Adding mixed numbers with like denominators worksheets.

Answer:

Step-by-step explanation:

volume of cone =(1/3)*3.14*r^2h

radius of cone=(11/2)=5.5in

height (h)=26in

volume=826.62in³

A z-statistic reports how many standard deviations an observed value is from the expected value, where the expected value is calculated using the population mean and standard deviation.

To calculate the z-statistic, use the following formula:

Where:

x = the observed value

μ = the population mean

σ = the population standard deviation

n = the sample size

The z-statistic tells us how many standard deviations an observed value is from the expected value, which is the population mean. A positive z-score indicates that the observed value is above the expected value, while a negative z-score indicates that the observed value is below the expected value. A z-score of 0 indicates that the observed value is equal to the expected value. By calculating the z-statistic, we can determine how unusual or significant an observed value is relative to the population.

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Complete Question:

a z-statistic reports how many sds an observed value is from the expected value, where the expected value is calculated using the___.

Part (a)  K is a normal extension of E, but not of F.

Part (b) L is a normal extension of F, but L' is not.

Part (a)

Let E = KG(K,F). Since E is a field extension of F and K is a finite extension of F, then K is a normal extension of E. This is because F[K] is a finite separable extension and hence normal.

However, K is not a normal extension of any field F ⊆ E. This is because K is not a normal extension of F by assumption.

Part (b)

(a) L is a normal extension of F. This is because K is a normal extension of F (by assumption) and L is a splitting field of a polynomial over F, and so is a finite extension of K. Hence, L is a normal extension of F.

(b) Let L be any field such that K ⊆ L and L ≠ L'. Then L is not a normal extension of F. This is because K is a normal extension of F, and if L were a normal extension of F, then L' would also be a normal extension of F by transitivity. Because  L ≠ L', L is not a typical extension of F.

Complete Question:

Suppose that the characteristic of F is 0, and that K is a finite, but not normal, extension of F. (a) Let E = KG(K,F). Show that K s a normal extension of E, but not of any field F ES E. Solution. Proof (b) Suppose K -F(a) for some aE K. Let L be the splitting field of the minimal polynomial of a over F, with K C L. Prove that: (a) L is a normal extension of F Solution. Put your answer here... (b) For any field L, such that K L, Ç L, L' is not a normal extension of F. (L is called the normal closure of F in K.) Solution. Put your answer here...

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The product of the numbers (−0. 64) and (−2. 5) is

1.6.

To multiply (-0.64)(-2.5), we can accomplish the task using the following steps:

Multiply the absolute values of the numbers:

0.64 x 2.5 = 1.6

Determine the sign of the product: Since we are multiplying two negative numbers, the product is positive.

Add the sign to the product: (-0.64)(-2.5) = 1.6

hence, (-0.64) x (-2.5) = 1.6.

These can also be solved using calculator

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The probability of selecting a white bead, replacing it, and then selecting a red bead is 0.15 and this is an independent event.

To fill in the blanks:

1)The probability of selecting a white bead, replacing it, and then selecting a red bead is:

P(white, then red) = P(white) * P(red | white) = (3/10) * (5/10) = 0.15

Here, P(white) is the probability of selecting a white bead on the first draw, which is 3/10

And P(red | white) is the probability of selecting a red bead on the second draw, given that a white bead was selected on the first draw.

Since the first bead is replaced before the second draw, the probability of selecting a red bead on the second draw is still 5/10.

2)This is an independent event.

This is because the first draw does not affect the probability of the second draw, since the bead is replaced.

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Answer: Type I error and Type II error are associated with hypothesis testing, where we test a hypothesis by collecting data and analyzing it.

For the given hypothesis, we can set up the null hypothesis as follows:

H0: The percentage of households with Internet access is less than or equal to 60%.

And the alternative hypothesis as:

Ha: The percentage of households with Internet access is greater than 60%.

Now, a Type I error occurs when we reject the null hypothesis (i.e., conclude that the percentage of households with Internet access is greater than 60%) when it is actually true. This means that we would be making a false claim that the percentage of households with Internet access is greater than 60%, when it is not.

On the other hand, a Type II error occurs when we fail to reject the null hypothesis (i.e., conclude that the percentage of households with Internet access is less than or equal to 60%) when it is actually false. This means that we would be missing the truth that the percentage of households with Internet access is greater than 60%.

So, in the context of the given hypothesis, a Type I error would be to conclude that the percentage of households with Internet access is greater than 60% when it is actually less than or equal to 60%, and a Type II error would be to fail to conclude that the percentage of households with Internet access is greater than 60% when it is actually greater than 60%.

For a NaCl crystal has six slip systems,

A) The exact slip planes are [tex][\bar 1 1 0] [/tex], [tex][0 \bar1 1] [/tex], [tex][\bar 1 0 1] [/tex], [ 1 1 0], [0 1 1] and [ 1 0 1] and slip direction are (110), (011), ( 101) , [tex]( \bar 1 1 0) [/tex], [tex](0 \bar 11 ) [/tex] and [tex]( \bar1 0 1) [/tex] of te six systems.

B) The sketch of the slip plane and slip direction for each slip system is present in attached figure.

C) Slip systems (ii), (iii), (v) & (vi) are the suitable ones.

We have a NaCl crystals has slip on {110} < 110 > slip systems. Total number of possible systems = 6

The slip plane refers to the plane of maximum atomic density, and the slip direction is the closest folded direction in the slip plane.

A) For six possible systems, slip planes and slip directions are the following:

plane : [tex][\bar 1 1 0] [/tex].

Plane : [tex][0 \bar1 1] [/tex]

plane : [tex][\bar 1 0 1] [/tex]

plane : [ 1 1 0]

plane : [0 1 1]

plane : [ 1 0 1]

B) Using the standard cubic representations, the slip plane and slip direction of each system is present in attached figure.

C) Shear is y stress would at assis ; suitable 21 be highest in the direction which either x axis and y axis parallel to system (ii), (iii), (v) & (vi) are the suitable ones. Hence, we resolved all parts.

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The degree of freedom is 2 and the p-value is between 0.0005 and 0.001.

To calculate the degrees of freedom, we need to first determine the number of rows and columns in the table. Here, we have 3 rows and 2 columns, so the degrees of freedom will be (number of rows - 1) x (number of columns - 1) = 2 x 1 = 2.

To calculate the chi-squared statistic, we need to first calculate the expected values for each cell assuming the null hypothesis of independence. The expected value for the cell in the first row and first column can be calculated as follows:

Expected value = (row total x column total) / grand total

Expected value = (172 x 147) / 291

Expected value = 86.93

Similarly, we can calculate the expected values for the other cells:

Expected value for cell (1,2) = (172 x 144) / 291 = 85.07

Expected value for cell (2,1) = (171 x 147) / 291 = 86.07

Expected value for cell (2,2) = (171 x 144) / 291 = 84.93

Expected value for cell (3,1) = (167 x 147) / 291 = 84.93

Expected value for cell (3,2) = (167 x 144) / 291 = 82.07

Using these expected values, we can calculate the chi-squared statistic as follows:

[tex]X^{2}[/tex] = Σ[[tex](O-E)^{2}[/tex] / E]

[tex]X^{2}[/tex]  = [[tex](79-86.93)^{2}[/tex] / 86.93] + [[tex](93-85.07)^{2}[/tex] / 85.07] + [[tex](103-86.07)^{2}[/tex] / 86.07] + [[tex](44-84.93)^{2}[/tex] / 84.93] + [[tex](68-84.93)^{2}[/tex] / 84.93] + [[tex](99-82.07)^{2}[/tex] / 82.07]

[tex]X^{2}[/tex]  = 19.46

To find the exact P-value, we can use a chi-squared distribution table with 2 degrees of freedom. From the table, we see that the probability of getting a chi-squared value of 19.46 or higher with 2 degrees of freedom is between 0.0005 and 0.001. Therefore, the exact P-value for this test is between 0.0005 and 0.001.

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Answer: The volume of the light sphere is 3052in³

Step-by-step explanation; diameter of sphere= 18in

therefore, the radius of the sphere = diameter/2

R = 18/2 = 9in

now, the volume of the light sphere = 4/3 πr³

Therefore volume = 4/3*3.14*9*9*9

                              =3052.08 in³  ≈ 3052in³

Answer:

The real solutions are:

x = -5, -3, 4, 8

The critical value of z for a two-tailed test at 12.66% significance level is approximately ±1.88.

For a two-tailed test at 12.66% significance level, we need to find the critical value of z. Here are the steps to find the critical value:

1. Since it is a two-tailed test, we need to divide the significance level by 2. This is because the rejection region is distributed equally in both tails of the distribution. So, 12.66% ÷ 2 = 6.33%.

2. Now, we need to convert the percentage to a decimal: 6.33% = 0.0633.

3. Next, we need to find the z-score that corresponds to the given probability (0.0633) in the standard normal distribution table. To do this, look for the closest probability value to 0.0633 in the body of the table.

4. After finding the closest probability value, locate the corresponding z-score. For a two-tailed test, you will have both a positive and negative z-score, which are symmetric around the mean (0).

After following these steps, you will find that the critical value of z for a two-tailed test at 12.66% significance level is approximately ±1.88.

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The orthogonal projection of vector v = (7,-4) onto the subspace w spanned by vector u1 = (1,1) is (3/2, 3/2).

To find the orthogonal projection of v onto w, we need to find the projection vector p, where p is the closest vector in w to v.

Since the vectors ui are orthogonal, the subspace w is the span of u1 only. Therefore, we need to find the scalar projection of v onto u1

proj_u1(v) = ((v·u1) / ||u1||²) * u1

where · denotes the dot product, and ||u1|| denotes the norm of u1.

We have

v·u1 = (7)(1) + (-4)(1) = 3

||u1||² = (1)² + (1)² = 2

So,

proj_u1(v) = (3 / 2) * u1 = (3/2, 3/2)

Therefore, the orthogonal projection of v onto w is p = proj_u1(v) = (3/2, 3/2).

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Note that the estimated maximum number of employees going to the gym at least twice a week is 675.

The sample size is 500 people, and 25% of them go to the gym at least twice a week, resulting in 0.25*500 = 125 employees.

Because the margin of error is 2%, the actual percentage of employees who go to the gym at least twice a week could range between 23% and 27%.

To compute the expected maximum number of workers who go to the gym at least twice a week, we may assume that all 2,500 employees have been polled and use the upper bound of the confidence interval

675 workers are equal to 0.27 x 2,500.

As a result, the maximum number of employees who go to the gym at least twice a week is 675.

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The correct answer is option A, "smoothing constant".

What is statistics?

Statistics is a branch of mathematics that deals with the collection, analysis, interpretation, presentation, and organization of numerical data.

Exponential smoothing is a widely used forecasting technique that involves giving different weights to the past observations in a time series, with the most recent observation being given the highest weight. The parameter that determines the weight given to the most recent observation is known as the "smoothing constant".

The smoothing constant is a value between 0 and 1, and it represents the rate at which the influence of past observations on the forecast decreases over time. A larger value for the smoothing constant gives more weight to recent observations, while a smaller value gives more weight to older observations.

The parameter of the exponential smoothing model that provides the weight given to the most recent time series value in the calculation of the forecast value is known as the "smoothing constant".

Therefore, the correct answer is option A, "smoothing constant".

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To find a parametrization of the surface 3x^2 + 8xy, we can use two parameters u and v, such that x = u and y = v.

Substituting these values into the equation gives us z = 3u^2 + 8uv. Therefore, a parametrization of the surface is (u, v, 3u^2 + 8uv).

To find the tangent plane at x = 1, y = 0, z = 3, we need to first find the partial derivatives of the parametric equation with respect to u and v. These are:

∂/∂u (u, v, 3u^2 + 8uv) = (1, 0, 6u + 8v)
∂/∂v (u, v, 3u^2 + 8uv) = (0, 1, 8u)

Evaluating these partial derivatives at the point (1, 0, 3), we get:

∂/∂u (1, 0, 3) = (1, 0, 6)
∂/∂v (1, 0, 3) = (0, 1, 8)

Using the cross product of these two vectors, we can find the normal vector to the tangent plane:

(1, 0, 6) x (0, 1, 8) = (-6, -8, 1)

Therefore, the equation of the tangent plane is:

-6(x-1) - 8y + (z-3) = 0

To compare this with the graph, we can plot the surface using a software such as WolframAlpha, and then plot the tangent plane at the given point.

The tangent plane should appear as a flat surface tangent to the original surface at that point.

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The zeroes of the function that models the height of a ball after it is thrown are t = 1 and t = - 1 / 2.

We are given  h ( t ) =  ( 1 - t ) ( 8 + 16 t)

We can find the zeros by setting them to zero :

( 1 - t ) = 0

t = 1

( 8 + 16 t ) = 0

16 t = - 8

t = - 8 / 16

t = -1 / 2

The zeroes signify moments when the ball's height is at or below ground level. Solely one zero possesses significance in a physical sense since negative time lacks relation to reality, t = 1 marking the point at which the ball arrives at the ground, denoting a period spanning only one second.

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Answer: 38 3/4 or 38.75

Step-by-step explanation:

multiply 3 1/2 * 5 = 17 1/2

multiply 4 1/4 * 5 = 21 1/4

add the 2

=38 3/4

Answer:

To find the central angle in radians, you can use the formula:

θ = s/r

where θ is the central angle in radians, s is the length of the arc, and r is the radius of the circle.

Substituting the given values, we get:

θ = 27.1434/9

θ = 3.01604 radians

To find the central angle as a percentage of 2π, we can use the formula:

θ/2π x 100%

Substituting the value of θ, we get:

(3.01604/2π) x 100%

≈ 48.04%

Therefore, the central angle is approximately 48.04% of 2π.

Step-by-step explanation:

a.  P(X ≤ 2) = 0.77

b.  P(X = 1 or X = 2) = 0.60

c.  P(X > 0) = 0.83

d. The cumulative probability distribution function for X is given in the table.

(a) The probability of a randomly selected student holding two or fewer jobs during the past year is given by the sum of the probabilities of holding 0, 1, or 2 jobs:

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.17 + 0.35 + 0.25 = 0.77

Therefore, P(X ≤ 2) = 0.77.

(b) The probability of a randomly selected student holding either one or two jobs during the past year is given by the sum of the probabilities of holding 1 or 2 jobs:

P(X = 1 or X = 2) = P(X = 1) + P(X = 2) = 0.35 + 0.25 = 0.60

Therefore, P = 0.60.

(c) The probability of a randomly selected student holding at least one job during the past year is the complement of the probability of holding no jobs:

P(X > 0) = 1 - P(X = 0) = 1 - 0.17 = 0.83

Therefore, P(X > 0) = 0.83.

(d) The cumulative probability distribution function for X is the sum of the probabilities up to and including the value k:

P(X ≤ k) = ∑P(X = i) for i = 0 to k

Using the given probabilities, we can fill in the table:

k   0    1    2    3    4

P(X ≤ k) 0.17 0.52 0.77 0.93 1.00

Therefore, the cumulative probability distribution function for X is:

P(X ≤ k) = 0.17 for k = 0

P(X ≤ k) = 0.52 for k = 1

P(X ≤ k) = 0.77 for k = 2

P(X ≤ k) = 0.93 for k = 3

P(X ≤ k) = 1.00 for k = 4

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The probability that at least 1 student comes is therefore 1 - 0.05 = 0.95. So the probability that at least 1 student comes is 1.00 - 0.05 = 0.95.

To calculate the probability that at least 1 student comes to office hours on Wednesday, we need to add up the probabilities of all the possible scenarios in which at least 1 student comes.

First, we can calculate the probability that no student comes, which is given by the last number in the list, 0.05.

The probability that at least 1 student comes is therefore 1 - 0.05 = 0.95.

Alternatively, we could add up the probabilities of all the scenarios where at least 1 student comes, which are given by the first 9 numbers in the list:

0.00 + 0.10 + 0.20 + 0.50 + 0.15 + 0.05 + 0.00 + 0.00 + 0.00 = 1.00

So the probability that at least 1 student comes is 1.00 - 0.05 = 0.95.

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A simple (1-for-1) substitution would provide a direct replacement of one element or variable with another element or variable.

This can be useful in simplifying equations or formulas by replacing complex expressions with simpler ones. However, it may not always be applicable or accurate in more complex situations.

A simple 1-for-1 substitution provides a straightforward replacement of one element with another in a given context. This can be applied in various scenarios, such as replacing letters in cryptography, swapping ingredients in a recipe, or substituting variables in mathematical equations. The primary purpose of this substitution is to maintain the overall structure while changing a specific component.

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The average value of the function [tex]f(x) = x^{(1/2)[/tex] on the interval [0, 25] is 50.

An input and an output are connected by a function. It functions similarly to a machine with an input and an output. Additionally, the input and output are somehow connected. The traditional format for writing a function is f(x) "f(x) =... "

To find the average value fave of the function [tex]f(x) = x^{(1/2)[/tex] on the interval [0, 25], we need to use the following formula:

fave = (1/(b-a)) * ∫(a to b) f(x) dx

where a and b are the endpoints of the interval [0, 25].

Substituting the values for a, b, and f(x), we get:

fave = (1/(25-0)) * ∫(0 to 25) [tex]x^{(1/2)[/tex] dx

Using the power rule of integration, we can simplify the integral:

fave = (2/50) * [[tex]x^{(3/2)/(3/2)[/tex]] from 0 to 25

fave = (4/50) * [[tex]25^{(3/2)[/tex] - [tex]0^{(3/2)[/tex]]

fave = (4/50) * ([tex]25^{(3/2)[/tex])

fave = 10√25

fave = 10(5)

fave = 50

Therefore, the average value of the function [tex]f(x) = x^{(1/2)[/tex] on the interval [0, 25] is 50.

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FALSE. The number of dots in a dotplot of a bootstrap distribution is equal to the size of the bootstrap sample, not the original sample.

In a bootstrap analysis, multiple samples of the same size are drawn from the original sample with replacement, creating a distribution of sample statistics. The dotplot of this distribution represents the frequency of the sample statistics and can help us understand the variability of the original sample.

The number of dots in the dotplot reflects the number of bootstrap samples, which can be much larger than the original sample size. Therefore, the number of dots in the bootstrap distribution can be different from the size of the original sample.

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Answer:Any number that we can write as a decimal number, between 1.0 and 10.0, multiplied by a power of 10, is said to be in standard form. 1.23 × 108; If you observe carefully, 1.23 is a decimal number between 1.0 and 10.0 and so we have the standard form of 123,000,000 as 1.23 × 10 8.

Step-by-step explanation:

P.S i am emo

There is not sufficient evidence to reject the claim that the mean weight of the cereal packets is at least 14 oz.

Η0 : μ = 14 (Null hypothesis)

H1 : μ < 14 (Alternative hypothesis)

At least sign is ≥ therefore, it should be in null hypothesis. Null Hypothesis, the mean weight of the individual serving boxes of the cereal company is 14 oz or less.

But, we can consider = sign instead of ≥ : Alternative Hypothesis, the mean weight of the individual serving boxes of the cereal company is greater than 14 oz.

Therefore, There is not sufficient evidence to reject the claim that the mean weight of the cereal packets is at least 14 oz.

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Complete question:

A cereal company claims that the mean weight of the cereal in its packets is at least 14 oz. Express the null and alternative hypotheses in symbolic form.

The variance is approximately 177.5 and the standard deviation is approximately 13.32.

The standard deviation (SD, also written as the Greek symbol sigma or the Latin letter s) is a statistic that is used to express how much a group of data values vary from one another.

To find the variance and standard deviation of the data set {91, 101, 75, 90, 88, 87, 90, 60}, we first need to find the mean:

mean = (91 + 101 + 75 + 90 + 88 + 87 + 90 + 60) / 8 = 85.5

Next, we need to calculate the deviation of each data point from the mean:

91 - 85.5 = 5.5

101 - 85.5 = 15.5

75 - 85.5 = -10.5

90 - 85.5 = 4.5

88 - 85.5 = 2.5

87 - 85.5 = 1.5

90 - 85.5 = 4.5

60 - 85.5 = -25.5

Then, we square each deviation:

5.5² = 30.25

15.5² = 240.25

(-10.5)² = 110.25

4.5² = 20.25

2.5² = 6.25

1.5² = 2.25

4.5² = 20.25

(-25.5)² = 650.25

The variance is the average of these squared deviations:

variance = (30.25 + 240.25 + 110.25 + 20.25 + 6.25 + 2.25 + 20.25 + 650.25) / 8 = 177.5

Finally, the standard deviation is the square root of the variance:

standard deviation = sqrt(variance) = √(177.5) = 13.32 (rounded to the nearest hundredth)

Therefore, the variance is approximately 177.5 and the standard deviation is approximately 13.32.

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When talking about a scale drawing or model, the scale factor is determined by the ratio of a given length on the drawing or model to its corresponding length on the actual object.

This can be written as a fraction, with the length on the drawing or model as the numerator and the corresponding length on the actual object as the denominator. For example, if a drawing of a building has a scale factor of 1:100 and the length of a wall on the drawing is 5 cm, the corresponding length on the actual building would be 500 cm (5 cm x 100). Therefore, the scale factor can be written as 5/500 or simplified to 1/100.

When talking about a scale drawing or model, the term you're looking for is "scale factor." It is determined by the ratio of a given length on the drawing or model to its corresponding length on the actual object. To write this as a fraction, you would place the length on the drawing or model as the numerator and the corresponding length on the actual object as the denominator. For example, if the scale factor is 1:10, you would write it as 1/10.

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The term "distance" can refer to different types of measuring  depending on the context. In general, distance is a numerical value that indicates the amount of separation between two objects or points in space.

In mathematics, distance usually refers to the length of a line segment connecting two points in a Euclidean space. The distance can be measured using various formulas, such as the Pythagorean theorem, which computes the length of the hypotenuse of a right triangle formed by the two points and the origin.

In physics, distance can refer to the separation between two objects in three-dimensional space, often measured in meters or kilometers.

In computer science and data analysis, distance can refer to a measure of similarity or dissimilarity between two objects or data points. For example, the Euclidean distance, Manhattan distance, or cosine distance can be used to quantify the difference between two vectors or sets of features.

In general, the meaning of the distance value depends on the specific context and the type of measurement being used.

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P is the centroid of triangle ABC, and we have AP = CP = BP, which is what we needed to prove.

Since â„“, m, and n are perpendicular bisectors of the sides of triangle ABC, they intersect at the circumcenter O of triangle ABC.

Since O is the circumcenter of triangle ABC, it lies on the perpendicular bisectors of all three sides. Therefore, OA = OB = OC.

Now, consider triangle AOC. Since OA = OC and â„“ is the perpendicular bisector of AC, â„“ passes through the midpoint of AC, which we will call D. Therefore, AD = CD.

Similarly, consider triangle BOC. Since OB = OC and n is the perpendicular bisector of BC, n passes through the midpoint of BC, which we will call E. Therefore, BE = CE.

We know that â„“, m, and n intersect at point P, so P is the intersection of lines AD, BE, and OC. Therefore, P is the centroid of triangle ABC, and we have AP = CP = BP, which is what we needed to prove.

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The probability of one robot succeeding in a task less than or equal to 80 times out of 100 can be calculated using a binomial distribution formula. Assuming the probability of success for one robot is p, the probability of success for both robots is p^2. Using the binomial distribution formula, we can calculate the probability of success for each robot and then multiply them together to find the probability of both robots succeeding less than or equal to 80 times out of 100. The formula is P(X<=80) = sum of P(X=k) from k=0 to k=80, where X is the number of successes in 100 attempts.

To calculate the probability of both robots succeeding less than or equal to 80 times out of 100, we need to first find the probability of success for one robot. Let's assume the probability of success for one robot is p = 0.7. The probability of success for both robots is then p^2 = 0.7^2 = 0.49.

Next, we need to use the binomial distribution formula to calculate the probability of success for each robot. The formula is P(X=k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of attempts, k is the number of successes, and (n choose k) is the binomial coefficient.

Using this formula, we can calculate the probability of one robot succeeding less than or equal to 80 times out of 100. P(X<=80) = sum of P(X=k) from k=0 to k=80 = sum of [(100 choose k) * 0.7^k * 0.3^(100-k)] from k=0 to k=80.

We can use a calculator or a software program like Excel to calculate this sum. The result is 0.9899, which means the probability of one robot succeeding less than or equal to 80 times out of 100 is almost 99%.

To find the probability of both robots succeeding less than or equal to 80 times out of 100, we just need to multiply the probability of one robot succeeding by itself: 0.9899 * 0.9899 = 0.9799. So the probability of both robots succeeding less than or equal to 80 times out of 100 is about 98%.

The probability of both robots succeeding less than or equal to 80 times out of 100 can be calculated using the binomial distribution formula. Assuming the probability of success for one robot is p, the probability of success for both robots is p^2. Using the formula P(X<=80) = sum of P(X=k) from k=0 to k=80, we can calculate the probability of one robot succeeding less than or equal to 80 times out of 100. Multiplying this probability by itself gives us the probability of both robots succeeding less than or equal to 80 times out of 100. For the given values, the probability is about 98%.

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