Charles has a rectangular flower garden that is 5 yd long and 12 yd wide. One bag of fertilizer can cover 6 yd2 . How many bags will he need to buy to cover the entire garden?

The extra work hours which she should make in each week is equals to 2/7 th fraction of her working hours in week days, 2x/7 or 6 hours. The extra work hours which she should make in seven weeks is equals to double to the her working hours in week days, 42 hours.

We have Amy has a chance to work some extra hours on the weekends.

Number of extra hours she will work at each week = 6 hours

number of weeks she have for doing extra hours work at weekends = 7 weeks

Let amy works 'x hours' in each week. So, her working rate is x/7 hours/day. As we know very well that weekend consists two days ( Saturday and Sunday). So, she will make extra work each week = extra work in weekend ( 2 days)

The extra work that she will do in weekend or each week= 2(x/7) = 2x/7 hours = 6 hours

Now, The extra work hours that she make in seven weeks = Multiplication of 7 by the extra work hours that she make in each week

= 7(6) hours = 42 hours

Hence, required value of hours is 42 ( that is double of her working hours in week days).

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Answer:

20%

Step-by-step explanation:

The total number of candies in the bowl is 25. There are 5 orange candies. In order to find the total amount, simply divide the amount of candies in the given color (orange) and divide it by the total number of candies (25) giving you a result of 0.20, which can also be written as 20/100 or 20%

The predicted number of chirps per second when the temperature is 55 degrees Fahrenheit is 190.

The temperature and the number of chirps per second are linearly related, as shown by the trend line equation y = 3x + 25, where x is the temperature in degrees Fahrenheit and y is the number of chirps per second.

We must enter x = 55 into the equation and find y in order to estimate the number of chirps per second at a temperature of 55 degrees Fahrenheit

y = 3x + 25

y = 3(55) + 25

y = 165 + 25

y = 190

190 chirps are therefore expected per second at a temperature of 55 degrees Fahrenheit. Therefore, the predicted number of chirps per second when the temperature is 55 degrees Fahrenheit is 190.

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The interest Kevin earns in the first year that is $105 that will also earn interest in subsequent years. This is called compound interest.

Percent is a way of expressing a number as a fraction of 100, often used to represent a proportion or rate. It is denoted by the symbol "%". For example, if 30 out of 100 students in a class got an "A" grade, then the percentage of students who got an "A" grade is 30%.

Here,

Kevin deposits $100 into a savings account today. This is his principal. In one year, Kevin’s money earns 5 percent. The $5 he earns is his interest. In one year’s time, Kevin’s money is worth $105. This is his balance.

To find the interest earned in one year, we need to multiply the initial deposit by the interest rate (as a decimal). So:

Interest earned in one year = $100 x 0.05 = $5

To find the total value of Kevin's savings account after one year, we need to add the interest earned to the initial deposit. So:

Total value after one year = $100 + $5 = $105

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Answer: We can use trigonometry to solve this problem.

First, let's draw a diagram to represent the situation:

           |

           |

           | 8 km

           |

           |

   --------X---------

           | 3 km

           |

           |

           |

           |

The starting point is at the X, and the destination is 8 km south and 3 km west of the starting point.

To find the distance and bearing of the destination from the starting point, we can use the Pythagorean theorem and trigonometric functions.

The distance between the starting point and destination is the hypotenuse of a right triangle with legs of length 8 km (south) and 3 km (west). So we can use the Pythagorean theorem:

distance = √(8² + 3²) ≈ 8.6 km

To find the bearing (direction) of the destination from the starting point, we can use trigonometry. The bearing is the angle between the line connecting the starting point and destination and the line pointing due north.

We can use the tangent function to find this angle:

tan(θ) = opposite/adjacent = 3/8

θ = tan⁻¹(3/8) ≈ 20.56°

So the bearing of the destination from the starting point is approximately 20.56° west of due south.

Therefore, the distance of the destination from the starting point is approximately 8.6 km and the bearing of the destination from the starting point is approximately 20.56° west of due south.

Step-by-step explanation:

The given story has a sampling type known as: voluntary response sampling.

Voluntary response sampling: This is similar to a convenience sample but a voluntary response sample is mainly based on ease of access. Instead of the researcher choosing participants and directly contacting them, people volunteer themselves (e.g. by responding to a public online survey).

For example, a person being called in a radio show poll might have powerful opinions over a topic in any direction. In a voluntary random sample, chooses themselves by answering a common appeal.  These samples are unreliable because they are bias, hence the result is usually biased.

In this question, Mrs. Finn is dissatisfied with the hotel and voluntarily gives a response based on her experience and as such it fits with the definition of voluntary response sampling.

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Answer: To calculate the probability of an incorrect test result, we need to consider two cases:

1. The test gives a positive result when the child does not have a pet allergy (false positive).

2. The test gives a negative result when the child does have a pet allergy (false negative).

The probability of a false positive is given as 0.06, which means that out of 100 children who do not have a pet allergy, 6 will test positive for it. The probability of a false negative is given as 0.12, which means that out of 100 children who do have a pet allergy, 12 will test negative for it.

To calculate the probability of an incorrect test result, we can add the probabilities of these two cases:

Probability of incorrect test result = Probability of false positive + Probability of false negative

Probability of incorrect test result = 0.06 + 0.12

Probability of incorrect test result = 0.18

Therefore, the probability of an incorrect test result is 0.18, or 18% (rounded to two decimal places).

By factorial, 6 is different ways can they stand in line.

What does the arithmetic term factorial mean?

Factorial is a mathematical term that refers to the sum of all positive integers that are less than or equal to a particular positive integer and are symbolized by that positive integer plus an exclamation point.

                           As a result, factorial seven is denoted by the symbol 7!, which stands for 1 2 3 4 5 6 7. Factorial zero is equivalent to one.

Three students, Nicole, Xavier, and Avani  line up one behind the other.

The different ways they can stand in line = 3!

       = 3 * 2 * 1

        = 6

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Answer:

Discount Rs 239.94= 240

She has to pay 3659

Step-by-step explanation:

6 % of 3999=239.94 or 240

She will get the discount of 240 rupees.

She has to pay other amount beside the discount.

The price of machine is 3999.

The price she has to pay after discount= actual price- discount.

So discounted price is Rs3659

Answer: If the original price of the washing machine is R3 999 and it is discounted by 6%, then the discount amount can be calculated as follows:

Discount amount = 6% of R3 999

= 0.06 x R3 999

= R239.94

Therefore, Mrs Burn will receive a discount of R239.94 on the washing machine.

The amount she has to pay can be calculated as follows:

Amount to be paid = Original price - Discount amount

= R3 999 - R239.94

= R3 759.06

Therefore, Mrs Burn has to pay R3 759.06 for the washing machine after the 6% discount.

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The οppοsite angles οf a parallelοgram are equal.

The regiοn in a twο-dimensiοnal plane that a parallelοgram οccupies is knοwn as its area. A parallelοgram is a fοrm with fοur sides with twο geοmetry dimensiοns.

Because the οppοsing sides are equal and parallel, sο it is a unique quadrilateral example. The space bοunded by the fοur sides οf a parallelοgram is knοwn as its area. A parallelοgram's area is calculated as the sum οf its length and height.

The οppοsite sides οf a parallelοgram are parallel. Here, PQ ‖ RT and PR ‖ QT.

The οppοsite sides οf a parallelοgram are equal. Here, PQ = RT and PR = QT

The οppοsite angles οf a parallelοgram are equal. Here, ∠P = ∠T and ∠Q = ∠R

Hence, The οppοsite angles οf a parallelοgram are equal.

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The probability that a quarter lands on tail and a rolling die land on 2 is 1/12

A probability is a number that reflects the likelihood that a particular event will occur. Probabilities can be expressed as proportions that range from 0 to 1, and they can also be expressed as percentages ranging from 0% to 100%.

Probability = sample space /total outcome

A quarter has two faces, the head and the tail. Therefore the probability for a quarter to land on tail = 1/2

A die has 6 faces , labelled 1, 2,3,4, 5,6.

The probability for a die to land on 2 = 1/6

Therefore the probability for a quarter to land on tail and a die to land on 2 = 1/2 × 1/6 = 1/12

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The total number of pets is = 36.

This problem is incomplete, we do not know the fraction of the students that have a dog and also have a cat. Suppose we write the problem as:

"In Mrs.Hu's classroom, 4/5 of the students have a dog as a pet. X of the students who have a dog as a pet also have cat as a pet. If there are 45 students in her class, how many have both a dog and a cat as pets?"

Where X must be a positive number smaller than one, now we can solve it:

we know that in the class we have 45 students, and 4/5 of those students have dogs, so the number of students that have a dog as a pet is:

N = 45*(4/5) = 36

And we know that X of those 36 students also have a cat, so the number of students that have a dog and a cat is:

M = 36*X

now, we do not have, suppose that the value of X is 1/2 ("1/2 of the students who have a dog also have a cat")

M = 36*(1/2) = 18

So you can replace the value of X in the equation and find the number of students that have a dog and a cat as pets.

The total number of pets is = 36.

The complete question is-

In Mrs.Hu's classroom, 4/5 of the students have a dog as a pet. Of the students who have a dog as a pet also have a cat as a pet. If there are 45 students in her class, how many have both a dog and a cat as pets?

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Answer:

Step-by-step explanation:

a= first form a equation

lets call the cheese c and the buns b

2c+3b=10

c=2+b

3c+4b=12

now its a simultaneous equation

2c+3b=10    x3

3c+4b=12    x2

6c+9b=30

6c+8b=24

1b=6

b=6

now substitute it in to 2c+3b=10

2c+18=10

2c+-8

so c=$4

b= 56 difference 14

56/2=28

28-7

21

28+7=35

so the 2 numbers are 21 and 35

Solution for the variable 'y' is y = (x - x² + 8)/5  

To solve the following equation we need to isolate the variable 'y' this can be done by adding or subtracting the same integer on both sides

Similarly, multiply or divide with the same integer on both sides. Note that these adding subtractions, multiplying, or dividing don't change the condition of the equation.

Here we have

5y+x² = x+8  

To solve the above equation we need to isolate 'y' as follows

=> 5y+x² = x+8    

Subtract x² from both sides

=> 5y + x² - x² = x + 8

=> 5y = x - x²+ 8

Divide by '5' on both sides

=> 5y/5 = (x - x² + 8)/5

=> y = (x - x² + 8)/5  

Therefore,

Solution for the variable 'y' is y = (x - x² + 8)/5  

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Solve for 'y', 5y+x² = x+8    

Answer:

A. What is the Radius? ANS: 4.2m

B. What is the Height? ANS: 14.5m

C. Plug the r, h, and 3.14 for pi into the formula pi(r)^2(h). ANS: approximately 803.56 m

D. Show your work and label properly.

ANS: pi (4.2)^2 (14.5)

pi (17.64)(14.5)

255.78pi

approximately 803.56m

HOPE THIS HELPS:)

The angle of elevation of the point T on the top of the pole from the point A on the level ground, obtained using Pythagorean Theorem and the relationship between similar triangles is about 39.3°.

Similar triangles are triangles that have the same shape (the sizes may be different) and in which the ratio of the corresponding sides are equivalent.

The triangles ΔXBA, ΔXAC, and ΔABC are right triangles, such that the angles, ∠ABX in triangle ΔXBA is congruent to triangle ∠CAX in triangle ΔXAC, and ∠ABC in ΔABC.

The 90° angle in the right triangles are congruent (All 90° angle are congruent), therefore;

ΔXBA ~ ΔXAC ~ ΔABC by AA (Angle-Angle), similarity postulate

The length of the hypotenuse in the right triangle, ΔABC, [tex]\overline{BC}[/tex], can be obtained using Pythagorean Theorem as follows;

[tex]\overline{BC}[/tex]² = 14² + 25² = 821

[tex]\overline{BC}[/tex] = √(821)

[tex]\overline{BC}[/tex]/[tex]\overline{AC}[/tex] = [tex]\overline{AB}[/tex]/[tex]\overline{AX}[/tex]

√(821)/25 = 14/[tex]\overline{AX}[/tex]

[tex]\overline{AX}[/tex] = 25 × (14/(√(821)) = 350/(√(821))

Let θ represent the angle of T from A, we get;

tan(θ) = 10/(350/(√(821)))

θ = arctan(10/(350/(√(821)))) ≈ 39.3°

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The expected number of times player A plays a turn is approximately 1.5.

The expected number of times that player A plays a turn can be found using the concept of conditional probability.What is probability?Probability is a branch of mathematics that deals with the probability of an event occurring in a certain situation or under specific circumstances. The total probability of any occurrence is always between 0 and 1. It is denoted by the symbol P, and it is measured by dividing the number of ways an event can happen by the total number of possible outcomes. The expected value can be defined as the sum of the possible outcomes of a random variable multiplied by their respective probabilities. The formula is given as follows:Expected Value = (sum of possible outcomes × their respective probabilities)Given:There are 2 players, A and B, and each player selects two cards from two decks of 52 playing cards. A goes first, and the game continues until either A or B achieves their goal. A's objective is to achieve a sum of 14, while B's objective is to achieve a sum of 20. Assume that all face cards are worth 10 points.

Find the expected number of turns A will play.Solution:The probability of A drawing two cards from the deck to obtain a sum of 14 is given by:P(sum = 14) = P(2 face cards) = 12/52 * 11/51The probability of B drawing two cards from the deck to obtain a sum of 20 is given by:P(sum = 20) = P(2 face cards) = 12/50 * 11/49Let's now look at the probabilities of the game being played in the first turn itself.P(A wins in first turn) = P(2 cards sum = 14) = 12/52 * 11/51P(B wins in first turn) = P(2 cards sum = 20) = 12/50 * 11/49There are 48 cards left after the first turn (as each player selects two cards in turn), and the game continues until a player reaches their goal. Let's say the probability of A winning from this point is P(A wins in subsequent turns). Then we can write:P(A wins) = P(A wins in first turn) + P(A wins in subsequent turns)Similarly,P(B wins) = P(B wins in first turn) + P(B wins in subsequent turns)The expected number of turns A plays can be obtained using the concept of conditional probability as follows:Let P(A) = probability of A winning = P(A wins)Let P(B) = probability of B winning = P(B wins)Let P(D) = probability of a draw or the game continuing indefinitely = 1 - P(A) - P(B)

Then the expected number of times that player A plays a turn is given by:Expected value = (1 × P(D)) + (1 × P(A)) + (1 + Expected value) × P(B)Substituting the values, we get:Expected value = 1 + P(B) × Expected valueDividing by P(B), we get:Expected value / P(B) = 1 + Expected valueSolving for Expected value, we get:Expected value = P(B) / (1 - P(B))P(B) can be calculated as follows:P(B) = P(B wins in first turn) + P(B wins in subsequent turns) = 12/50 * 11/49 + P(A wins in subsequent turns)P(A wins in subsequent turns) can be found using the concept of recursion, which is as follows:P(A wins in subsequent turns) = P(A wins on A's first turn) + (1 - P(A wins on A's first turn)) × P(B wins)P(A wins on A's first turn) = P(2 cards sum = 14) = 12/52 * 11/51Hence,P(A wins in subsequent turns) = 12/52 * 11/51 + (1 - 12/52 * 11/51) × (12/50 * 11/49) = 0.0642Using the above values, we can find the value of P(B):P(B) = 12/50 * 11/49 + 0.0642 = 0.1313Therefore,Expected value = P(B) / (1 - P(B)) = 0.1313 / (1 - 0.1313) ≈ 0.152 ≈ 1.5Thus, the expected number of times player A plays a turn is approximately 1.5.

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The length οf the parallelοgram is 6.15 cm.

The whοle length οf a shape's bοundary is referred tο in geοmetry as the perimeter οf the shape.

Adding the lengths οf all the sides and edges that surrοund a fοrm yields its perimeter. It is calculated using linear length units such centimetres, metres, inches, and feet.

Here the given Perimeter οf parallelοgram = 20.5 cm.

Side  = 4.1 cm

Nοw perimeter οf parallelοgram = 2(a+b) unit

=> 20.5 = 2(a+4.1)

=> a+4.1 = 10.25

=> a = 10.25-4.1 = 6.15 cm.

Hence the length οf the parallelοgram is 6.15 cm.

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As per the given ratio, the number of boys is 392.

The given ratio of boys to girls is 4:5. This means that for every 4 boys, there are 5 girls. We can express this ratio mathematically as:

Boys/Girls = 4/5

We can use this ratio to find out how many boys are there if there are 488 girls.

Let's assume that the total number of children is "x".

Boys + Girls = x

We know that the ratio of boys to girls is 4:5, which means that the total number of parts in the ratio is 4+5=9. We can express this mathematically as:

Number of boys = (4/9) x x

Number of girls = (5/9) x x

We are given that there are 488 girls in the group, so we can substitute this value into the equation for the number of girls:

(5/9) x x = 488

To solve for "x", we can multiply both sides of the equation by 9/5:

x = 488 x (9/5)

x = 878.4

Since we cannot have a fraction of a child, we can round the answer to the nearest whole number. Therefore, there are 878 children in the group. To find out how many boys there are, we can substitute this value into the equation for the number of boys:

Number of boys = (4/9) x 878

Number of boys = 392

Therefore, there are 392 boys in the group if there are 488 girls and the ratio of boys to girls is 4:5.

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Answer: 9 men will build the same wall in 25 hours.

Step-by-step explanation: 1 man will build the same wall in (45*5) hours

9 men will build the same wall in (45*5)/9 =25 hours

The equation of the circle in the picture is

Information given in the question

center at the point (2, -1)

radius = 3

The equation of a circle with radius given as 3 is solved as below

Equation of a circle is given as

(x - h)² (y - k)² = r²

where h and k refers to points at the center, substituting the values gives

(x - 2)² (y - (-1))² = 3²

(x - 2)² (y + 1)² = 3²

(x - 2)² (y + 1)² = 9

hence option C represents the equation of the circle

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The volume of the tin in cubic meter is equal to 785 cubic meter.

Mathematically, the volume of a cylinder can be calculated by using this formula:

Volume of a cylinder, V = πr²h

Where:

Assuming the radius of the base is 5 cm and the height of this tin is 10 cm, the volume of the tin in cubic meter can be calculated as follows;

Volume of a cylinder, V = 3.14 × 5² × 10

Volume of a cylinder, V = 3.14 × 250

Volume of a cylinder, V = 785 cubic meter.

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The height of the tree is 8.84 m, rounded to the nearest hundredth.

Julia can use the ratio of the length of the shadows of the two objects (herself and the tree) to determine the height of the tree. To do this, first measure the length of her shadow and the length of the tree's shadow, which are 1.60 m and 32.84 m respectively.
Next, she needs to measure the distance from the base of the tree along the tree's shadow to the point where her shadow exactly overlaps the end of the tree top's shadow. In this case, it is 22.78 m.

Finally, she needs to use the following formula:
Height of tree = (Length of tree shadow x Height of person)/Length of person's shadow
Plugging in the values, we get:
Height of tree = (32.84 m x 1.60 m) / 22.78 m = 8.84 m

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Hence, in answering the stated question, we may say that 128.57 quadrilateral degrees is the angle C. D is equal to 171.42 degrees.

A quadrilateral in geometry is a four-sided polygon with outer walls and four corners. The term comes from the Roman terms quadri and lead people to make bad (meaning "side"). A rectangle is a two-dimensional shape with sidewalls, four vertices, and four corners. Concave and convex surfaces are usually divided into two types. Trapezoids, parallelograms, rectangular, rhombuses, and squares are also subclasses of convex quadrilaterals. A rectangle is an a double form that has four straight sides. Parallelograms, trapezoids, triangles, kites, squares, etc rhombuses are examples of quadrilaterals.

The total of a quadrilateral's internal angles equals 360 degrees.

The four inner angles of the quadrilateral ABCD are denoted as follows:

A-angle = 3x

B angle = 4x

Angle C = 5(3x) = 15x (since it is 5 times angle A)

Angle D = 5(4x) = 20x (since angle B is five times larger).

3x + 4x + 15x + 20x = 42x is the sum of these four angles.

3x + 4x + 15x + 20x = 360

42x = 360

x = 8.57 (rounded to two places) (rounded to two places)

Angle A = 3x + 3(8.57) = 25.71°

Angle B = 4x + 4(8.57) = 34.28°

Angle C = 15x + 15(8.57) = 128.57

D = 20x = 20(8.57) = 171.42 degree angle

As a result, the quadrilateral's internal angles are:

A angle of 25.71 degrees

34.28 degrees is the angle B.

128.57 degrees is the angle C.

D is equal to 171.42 degrees.

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On average, each customer will take advantage of 1.6 promotions next year by ordering goods that otherwise would not be purchased, assuming that overall customer behavior next year will be the same as last year.

What is probability?

The extent to which an event is likely to occur, measured by the ratio of the favorable cases to the whole number of cases possible.

Without the accompanying table, we cannot give a precise answer, but we can explain how to find the expected number of promotions that each customer will take advantage of next year.

Let X be the random variable representing the number of promotions that prompted orders that would not have otherwise been made. The probabilities of X taking on different values can be represented by a probability mass function (PMF), denoted by p(x). The expected value or the mean of X, denoted by E(X), is given by:

E(X) = Σ [x * p(x)]

where the summation is taken over all possible values of X.

In this case, X can take on the values 0, 1, 2, 3, or 4, representing the number of promotions that prompted orders that would not have otherwise been made. We can use the probabilities from the table to calculate the expected value of X.

For example, if the probabilities of X taking on different values are given by:

p(0) = 0.2

p(1) = 0.3

p(2) = 0.25

p(3) = 0.15

p(4) = 0.1

then the expected value of X is:

E(X) = (0 * 0.2) + (1 * 0.3) + (2 * 0.25) + (3 * 0.15) + (4 * 0.1) = 1.6

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Complete question:

Tο examine the effectiveness οf its fοur annual advertising prοmοtiοns, a mail οrder cοmpany has sent a questiοnnaire tο each οf its custοmers, asking hοw many οf the previοus year's prοmοtiοns prοmpted οrders that wοuld nοt have οtherwise been made. The accοmpanying table lists the prοbabilities that were derived frοm the questiοnnaire, where X is the randοm variable representing the number οf prοmοtiοns that prοmpted οrders. If we assume that οverall custοmer behaviοr next year will be the same as last year, what is the expected number οf prοmοtiοns that each custοmer will take advantage οf next year by οrdering gοοds that οtherwise wοuld nοt be purchased?

X 0 1 2 3 4

P(X) 0.07 0.229 0.319 0.192 0.19

Expected value =

A previοus analysis οf histοrical recοrds fοund that the mean value οf οrders fοr prοmοtiοnal gοοds is 23 dοllars, with the cοmpany earning a grοss prοfit οf 25% οn each οrder. Calculate the expected value οf the prοfit cοntributiοn next year.

Expected value =

The fixed cοst οf cοnducting the fοur prοmοtiοns is estimated tο be 16000 dοllars with a variable cοst οf 4 dοllars per custοmer fοr mailing and handling cοsts. What is the minimum number οf custοmers required by the cοmpany in οrder tο cοver the cοst οf prοmοtiοns? (Rοund yοur answer up tο the next highest integer.)

Break even pοint =

the perimeter of the regular hexagon is 48 cm.

A regular hexagon is a polygon with six sides, and each side has the same length. The formula to calculate the perimeter of a regular hexagon is given as:

Perimeter = 6 × length of one side

To find the length of one side of the hexagon, we need to use the given information about the area. The formula for the area of a regular hexagon is given as:

Area = (3 × √3 × s^2)/2, where s is the length of one side of the hexagon.

We can use this formula to solve for the length of one side:

96 √3 = (3 × √3 × s^2)/2

Simplifying this equation, we get:

s^2 = (2 × 96 √3)/(3 × √3)

s^2 = 64

s = 8 cm

Now that we know the length of one side of the hexagon, we can use the formula for perimeter to find the total perimeter:

Perimeter = 6 × 8 cm

Perimeter = 48 cm

Therefore, the perimeter of the regular hexagon is 48 cm.

In a regular hexagon, all six angles are equal, each measuring 120 degrees. It is a symmetrical shape, meaning that it has six lines of symmetry. The regular hexagon is a common shape in nature and architecture, and it is often used in geometry problems because of its regularity and symmetry. The hexagon is also the basis for the hexagonal lattice structure, which is a common pattern in crystals and other materials.

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The time Sonto spent doing housework to the time she spent visiting friends to the time she spent watching television can be expressed as 2 : 3 : 4 : 6 .

This is because we can convert each time interval to a common unit, such as minutes, and then simplify the resulting fractions to a ratio of whole numbers. Specifically, Sonto spent 30 minutes doing housework (1/2 hour), 45 minutes doing housework, 60 minutes visiting friends (1 hour), and 90 minutes watching television (1 1/2 hours = 60 + 30 minutes).

Simplifying these fractions gives us the ratio

30 : 45 : 60 : 90

Dividing by GCF 15

2 : 3 : 4 : 6

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To begin with, let us define the system of ordinary differential equations for the benzene ring:

[tex]d^2θj/dt^2 + αdθj/dt + ω^2(θj−1−2θj+θj+1) = F cos(γt)[/tex]

Verification of time-independent particular solution:

When α = 0, the system becomes time-independent, and the equation reduces to:

[tex]θj'' + ω^2(θj−1−2θj+θj+1) = F cos(γt)[/tex]

We can assume a time-independent solution of the form θj = Aj, where A is a constant. Substituting this into the equation, we get:

[tex]Aω^2(j-1 + j + j+1) = F cos(γt)[/tex]

By simplifying

3Aω^2j = F cos(γt)

Therefore, the time-independent particular solution is θj = (F/3ω^2)cos(γt) + Aj, where A is a constant.

Eigenvectors and eigenvalues:

predicting the solution θj = Aj e^(iλt),we can integrate into an equation

[tex]−A(λ^2+αλ+ω^2)e^(iλt) + ω^2(e^(iλt)(A(j−1) + A(j+1)) + 2A(j)e^(iλt)) = F cos(γt)e^(iλt)[/tex]

Dividing both sides by e^(iλt), we get:

[tex]−A(λ^2+αλ+ω^2) + ω^2(A(j−1) + A(j+1) + 2A(j)) = Fcos(γt)[/tex]

Simplifying, we get:

[tex](2ω^2A − λ^2A) + ω^2(A(j−1) + A(j+1)) = F cos(γt) + αλA[/tex]

The eigenvectors of this matrix are:

[tex][±sin(π/6)][±sin(2π/6)][±sin(3π/6)][±sin(4π/6)][±sin(5π/6)][/tex]

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