Weekly wages at a certain factory are normally distributed with a mean of $400 and a standard deviation of $50. Find the probability that a worker selected at random makes between $450 and $550.

The prοbability that mοre than 2 animals need tο stay οvernight is apprοximately 0.9124, οr 91.24%.

The binοmial distributiοn is a prοbability distributiοn that describes the number οf successes in a fixed number οf independent trials, each with the same prοbability οf success. The distributiοn is characterized by twο parameters: the number οf trials (n) and the prοbability οf success (p) fοr each trial.

Tο sοlve this prοblem, we need tο use the binοmial distributiοn since we have a fixed number οf trials (25 animals brοught in) and each trial (animal) can either be a success (needs tο stay οvernight) οr a failure (dοesn't need tο stay οvernight).

Let X be the number οf animals that need tο stay οvernight. Then X fοllοws a binοmial distributiοn with n = 25 trials and p = 0.2 prοbability οf success (animal needing tο stay οvernight).

We want tο find the prοbability that mοre than 2 animals need tο stay οvernight, which can be expressed as:

nοw, P(X > 2) = 1 - P(X ≤ 2)

Tο calculate P(X ≤ 2), we can use the binοmial cumulative distributiοn functiοn (CDF) οr simply add up the prοbabilities οf X = 0, X = 1, and X = 2:

similarly, P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)

[tex]= (0.8)^{25} + 25(0.2)(0.8)^{24} + (25\ \text{choose}\ 2)(0.2)^{2}(0.8)^{23}[/tex]

≈ 0.0876

Therefοre, P(X > 2) = 1 - P(X ≤ 2) ≈ 1 - 0.0876 = 0.9124

Sο the prοbability that mοre than 2 animals need tο stay οvernight is apprοximately 0.9124, οr 91.24%.

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Answer:

0.01 cm/min

Step-by-step explanation:

The volume of oil in the tank is given by the formula:

V = l × w × h

where l, w, and h are the length, width, and height of the tank respectively.

Substituting the given values, we get:

V = 60 cm × 100 cm × 100 cm = 600,000 cm³

The rate at which the oil is being lost is given as 3.6 litres per hour. Since 1 litre = 1000 cm³, we can convert this to cubic centimeters per hour as follows:

3.6 litres/hour × 1000 cm³/litre = 3600 cm³/hour

To find the rate at which the oil level is falling, we need to divide the rate of oil loss by the surface area of the oil in the tank. Since the tank is rectangular and has a flat bottom, the surface area of the oil is simply the area of the top of the tank, which is:

A = l × w = 60 cm × 100 cm = 6000 cm²

Now, we can calculate the rate at which the oil level is falling as follows:

rate of oil loss ÷ surface area of oil = (3600 cm³/hour) ÷ (6000 cm²)

= 0.6 cm/hour

To convert this to cm/min, we simply divide by 60:

0.6 cm/hour ÷ 60 min/hour = 0.01 cm/min

Therefore, the rate at which the oil level is falling is 0.01 cm/min

Answer:

Step-by-step explanation:

8+4x

Step-by-step explanation:

We can also solve the equation x² + 4x + 4 = 8 by rearranging the terms and using the quadratic formula:

x² + 4x + 4 - 8 = 0

x² + 4x - 4 = 0

Applying the quadratic formula, we have:

x = (-b ± √(b² - 4ac)) / 2a

where a = 1, b = 4, and c = -4. Substituting these values, we get:

x = (-4 ± √(4² - 4(1)(-4))) / 2(1)

x = (-4 ± √32) / 2

Simplifying the square root of 32, we get:

x = (-4 ± 4√2) / 2

x = -2 ± 2√2

Therefore, the solutions to the equation x² + 4x + 4 = 8 are x = -2 + 2√2 and x = -2 - 2√2.

The expected time taken by the man to leave the maze is 15 minutes.

To find the expected time taken by the man to leave the maze, we'll first calculate the conditional expectation of time taken given that he chose each of the exits, and then use these conditional expectations to calculate the overall expectation.

Step 1: Calculate the conditional expectations :
- If he chooses Exit 1 (probability 1/3), he leaves the maze after 3 minutes.
- If he chooses Exit 2 (probability 1/3), he returns to the same room after 5 minutes and starts again. So, the expected time in this case is 5 + E(X).
- If he chooses Exit 3 (probability 1/3), he returns to the same room after 7 minutes and starts again. So, the expected time in this case is 7 + E(X).

Step 2: Calculate the overall expectation :
E(X) = (1/3)*(3) + (1/3)*(5 + E(X)) + (1/3)*(7 + E(X))

Now, we'll solve for E(X):
3E(X) = 3 + 5 + 7 + 2E(X)
E(X) = 15 minutes

The expected time taken by the man to leave the maze is 15 minutes.

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According to the problem, there are three possible exits (1, 2, and 3) from the room in the center of the maze. The probabilities of choosing each of these exits are equal.

Exit 1 leads to the outside of the maze, and it takes 3 minutes on average to reach it. Exit 2 leads back to the same room, so the man will need to start over again. Exit 3 also leads back to the same room, and it takes longer than exit 2 to get there (7 minutes).Let X be the time taken by the man to leave the maze from this room. Let Y be the exit he chooses first. Y belongs to {1, 2, 3}. Calculate the conditional expectation of the time taken to leave the maze given that he chose each of the exits. Then use these conditional expectations to calculate the expectation of the time taken to leave the maze.The expected value of X can be calculated as follows:() = ( | = 1) × ( = 1) + ( | = 2) × ( = 2) + ( | = 3) × ( = 3)Expected time to leave the maze through exit 1:( | = 1) = 3Expected time to leave the maze through exit 2:( | = 2) = 5 + ()Expected time to leave the maze through exit 3:( | = 3) = 7 + ()The probability of choosing each exit is 1/3, so:P(Y = 1) = 1/3P(Y = 2) = 1/3P(Y = 3) = 1/3Substituting these values into the equation for ():() = 3(1/3) + (5 + ())(1/3) + (7 + ())(1/3)() = 5 + (2/3)() + (7/3)()() = 15 minutes. Therefore, the expected time taken by the man to leave the maze is 15 minutes.

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Based on the information, the area of the figure would be: 86.13 units ²

To find the area of the figure we must perform the following procedure:

1. Find the area of the triangle with the following formula:

height * base / 2 = area of the triangle

6 * 6 / 2 = area of the triangle

18 = area of the triangle

2. Find the area of the rectangle with the following formula:

height * base = area of rectangle

6 * 9 = area of the rectangle

54 = area of rectangle

3. Find the area of the semicircle with the following formula:

[tex]\pi[/tex]  * r² / 2 = area of the semicircle

[tex]\pi[/tex]  * 3² / 2 = area of the semicircle

[tex]\pi[/tex]  * 9 / 2 = area of the semicircle

28.27 / 2 = area of the semicircle

14.13 = area of the semicircle

4. We must add the area of all the figures to find the total area.

14.13 + 54 + 18 = 86.13 units ²

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In order to solve for x in this equation, we must divide 75 by 9. Dividing 75 by 9 gives us 8.3 as the answer. So, x = 8.3.

An equation is a mathematical statement that expresses the equality of two expressions, usually containing one or more variables. It is used to describe the relationships between different variables, such as the area of a circle or the slope of a line. Equations can be used to describe the behavior of physical systems, calculate the solutions to problems, and make predictions about future events.

This is happening because when we divide 75 by 9, we are essentially asking "how many 9s go into 75?". 75 divided by 9 is equal to 8.3, meaning 8 and 3/9ths of 9 go into 75. Therefore, x = 8.3.

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In order to solve for x the given equation, we must divide 75 by 9. Dividing 75 by 9 gives us 8.3. So, x = 8.3.

An equation is a mathematical statement that expresses the equality of two expressions, usually involving one or more variables. Used to describe relationships between different variables. Area of ​​circle or slope of line. Equations can be used to describe the behavior of physical systems, calculate solutions to problems, and predict future events.

That's because when you divide 75 by 9, you're essentially asking, "How many 9s are there in 75?" 75 divided by 9 is 8.3. So 3/9 of 8 and 9 is 75. So x = 8.3.  

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The complete question is as follows:

Solve for x. Round to the nearest tenth, if necessary.

75 = 9x

A student should score 598 in order to win the award.

To calculate how high should a student score be to win this award, we use the z-score formula.

z=(x-μ)/σ

Where,

μ= Mean of the scores on this exam

σ= Standard deviation of the scores on this exam

z= Z-score

x= Scores on this exam

By substituting the given values in the formula, we get

z=(x-μ)/σ

[tex]= (x-538)/30[/tex]

To find the highest 2.5%, we use the normal distribution table which gives us the Z-score corresponding to 0.975. The highest 2.5% is on each side of the normal distribution curve. Therefore, we have to subtract the area from the highest score, then divide by two.

Subtracting 0.975 from 1 gives us the area to the left of this point on the table which is equal to 0.025.Z = 1.96.

So the score should be one standard deviation above the mean. Thus, the score a student needs to get in the top 2.5% of statewide scores is

   [tex]Z = (x - 538) / 30[/tex]

[tex]1.96 = (x - 538) / 30x - 538 = 1.96 * 30x - 538 = 58.8 + 538x = 597.8[/tex]

The score a student needs to get in the top 2.5% of statewide scores is 598, rounded up to the next integer.

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The expressions 6 × 3+6 × 8 and 6(3 + 8), respectively, indicate the combined size of the two rooms.

A plane figure's overall area is the sum of all of the distinct area shapes that make up that plane figure.

We have two regions that take on the characteristics of a rectangle based on the information provided.

The office area = 18 square feet

There are 48 square feet in the sitting room.

The two chambers now have a combined size of 18 + 48 = 66 square feet.

Now, the expressions that better represent the total area of the two rooms can be written as 6 × 3+6 × 8 and 6(3 + 8).

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Answer:

2x + 18y

Step-by-step explanation:

-5x - 3y + 7x + 21y  ----> (combine like terms)

2x - 3y + 21 y   ---> (combine like terms)

2x + 18y

Answer:

[tex]\huge\boxed{\sf 2(x + 9y)}[/tex]

Step-by-step explanation:

= -5x - 3y + 7x + 21y

= -5x + 7x - 3y + 21y

= 2x + 18y

So, take 2 as a common factor

[tex]\rule[225]{225}{2}[/tex]

Player A's weight was 0.208 standard deviations below the mean weight of the football team.

To calculate the standard deviation, we first need to calculate the mean weight of the football team

Mean weight = (sum of all weights) / (number of team members)

(174+176+178+184+185+185+185+185+188+190+200+202+205+206+210+211+211+212+212+215+215+220+223+228+230+232+241+241+242+245+247+250+250+259+260+260+265+265+270+272+273+275+276+278+280+280+285+285+286+290+290+295+302) / 52

= 225.5 pounds

Now we can calculate the standard deviation using the following formula:

Standard deviation = sqrt((sum of (x - mean)^2) / N)

Where x is the weight of a team member, N is the total number of team members.

We can simplify this formula by calculating the variance first:

Variance = (sum of (x - mean)^2) / N

So we have

Variance = ((174-225.5)^2 + (176-225.5)^2 + ... + (302-225.5)^2) / 52

= 10764.35

Now we can calculate the standard deviation

Standard deviation = sqrt(Variance)

= sqrt(10764.35)

= 103.76

To find out how many standard deviations above or below the mean Player A's weight was, we can use the following formula

Z-score = (x - mean) / standard deviation

Where x is Player A's weight, mean is the mean weight of the team, and standard deviation is the standard deviation we just calculated.

So we have

Z-score = (204 - 225.5) / 103.76

= -0.208

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The given question is incomplete, the complete question is:

Following are the published weights (in pounds) of all of the team members of Football Team A from a previous year.

174, 176, 178, 184, 185, 185, 185, 185, 188, 190, 200, 202, 205, 206, 210, 211, 211, 212, 212, 215, 215, 220, 223, 228, 230, 232, 241, 241, 242, 245, 247, 250, 250, 259, 260, 260, 265, 265, 270, 272, 273, 275, 276, 278, 280, 280, 285, 285, 286, 290, 290, 295, 302

median = 241

the first quartile = 205.5

the third quartile = 272.5

Assume the population was Football Team A. When Player A played football, he weighed 204pounds. How many standard deviations above or below the mean was he?

As a general rule, a Z-score of - 3.0 to 2.0 recommends that a stock is exchanged inside three standard deviations of its mean.

A Z-Score is a statistical estimation of a score's relationship to the mean in a gathering of scores.

A Z-score can uncover to a merchant on the off chance that worth is normal for a predefined informational collection or on the other hand assuming it is abnormal.

As a general rule, a Z-score of - 3.0 to 2.0 recommends that a stock is exchanged inside three standard deviations of its mean.

Merchants have created numerous techniques that utilize z-score to distinguish connections between's exchanges, and exchanging positions, and assess exchanging systems.

The higher (or lower) a z-score is, the further away from the mean the fact is. This isn't really positive or negative; it just shows where the information lies in a regularly conveyed test. This implies it comes down to inclination while assessing speculation or opportunity. For instance, a few financial backers utilize a z-score scope of - 3.0 to 3.0 on the grounds that 99.7% of regularly conveyed information falls here, while others could utilize - 1.5 to 1.5 in light of the fact that they favor scores nearer to the mean.

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Option C, 12, is the correct answer. To spray the vegetables 5 times each hour, we need to determine how often the spray should occur.

Since there are 60 minutes in an hour, and we want the vegetables to be sprayed 5 times in that hour, we can divide 60 by 5 to get the interval between each spray:

60 ÷ 5 = 12

This means that the interval between each spray should be 12 minutes. Therefore, we should set the timer for 12 minutes in order to spray the vegetables 5 times each hour.

Option C, 12, is the correct answer.

It is important to note that this assumes the vegetable display is in operation for the entire hour without interruption. If the display is turned off or there are other factors that interrupt the timing, the frequency of sprays may be affected. Additionally, the optimal frequency of sprays may vary depending on the specific vegetables and the environment they are in.

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The sum of the matrices A and C from the list of options is the matrix B

Given the following matrices

Matrix A

| 0  6  2 |

| 1  5  -2 |

Matrix C

| -2  5  3 |

| -1  -7  3|

To find the sum of matrices A and C, we add the corresponding elements in each matrix:

So, we have: A + C

| 0 - 2  6 + 5  2 + 3 |

| 1 - 1    5 - 7  -2 + 3|

Evaluate the sum

| -2  11  5 |

| 0   -2  1 |

This represents option B

Therefore, the sum of matrices A and C is (B)

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Complete question

What is the sum of A+C?

Matrix A

| 0  6  2 |

| 1  5  -2 |

Matrix C

| -2  5  3 |

| -1  -7  3|

The result is a very large negative number, specifically -2,176,782,192. Therefore: 4a²−b⁶ = -2,176,782,192, when a=6 and b=36.

An equation is a mathematical statement that shows the equality between two expressions. It usually consists of two sides, the left-hand side (LHS) and the right-hand side (RHS), separated by an equal sign (=).

The expressions on both sides can contain variables, constants, and mathematical operations such as addition (+), subtraction (-), multiplication (*), division (/), exponentiation (^), and others. The goal of an equation is to find the values of the variables that make both sides equal.

For example, the equation "2x + 3 = 7" means that the sum of 2 times the variable x and 3 is equal to 7. The solution to this equation is x = 2, because when we substitute x = 2 into the equation, we get 2(2) + 3 = 7, which is a true statement.

by the question.

To evaluate the expression 4a²−b⁶ when a=6 and b=36, we substitute the values of a and b into the expression:

[tex]4(6)^{2} - (36)^{6}[/tex]

Simplifying the expression:

[tex]4(36) - 2,176,782,336\\144 - 2,176,782,336[/tex]e

[tex]= -2,176,782,192,[/tex]

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The amount of work done when winding up a 50-foot long cable with a weight density of 8 pounds per foot is 14 feet is 5600 foot-pounds.

To explain this answer in more detail, work is a measure of energy and is calculated by multiplying force and distance. In this case, the force is the weight-density of the cable, which is 8 pounds per foot. The distance is the amount of cable that was wound up, which is 14 feet. Multiplying 8 and 14 together gives us the force of 112 pounds. Multiplying 112 by the total length of the cable, which is 50 feet, gives us the answer of 5600 foot-pounds.

To further explain, it is important to remember that 1 foot-pound is equal to 1 pound-foot. That means that 1 pound of force must be applied to move an object 1 foot in order to do 1 foot-pound of work. When this is applied to the example above, we can see that 8 pounds of force must be applied to the cable in order to move it 1 foot. Multiplying this by the length of the cable (50 feet) and the amount of cable wound up (14 feet) gives us the total amount of work done, which is 5600 foot-pounds.

In summary, the amount of work done when winding up a 50-foot long cable with a weight density of 8 pounds per foot is 14 feet is 14 x 8 x 50 = 5600 foot-pounds.

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Answer:

x ≥ 2

Step-by-step explanation:

8x + 3 ≥ 19

8x ≥ 19 - 3

8x / 8 ≥ 16 / 8

x ≥ 2

Answer:

Sure, I can solve for x:8x + 3 ≥ 19Subtracting 3 from both sides:8x ≥ 16Dividing both sides by 8:x ≥ 2Therefore, the solution is x ≥ 2.

Destiny sold 23 cups of espresso and 53 cups of cappuccino on Friday, for a total sales amount of $178.50.

Let's start by defining our variables. Let x be the number of cups of espresso sold and y be the number of cups of cappuccino sold. We know that the price of espresso is $2 per cup and the price of cappuccino is $2.50 per cup. We also know that Destiny sold 30 more cups of cappuccino than espresso and the total sales amount was $178.50.

Using this information, we can create a system of equations to solve for x and y. The first equation comes from the fact that Destiny sold 30 more cups of cappuccino than espresso:

y = x + 30

The second equation comes from the total sales amount:

2x + 2.5y = 178.5

Now we can substitute the first equation into the second equation and solve for x:

2x + 2.5(x + 30) = 178.5

2x + 2.5x + 75 = 178.5

4.5x = 103.5

x = 23        

So Destiny sold 23 cups of espresso. Using the first equation, we can find that she sold 53 cups of cappuccino:

y = x + 30

y = 23 + 30    

y = 53

Therefore, Destiny sold 23 cups of espresso and 53 cups of cappuccino on Friday, for a total sales amount of $178.50.

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Answer: 63.8

Step-by-step explanation:

The area of the figure rounded to the nearest square meter is 97 m². So correct option is A.

With five equal sides and five equal angles, a regular pentagon is a polygon. In other words, a regular pentagon's sides and angles are all congruent. It is a closed, two-dimensional object that is categorized as a sort of polygon since it is constructed of straight line segments.

Regular pentagons have a number of distinctive features, including:

A regular pentagon has 108 degree internal angles that are all equal.

A standard pentagon's outside angles are all 72 degrees in angle.

A regular pentagon is divided along its diagonals into three smaller triangles, two of which are congruent.

Regular pentagons are prevalent in a variety of fields, including geometry, architecture, and the arts.

First, you would need to determine the side length of the regular hexagon. This can be done using the radius of the circumscribed circle, which is given as 5 meters. The side length of the hexagon can be calculated using the formula:

s = r * √(3)

where s is the side length and r is the radius of the circumscribed circle. Plugging in the given value for r, we get:

s = 5 * √(3)

simplifying, we get:

s ≈ 8.7 meters

Next, you would need to calculate the area of one of the congruent triangles. This can be done using the formula:

A = (1/2) * b * h

where A is the area of the triangle, b is the base, and h is the height. The base of the triangle is equal to one side of the hexagon, which we just calculated to be approximately 8.7 meters. To find the height, we can draw an altitude from one vertex of the triangle to the opposite side, creating a 30-60-90 right triangle. The height is equal to the shorter leg of this triangle, which can be calculated using the formula:

h = (1/2) * s * √(3)

Plugging in the value for s, we get:

h ≈ 7.5 meters

Now we can calculate the area of one of the triangles:

A = (1/2) * 8.7 * 7.5 ≈ 32.6 square meters

Finally, we can calculate the total area of the figure by adding the area of the hexagon to three times the area of one of the triangles:

A = 6 * (s²) * √(3)/4 + 3 * A

Plugging in the values we calculated, we get:

A ≈ 96.9 square meters

Rounding to the nearest square meter, we get:

A ≈ 97 square meters

Therefore, the answer is A. 97 m².

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Answer: B

Step-by-step explanation: because

Answer:

1/2

Step-by-step explanation:

options in a 6 sided die= 1,2,3,4,5,6

greater than 3 = 4,5,6

so three  sides  out of a 6 sided die is greater than three

P(not greater than 3) = [tex]\frac{6-3}{6} = \frac{1}{2}[/tex]

Answer:

1/2 or 0.5

Step-by-step explanation:

The probability of rolling a number greater than 3 on a six-sided die is 3/6 or 1/2, since there are three numbers (4, 5, 6) out of six that are greater than 3.

To find the probability of not rolling a number greater than 3, we can subtract the probability of rolling a number greater than 3 from 1:

P(not greater than 3) = 1 - P(greater than 3)

P(not greater than 3) = 1 - 1/2

P(not greater than 3) = 1/2

Therefore, the probability of not rolling a number greater than 3 is 1/2 or 0.5.

Ashley's tower is 30 centimeters taller than Mason's tower. Ashley's tower is 10 x (1/10) = 1 meter tall. Mason's tower, on the other hand, is 7 x (1/10) = 0.7 meters tall.

Ashley's tower is 10 blocks high and each block has a height of 0.1 meters since 1 meter is divided into 10 equal parts. Therefore, the total height of Ashley's tower is 10 x 0.1 = 1 meter. Mason's tower, on the other hand, is 7 blocks high and has a total height of 7 x 0.1 = 0.7 meters. The difference between Ashley's tower and Mason's tower is the height of Ashley's tower minus the height of Mason's tower, which is 1 meter - 0.7 meters = 0.3 meters. To convert meters to centimeters, we need to multiply by 100, so Ashley's tower is 0.3 x 100 = 30 centimeters taller than Mason's tower.

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Answer: 21

Step-by-step explanation:because 21 savage is #1

Yes, the given values constitute a possible bootstrap sample from the original sample.

Bootstrap samples are taken with replacement from the original data set. In other words, it means that each sample consists of a random sample of the same size as the original data set taken from the original data set.

Hence, a given set of values can be a bootstrap sample from the original sample as long as the values are randomly selected from the original sample and with replacement.

The sample is said to be a bootstrap sample from the original data set if it satisfies the following properties:

1. The bootstrap sample must be of the same size as the original data set.

2. Each observation from the original data set must have an equal chance of being selected in each bootstrap sample.3. The bootstrap samples must be independent of each other.

In the given problem, the sample consists of 5 values, namely 84, 71, 79, 96, and 87.

These values can be a bootstrap sample from the original sample as long as the values were randomly selected with replacement from the original sample.

Since the problem does not provide any information on how the sample was obtained, it cannot be determined whether or not it is a bootstrap sample.

Therefore, the answer is that the given values constitute a possible bootstrap sample from the original sample.

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The exact value of side lengths of triangle are d= [tex]\frac{8}{\sqrt{3} }[/tex] and h=[tex]\frac{5\sqrt{2} }{2}[/tex]

A triangle is a closed, two-dimensional shape that consists of three straight sides and three angles. It is one of the basic shapes in geometry and is often used in various mathematical and scientific contexts.

The three sides of a triangle are usually named using lowercase letters a, b, and c, and the three angles are named using uppercase letters A, B, and C, with the opposite angles and sides having the same letter. The sum of the angles in a triangle is always 180 degrees, and this property is known as the Angle Sum Theorem.

Triangles can be classified based on their side lengths and angles. Based on the side lengths, triangles can be classified as:

Scalene triangle: A triangle in which all three sides have different lengths.

Isosceles triangle: A triangle in which two sides have the same length, and the third side has a different length.

Equilateral triangle: A triangle in which all three sides have the same length.

Let's start with the first triangle. We can use the trigonometric ratios for a 30-60-90 triangle:

sin(60) = opposite / hypotenuse = d / (2d) = [tex]\frac{1}{2}[/tex]

cos(60) = adjacent / hypotenuse = 4 / (2d) = [tex]\frac{1}{2\sqrt{3} }[/tex]

Solving for d:

d = 2 * sin(60) = 2 *[tex]\frac{1}{2}[/tex]  = 1

2d = 4 / cos(60) = [tex]\frac{4}{\frac{1}{2\sqrt{3} }}[/tex] = [tex]\frac{8}{\sqrt{3} }[/tex]

So d = 1 and 2d = [tex]\frac{8}{\sqrt{3} }[/tex] in the first triangle.

For the second triangle, we can use the trigonometric ratios for a 45-45-90 triangle:

sin(45) = opposite / hypotenuse = [tex]\frac{h}{5}[/tex]

cos(45) = adjacent / hypotenuse = [tex]\frac{h}{5}[/tex]

Since sin(45) = cos(45) = [tex]\frac{1}{\sqrt{2} }[/tex], we can solve for h:

h = 5 * sin(45) = 5 * [tex]\frac{1}{\sqrt{2} }[/tex] = [tex]\frac{5\sqrt{2} }{2}[/tex]

So h =  [tex]\frac{5\sqrt{2} }{2}[/tex] in the second triangle.

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The probability of getting at least one 2 given that the sum of the numbers is 7 is 2/6 or 1/3.

To find the probability that the number 2 has appeared at least once given that the sum of the numbers is 7, we need to use the concept of conditional probability.

Let's consider the possible outcomes when two dice are rolled. The total number of outcomes is 36, as each die has six possible outcomes.

Out of these 36 outcomes, there are six outcomes in which the sum of the numbers appearing on the upper faces is 7. These outcomes are (1,6), (2,5), (3,4), (4,3), (5,2), and (6,1).

Out of these six outcomes, there are two outcomes in which the number 2 appears at least once: (1,6) and (2,5).

We can use the formula for conditional probability to verify our answer:

P(2 appears at least once | sum is 7) = P(2 appears at least once and sum is 7) / P(sum is 7)

P(2 appears at least once and sum is 7) = 2/36 = 1/18 (as there are two outcomes with a sum of 7 that have a 2 in them)

P(sum is 7) = 6/36 = 1/6

So, P(2 appears at least once | sum is 7) = (1/18) / (1/6) = 1/3, which is consistent with our previous answer.

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