(6) Show your work. (a) Throw a fair die 500 times. What is the approximate probability that you will get the sides numbered either "5" or "6" at least 150 times (inclusive)? (b) The following are 7 randomly selected observations from an exponential distribution with p. D. F. F(x) 1 e-2/0 17, 14, 27, 8, 12, 19, 12 Give a maximum likelihood estimate of the parameter 1

Answer:

Step-by-step explanation:To solve the system of equations, we can set the two expressions for y equal to each other:

−x + 2 = 3x + 1

Solving for x, we get:

4x = 1

x = 1/4

Substituting this value of x into either of the original equations, we get:

y = −(1/4) + 2

y = 7/4

Therefore, the solution to the system of equations is the point (1/4, 7/4), which is the point of intersection of the two lines.

So the correct description of the solution to the system of equations is:

a

Line y = −x + 2 intersects line y = 3x + 1.

The value of x is 116°

Given that a circle, with two congruent chords, FG and DE we need to find the central angle x,

Here we will use the properties of circle,

Since, the chords are equal, so, the measure of the arc intercepted by them will be equal,

arc DE = arc FG = 116°

Also, we know that the central angle is equal to the measure of the arc intercepted by it,

Therefore, x = m arc DE

x = 116°

Hence, the value of x is 116°

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Confidence interval for the true mean weight is (964, 1043.8)

P-value= 0.5836≥0.5 then conclude that null hypothesis is rejected.

What is null hypothesis?

The assertion that there is no link between the two sets of data or variables under investigation is known as the null hypothesis. The word "null" refers to the idea that any empirically observed difference is entirely attributable to chance and that no underlying causal link exists.

Given,

Data= 956, 1012, 954, 988, 973, 1048, 1075, 1064, 856, 1026, 1031, 1064

n=12

Mean = sum(x)/n=sum(x)/12= 12047/12= 1003.9167

We get the standard deviation from the given data by using formula

Std. deviation= 62.7976

C=95%= 0.95

α= 1-0.95=0.05

Critical value= 2.201

The 95% confidence interval for the true mean value is=

=(1003.9167±2.201*62.7976/√12)

= (1003.9167±39.9)

= (964, 1043.8)

2)The null and alternative hypothesis,

H0:μ=1000

Hα: μ<1000

The statistic is (t)= (1003.9167-1000)/ 62.7976/√12

=0.216

The P-value from online calculator

P-value= 0.5836≥0.5

Then conclude that null hypothesis is rejected.

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The function that is represented by the graph is -|x - 1| + 8.

Option A is the correct answer.

We have,

The graph of the function - |x - 1| + 8 can be obtained by applying a series of transformations to the graph of the absolute value function, y = |x|.

First, the expression x - 1 inside the absolute value brackets shifts the entire graph of y = |x| one unit to the right.

This means that the "V" shape of the absolute value function is centered at x = 1 instead of x = 0.

Next, the negative sign in front of the absolute value reflects the graph of y = |x| across the x-axis.

This means that the "V" shape is now pointing downwards instead of upwards.

Finally, the entire graph is shifted vertically upward by 8 units due to the constant term + 8.

This means that the lowest point of the "V" shape is at y = 8 instead of

y = 0.

Thus,

The function that is represented by the graph is -|x - 1| + 8.

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Since we do not have information on a previous survey, we cannot calculate this percentage

i. To find the number of employees who prefer LEXUS only, we need to subtract the number of employees who preferred both LEXUS and MERCEDES, and the number of employees who preferred all three brands, from the total number of employees who prefer LEXUS:

LEXUS only = Total number of LEXUS preference - (Number of employees who prefer both LEXUS and MERCEDES) - (Number of employees who prefer all three brands)

LEXUS only = (8 + 4) - 8 - 4

LEXUS only = 8

Therefore, 8 employees prefer LEXUS only.

ii. To find the number of employees who prefer AMAROK only, we need to subtract the number of employees who preferred both MERCEDES and AMAROK, and the number of employees who preferred all three brands, from the total number of employees who prefer AMAROK:

AMAROK only = Total number of AMAROK preference - (Number of employees who prefer both MERCEDES and AMAROK) - (Number of employees who prefer all three brands)

AMAROK only = (8 + 4) - 4 - 4

AMAROK only = 4

Therefore, 4 employees prefer AMAROK only.

iii. We are given that 5 employees dislike all three brands. Since this information is only from the Private Sector, we cannot determine the number of employees who dislike all three brands in the total of 175 employees from both Public and Private Sectors.

iv. The percentage increase in the number of employees who preferred all three brands is the difference between the number of employees who preferred all three brands in the current survey and the number of employees who preferred all three brands in a previous survey, divided by the number of employees who preferred all three brands in the previous survey, multiplied by 100%. Since we do not have information on a previous survey, we cannot calculate this percentage.

v. The proportion of employees who preferred all three brands in the current survey is 4/23 or approximately 0.174. To compute the percentage decrease in the proportion of employees who preferred all three brands, we need to compare this proportion with the proportion of employees who preferred all three brands in a previous survey. Since we do not have information on a previous survey, we cannot calculate this percentage.

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The measure of ∠WOV is 60°. You would use complementary angles that are adjacent (∠WOV, and ∠XOW)

The measure of ∠YOZ is 60°. You would use the vertical angles that are non-adjacent (∠WOV, and ∠YOZ). These two angles are congruent so they would have the same measure. These angles combined also create supplementary angles

Another way to find the measure of ∠YOZ would be to make/write an equation and solve for x. For example, (3x+30)°=60°. x would equal 10 because 10x3=30+30=60°

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The solution of the fractions 1 / 4 × 20 / 21  is 5 / 21.

A fraction is number with numerator and denominator. Therefore, let's multiply the fractions.

1 / 4 × 20 / 21 = 20 / 84

Hence, let's reduce the fraction by dividing the numerator and denominator by 4.

20 / 84 ÷ 4 / 4

20 / 84 ÷ 4 / 4 = 5 / 21

Therefore, the fraction is 5 / 21.

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The population proportion (P) can be found by dividing the number of Soft Lilac bottles by the total number of bottles in the factory:
P = 1400 Soft Lilac bottles / 5300 total bottles
P ≈ 0.26 (rounded to two decimal places)

The sample proportion (p) can be found by dividing the number of Soft Lilac bottles in the sample by the total number of bottles in the sample:
p = 3 Soft Lilac bottles / 14 total bottles in the sample
p ≈ 0.21 (rounded to two decimal places)
So, the population proportion (P) of Soft Lilac bottles in Zahra's factory is approximately 0.26, and the sample proportion (p) of Soft Lilac bottles in the inventory manager's sample is approximately 0.21.

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a. A result between -0.08 and 0.08 has a probability of around 0.1562.

b. Using the standard normal distribution table the value of a is 19.4.

c. After using z-score to find the value of a, we got 28.2.

(a) To find P(19.6 < X < 20.4), we need to standardize the values and use the standard normal distribution table:

z1 = (19.6 - 20) / √(25) = -0.4 / 5 = -0.08

z2 = (20.4 - 20) / √(25) = 0.4 / 5 = 0.08

Using the standard normal distribution table, the probability of a value falling between -0.08 and 0.08 is approximately 0.1562. Therefore:

P(19.6 < X < 20.4) = 0.1562

(b) To find the value of a such that P(X > a) = 0.5517, we need to use the standard normal distribution table to find the z-score that corresponds to a cumulative probability of 0.5517:

P(X > a) = 1 - P(X ≤ a) = 0.5517

P(X ≤ a) = 1 - 0.5517 = 0.4483

Using the standard normal distribution table, the z-score that corresponds to a cumulative probability of 0.4483 is approximately -0.12. We can use this z-score to solve for a:

z = (a - μ) / σ

-0.12 = (a - 20) / 5

-0.6 = a - 20

a = 19.4

Therefore, a = 19.4.

(c) To find the value of a such that P(X > a) = 0.05, we need to use the standard normal distribution table to find the z-score that corresponds to a cumulative probability of 0.95:

P(X > a) = 0.05

P(X ≤ a) = 1 - 0.05 = 0.95

Using the standard normal distribution table, the z-score that corresponds to a cumulative probability of 0.95 is approximately 1.64. We can use this z-score to solve for a:

z = (a - μ) / σ

1.64 = (a - 20) / 5

8.2 = a - 20

a = 28.2

Therefore, a = 28.2.

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Answer:

[tex]3x - 12 = 4x - 24[/tex]

[tex]x = 12[/tex]

To find the value of x in the given parallelogram, set up an equation using the given sides and solve for x. The value of x is 12.

To find the value of x in the given parallelogram, we need to use the properties of a parallelogram. In a parallelogram, opposite sides are equal in length. So, we can set up an equation using the given sides:

3x - 12 = 4x - 24

Now, we can solve the equation to find the value of x:

3x - 4x = -24 + 12

-x = -12

x = 12

Therefore, the value of x is 12.

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Answer: I got you fam

Step-by-step explanation:

Y=-3x+10 is NEITHER

Y-5=-1/3(x+2) is perpendicular to P

Y=1;3x is NEITHER

-3+y=1 is parallel to P

For the depth function of submarine in water is, d = 500t - 4,500, where t is the time (in minutes), the intercepts say x-intercept and y-intercept values are -4500 and 9 respectively.

The x-intercept is the point where a line cross or meet the x-axis, and the y-intercept is the point where a line cross or meet the y-axis. For y-intercept we are setting x to zero and for x-intercept we are setting y = 0 and determining their corresponding values. We have a submarine is exploring the ocean floor and begins to ascend to the surface.

The depth of submarine in the water can be modeled by equation, d = 500t - 4,500, where t is the time (in minutes). We have to determine the x and y intercept values. As we know, equation of line in slope intercept form is y = mx + b

where, b--> y-intercept

m --> slope

In this case b = - 4500, m = 500

So, y-intercept= -4500 for t = 0. Now, for x-intercept that for t value, plug d = 0, 0 = 500t - 4500

=> 500 t = 4500

=> t = 9

So, x-intercept = 9

Hence, required value is 9.

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Complete question:

A submarine is exploring the ocean floor and begins to ascend to the surface. The depth of the submarine in the water can be modeled by the function `d=500t-4,500` where t is the time (in minutes) since the submarine began to ascend. Find the intercepts of the graph of the equation:

x-intercept:

y-intercept:

The volume of the figure is given as follows:

V = 132 ft³.

The volume of a triangular prism is given as half the multiplication of the dimensions of the triangle, as follows:

V = 0.5 x l x w x h.

The dimensions of the triangle in this problem are given as follows:

3 ft, 8 ft and 11 ft.

Hence the volume of the prism is given as follows:

V = 0.5 x 3 x 8 x 11

V = 132 ft³.

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The correct interpretation is option (a). It states that we are 90% certain that the difference in percentages of men and women experiencing food cravings falls within the specified interval.

Your question is asking for the correct interpretation of a 90% confidence interval for the difference in percentages of men and women experiencing food cravings. Here are the interpretations of each statement:

a. We are 90% certain that the difference in percentages of men and women experiencing food cravings falls within the specified interval.
b. We are 90% certain that the percentages of men and women experiencing food cravings both fall within the specified interval.
c. 90% of men and women feel that the difference report experiencing food cravings a percentage of the time, specified within the given interval.
d. The specified interval represents 90% of the data for men and women having food cravings.

This interpretation accurately reflects the confidence interval for the difference in percentages between the two groups.

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Using proportions, it is found that 8 more seventh graders are involved in the yearbook.

What is a proportion?

A proportion is a fraction of a total amount.

Researching the problem on the internet, it is found that 10% of the 6th graders and 18% of the 7th graders are on the yearbook, hence:

0.1 x 280 = 28 6th graders.

0.18 x 200 = 36 7th graders.

36 - 28 = 8

8 more seventh graders are involved in the yearbook.

Since the utility cost is lowest at t=0, you will conduct your research during period 0 as a quasi-hyperbolic discounter. Procrastination may be beaten if the painful job was linked to an enjoyable task.

a) As a naive quasi-hyperbolic discounter, you must choose the period to study based on your discount factors (β = 0.75, δ = 1) and the utility costs. To determine this, you must compare the present value of the utility costs of studying in each period:

At t = 0: Utility cost = 8 (since you're studying now, no discounting is applied)

At t = 1: Utility cost = 10 x β = 10 x 0.75 = 7.5

At t = 2: Utility cost = 12 x β x δ = 12 x 0.75 x 1 = 9

Since the utility cost is lowest at t=0 (8), you will study during period 0.

b) By binding the unpleasant task (studying) with the pleasant task (eating chocolate cake with utility of 6), it may help overcome procrastination if the combined utility is less than the utility cost of studying in later periods. In this case:

At t = 0: Combined utility cost = 8 - 6 = 2

Since the combined utility cost at t=0 (2) is lower than the utility costs of studying in periods 1 and 2 (7.5 and 9), binding these tasks could help overcome procrastination.

c) As a sophisticated quasi-hyperbolic discounter, you're now aware of your present bias. However, since there is no cake involved, the utility costs remain the same as in part (a):

At t = 0: Utility cost = 8

At t = 1: Utility cost = 7.5

At t = 2: Utility cost = 9

Even though you're now sophisticated, the period in which you study does not change. You will still study during period 0, as it has the lowest utility cost.

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The transformation that would result in a similar shape to XYZ but one that is not congruent is D. a dilation centered at the origin with a scale factor of​ x.

When a triangle undergoes dilation, there is an alteration to its size while still conserving its shape. In scenarios where the scale factor of the triangular dilation happens to exceed 1, it leads to a bigger triangle than the original.

However, if this factor falls below 1, then the resulting triangle will be smaller compared to its original. The similarity between these triangles exists because they harbour identical shapes, but their incongruency emerges from their diverse magnitudes post-dilation.

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Options for this question are:

A. a translation 3 units to the right

B. a reflection across the line y = - 2x + 3

C. a rotation 90° clockwise around the origin

D. a dilation centered at the origin with a scale factor of​ x.

(a) According to Chebyshev's theorem, at least 36% of the measurements lie between 122.7 mmHg and 145.5 mmHg. (b) According to Chebyshev's theorem, at least 56% measurements lie between 122.7 mmHg and 147.2. So, the correct option is 56%.

(a) Using Chebyshev's theorem to find how much data falls within a certain number of standard deviations from the mean.

Using k = 2,  to capture at least 75% of the data (which is 1 - 1/2^2 = 0.75).

Using k = 2, we can say that at least 75% of the data falls within the range of 134.1 - 2(5.7) = 122.7 mmHg and 134.1 + 2(5.7) = 145.5 mmHg.

The percentage of data that falls outside of this range is (1 - 0.75)/2 = 0.125, or 12.5%.

Therefore, at least 12.5% of the data falls in each, below 122.7 mmHg and above 145.5 mmHg. This means that at least 36% of the data falls within the range of 122.7 mmHg and 145.5 mmHg.

(b) We can use Chebyshev's theorem again, this time with k = 2.5, since we want to capture at least 56% of the data (which is 1 - 1/2.5^2 = 0.64).

Using the same calculations as in part (a), we find that at least 64% of the data falls within the range of 121.0 mmHg and 147.2 mmHg.

Therefore, we can say that at least 56% of the data falls within this range, since 56% is less than 64%.

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The probability that the 20th ticket purchased is the second winning ticket is 0.10

Calculating the likelihood of experiments happening is one of the branches of mathematics known as probability. We can determine everything from the likelihood of receiving heads or tails when tossing a coin to the likelihood of making a research blunder, for instance, using a probability.

The probability of a winning ticket is = 0. 10

Thus,

This means there is a 10% chance of winning with each ticket purchased.

The likelihood that the 20th ticket will be the second winning ticket is the same as the chance that any other ticket will be the second winning ticket, which is 0.10. This is because each ticket purchase is autonomous and has no bearing on the results of other tickets. Therefore, the probability of the 20th ticket purchased being the second winning ticket is also 0.10 or 10%

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The solution of the Cauchy problem is y = (sina - 1/(2u - 1))e^(ux)cos(sqrt(1 - u^2)x) + (1/(2u - 1))cos(x).

To solve the Cauchy problem 2ux + y = cos x, we first need to find the general solution of the corresponding homogeneous equation 2ux + y = 0.

The characteristic equation is r^2 - 2ur + 1 = 0, which has roots r = u ± sqrt(u^2 - 1).

Case 1: u^2 < 1

In this case, the roots are complex conjugates, so the general solution of the homogeneous equation is

y = c₁e^(ux)cos(sqrt(1 - u^2)x) + c₂e^(ux)sin(sqrt(1 - u^2)x).

Case 2: u^2 > 1

In this case, the roots are real and distinct, so the general solution of the homogeneous equation is

y = c₁e^(r1x) + c₂e^(r2x),

where r1 = u + sqrt(u^2 - 1) and r2 = u - sqrt(u^2 - 1).

Case 3: u^2 = 1

In this case, the root is r = u, so the general solution of the homogeneous equation is

y = c₁e^(ux) + c₂xe^(ux).

Now, we can find the particular solution of the non-homogeneous equation using the method of undetermined coefficients.

Assuming a particular solution of the form y = Asin(x) + Bcos(x), we have

2uB - Asin(x) - Bcos(x) = cos(x).

Matching coefficients, we get A = 0 and 2uB - B = 1, so B = 1/(2u - 1).

Therefore, the particular solution is y = (1/(2u - 1))cos(x).

The general solution to the Cauchy problem is then

y = c₁e^(ux)cos(sqrt(1 - u^2)x) + c₂e^(ux)sin(sqrt(1 - u^2)x) + (1/(2u - 1))cos(x).

To determine the constants c₁ and c₂, we use the initial condition y(2,0) = sina.

Substituting x = 0, we get

c₁ + (1/(2u - 1)) = sina.

Substituting x = pi/2sqrt(1 - u^2), we get

c₂sqrt(1 - u^2) = 0.

Since sqrt(1 - u^2) ≠ 0, we have c₂ = 0.

Therefore, c₁ = sina - 1/(2u - 1), and the solution of the Cauchy problem is

y = (sina - 1/(2u - 1))e^(ux)cos(sqrt(1 - u^2)x) + (1/(2u - 1))cos(x).

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Option (c) is the correct formula for the Newton-Raphson method to find the square root of a real number r from the equation [tex]x^2 - r = 0[/tex]

The Newton-Raphson method is an iterative method for finding the root of a function. To use this method to find the square root of a real number r from the equation [tex]x^2 - r = 0[/tex], we first define a function [tex]f(x) = x^2 - r[/tex]. The root of this function is the square root of r.

The Newton-Raphson method starts with an initial guess x0 for the root and then iteratively improves the guess using the formula:

[tex]xi+1 = xi - f(xi)/f'(xi)[/tex]

where f'(x) is the derivative of f(x). In this case,[tex]f(x) = x^2 - r[/tex], so f'(x) = 2x.

Substituting these values, we get:

[tex]xi+1 = xi - (xi^2 - r)/(2xi)[/tex]

Simplifying, we get:

[tex]xi+1 = xi - (xi/2 - r/2xi)[/tex]

[tex]xi+1 = 1/2(xi + r/xi)[/tex]

Therefore, the formula for the Newton-Raphson method to find the square root of a real number r from the equation [tex]x^2 - r = 0[/tex] is:

[tex]xi+1 = 1/2(xi + r/xi)[/tex]

Option (c) is the correct formula for the Newton-Raphson method to find the square root of a real number r from the equation[tex]x^2 - r = 0[/tex]. This formula involves taking the average of the current guess xi and r/xi as the next guess. This method is efficient and converges quickly to the square root of r.

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Answer: 50

Step-by-step explanation:

Answer:

50?

Step-by-step explanation:

10-5 equals 5

5+7 equals 12

12-2 equals 10

10-9 equals 1

1+2 equals 3

3-5 equals -2

-2+12 equals 10

10-3 equals 7

7+7 equals 14

14-3 equals 11

11+9 equals 20

20-10 equals 10

10-6 equals 4

4+8 equals 12

12-2 equals 10

10+60 equals 70

70-20 equals 50

Therefore, the apparent nth term of the expression is: aⁿ = 5n² - 3n - 2.

The given sequence is not an arithmetic or geometric sequence. However, we can notice that the sequence of differences between consecutive terms is an arithmetic sequence.

The sequence of differences is: 7, 19, 37, 61,...

To find the nth term of this sequence, we can use the formula for the nth term of an arithmetic sequence:

dn = a1 + (n-1) * d

where dn is the nth term of the sequence of differences, a1 is the first term of the sequence of differences, d is the common difference of the sequence of differences, and n is the index of the term we want to find.

So, we have:

dn = 7 + (n-1) * 12

Simplifying this expression, we get:

dn = 5n - 3

Now, we can use this formula to find the nth term of the original sequence. Let's call the nth term an:

an = an-1 + dn-1

where an-1 is the (n-1)th term of the original sequence and dn-1 is the (n-1)th term of the sequence of differences.

We know that a1 = 2 and d1 = 7, so we can use the above formula to find the next terms:

a2 = a1 + d1 = 2 + 7 = 9

a3 = a2 + d2 = 9 + 19 = 28

a4 = a3 + d3 = 28 + 37 = 65

a5 = a4 + d4 = 65 + 61 = 126

Therefore, the apparent nth term of the sequence is: aⁿ = 5n² - 3n - 2.

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Answer:

The answer would be D. [tex]\frac{5\sqrt{11} }{11}[/tex]

For theoretical probability of each letter is 0.2, the experimental probability of drawing a card with the letter for which the experimental probability is closest to the theoretical probability is equals the 0.5.

Theoretical Probability: The theoretical probability of an event is the probability of a particular event based on mathematical calculations, assuming ideal conditions. The theoretical probabilities do not take into account the flaws of the system.

Experimental probability: The experimental probability of an event is the actual probability of an event occurring based on the results of facts rather than mathematical calculations

The theoretical probability of each letter = 0.2 = 2/10

We have to determine the experimental probability of drawing a card with the letter for which the experimental probability is near to the theoretical probability. Now, total possible outcomes = 10 and each letter comes two times

so, the experimental probability = 10/20

= 0.5

Hence, required value is 0.5.

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The correct equation representing the total number, s, of stickers is option C. s/3 = 4.

The formula or relation that can accurately indicate the total number of stickers, number of girls and number of stickers received by girl is as follows -

Total number of stickers/number of girls = number of stickers received by each girl

(We will perform division as number of stickers must be more than number of girls)

Keep the representative value of each

s/3 = 4

Solving it to know the number of stickers

s = 4 × 3

s = 12

Hence, there were 12 stickers divided four each among three girls. Thus, the correct answer is C. s/3 = 4.

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The complete question is -

Elizabeth has some stickers. She divide her stickers equally amiung herself and two friends. Each person get 4 stickers. Which equation repersents the total number,s,of stickers?

A. s + 3 = 4

B. s 3 = 4

C. s/3 = 4

D. 3s = 4

The values of angles in the diagram are ∠DAE = 53⁰, ∠DAE = 48⁰, ∠ACB = 102⁰, ∠ABC = 56⁰.

The value of angles is calculated as follows;

∠DAE = 90 - 37 (complementary angles add up to 90⁰ )

∠DAE = 53⁰

∠DBE = 90 - 42 (complementary angles add up to 90⁰ )

∠DAE = 48⁰

∠ACB = 180 - 78 (sum of angles on a straight line )

∠ACB = 102⁰

∠ABC = 180 - (22 + 102) (sum of angles in a triangle )

∠ABC = 56⁰

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Answer:

11.4583 = 114583 / 10000

Step-by-step explanation:

The diver reaches a height of 72 feet after 1.5 seconds.

Given that, a diver leaps off of cliff, modeled by the equation,

y = -16x²- 8x +120.

We need to time taken by the diver to reach a height of 72 ft.

72 = -16x²- 8x + 120.

-16x²- 8x + 48 = 0

2x² + x - 6 = 0

Factorizing,

2x² + 4x -3x - 6 = 0

2x(x+2) -3(x+2) = 0

(2x-3)(x+2) = 0

x = -2, x = 1.5

Since, time cannot be negative so neglecting x = -2

Hence, the diver reaches a height of 72 feet after 1.5 seconds.

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The solution to the system of equations is :

x1 = 3, x2 = -1, and x3 = 2.

To use the method of Cholesky factorization to solve this system of equations, we need to first write the system in matrix form as AX = B, where:

A = 1 -2 2
-2 5 -3
2 -3 6

X = x1
x2
x3

B = 4
-7
10

Next, we need to perform Cholesky factorization on matrix A. This involves finding a lower triangular matrix L such that A = LL^T, where L^T is the transpose of L. To do this, we use the following algorithm:

1. Set L11 = sqrt(A11).
2. For i = 2 to n, do the following:
a. Set Lii = sqrt(Aii - sum(Lik^2, k=1 to i-1)).
b. For j = i+1 to n, set Lij = (Aij - sum(Lik*Ljk, k=1 to i-1)) / Lii.

Applying this algorithm to matrix A, we get:

L = 1 0 0
-2 1 0
2 -1 1

Now we can solve for X by first solving L*Y = B for Y using forward substitution, and then solving L^T*X = Y for X using backward substitution. The solutions for Y and X are:

Y = 4
1
3

X = 3
-1
2

Therefore, the solution to the system of equations is x1 = 3, x2 = -1, and x3 = 2.

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