmetric, zero etc..

Therefore , the solution of the given problem of expressions comes out to be cost $357.

It is preferable to use moving integer variables, which can be rising, decreasing, or blocking, rather than estimates that are generated at random. They could only assist one another by exchanging resources, knowledge, or answers to problems. A declaration of truth equation may include the justifications, components, and mathematical comments for strategies like extra disapproval, production, and mixture.

Here,

Let's say Blair's previous PC cost "x" dollars.

Her new computer cost $5 less than twice as much as her previous one, according to the information provided, which can be written as:

=> Cost of Blair's new computer =  2x - 5

She also purchased a new PC, which cost her $709. In light of these two bits of knowledge, we can construct the following equation:

=> 2x - 5 = 709

In order to find "x," add 5 to both parts, divide by 2, and we get:

=> 2x = 714

=> x = 357

Blair's previous PC therefore cost $357.

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To verify our response, we can multiply it by 5 to the power of see if we get 65 with our response:(log₅65) ≈ 2.737

Several fields, including mathematics, physics, engineering, and others, use logarithms extensively. These are a few instances:

1. Sound intensity: Logarithms serve as the foundation for the decibel scale that measures sound intensity.

2. Earthquakes: The Richter scale, which gauges earthquake magnitude using logarithms, was developed.

3. pH scale: A solution's acidity or basicity is determined by the pH scale, which is based on logarithms.

To find x, we can use logarithmic properties. Using base 5 to calculate the logarithm of both sides, we obtain:

(5x) log5 = log₅65

By applying the logarithm power rule, we may simplify the left-hand side as follows:

5x log5 = 65x log5

Since log5  = 1, we can further reduce to obtain:

x = log₅ 65

We can determine that using a calculator or software that:

log₅ 65 ≈ 2.737

Hence, log₅ 65=  2.737.

To verify our response, we can multiply it by 5 to the power of see if we get 65 with our response:

(log₅ 65) ≈ 2.737

This demonstrates that our response is accurate.

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Answer:

To compare the two functions, it is necessary to find the x values for which each function is positive.

For f(x), we can solve for when it is positive by setting the equation equal to zero and finding the x-intercepts.

-x^2 + 2x = 0

-x(x - 2) = 0

-x = 0 and x = 2

This tells us that f(x) is positive for (-inf, 0) and (2, inf).

Next, for g(x), we can solve for when it is positive by setting the equation equal to zero and finding the x-intercept.

log(2x+1) = 0

2x + 1 = 1

x = 0

This tells us that g(x) is positive for (0, inf).

Therefore, the interval on which both functions are positive is (0, 2).

In hire-assistant, assuming that the candidates are presented in a random order, the probability of hiring exactly one time is 1/n, while the probability of hiring exactly n times is 1/n!.

In the Hire-Assistant problem, the objective is to hire the best candidate among a pool of candidates. The candidates are presented in random order, and you must decide whether to hire or reject a candidate after each interview. The hiring process ends when the best candidate is hired.

To answer the student question, let's first consider the probability of hiring exactly one time. This happens when the best candidate is the first one interviewed since no other candidate can be better than them. Since the candidates are in random order, the probability of this occurring is 1/n, where n is the total number of candidates.

Now let's consider the probability of hiring exactly n times. This means you would hire every candidate you interview. This only happens if the candidates are presented in ascending order of quality, so that each new candidate is better than the previous one. There is only 1 possible ordering like this out of the total n! possible orderings of candidates. So, the probability of hiring exactly n times is 1/n!.

In summary, the probability of hiring exactly one time is 1/n, while the probability of hiring exactly n times is 1/n!. These probabilities are derived from the random order of candidate presentations and the decision-making process of the Hire-Assistant problem.

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0.0027 or 0.27% is the probability that from a sample of 40 Indianapolis residents, less than 95% were rooting for the Giants in Super Bowl XLV

To find the probability that fewer than 95% were rooting for the Giants, we need to sum the probabilities of having 0 to 19 residents rooting for the Giants out of 40, because if 19 or more residents were rooting for the Giants, then the percentage would be at least 95%.

Using the binomial distribution formula:

P(X < 0.95n) = Σ (n choose i) * p^i * (1-p)^(n-i), i = 0 to 19

where:

n = 40 (sample size)

p = 0.95 (probability of rooting for the Giants)

X = number of residents rooting for the Giants

Using a binomial distribution calculator or software, we get P(X < 0.95n) = 0.0027 or 0.27%.

Therefore, the probability that from a sample of 40 Indianapolis residents, fewer than 95% were rooting for the Giants in Super Bowl XLVI is approximately 0.0027 or 0.27%.

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Answer:

Step-by-step explanation:

Weight of wood block is about 15.5 ounces.

Given that:

Height of block [tex]= 4 \ \text{inches}[/tex]

Diameter of block [tex]= 3 \ \text{inches}[/tex]

1 cubic inch of wood [tex]= 0.55 \ \text{ounces}[/tex]

Find:

Weight of wood block

Computation:

Radius of block [tex]= 3 \div 2[/tex]

Radius of block [tex]= 1.5 \ \text{inches}[/tex]

Volume of toy [tex]= \pi r^2h[/tex]

Volume of toy [tex]= (3.14)(1.5)^2(4)[/tex]

Volume of toy [tex]= (3.14)(2.25)(4)[/tex]

Volume of toy [tex]= 28.26 \ \text{cubic inch}[/tex]

Weight of wood block [tex]= \text{Volume of toy} \times \ \text{0.55 ounces}[/tex]

Weight of wood block [tex]= 28.26 \times 0.55 \ \text{ounces}[/tex]

Weight of wood block [tex]= 15.5 \ \text{ounces}[/tex]

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The value of x and y in equation (a) is x = 2 and y = 41

The value of x and y in equation (b) is x = 3 and y = 4

A parallelogram is a quadrilateral with opposite sides parallel (and therefore opposite angles equal). A quadrilateral with equal sides is called a rhombus, and a parallelogram whose angles are all right angles is called a rectangle.

To solve the first quadrilateral

Let ABCD is a quadrilateral

given that,

AB = 2x + 7

DC = x + 7

∠A = (2y - 5)°

∠D = (2y + 21)°

To find the value of x and y

2x + 7 = x + 9

2x - x = 9-7

x = 2

put the value of x in equation

2y - 5 + 2y +21 = 180°

4y + 16 = 180°

4y = 180° - 16

y = 164/4

y = 41

Hence,the value of x and y is 2 and 41

In second quadrilateral we have given that,

ABCD is a parallelogram

As we know that the diagonals of a parallelogram bisect each other

so, OD = OB ana OA = OC

2y + 13 = 5y + 4

13 - 4 = 5y - 2y

9 = 3y

y = 9/3

y = 3

now, OA = OC

x + 8 = 16 - x

x + x = 16 - 8

2x = 8

x = 8/2

x = 4

Hence, the value of x and y is 4 and 3

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0.625 is a rational number. A rational-number is the number that can be expressed as ratio of two integers with a denominator (not zero).

What is rational and irrational number?

A rational number is a number which can be expressed as the product of two integers with non-zero denominators.In other words, it is a number that can be written in the form of p/q,

where:

p and q -> integers

and q -> not equal to 0.

An irrational number, is just opposite, it cannot be expressed as the ratio of 2 integers.It is a number that cannot be written as a simple fraction, and its decimal expansion continues indefinitely. The most famous examples of irrational numbers are π and √2.

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Answer:

Step-by-step explanation:

Using Pythagorean Theorem : A² = B²+C².

→x² = 5²+12² = 25+144 = 169.

→x = √169 = 13.

a. n=100

The probability that a sample proportion will be within +/-6% of the population proportion for a sample of size 100 is 0.7469.

b. n=200  

The probability that a sample proportion will be within +/-6% of the population proportion for a sample of size 200 is 0.8214.

c. n=500  

The probability that a sample proportion will be within +/-6% of the population proportion for a sample of size 500 is 0.8977.

d. n=1000 Th

The probability that a sample proportion will be within +/-6% of the population proportion for a sample of size 1000 is 0.9227.

"With a larger sample size, there is a higher probability that the sample proportion will be within a certain range of the population proportion" is the advantage of having larger sample size.

To solve this problem, we need to use the formula for the standard error of the sample proportion:

SE = [tex]\sqrt{ p \times (1 - p) / n }[/tex]

where p is the population proportion, and n is the sample size.

a) For n = 100:

SE = [tex]\sqrt{0.45 \times (1 - 0.45) / 100 }[/tex] = 0.0499

The margin of error is +/-6%, which translates to +/-0.06 in proportion units.

We can find the z-scores corresponding to these margins of error using a standard normal distribution table:

[tex]z_1[/tex] = (0.45 - 0.06 - 0.45) / 0.0499 = -1.2024

[tex]z_2[/tex] = (0.45 + 0.06 - 0.45) / 0.0499 = 1.2024

The probability of a sample proportion within +/-6% of the population proportion is the area under the normal curve between these two z-scores:

P(-1.2024 < z < 1.2024) = 0.7469

b) For n = 200:

SE = [tex]\sqrt{ 0.45 \times (1 - 0.45) / 200 }[/tex] = 0.0353

[tex]z_1[/tex] = (0.45 - 0.06 - 0.45) / 0.0353 = -1.691

[tex]z_2[/tex] = (0.45 + 0.06 - 0.45) / 0.0353 = 1.691

P(-1.691 < z < 1.691) = 0.8214

c) For n = 500:

SE = [tex]\sqrt{ 0.45 \times (1 - 0.45) / 500 }[/tex] = 0.0249

[tex]z_1[/tex] = (0.45 - 0.06 - 0.45) / 0.0249 = -2.2088

[tex]z_2[/tex] = (0.45 + 0.06 - 0.45) / 0.0249 = 2.2088

P(-2.2088 < z < 2.2088) = 0.8977

d) For n = 1000:

SE = [tex]\sqrt{ 0.45 \times (1 - 0.45) / 1000 }[/tex] = 0.0177

[tex]z_1[/tex] = (0.45 - 0.06 - 0.45) / 0.0177 = -2.5424

[tex]z_2[/tex] = (0.45 + 0.06 - 0.45) / 0.0177 = 2.5424

P(-2.5424 < z < 2.5424) = 0.9227
Advantage of a larger sample size:
With a larger sample size, there is a higher probability that the sample proportion will be within a certain range of the population proportion.

This is because a larger sample size reduces the standard error, resulting in a more precise estimate of the population proportion.

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Question:-

The population proportion is 0.45 . What is the probability that a sample proportion will be within +/-6 of the population proportion for each of the following sample sizes? Round your answers to 4 decimal places. Use z-table.

a.  n=100

b.  n=200

c.  n=500

d.   n=1000

What is the advantage of a larger sample size?

With a larger sample, there is a - Select your answer (lower or higher) Item 5 probability will be within of the population proportion .

The length οf diameter οf the given sphere is 18m.

A sphere is a three-dimensiοnal οbject which is rοund in shape. It has nο edges οr vertices. The sphere can be defined in three axes, thοse are x-axis, y-axis and z-axis.

The given diagram is a sphere.

Given that,

The surface area οf the sphere is  1017.36 m²

Fοrmula fοr surface area οf the sphere = 4πr² where r is the radius and π=3.14

Equating both we get,

4πr² = 1017.36

[ dividing bοth sides by 4 ]

⇒πr² = (1017.36)/4

⇒πr² = 254.34

[putting the value of π]

⇒r² = 254.34/ 3.14

⇒r² = 81

⇒r =± 9

As radius cannot be a negative value so we take r=9

Sο the radius οf the sphere is 9 m

Diameter = 2× radius =2×9= 18m

Hence, diameter οf the given sphere is 18m.

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Answer: x=21

Step-by-step explanation:

The shape is a quadrilateral. (Quadrilateral= 180 degrees)

5x+18+2x+15=180 - combine like terms

7x+33=180 - subtract 33

7x=147 - divide by 7

x=21

A scatterplot can help us visualize the relationship between two variables and determine if there is a correlation between them. If the scatterplot shows a clear pattern or trend, conclude that there is a relationship between the two variables.

Without access to the scatterplot or the newspaper report, it is difficult for me to provide a definitive answer. However, in general, to determine if the scatterplot supports the newspaper report about the number of semesters and starting salary, we need to examine the pattern in the scatterplot.

Suppose we see a clear trend or pattern, such as a linear relationship where an increasing number of semesters is associated with an increased starting salary or a non-linear relationship. In that case, we can say that the scatterplot supports the newspaper report.

However, if we see no discernible pattern or trend or significant variability in the data points makes it is difficult to draw any meaningful conclusions, then we cannot say that the scatterplot supports the newspaper report.

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If at least one card is a heart then P is three-fοurths.

The study οf prοbability examines the likelihοοds οf results οccurring and is based οn the ratiο οf likely and imprοbable scenariοs.

Tο determine the likelihοοd οf at least οne card being a heart, we can determine the likelihοοd that neither card will be a heart and subtract that prοbability frοm 1.

The prοbability οf the first card nοt being a heart is 26/52 = 1/2, since there are 26 nοn-heart cards in the deck οut οf a tοtal οf 52 cards. Since the card is replaced, the prοbability οf the secοnd card nοt being a heart is alsο 1/2.

Therefοre, the prοbability οf neither card being a heart is (1/2) x (1/2) = 1/4.

The prοbability οf at least οne card being a heart is 1 - (1/4) = 3/4.

Hence, the answer is: three-fοurths.

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The probability of getting a ticket when parking for a random amount of time that is uniformly distributed on .0; 4/ hours is 1/2.

Since the parking officials pass the same point every two hours, we can divide the possible parking times into two-hour intervals. If a car is parked for less than two hours, it will not be marked with chalk. If a car is parked for more than two hours, it will be marked with chalk and will receive a ticket if it remains parked for another two hours.

Since the possible parking times are uniformly distributed on .0; 4/ hours, the probability of parking for less than two hours is 2/4 = 1/2, and the probability of parking for more than two hours is also 1/2. Therefore, the probability of getting a ticket is the probability of parking for more than two hours, which is 1/2.

Alternatively, we can calculate the probability of getting a ticket by finding the area of the part of the uniform distribution that corresponds to parking for more than two hours. This area is a triangle with base 2 and height 1/2, so its area is (1/2)(2)(1/2) = 1/2. Therefore, the probability of getting a ticket is also 1/2.

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Answer:

  d.  9.4 feet

Step-by-step explanation:

You want the measure of long side XY of the right triangle that has its altitude dividing the hypotenuse into segments 3 and 8 feet in length.

In this geometry, all of the right triangles are similar. That means the ratio of long side to hypotenuse is the same for all of the triangles:

  XY/(3+8) = 8/XY

  XY² = (3+8)(8)

  XY = √88 ≈ 9.4

The length of segment XY is about 9.4 feet.

__

Additional comment

As we see here, the long side of the large triangle is the root of the product of the long segment of the hypotenuse with the whole hypotenuse:

  XY=√(8(8+3)) = √88

The same relation works on the other side of the triangle. The short side of the large triangle is the root of the product of the short segment of the hypotenuse and the whole:

  short side = √(3(11)) = √33

The relation for the altitude of the triangle is that it is the root of the product of the two hypotenuse segments:

  altitude = √(3·8) = √24

An equation of the quadratic function that passes through the points (-2, 0), (2, - 16) and (6, 0) is: D. f(x) = (x - 2)² - 16.

In this exercise, you are required to determine the factored form of the given quadratic function that passes through the points (-2, 0), (2, - 16) and (6, 0).

In Mathematics, the vertex form of a quadratic function is represented by the following mathematical equation:

f(x) = a(x - h)² + k

Where:

Based on the information provided in the graph above, we can determine the value of a as follows:

f(x) = a(x - h)² + k

0 = a(-2 - 2)² - 16

0 = 16a - 16

16 = 16a

a = 1

Therefore, the required quadratic function is given by:

f(x) = a(x - h)² + k

f(x) = y = (x - 2)² - 16

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Based on the information in the graph, we can establish that a person who studied for six hours could have a grade of 204.

To calculate the qualification that a person who studied 6 hours would have, we must carry out the following mathematical procedure: Rule of three.

In this case, if a person who studied for two hours had a result of 68 points, then we can make the projection to find out how much someone who studied for six hours obtained.

According to the above, this person got 204 points after studying for 6 hours.

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The probability that the rack ends up in alphabetical order is 1/9! or approximately 1.46 x 10⁻⁷.

The problem involves arranging 9 CDs out of 16 in alphabetical order in a CD rack. The total number of possible ways to arrange 9 CDs out of 16 is 16 choose 9, which equals 11440. Only one of these arrangements is in alphabetical order. Therefore, the probability of randomly selecting an alphabetical order is 1/11440, which is approximately 0.0000874 or 0.00874%. This is a very small probability, indicating that it is highly unlikely that the CDs will end up in alphabetical order by chance.

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