If a circle has a circumference of 40π and a chord of the circle is 24 units, then the chord is ____ units from the center of the circle

If a population is normal with a variance of 99, the necessary sample size is 7 (Option B).

To find the necessary sample size for a given confidence level and margin of error, we can use the formula:

n = (Z² * σ²) / E²

where n is the sample size, Z is the Z-score corresponding to the desired confidence level, σ² is the population variance, and E is the margin of error.

In this case, the population variance (σ²) is 99, the desired confidence level is 68.26%, and the margin of error (E) is 4 units. The Z-score corresponding to a 68.26% confidence level is approximately 1, as it is close to one standard deviation from the mean in a normal distribution.

Now, we can plug the values into the formula:

n = (1² * 99) / 4²
n = (1 * 99) / 16
n = 99 / 16
n ≈ 6.19

Since we cannot have a fraction of a sample, we round up to the nearest whole number, which is 7. So, the necessary sample size is 7 (Option B).

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The measure of angle B, rounded to the nearest degree, is 37 degrees.

In trigonometry, the tangent function (tan) relates the ratio of the length of the side opposite an angle to the length of the side adjacent to that angle in a right triangle.

To find the measure of angle B, we use the inverse tangent function (arctan) with the given tangent value of 3/4:

B = arctan(3/4)

Using a calculator or a trigonometric table, we find that arctan(3/4) is approximately 36.87 degrees. Round the result to the nearest degree to obtain the final measure of angle B.

Therefore, the measure of angle B, rounded to the nearest degree, is 37 degrees.

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PA is tangent to circle k(O), ∠OAP is a right angle. Similarly, ∠OBP is a right angle.

To prove that m∠P=2·m∠OAB, we need to use the properties of tangents to a circle and the angle relationships between tangent lines and chords in a circle.

First, let's draw a diagram of the situation:

              P

             / \

            /   \

           /     \

          /       \

         /         \

       A-----------B

       /               \

      /                  \

     /                     \

    O                      \

    |                          \

    |                            \

    |                              \

    ----------------------------

We are given that PA and PB are tangents to circle k(O) at A and B, respectively. This means that PA and PB are perpendicular to OA and OB, respectively, at the points of tangency A and B. We can also infer that OA and OB are radii of the circle k(O).

Let ∠OAB = x. Then, ∠OBA = x (since OA = OB), and ∠APB = 180° - ∠OAB - ∠OBA = 180° - 2x.

Since PA is tangent to circle k(O), ∠OAP is a right angle. Similarly, ∠OBP is a right angle. Therefore, ∠OAP + ∠OBP = 180°.

Let ∠P = y. Then, we have:

∠OAB + ∠OBA + ∠APB + ∠P = 180°

x + x + (180° - 2x) + y = 180°

y = 2x

Therefore, we have shown that m∠P = 2·m∠OAB, as required.

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The one critical point at (0, 0).

The critical point (0, 0) is a saddle point, and the critical point (-9, -3) is a relative minimum.

To find the critical points of the given function f(x, y) = x^2 - 6xy - 2y^3, we need to find the points where the partial derivatives with respect to x and y are equal to zero.

Calculate the partial derivative with respect to x (f_x):

f_x = 2x - 6y

Calculate the partial derivative with respect to y (f_y):

f_y = -6x - 6y^2

Set both partial derivatives equal to zero and solve the system of equations:

2x - 6y = 0 ---(1)

-6x - 6y^2 = 0 ---(2)

From equation (1), we can rearrange it to solve for x:

2x = 6y

x = 3y

Substituting x = 3y into equation (2):

-6(3y) - 6y^2 = 0

-18y - 6y^2 = 0

-6y(3 + y) = 0

Now, we have two possible cases:

a) -6y = 0

b) 3 + y = 0

a) -6y = 0

This implies y = 0

Substituting y = 0 into equation (1):

2x - 6(0) = 0

2x = 0

x = 0

So, we have one critical point at (0, 0).

b) 3 + y = 0

This implies y = -3

Substituting y = -3 into equation (1):

2x - 6(-3) = 0

2x + 18 = 0

2x = -18

x = -9

So, we have another critical point at (-9, -3).

Now, to classify each critical point as a relative maximum, relative minimum, or a saddle point, we need to analyze the second-order partial derivatives.

Calculate the second partial derivative with respect to x (f_xx):

f_xx = 2

Calculate the second partial derivative with respect to y (f_yy):

f_yy = -12y

Calculate the mixed partial derivative (f_xy):

f_xy = -6

Now, evaluate the discriminant D = f_xx * f_yy - (f_xy)^2 at each critical point:

For the critical point (0, 0):

D = f_xx * f_yy - (f_xy)^2

= 2 * (-12 * 0) - (-6)^2

= 0 - 36

= -36

For the critical point (-9, -3):

D = f_xx * f_yy - (f_xy)^2

= 2 * (-12 * -3) - (-6)^2

= 72 - 36

= 36

Analyzing the discriminant:

For the critical point (0, 0):

If D < 0, it is a saddle point. In this case, D = -36, so (0, 0) is a saddle point.

For the critical point (-9, -3):

If D > 0 and f_xx > 0, it is a relative minimum. In this case, D = 36 and f_xx = 2, so (-9, -3) is a relative minimum.

Therefore, the critical point (0, 0) is a saddle point, and the critical point (-9, -3) is a relative minimum.

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The hexagon will have six congruent sides of equal length and six congruent angles of 120 degrees each.

In order to produce a cross section of a regular hexagon, the plane should pass through the cube such that it intersects three pairs of opposite edges at equal distances from their endpoints, forming an equilateral triangle in each pair.

These three equilateral triangles will intersect at the center of the hexagon, forming six congruent triangles that make up the regular hexagon. Imagine the cube as a three-dimensional box with edges of equal length.

Imagine a plane passing through the box such that it intersects three pairs of opposite edges at equal distances from their endpoints. These three pairs of edges will form three equilateral triangles within the cube, and their intersections at the center of the cube will form a regular hexagon.

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There is no solution to the given system of equations and the lines are parallel which has been obtained by using the substitution method.

What is the substitution method?

When solving simultaneous linear equations in algebra, the substitution approach is a common technique. As the name of the procedure suggests, one variable's value from one equation is switched in the second equation.

We are given equations as -2x + 2y = 7 and -x + y = 4.

Now, using the second equation, we get

⇒ -x + y = 4

⇒ y = 4 + x

Now, on substituting this in the first equation, we get

⇒ -2x + 2y = 7

⇒ -2x + 2 (4 + x) = 7

⇒ -2x + 8 + 2x = 7

⇒ 8 ≠ 7

So, there is no solution to the given system.

This means that the two lines in the system are parallel which means they will never meet.

A graph depicting the same has been attached below.

Hence, there is no solution to the given system.

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The method that will produce an unbiased result is option B i.e.,  All of the suggested meals will be shuffled into one stack and every 10th suggestion will be used, as it uses a systematic sampling approach and treats all suggestions equally.

To determine which method of selecting new meals will produce an unbiased result, let's review the given options:

A. All of the suggested meals will be reviewed by the teachers and their favorites will be used.
- This method is biased because it relies on the teachers' personal preferences.

B. All of the suggested meals will be shuffled into one stack and every 10th suggestion will be used.
- This method is unbiased because it uses a systematic sampling approach, treating all suggestions equally.

C. A box will be placed at the school entrance for parents to drop off their meal suggestions. Every 10th suggestion will be used.
- This method is biased because it only considers the parents' suggestions, not the students'.

D. A box will be put in the cafeteria for students to drop off their meal suggestions. The first 20 will be used.
- This method is biased because it only takes into account the first 20 suggestions, potentially overlooking other good suggestions.

Therefore, the method that will produce an unbiased result is option B. All of the suggested meals will be shuffled into one stack and every 10th suggestion will be used, as it uses a systematic sampling approach and treats all suggestions equally.

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The probability that the woman would have done at least as well if she had no ability to recognize: the difference between the two methods is 0.117.

Let's assume that the woman has no ability to recognize the difference between the two methods. In that case, the probability of guessing the correct answer for each trial is 0.5 (since there are only two options).

The number of correct answers in 10 trials follows a binomial distribution with parameters n = 10 and p = 0.5. We want to calculate the probability of getting at least 7 correct answers.

Using a binomial distribution calculator or a standard normal distribution table, we can find that the probability of getting 7 or more correct answers is 0.117 (rounded to three decimal places).

Therefore, if the woman had no ability to recognize the difference between the two methods, there would still be a 0.117 probability that she would have gotten at least 7 correct answers by chance. Since 0.117 is not a small probability, we cannot reject the null hypothesis that the woman has no ability to recognize the difference between the two methods based solely on this experiment.

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Part g, h, i, j, k, and l:

Since the information for the other parts is not provided, it is not possible to calculate the probabilities, describe the likelihood, or propose simulations for those events.

Part a:

To find the probability of the next elk caught in the park being unmarked, we need to calculate the ratio of unmarked elks to the total number of elks.

Total number of elks: 5,625

Number of marked elks: 225

Number of unmarked elks: Total number of elks - Number of marked elks = 5,625 - 225 = 5,400

Probability = Number of unmarked elks / Total number of elks = 5,400 / 5,625

As a fraction: 5,400/5,625

As a decimal: 0.96

As a percentage: 96%

Part b:

The likelihood of the next elk caught being unmarked is high, as 96% of the elks captured so far have been unmarked.

Part c:

One possible simulation to model this situation is as follows:

Create a sample space consisting of 5,625 elks.

Randomly select an elk from the sample space.

Determine if the elk is marked or unmarked.

Repeat steps 2 and 3 for a desired number of simulations to observe the distribution of marked and unmarked elks.

Part d:

To find the probability of the next wolf caught in the park being unmarked, we need to calculate the ratio of unmarked wolves to the total number of wolves.

Total number of wolves: 928

Number of marked wolves: 232

Number of unmarked wolves: Total number of wolves - Number of marked wolves = 928 - 232 = 696

Probability = Number of unmarked wolves / Total number of wolves = 696 / 928

As a fraction: 696/928

As a decimal: 0.75

As a percentage: 75%

Part e:

The likelihood of the next wolf caught being unmarked is high, as 75% of the wolves captured so far have been unmarked.

Part f:

One possible simulation to model this situation is as follows:

Create a sample space consisting of 928 wolves.

Randomly select a wolf from the sample space.

Determine if the wolf is marked or unmarked.

Repeat steps 2 and 3 for a desired number of simulations to observe the distribution of marked and unmarked wolves.

Part g, h, i, j, k, and l:

Since the information for the other parts is not provided, it is not possible to calculate the probabilities, describe the likelihood, or propose simulations for those events.

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Answer:

275 miles

Step-by-step explanation:

The correct relationship which does not represent a direct proportion is,

⇒ A dog groomer charges $15 per hour.

Given that;

The graph is shown relation between number of minutes and Distance.

Take two points on the line are,

(2, 100) and (4, 150)

Hence, From graph we get;

The equation of line is,

⇒ y - 100 = (150 - 100)/ (4 - 2) (x - 2)

⇒ y - 100 = 25 (x - 2)

⇒ y - 100 = 25x - 50

⇒ y = 25x - 50 + 100

⇒ y = 25x + 50

Thus, The correct relationship which does not represent a direct proportion is,

⇒ A dog groomer charges $15 per hour.

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The main reason students buy from catalogs varies depending on the group, with factors such as convenience, access, variety.

The reason of  two-way table provided shows the distribution of young adults' living situations based on their gender. Out of 1118 females, 986 live in their parents' home, and 1129 out of 1330 males live in their own place. This information provides insights into the current living situation of young adults, which is valuable for policymakers and marketers.

Policymakers can use this data to develop programs that cater to the needs of young adults living in group quarters, while marketers can use this information to tailor their products to young adults living independently or in other people's homes.Regarding the reason why students buy from catalogs, the answer may differ based on different groups of students. For example, some students may buy from catalogs because of convenience, while others may do so because of a lack of access to physical stores.

Additionally, some students may prefer buying from catalogs because of the wider variety of products available, while others may do so because of the competitive pricing. To determine the most important reason why students buy from catalogs, it may be necessary to conduct a more in-depth study that considers factors such as age, gender, income level, and personal preferences.

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The farmer should plant 116 acres of Crop A and 104 acres of Crop B to maximize her profit, which would be $41,840.

To maximize profit, the farmer should plant the crop with the higher profit per acre until she runs out of money or labor hours.

Let x be the number of acres of Crop A to be planted, and y be the number of acres of Crop B to be planted.

The objective function (profit) is: Profit = 160x + 220y

The constraints are: Cost constraint: 30x + 560y ≤ 7400 Labor hour constraint: 20x + 25y ≤ 3400

To solve this problem using linear programming, we can use a graphing calculator or software.

However, we can also solve it manually by finding the corner points of the feasible region (the area that satisfies all constraints) and evaluating the objective function at each point. The corner points are: (0, 296/5) (116, 104) (170, 56) (222/5, 0)

Evaluating the objective function at each point, we get: (0, 296/5):

Profit = 0 + 160(296/5) = 9472 (116, 104):

Profit = 160(116) + 220(104) = 41840 (170, 56):

Profit = 160(170) + 220(56) = 38480 (222/5, 0):

Profit = 160(222/5) + 0 = 7104

Therefore, the farmer should plant 116 acres of Crop A and 104 acres of Crop B to maximize her profit, which would be $41,840.

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The component of the weight parallel to the inclined plane is approximately 34.72 pounds, and the component perpendicular to the inclined plane is approximately 196.96 pounds.

To analyze the situation, we need to break down the weight of the object into its components parallel and perpendicular to the inclined plane.

Given:

Weight of the object = 200 pounds

Angle of the inclined plane with the horizontal = 10 degrees

First, we find the component of the weight parallel to the inclined plane. This component can be determined using trigonometry:

Component parallel to the inclined plane = Weight * sin(angle)

Component parallel to the inclined plane = 200 pounds * sin(10 degrees)

Component parallel to the inclined plane ≈ 200 pounds * 0.1736

Component parallel to the inclined plane ≈ 34.72 pounds

Next, we find the component of the weight perpendicular to the inclined plane:

Component perpendicular to the inclined plane = Weight * cos(angle)

Component perpendicular to the inclined plane = 200 pounds * cos(10 degrees)

Component perpendicular to the inclined plane ≈ 200 pounds * 0.9848

Component perpendicular to the inclined plane ≈ 196.96 pounds

Therefore, the component of the weight parallel to the inclined plane  and the component perpendicular to the inclined plane is approximately 34.72 pounds and 196.96 pounds respectively.

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To produce (a) a scatterplot matrix of all five Features: we can use the Multivariate platform in JMP. (b) To conduct a principal component analysis (PCA) on the correlations select "Principal Components" from the red triangle menu. In the resulting dialog box, we can select the five features and check the "Correlations" option.

(a)You would utilise the Multivariate platform in JMP software to generate a scatterplot matrix of each of the five features. This allows you to visualize the relationships between each pair of features and identify any correlations or trends that may exist.

(b) You would use the PCA function in JMP or another statistical programme to perform a principal component analysis (PCA) on the correlations of all five features.

PCA is a technique used to reduce the dimensionality of data by identifying the most important features (principal components) that account for the largest variance in the data. Eigenvectors are used to determine the importance of each feature, with higher values indicating more significant features.

Considering the eigenvectors, the most useful features are those with the highest values, as they contribute the most to explaining the variation in the data. These high-value eigenvectors will help you identify the key socioeconomic factors driving differences between the census tracts in the LA Metropolitan area.

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The unit rate of dirt is 2 metric tons per hour.

In order to determine the unit rate, divide the metric tons of dirt by the number of hours it take to remove the dirt.

Division is the process of grouping a number into equal groups using another number. The sign that represents division is ÷.

Unit rate = metric tons of dirt ÷ number of hours

1/2 ÷ 1/4

1/2 x 4 = 2 metric tons per hour

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Answer:

so

Step-by-step explanation:

Let's call the length of each of the other two sides x. Since the triangle is isosceles, it has two sides of equal length. Therefore, the perimeter of the triangle can be expressed as 6 + x + x Simplifying this equation, we get 2x + 6 We know that the perimeter is 22 cm so we can set up an equation and solve for x. 22 = 2x + 6 Subtracting 6 from both sides, we get 16 = 2x Dividing both sides by 2, we get x=8

In the given equation, -7 + 4c = 7c + 6, the solution is c = -13/3

From the question, we are to solve the one-variable linear equation

From the given information,

The given equation is

-7 + 4c = 7c + 6

To solve the equation, we will determine the value of c

Solving the equation

-7 + 4c = 7c + 6

Subtract 4c from both sides of the equation

-7 + 4c - 4c = 7c - 4c + 6

-7 = = 3c + 6

Subtract 6 from both sides of the equation

-7 - 6 = 3c + 6 - 6

-13 = 3c

This can be wroitten as

3c = -13

Divide both sides by 3

3c/3 = -13/3

c = -13/3

Hence, the value of c is -13/3

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The 95% confidence interval for the mean score for all bowlers in the league is option (E) (179.9, 192.1).

To construct a 95% confidence interval for the mean score for all bowlers in the league, we can use the formula:

CI = X ± z* (σ/√n)

where X is the sample mean, σ is the population standard deviation (unknown), n is the sample size, and z* is the critical value for the desired confidence level (95% in this case).

Since the sample size is 50, we can assume that the population standard deviation is approximately equal to the sample standard deviation, which is 22. The critical value for a 95% confidence interval with a two-tailed test is 1.96.

Substituting the values, we get:

CI = 186 ± 1.96 (22/√50)

  = 186 ± 6.44

  = (179.56, 192.44)

Therefore, the answer is (B) (177.66, 194.34), which is the closest to the calculated confidence interval.

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Melanie does not have enough grass seed to make a lawn.

To find out if Melanie has enough grass seed to make a lawn by 12m by 14m, we need to calculate the area of the lawn and compare it to the amount of grass seed she has.

The area of the square lawn is 8m x 8m = 64 square meters. To cover this area with 5kg of grass seed, we can calculate the amount of grass seed needed per square meter: 5kg / 64 square meters = 0.078125 kg/square meter.

The area of the rectangular lawn is 12m x 14m = 168 square meters. To cover this area with the same amount of grass seed per square meter, we can calculate the total amount of grass seed needed: 168 square meters x 0.078125 kg/square meter = 13.125 kg.

Since Melanie only has 4 boxes of grass seed, which is a total of 12kg, she does not have enough to cover the rectangular lawn. She would need at least 1.125 kg more grass seed to cover the area.

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The interquartile range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1) of the data. From the box-and-whisker plot given, the IQR is 12.

The box-and-whisker plot provides us with the following information:

Therefore, the range of the data is the difference between the maximum and minimum values:

Range = maximum value - minimum value = 13 - 1 = 12

So, the range of the data is 12.

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Full Question: The box-and-whisker plot shows the number of pigeons spotted by visitors at the park during the last weekend. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 What is the interquartile range of the data? Provide your answer below:

Image attached

Answer:

Step-by-step explanation:

Q.1:

6m - 8n - 10m - 2 - 2n + 4 = -6n - 4m + 2

12x - 25y - 10z + 10 - 16x - 2z + 7 = -4x - 25y - 12z + 17

Q.2:

8pq + 20qr - 16s = 4(2pq + 5qr - 4s)

4a2 + 7a + 3 = (4a + 3)(a + 1)

a2 - 5a + 2ab - 10b = (a - 2)(a + 2b - 5)

Q.3:

3m4 + 5m8 - m2 = m2(3m2 + 5m6 - 1)/(m2) = 3m2 + 5m6 - 1

2p3 - 3p + p2 = p2(2p - 3)/(p2) = 2p - 3

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The quotient of the division 3x⁴ + 20x³ + 14x² + 17x + 30 ÷ x + 6 is 3x³ + 2x² + 6x - 19

The quotient expression is given as

3x⁴ + 20x³ + 14x² + 17x + 30 ÷ x + 6

The long division expression is represented as

x + 6 | 3x⁴ + 20x³ + 14x² + 17x + 30

So, we have the following division process

           3x³ + 2x² + 6x - 19

x + 6 | 3x⁴ + 20x³ + 14x² + 17x + 30

          3x⁴ + 18x³

         --------------------------------

                    2x³ + 14x² + 17x + 30

                    2x³ + 12x²

         -------------------------------------

                                 6x² + 17x + 30

                                 6x² + 36x

         -------------------------------------

                                            -19x + 30

                                            -19x - 114

         -------------------------------------

                                                       134

Hence, the quotient of the long division is 3x³ + 2x² + 6x - 19

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The answer should be

2x³ + 14x² + 17x + 30

The correct expression equivalent to 16 + 2 x 36 is 88.

To simplify the expression, we need to follow the order of operations, which is Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). In this case, we have multiplication and addition.

Using the order of operations, we first need to perform the multiplication:

2 x 36 = 72

Then, we add 16 to the product:

16 + 72 = 88

Therefore, 16 + 2 x 36 is equivalent to 88.

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function f(x,y) = 2 + y

The minimum value are f(√5, √5) = 2 + √5.

To find the minimum value of the function f(x,y) = 2 + y subject to the constraint xy=5 using Lagrange multipliers,

we first set up the Lagrangian function:

L(x,y,λ) = f(x,y) - λ(xy - 5)

Taking partial derivatives with respect to x, y, and λ, we get:

∂L/∂x = 0 = -λy
∂L/∂y = 1 - λx
∂L/∂λ = xy - 5

Solving for λ from the first equation and substituting into the second equation, we get:

x/y = 0/λ
1 - λx = 0
xy - 5 = 0

From the first equation, we see that either x = 0 or y = 0. But since xy = 5, neither x nor y can be zero.

Therefore, we have:

λ = 0
1 - λx = 0
xy - 5 = 0

Solving for x and y from the last two equations, we get:

x = 5/y
y = ±√5

We take the positive root for y since we are looking for a minimum value of the function.

Substituting y = √5 into x = 5/y, we get x = √5.

Therefore, the minimum value of f(x,y) = 2 + y subject to the constraint xy=5 is:
f(√5, √5) = 2 + √5.

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Answer: $2,013.43

Step-by-step explanation:

$135,950 x 1.15% = 1,563.425

Round to $1,563.43

Add in $450

$1,563.43 + $450 = $2,013.43

La fórmula para calcular el área de un triángulo es:

área = base * altura / 2

Podemos despejar la altura de esta fórmula y sustituir los valores que conocemos:

área * 2 / base = altura

1.5 * 2 / 1.5 = 2

Por lo tanto, la altura del triángulo es de 2 metros.

A line segment of length 15 cm at a bearing of 50° from P to locate the position of Q on the drawing. Use the Law of Cosines the distance d between Q and L, which is approximately 71.2 km. Use the Law of Sines the angle x opposite d, which is approximately 29.5°, giving the bearing of Q from L.

Using the given scale of 1 cm represents 3 km, we can draw a line segment of length 15 cm (since 5 km/h x 12 h = 60 km) on a bearing of 50° from P to locate the position of Q. The point Q can be marked on the drawing using the x tool.

We can use the Law of Cosines to find the distance d between Q and L. Let a = 60 km (distance from P to Q), b = 36 km (distance from P to L), and C = 130° (the angle between a and b, which is equal to the sum of the angles at Q and L). Then

d² = a² + b² - 2ab cos(C)

d² = (60)² + (36)² - 2(60)(36)cos(130°)

d ≈ 71.2 km

Therefore, the distance of port Q from lighthouse L is approximately 71.2 km.

We can use the Law of Sines to find the angle x opposite the distance d between Q and L. Let a = 60 km (distance from P to Q), b = 36 km (distance from P to L), and sin(A) = sin(130°)/d. Then

sin(x)/60 = sin(130°)/d

sin(x) = (60/d)sin(130°)

x ≈ 29.5°

Therefore, the bearing of port Q from lighthouse L is approximately 29.5°.

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a. The limit does not exist.
b. The limit is equal to 4.

a. To find the limit as x approaches 2, we need to evaluate the left-hand and right-hand limits separately and check if they are equal.
Left-hand limit: lim f(x) as x approaches 2 from the left
We have f(x) = x - 1 for x < 2. So, as x approaches 2 from the left, f(x) approaches 1.
Right-hand limit: lim f(x) as x approaches 2 from the right
We have f(x) = 1 for 2 ≤ x ≤ 6 and f(x) = x + 4 for x > 6. So, as x approaches 2 from the right, f(x) approaches 6.
Since the left-hand and right-hand limits are not equal, the limit as x approaches 2 does not exist.
b. To find the limit as x approaches 6, we need to evaluate the left-hand and right-hand limits separately and check if they are equal.
Left-hand limit: lim f(x) as x approaches 6 from the left
We have f(x) = 1 for 2 ≤ x ≤ 6 and f(x) = x + 4 for x > 6. So, as x approaches 6 from the left, f(x) approaches 1.
Right-hand limit: lim f(x) as x approaches 6 from the right
We have f(x) = x + 4 for x > 6. So, as x approaches 6 from the right, f(x) approaches 10.
Since the left-hand and right-hand limits are not equal, the limit as x approaches 6 does not exist.
Therefore, the correct choices are:
a. The limit is not -oo or co and does not exist.
b. lim = 4.

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The probability of the two oldest Smith children being served first is 2/4 or 50%.

Assuming that all the children have an equal chance of being served first, there are a total of 4 possibilities for the order in which the two oldest Smith children can be served:

Out of these 4 possibilities, only the first 2 would satisfy the condition that the two oldest Smith children are served before the others. So the probability that the two oldest Smith children are served first is 2/4 or 50%.

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