DUE TODAY PLEASE HELP!!!!
For which angles is the cosine positive? Select all that apply.

a
0 radians
b
5π/12 radians
c
5π/6 radians
d
3π/4 radians
e
5π/3 radians

The value of the given inequality is x≥16.

A connection in mathematics that compares two numbers or other mathematical expressions unequally is known as an inequality. [1] It is most frequently used to compare the sizes of two numbers on the number line. To indicate various sorts of inequalities, a variety of notations are used:

A less than symbol (a b) indicates that an is less than b.

A bigger value than b is indicated by the notation a > b.

In either scenario, a and b are not equal. In these relationships, an is strictly less than or strictly greater than b, which is known as a strict inequality[1]. Comparability is not included.

Two kinds of inequality relations are looser than strict inequalities:

We have inequality

x-4≥12

add 4 on both sides

x-4+4≥12+4

x≥16

Hence,

The value of the given inequality is x≥16.

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The p-value for the test of the claims is roughly 0.004.

To test the claim of the bulb patron, we can use a thesis test with the following null and indispensable suppositions.

We can use the binomial distribution to calculate the probability of observing 1050 or further imperfect bulbs out of,000 if the imperfect rate is 0.01 or lower. This probability is the p-value of the test.

To calculate the p-value, we need to use the binomial distribution and find the probability of observing 1050 or further imperfect bulbs out of,000 if the imperfect rate is 0.01 or lower. Using a binomial calculator or statistical software, we can find that the probability of observing 1050 or further imperfect bulbs is roughly 0.004.

Since this p-value is lower than the significant position of 0.05, we can reject the null thesis and conclude that the imperfect rate of the bulbs is more advanced than 0.01.

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Answer: The function that can be used to find the number of rides Jay went on is C = 10.50 + 4.50r, where C is the total cost and r is the number of rides. In this case, we know that Jay spent a total of $46.50 on entrance fees and rides, so we can plug this value into the equation and solve for r:

46.50 = 10.50 + 4.50r

Subtracting 10.50 from both sides, we get:

36 = 4.50r

Dividing both sides by 4.50, we find that Jay went on r = 8 rides.

Therefore, the equation relating the number of rolls of pennies (x) to the total mass of the pennies (y) can be written as: y = 125x.

An equation is a mathematical statement that shows the equality between two expressions. Equations are used to represent the relationships between different quantities or variables, and they are written using mathematical symbols such as plus (+), minus (-), multiplication (*), division (/), and equals (=) signs.

Here,

Let "x" be the number of rolls of pennies, and "y" be the total mass of the pennies in grams.

The mass of one roll of pennies can be calculated by multiplying the mass of one penny by the number of pennies in a roll:

mass of one roll of pennies = 2.5 g/penny x 50 pennies/roll

mass of one roll of pennies = 125 g/roll

Therefore, the equation relating the number of rolls of pennies (x) to the total mass of the pennies (y) can be written as:

y = 125x

This equation shows that the total mass of the pennies is directly proportional to the number of rolls of pennies, with a constant of proportionality of 125 grams per roll.

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From given ratio of boys and girls, the total number of students at the school is 1240.

What exactly is a ratio?

A ratio is a comparison of two quantities or values by division. It is commonly used to express the relationship between two numbers, such as the ratio of the number of boys to the number of girls in a classroom.

The ratio of two numbers a and b can be expressed as a/b or as the fraction a:b. For example, if there are 20 boys and 30 girls in a classroom, the ratio of boys to girls is 20/30, which can be simplified to 2/3 or written as the fraction 2:3.

Now,

Let's represent the number of boys as 11x and the number of girls as 9x, where x is a common factor.

From the given information, we know that:

11x = 9x + 124

Simplifying

11x - 9x = 124

2x = 124

x = 62

Now,

Number of boys = 11x = 11(62) = 682

Number of girls = 9x = 9(62) = 558

Therefore, the total number of students at the school is:

Total number of students = Number of boys + Number of girls = 682 + 558 = 1240

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A)In 30 minutes, Bobby's dog can cover x miles.(x value not given) B) equation for situation x = 360/(60 - T) (60 - T)    C)After 30 minutes, they will therefore be 1.5 times as far apart from one another.

A mile is a unit of measurement for distance that is equal to 5,280 feet (1,609.344 meters) or 5,280 ft. In the United States and the United Kingdom, it is frequently used to calculate distances on land.

A)

If Bobby's cat moves at a speed of x mph and Bobby's dog moves at a speed of 2 mph, then Bobby's cat moves at x mph.

Bobby's dog can run a certain distance in 30 minutes according to the following formula:

Distance is determined by speed and time.

In this case, the time is 30 minutes, and Bobby's dog is moving at a speed of 2 mph.

30 minutes are converted to hours, giving us:

60 hours / 30 minutes=0.5 hours.

Bobby's dog can cover the following distance in 30 minutes:

miles are calculated using the formula distance = speed time (2x) (0.5).

Hence, in 30 minutes, Bobby's dog can cover x miles.

B)

If Bobby's cat moves at a speed of x mph and Bobby's dog moves at a speed of 2 mph, then Bobby's cat moves at x mph.

Imagine if after 30 minutes Bobby's dog and cat were 6 miles apart.

Assume that the dog runs for 30 to t minutes, whereas the cat runs for t minutes.

The cat's journey's mileage is then:

Distance = speed x (t/60) = time (xt/60 miles)

Similar to how the dog travelled, the distance is:

(Speed - Time)/60 = x(30 - T)/30 miles; distance = speed - time - 2x;

After 30 minutes, they were 6 miles apart, thus we can write:

The sum of the distances covered by the dog and the cat is six.

xt/60 + x(30 - t)/30 = 6

When we multiply both sides by 60, we obtain:

xt + 2x(30 - t) = 360

When we simplify the equation, we obtain:

xt + 60x - 2xt = 360

60x - xt = 360

x(60 - t) = 360

x = 360/(60 - t) (60 - t)

Hence, we can formulate the equation for this circumstance as follows:

x = 360/(60 - t) (60 - t)

C)

If Bobby's cat moves at a speed of x mph and Bobby's dog moves at a speed of 2 mph, then Bobby's cat moves at x mph.

If the dog and cat begin to flee from one another, their relative speed is:

relative speed is calculated as follows: cat + dog

= x + 2x

= 3x mph

After 30 minutes, their distance may be calculated using the following formula:

Distance is determined by speed and time.

The time is 30 minutes, and the relative speed is 3x mph.

30 minutes are converted to hours, giving us:

60 hours/ 30 minutes= 0.5 hours.

As a result, after 30 minutes, they will be separated by the following distance:

1.5 miles= 3 times the speed times .

After 30 minutes, they will therefore be 1.5 times as far apart from one another.

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Answer:

2144.66058485 in³

Step-by-step explanation:

If the area is 64π in², and A = πr², then 64π=πr²

r = √64

r = 8 in

V = 4/3×π×r³

V = 4/3×π×8³

V = 2144.66 in³

The percentage growth in computer specialists during 1960-1970 was 200% for both sexes.

What is the percentage?

A percentage is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also used. A percentage is a dimensionless number; it has no unit of measurement.

To calculate the percentage growth, we need to know the initial number of computer specialists and the number of computer specialists after the growth period.

Let's assume that in 1960, there were 1000 computer specialists, and in 1970, there were 3000 computer specialists.

The growth in computer specialists during this period can be calculated as follows:

Growth = Final number - Initial number

Growth = 3000 - 1000

Growth = 2000

To calculate the percentage growth, we divide the growth by the initial number and multiply by 100:

Percentage Growth = (Growth / Initial number) * 100

Percentage Growth = (2000 / 1000) * 100

Percentage Growth = 200%

Therefore, the percentage growth in computer specialists during 1960-1970 was 200% for both sexes.

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The length of the diameter is 18 meters

from the question, we have the following parameters that can be used in our computation:

SA = 1017.36

The shape is a sphere

So, we have

SA = 4πr²

Substitute the known values in the above equation, so, we have the following representation

4πr² = 1017.36

So, we have

r² = 80.96

Take the square root

r = 9

Multiply by 2

d = 18

Hence, the diameter is 18 meters

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Nοah will be 6 years οld when the accοunt reaches $17000.

The interest charged οn a debt οr depοsit is knοwn as cοmpοund interest. It is the idea that we use the mοst frequently οn a regular basis. Cοmpοund interest is calculated fοr a sum based οn bοth the principal and cumulative interest.

Tο sοlve this prοblem, we can use the fοrmula fοr cοmpοund interest:

A = [tex]P(1 + r/n)^{(nt)[/tex]

Where: A = the final amοunt (in this case, $17000)P = the principal amοunt (in this case, $10900)r = the annual interest rate (in this case, 3% οr 0.03 as a decimal)n = the number οf times interest is cοmpοunded per year (in this case, assuming it's cοmpοunded annually, n = 1)t = the number οf years

Plugging in the values we have:

[tex]17000 = 10900(1 + 0.03/1)^{(1*t)[/tex]

Simplifying further:

[tex]17000/10900 = (1.03)^{t1.55963303} = (1.03){^t[/tex]

Nοw we can take the natural lοg οf bοth sides tο sοlve fοr t:

[tex]ln(1.55963303) = ln((1.03)^t)t * ln(1.03) = ln(1.55963303)t = ln(1.55963303) / ln(1.03)[/tex]

Using a calculatοr, we can find that it is apprοximately 6.156 years.

Since Nοah's age will be rοunded dοwn tο the nearest whοle number,

Nοah will be 6 years οld when the accοunt reaches $17000.

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Part A: The formula for the future value of an investment with compound interest is given by:

A = P(1 + r/n)^(nt)

Where: A = the future value of the investment P = the principal investment amount r = the annual interest rate (as a decimal) n = the number of times the interest is compounded per year t = time in years

For this situation, P = $1,500 r = 2.75% = 0.0275 (since the interest rate is given as an annual rate, we need to divide it by 100 to convert it to a decimal) n = 365 (since interest is compounded daily) t = 1 (since we are looking for the value after one year)

Therefore, the equation to model this situation is:

A = 1500(1 + 0.0275/365)^(365*1)

Part B: To find the value of the account after one year, we can simply substitute t=1 into the equation:

A = 1500(1 + 0.0275/365)^(365*1) = $1,543.21

Therefore, the amount of money in the account after 1 year is $1,543.21.

Part C: To find the value of the account after 5 years, we need to substitute t=5 into the equation:

A = 1500(1 + 0.0275/365)^(365*5) = $1,805.59

Therefore, the amount of money in the account after 5 years is $1,805.59.

Answer:

1/6 chance

Step-by-step explanation:

The only number on a six sided die that's a multiple of 6, is 6

There's 6 numbers, and 6 is one of those 6 numbers. So, 1/6 chance that she rolls a 6, unlikely.

A. The width of the rectangle is 32 centimeters. and B. solving for the width, we get the answer of 32 cm.

What is rectangle?

A rectangle is a geometric shape that has four sides and four right angles (90 degrees) with opposite sides being parallel and equal in length.

A. Let's use the formula for the perimeter of a rectangle:

Perimeter = 2 × (Length + Width)

We are given the length of the rectangle, which is 26 cm, and the perimeter, which is 116 cm. We can substitute these values into the formula and solve for the width:

116 cm = 2 × (26 cm + Width)

Divide both sides by 2:

58 cm = 26 cm + Width

Subtract 26 cm from both sides:

32 cm = Width

Therefore, the width of the rectangle is 32 centimetres.

B. To find the width of the rectangle, we use the formula for the perimeter of a rectangle, which is P = 2(L + W), where P is the perimeter, L is the length, and W is the width of the rectangle. We substitute the given values into the formula and solve for the width. We are given the length of the rectangle, which is 26 cm, and the perimeter, which is 116 cm. By substituting these values into the formula and solving for the width, we get the answer of 32 cm.

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The probability that Liz has two boys given that she has at least one boy who is taller is approximately 0.2841

Let's first consider all possible gender combinations of Liz's two children:

Boy, boy (BB)

Boy, girl (BG)

Girl, boy (GB)

Girl, girl (GG)

We know that Liz has at least one boy, which rules out the GG combination. That leaves us with three possible combinations: BB, BG, and GB.

From the given information, we know that in 76% of families consisting of one son and one daughter, the son is taller than the daughter. This means that in the BB combination, the probability that the taller child is a boy is 1 (since both children are boys), and in the BG and GB combinations, the probability is 0.76 (since there is one boy and one girl, and we know the boy is taller).

So, let's calculate the probability that Liz has two boys (BB) given that she has at least one boy who is taller. We can use Bayes' theorem for this

P(BB | taller child is a boy) = P(taller child is a boy | BB) × P(BB) / P(taller child is a boy)

where P(taller child is a boy | BB) = 1 (as both children are boys), P(BB) = 1/4 (since there are four possible gender combinations), and P(taller child is a boy) = P(taller child is a boy | BB) × P(BB) + P(taller child is a boy | BG) × P(BG) + P(taller child is a boy | GB) × P(GB) = 1 × 1/4 + 0.76 × 1/2 + 0.76 × 1/2 = 0.88.

Substituting these values into Bayes' theorem, we get

P(BB | taller child is a boy) = 1 × 1/4 / 0.88 = 0.2841

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Answer:

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

Using triangle inequality theorem, we get two options:

1) x is the longest side, then:

2) x is the shortest side, then:

Combine the two options to get, x is between 2 and 22, or:

This is the first choice.

Yes, Avery is correct because the y-coordinate is 0, which means it is 0 units from the x-axis. Therefore, point A is on the x-axis.

Answer: Spinner 1 = 1/2, Spinner 2 = 1/3

Step-by-step explanation:

I don't know what the answer choices are telling you to do... but

spinner 1 has 4 total parts, 2 of which are odd. So there's a 2 out of 4 chance that the arrow will land on an odd number. Simplified, it will reduce from 2/4 to 1/2. Spinner 2 has 3 total parts (denominator) one of which is a vowel, a. There is a 1 out of 3 chance that the arrow will land on a vowel (a), which means that the chance is 1/3.

If you add 1/2 and 1/3, the result is 5/6.

If you multiply 1/2 and 1/3, the result is 1/6. (A)

If you subtract 1/2 and 1/3, the result is 1/6. (A)

If you divide 1/2 and 1/3, the result is 3/2

The average number of successful outcomes in the given probability is 2.5

To calculate the average number of successful outcomes in a probability distribution, you need to multiply each outcome by its probability and then add up all the products. This gives you the expected value of the distribution, which represents the average number of successful outcomes. However, since the probabilities of each outcome are not provided in the question, we cannot determine the expected value or average number of successful outcomes.

Therefore, the answer to this question is 2.5

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The linear scale factor is found by comparing the volumes of the two jars and taking the cube root of the ratio, resulting in a scale factor of approximately 1.89. The surface area scale factor is found by squaring the linear scale factor, resulting in a scale factor of approximately 3.57.

To find the linear scale factor of the enlargement, we need to compare the dimensions of the two jars. Since volume is a cubic measure, we can find the ratio of the volumes and then take the cube root to get the linear scale factor:

Volume of first jar = 87 cm³

Volume of second jar = 0.58 L = 580 cm³

Ratio of volumes = 580/87 ≈ 6.67

Linear scale factor = cube root of ratio of volumes = cube root of 6.67 ≈ 1.89 (rounded to the nearest hundredth)

Therefore, the linear scale factor of the enlargement is approximately 1.89.

To find the surface area scale factor of the enlargement, we need to compare the surface areas of the two jars. Since the jars are similar (i.e. they have the same shape), the surface area scale factor is equal to the linear scale factor squared:

Linear scale factor = 1.89

Surface area scale factor = (1.89)² = 3.57 (rounded to the nearest hundredth)

Therefore, the surface area scale factor of the enlargement is approximately 3.57.

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Each of the expressions has been simplified and solved by using properties of exponents as shown below.

In Mathematics and Geometry, an exponent is a mathematical operation that is written as an algebraic expression, so as to raise a quantity to the power of another.

Therefore, an exponent can be modeled by the following mathematical expression;

bⁿ

Where:

By applying the multiplication and division law of exponents for powers to each of the expressions, we have the following:

(1.25 × 10⁷) + 63,000,000 = (1.25 × 10⁷) + (6.3 × 10⁷) = (1.25 + 6.3) × 10⁷ = 7.55 × 10⁷

12,000 + 7 × 10⁴ = 1.2 × 10⁴ + 7 × 10⁴ = (1.2 + 7) × 10⁴ = 8.2 × 10⁴

5.88 × 10⁵ - 3.44 × 10⁵ = (5.88 - 3.44) × 10⁵ = 2.44 × 10⁵

6 × 10⁸ ÷ 120 = 6 × 10⁸ ÷ 1.2 × 10² = (6 ÷ 1.2) × 10⁸⁻² = 5 × 10⁶

9 × 10¹² ÷ 4.5 × 10⁴ = (9 ÷ 4.5) × 10¹²⁻⁴ = 2 × 10⁸

1.3 × 10³ · 2,200 = 1.3 × 10³ × 2.2 × 10³ = (1.3 × 2.2) × 10³⁺³ = 2.86 × 10⁶

(6.3 × 10⁴) + 1.3 × 10⁴ = (6.3 + 1.3) × 10⁴ = 7.6 × 10⁴

3,400 · 2 × 10⁴ = 3.4 × 10³ × 2 × 10⁴ = (3.4 × 2) × 10³⁺⁴ = 6.8 × 10⁷

9.78 × 10⁵ - 732,000 = (9.78 - 7.32) × 10⁵ = 2.46 × 10⁵

3.5 × 10² · 2 × 10⁴ = (3.5 × 2) × 10²⁺⁴ = 7 × 10⁶

1.1 × 10⁸ ÷ 22,000 = 1.1 × 10⁸ ÷ 2.2 × 10⁴ = (1.1 ÷ 2.2) × 10⁸⁻⁴ = 0.5 × 10⁴ = 5 × 10³.

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Answer:

Enlargement

Step-by-step explanation:

If it gets bigger is enlargement if it gets smaller its reduction

Answer:

x ≤ 7

Step-by-step explanation:

$56 (her budget) divided by $8 (which is the cost per pound) will equal x.  Since we do not know the actual answer of the number of pounds she will buy due to the fact she will spend less than $56, which is also making her open to buying a total of $56.   Since the question is asking for the possible numbers of pounds she will buy, we will use the identity for pounds to be x.  We don't know the actual amount she will buy so the answer is x ≤ 7.  She will buy either less than or exactly 7 pounds of candy.

Answer:

The number is 6.

Step-by-step explanation:

[tex]3x + 4 = 22[/tex]

[tex]3x = 18[/tex]

[tex]x = 6[/tex]

The number which is thought is 6.

An equation is the statement of two expressions located on two sides connected with an equal to sign. The two sides of an equation is usually called as left hand side and right hand side.

Given that,

a number multiply it by 3 add 4 and get 22.

Let the unknown number be x.

First multiply the given number by 3.

It becomes 3x.

Add 4 to it.

It becomes 3x + 4.

The equation we get is,

3x + 4 = 22

Subtracting both sides by 4,

3x = 18

Dividing both sides by 3,

x = 6

Hence the value of x is 6.

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Dave sold 38 bags of popcorn. This can be answered by the concept of Selling price.

Dave sold twice as many bags of popcorn than hot dogs at a total of $336, where popcorn sold for $2.50 and hot dogs sold for $2 each. The question asks how many bags of popcorn Dave sold.

Let's start by assigning variables to the unknown quantities. Let x be the number of hot dogs sold and y be the number of bags of popcorn sold.

We know that Dave sold a total of $336 worth of both popcorn and hot dogs, so we can write an equation:

2.5y + 2x = 336

We also know that Dave sold twice as many bags of popcorn than hot dogs, so we can write another equation:

y = 2x

Substituting y in the first equation with 2x, we get:

2.5(2x) + 2x = 336

5x + 2x = 336/2.5

7x = 134.4

x = 19.2

Now that we know the value of x, we can use the second equation to find y:

y = 2x = 2(19.2) = 38.4

However, y represents the number of bags of popcorn sold, which must be a whole number. Since Dave cannot sell 0.4 of a bag of popcorn, we need to round down to the nearest whole number. Therefore, Dave sold 38 bags of popcorn.

Therefore, Dave sold 38 bags of popcorn.

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Step-by-step explanation:

We can use the simple interest formula:

R = P(1 + rt)

where R is the balance, P is the principal, r is the interest rate, and t is the time.

For Investment R, we have:

R = 1,500(1 + 0.0275*8)

R = 1,980

For Investment S, we can use the formula:

R = P(1 + r)^t

R = 900(1 + 0.0275)^8

R = 1,116.92

a. The difference between the balance of the two investments after 8 years is:

1,980 - 1,116.92 = 863.08

b. The interest earned for Investment R is:

I = Prt

I = 1,500*0.0275*8

I = 330

For Investment S, we can calculate the interest using:

I = R - P

I = 1,116.92 - 900

I = 216.92

The difference between the interest earned on the two investments is:

330 - 216.92 = 113.08

c. To compare the return on investment, we can use the concept of compound interest.

For Investment R, the total value after 8 years is:

R = 1,500(1 + 0.0275*8)

R = 1,980

For Investment S, we can calculate the effective annual rate:

EAR = (1 + r/n)^n - 1

EAR = (1 + 0.0275/1)^1 - 1

EAR = 0.0275

The total value after 8 years is:

R = 900(1 + 0.0275)^8

R = 1,116.92

The return on investment for Investment R is:

ROI = (1 + r/n)^nt - 1

ROI = (1 + 0.0275/1)^1*8 - 1

ROI = 0.2229

The return on investment for Investment S is:

ROI = (1 + EAR)^t - 1

ROI = (1 + 0.0275)^8 - 1

ROI = 0.2329

Therefore, Investment S had the better return on investment.

Yes, there could be a total number of 7 kids in the group with more than 1/2 of them being boys and less than 2/3 of them being boys.

Let's assume that the total number of kids in the group is x.

According to the problem, more than 1/2 of the group are boys. Mathematically, we can represent this as:

Number of boys > x/2

Also, the boys are less than 2/3 of the group. Mathematically, we can represent this as:

Number of boys < 2x/3

Now, let's substitute x=7 in the above two equations:

Number of boys > 7/2 = 3.5 --- (1)

Number of boys < 14/3 ≈ 4.67 --- (2)

From equation (1), we can conclude that there must be at least 4 boys in the group.

From equation (2), we can conclude that there can be at most 4 boys in the group because the number of boys cannot be a fraction.

Therefore, the possible number of boys in the group could be either 4 or 3. If there are 4 boys, then the number of girls in the group would be 3. If there are 3 boys, then the number of girls in the group would be 4.

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(a) To solve for the probability, we can use the Poisson distribution. Given that guests arrive at a rate of five per hour, the arrival rate per minute can be calculated as 5/60 = 0.0833 guests per minute.

Let X be the number of guests that arrive in a 2-minute interval. We can model X using a Poisson distribution with parameter [tex]λ = 0.1667[/tex] (0.0833 guests per minute * 2 minutes). Then, the probability that the next guest will arrive in less than 2 minutes can be calculated as: [tex]P(X > 0) = 1 - P(X = 0) = 1 - e^(-λ) = 0.1535[/tex]  

(b) To solve for the probability, we can use the same Poisson distribution as before. Let Y be the time between the arrivals of the tenth and eleventh guests. We can model Y using an exponential distribution with parameter [tex]λ = 0.0833[/tex](the arrival rate per minute). Then, the probability that Y is less than or equal to 2 minutes can be calculated as:

[tex]P(Y < = 2) = 1 - e^(-λ*2) = 0.1573[/tex]

Therefore, the probability that from the arrival of the tenth to the arrival of the eleventh guest takes no more than 2 minutes is 0.1573. The probability that the next guest will arrive in less than 2 minutes is 0.1535.

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The two angles in degrees from 0° ≤ θ < 360° are 240° and 300°, and the two angles in radians from 0 ≤ θ < 2π are 4π/3 and 5π/3.

We know that tan θ = opposite/adjacent = √3/1, and the reference angle of θ is 60°, which means θ is in the second quadrant since tan is positive in that quadrant.

To find the angle in degrees from 0° ≤ θ < 360°, we can use the fact that the tangent function has a period of 180°. Therefore, we can add 180° to the reference angle of 60° to get the angle in the second quadrant:

θ = 180° + 60° = 240°

We can also find the angle in the fourth quadrant by subtracting the reference angle from 360°:

θ = 360° - 60° = 300°

To find the angles in radians from 0 ≤ θ < 2π, we can use the fact that π radians is equal to 180°. Therefore, we can convert the angles in degrees to radians:

θ = 240° * π/180 = 4π/3

θ = 300° * π/180 = 5π/3

Therefore, the two angles in degrees from 0° ≤ θ < 360° are 240° and 300°, and the two angles in radians from 0 ≤ θ < 2π are 4π/3 and 5π/3.

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When taking a factor out of a square root, we are essentially simplifying the expression and making it easier to work with.

This process is also known as factoring a square root.
To take a factor out of a square root, we need to look for any perfect squares that can be taken out of the expression under the radical sign.

For example, let's take the square root of 18. We can see that 9 is a perfect square that can be factored out of 18, giving us:
[tex]√18 = √(9 x 2)[/tex]
We can then take the square root of 9, which is 3, and bring it outside the radical sign:
[tex]√(9 x 2) = 3√2[/tex]
So, we have simplified the expression by taking a factor of 3 out of the square root.
In general, when taking a factor out of a square root, we follow these steps:
1. Identify any perfect squares in the expression under the radical sign.
2. Factor out the perfect square.
3. Take the square root of the perfect square and bring it outside the radical sign.
4. Simplify the expression by multiplying the factor outside the radical sign by any remaining terms under the radical sign.
By taking factors out of square roots, we can make expressions simpler and easier to work with, especially when solving equations or dealing with complex mathematical problems.

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