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In a circle, an angle measuring π radians intercepts an arc of length 9π. Find the radius of the circle in simplest form.

The volume of the larger cone will be 48 cm³.

This is because when a shape is enlarged by a scale factor of 2, the volume is increased by a factor of 2³ (or 8). So, 6 x 8 = 48.


When we enlarge a shape by a scale factor, we are multiplying all of its dimensions by that factor. In the case of a cone, this means that we are increasing the radius and the height of the cone by a factor of 2.

We can use the formula for the volume of a cone to find out how the volume changes when we enlarge it. The formula for the volume of a cone is V = (1/3)πr²h, where r is the radius and h is the height.

If we multiply the radius and the height of the cone by 2, we get a new cone with a radius of 2r and a height of 2h. Plugging these new values into the formula for the volume of a cone, we get:

V' = (1/3)π(2r)²(2h) = (1/3)π(4r²)(2h) = (8/3)πr²h

We can simplify this expression by multiplying the original volume by 8/3:

V' = (8/3) x 6 = 48

So the volume of the larger cone is 48 cubic centimeters.

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The unit price in dollars per centimeter is 80 cents.

:: Total wire length = 19.5 cm

:: Total amount paid = $15.60

Per unit price = [ (total amount paid) / (total wire length) ]

Per unit price = (15.60 / 19.5) $/cm

Per unit price = 0.8 ($/cm)

And as we know, $1 = 100 cents,

So,

$0.8 = 0.8 x 100 cents = 80 cents.

Therefore, the unit price in dollars per centimeter is 80 cents.

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Final Answer:  a. The number of units that should be sold in order to maximize the profit is 7 thousand units.

b. The  maximum profit is approximately $5.51

Conceptual part:   a. In order to find maximum profit we need to differentiate the profit function

[tex]dp/dx = (-3x^2+18x+21)/-x^3+9x^2+21x+1[/tex] = 0

[tex]-3x^2+18x+21=0[/tex]

[tex](x-7) (x+1) = 0[/tex]

as profit can't be negative.

hence x=7.

b. We can determine the maximum profit by substituting x=7 in profit function.

[tex]p(7) = ln(-7^3+9*7+21*7+1)[/tex]

[tex]p(7) = ln(246)[/tex]

p(7) = 5.51

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1/70 is the probability of having slips numbered 1, 2, 3, and 4 drawn in order from the box.

To calculate the probability of drawing slips numbered 1, 2, 3, and 4 in order from a box containing slips numbered 1 through 8, we need to first find out the total number of possible outcomes when drawing four slips without replacement from the box.

The number of ways to draw 4 slips from a set of 8 slips without replacement is given by the combination formula:

= 8!/4!(8-4)! = 70

This means there are 70 possible outcomes when drawing four slips from the box.

To calculate the probability of drawing slips 1, 2, 3, and 4 in that order, we need to consider that there is only one way to draw the slips in that specific order, out of the 70 possible outcomes.

Therefore, the probability of drawing slips 1, 2, 3, and 4 in order is:

P(1,2,3,4 in order) =  number of favorable outcomes/total number of possible outcomes = 1/70

So the probability of drawing slips numbered 1, 2, 3, and 4 in order from the box is 1/70.

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The correct response is 12, as finding the predicted smoking and asbestos exposure group requires finding x, and in that case, the ratios in the rows and columns are equal.

4/2 = x/6 Or 6/2= x/4

This suggests that x = 12.

Smoking refers to the act of inhaling and exhaling the smoke produced by burning tobacco or other substances such as marijuana or hookah. It is a highly addictive habit that poses significant health risks to both smokers and those exposed to secondhand smoke. Smoking can lead to a variety of illnesses and diseases, including lung cancer, heart disease, stroke, chronic obstructive pulmonary disease (COPD), and various other cancers. It is estimated that smoking is responsible for nearly 8 million deaths globally each year.

The chemicals in tobacco smoke can also harm the environment, contributing to air pollution and litter. Despite the well-known risks associated with smoking, many people continue to smoke due to addiction, peer pressure, or a lack of understanding about the long-term consequences.

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The location of point B is (4, −2). So the answer is (4, −2).

To find the location of point B, we can use the midpoint formula, which states that the coordinates of the midpoint of a line segment with endpoints (x1, y1) and (x2, y2) are:

((x1 + x2)/2, (y1 + y2)/2)

In this case, we know that point M is the midpoint of segment AB, and we know the coordinates of point M. We also know the coordinates of point A. So we can use the midpoint formula to solve for the coordinates of point B.

Let's call the coordinates of point B (x, y). We know that point M is the midpoint of segment AB, so we can set up the following equation:

((−2 + x)/2, (2 + y)/2) = (1, 0)

Simplifying this equation, we get:

(−2 + x)/2 = 1 and (2 + y)/2 = 0

Solving for x and y, we get:

x = 4 and y = −2

Therefore, the location of point B is (4, −2). So the answer is (4, −2).

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The height of the candle after x hours is given by the equation y = 2x + 14.

Let's first convert the time interval to hours. There are 60 minutes in an hour, so 30 minutes is equivalent to 0.5 hours.

After x hours, the candle will have burned down by 2x inches (since it burns 1 inch every 0.5 hours).

If y is the total height of the candle, then we can write the equation:

y - 2x = 14

We can solve for y:

y = 2x + 14

So the height of the candle after x hours is given by the equation y = 2x + 14.

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The group of natural numbers is a subset of the whole numbers.

What is Whole number ?

Whole numbers are a set of numbers that includes all positive integers (1, 2, 3, ...) and zero (0). They do not include negative numbers or fractions. Whole numbers are often used to count objects or represent quantities that cannot be divided into smaller parts.

Out of the given options, the group of natural numbers is the only one that is a subset of whole numbers. The other options - Rational numbers, Real numbers, Irrational numbers, and Integers - are not subsets of the whole numbers.

Rational numbers include fractions and decimal numbers, which can be expressed as a ratio of two integers, including non-whole numbers.

Real numbers include all rational and irrational numbers, including non-whole numbers. Irrational numbers are non-repeating, non-terminating decimals that cannot be expressed as a fraction. Integers include both positive and negative whole numbers, as well as zero.

The subset of whole numbers is called natural numbers.

Therefore, the group of natural numbers is a subset of the whole numbers.

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The solution to the inequality given is false.

To determine whether x = 5 is a solution of the inequality x + 3 > 12, we substitute x = 5 into the inequality and see if the resulting statement is true or false:

5 + 3 > 12

8 > 12

In this case, when we substitute x = 5 into the inequality, we get the statement 5 + 3 > 12, which simplifies to 8 > 12. This statement is false, since 8 is not greater than 12. Therefore, x = 5 is not a solution of the inequality x + 3 > 12.

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The full question is:

Tell whether the given value is a solution of the inequality X + 3 > 12; x = 5

If the fifth and tenth terms of an arithmetic sequence, respectively, are -2 and 53, the seventh term of the arithmetic sequence is 20.

To find the seventh term of the arithmetic sequence, we need to first find the common difference (d) of the sequence. We know that the fifth term is -2 and the tenth term is 53.

The formula for the nth term of an arithmetic sequence is: an = a1 + (n-1)d

Using this formula, we can set up two equations:

-2 = a1 + 4d  (since the fifth term is a1 + 4d)
53 = a1 + 9d  (since the tenth term is a1 + 9d)

We now have two equations with two variables (a1 and d). We can solve for either variable using substitution or elimination. I'll use elimination:

-2 = a1 + 4d
53 = a1 + 9d

Subtracting the first equation from the second equation, we get: 55 = 5d

Therefore, d = 11

Now that we know the common difference is 11, we can use the formula for the nth term again to find the seventh term:

a7 = a1 + (7-1)d
a7 = a1 + 6d

We still don't know a1, but we can solve for it using one of the previous equations:

-2 = a1 + 4d
-2 = a1 + 4(11)
-2 = a1 + 44
a1 = -46

Now we can substitute a1 and d into the formula for the seventh term:

a7 = -46 + 6(11)
a7 = -46 + 66
a7 = 20

Therefore, the seventh term of the arithmetic sequence is 20.

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The plant's height at 12 months would be 38 inches.

Assuming the plant's height follows a linear pattern, Robin can use the equation y = mx + b, where y is the height of the plant, x is the number of months, m is the slope or rate of growth, and b is the y-intercept or initial height.

To find the equation, Robin can use the data from the three months:

Using these data points, Robin can calculate the slope m:

m = (11-5)/(3-1) = 3 inches/month

Then, using the point-slope form of the equation, Robin can find the y-intercept b:

y - 5 = 3(x - 1)

y = 3x + 2

Therefore, the algebraic expression to find the plant's height for any month x is y = 3x + 2.

To find the plant's height at 12 months, Robin can substitute x = 12 into the equation:

y = 3(12) + 2 = 38 inches

So the plant's height at 12 months would be 38 inches.

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The best point estimate for the population mean score on the nationwide examination in psychology is the sample mean of 492.

When we take a sample from a population, the sample mean is a point estimate of the population mean. A point estimate is an estimate of a population parameter based on a single value or point in the sample. In this case, the sample mean of 492 is the best point estimate for the population mean, because it is an unbiased estimator.

An estimator is unbiased if it is expected to be equal to the true population parameter. In this case, the expected value of the sample mean is equal to the population mean. This means that if we were to take many different samples from the population and calculate the sample mean for each sample, the average of all these sample means would be equal to the population mean.

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The given question is incomplete, the complete question is:

You would like to construct a 95% confidence interval to estimate the population mean score on a nationwide examination in psychology, and for this purpose we choose a random sample of exam scores. The sample we choose has a mean of 492 and a standard deviation of 78. What is the best point estimate, based on the sample, to use for the population mean?

The 90% confidence interval for the amount of time that teens spend online each week is (30.761, 32.839) hours

To find the 90% confidence interval for the amount of time that teens spend online each week, we can use the following formula

CI = x ± z*(σ/√n)

where

x is the sample mean

σ is the population standard deviation

n is the sample size

z is the z-score associated with the desired confidence level

In this case, we have

x = 31.8 hours

σ = 5.91 hours

n = 85

z = 1.645 (for a 90% confidence level, using a z-table or calculator)

Plugging in these values, we get

CI = 31.8 ± 1.645*(5.91/√85)

Simplifying, we get

CI = 31.8 ± 1.039

We can interpret this interval as saying that if we were to take many random samples of 85 teens and compute the confidence interval for each sample, about 90% of these intervals would contain the true population mean.

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Answer:

(4),(3),(1),(2),(2),(1)

Step-by-step explanation:

Mnemonic to memorize trigonometric ratios: SOH CAH TOA

(S=O/H C=A/H T=O/A)

Sine, Cosine, Tangent, Opposite to x, Adjacent to x, Hypotenuse

Cosecant = 1/Sine, Secant = 1/Cosine, Cotangent = 1/Tangent

The ordered pair that is a reflection over the x-axis for the point shown include the following: A. (6, 4)

In Mathematics and Geometry, a reflection over or across the x-axis is represented by this transformation rule (x, y) → (x, -y).

This ultimately implies that, a reflection over or across the x-axis would maintain the same x-coordinate while the sign of the y-coordinate changes from positive to negative or negative to positive.

Next, we would apply a reflection over or across the x-axis to the point;

(x, y)                →      (x, -y)

(6, -4)                →      (6, -(-4)) = (6, 4)

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Answer:

3

Step-by-step explanation:

2x2=4

4x2=8

8x2=16

The maximum number of students allowed to attend the field trip is 59.

To determine the maximum number of students allowed to attend the field trip, given the ticket cost and money raised, we'll need to follow these steps:
1. Subtract the adult ticket cost from the total money raised.
2. Divide the remaining amount by the cost of a student ticket.
Subtract the adult ticket cost from the total money raised.
$580 (money raised) - $75 (adult ticket cost) = $505 (remaining amount)
Divide the remaining amount by the cost of a student ticket.
$505 (remaining amount) / $8.50 (student ticket cost) = 59.41
Since we can't have a fraction of a student, we round down to the nearest whole number.
The maximum number of students allowed to attend the field trip is 59.

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Answer:

Step-by-step explanation:

Find area of all shapes and add

A(biggest rectangle) = Lw =  6*3 =18   there are 2 of those A=36

A(long skinny rectangle = Lw = 2*6 =12  there are 2 of those A=24

A(smallest) = Lw = 2*3=6 there are 2 of those A=12

Add it all up

A=36+24+12 = 72

I checked it 5 times.  I keep getting 72

Note that the probability that he gives the lifejackets to just two of the three non-swimmers​ is 3/250 or 0.012

Let's define the following events...

A: Two of the three non-swimmers get lifejackets

B: Three lifejackets are given to two non-swimmers and one swimmer

We want to calculate P(A), the probability that two of the three non-swimmers get lifejackets. We can do this using the formula/...

P(A) = P(A|B) * P(B) + P(A|not B) * P(   not B)

P (B), the probability that the Captain gives the lifejackets to two non-swimmers and one swimmer....

P(B )  = (3/5 ) x (2/4) x (1/3) = 1/ 10

Note that  he number of ways to choose 2 non-swimmers from 3 is 3, and the number of ways to choose 1 swimmer from 2 is 2.

The total No.  of ways to choose 3 passengers from 5 is 10, hence

P(A | B) = (3 choose 2) x (2 choose 1) / (10 choose 3) = 6 /50

The No. of ways to choose 2 non-swimmers from 2 is 1, and the number of ways to choose 1 swimmer from 3 is 3. The total number of ways to choose 3   passengers from 5 is 10

P(A|not B) = (1 choose 2) x (3 choose 1) / (10 choose 3) = 0

Plugging in the values, we get....the followint

P(A) = (6/50) * (1/10) + (0) * (9/10) = 3/ 250

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Answer:

Step-by-step explanation:

We can use the Pythagorean theorem to solve this problem. Let's call the length of the ladder "L". The ladder, the wall of the house, and the ground form a right triangle. The distance between the ladder and the house is the base of the triangle, which is 15 feet. The height of the triangle is the distance from the ground to the spotlight, which is 20 feet. The length of the ladder is the hypotenuse of the triangle.

Using the Pythagorean theorem, we have:

L^2 = 15^2 + 20^2

L^2 = 225 + 400

L^2 = 625

L = sqrt(625)

L = 25

Therefore, a ladder of at least 25 feet is needed for the electrician to reach the place where the light is mounted.

When the pretax financial income is negative (indicating a loss), the taxable income will also be negative. This means that the company can use the loss to offset future profits and reduce its tax liability.

To calculate the taxable income (loss) for each year, we need to apply the corresponding tax rate to the pretax financial income (or loss) figures. Here's the breakdown:

2017:

Taxable income = Pretax financial income * Tax rate

Taxable income = $77,000 * 0.25

Taxable income = $19,250

2018:

Taxable income = Pretax financial income * Tax rate

Taxable income = ($38,000) * 0.20

Taxable income = ($7,600)

2019:

Taxable income = Pretax financial income * Tax rate

Taxable income = ($33,000) * 0.20

Taxable income = ($6,600)

2020:

Taxable income = Pretax financial income * Tax rate

Taxable income = $122,000 * 0.20

Taxable income = $24,400

2021:

Taxable income = Pretax financial income * Tax rate

Taxable income = $90,000 * 0.20

Taxable income = $18,000

Please note that when the pretax financial income is negative (indicating a loss), the taxable income will also be negative. This means that the company can use the loss to offset future profits and reduce its tax liability.

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The first derivative of g(a) is [tex]g'(x) = 18x^2 - 18x - 36.[/tex]

To find the first derivative of g(a), we need to use the power rule and the constant multiple rule.

First, we use the power rule to take the derivative of each term:

[tex]- The derivative of 6x^3 is 18x^2

- The derivative of -9x^2 is -18x

- The derivative of -36x is -36[/tex]

Next, we use the constant multiple rule to combine these derivatives:

g'(a) = 18x^2 - 18x - 36

Therefore, the first derivative of g(a) is [tex]g'(x) = 18x^2 - 18x - 36.[/tex]

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To find the height of the rocket after 2.5 seconds,

you'll need to plug in t = 2.5 into the given equation h(t) = 160t - 16t².

h(2.5) = 160(2.5) - 16(2.5)²
h(2.5) = 400 - 100
h(2.5) = 300 meters

The height of the rocket after 2.5 seconds is 300 meters.

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The p-value for a t-score of -0.92 is approximately 0.18 and since it is greater than the significant level, the null hypothesis is rejected.

The first step in testing this hypothesis is to calculate the test statistic, which in this case is a t-score. The formula for the t-score is (y - mu) / (s / sqrt(n)), where y is the sample mean, mu is the hypothesized population mean, s is the sample standard deviation, and n is the sample size.

In this case, the sample mean is 25.9, the hypothesized population mean is 26, the sample standard deviation is 7.6, and the sample size is 50. Plugging these values into the formula, we get a t-score of -0.92.

Next, we need to find the p-value associated with this t-score. We can use a t-table or a calculator to do this. Using a t-table with 49 degrees of freedom (since we have a sample size of 50 and one parameter estimated from the sample), we find that the p-value for a t-score of -0.92 is approximately 0.18.

Since the p-value is greater than the significance level of 0.05, we fail to reject the null hypothesis. In other words, we do not have substantial evidence to conclude that the population mean is less than 26. However, it is important to note that the sample mean is slightly below the hypothesized population mean, and the p-value is relatively close to the significance level. Therefore, it may be worthwhile to conduct additional studies with larger sample sizes or different populations to further investigate this question.

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If regular hexagon, regular octagon are inscribed in circle, perimeter of hexagon is greater than perimeter of octagon, each perimeter is greater than circumference of circle. Therefore, statement B is true.

The perimeter of a polygon is the sum of the lengths of all its sides. In a regular polygon, all sides have equal length, so the perimeter is simply the number of sides multiplied by the length of one side. The circumference of a circle is the distance around its outer edge.

Since both polygons are inscribed in the same circle, they have the same circumcircle, which means that their perimeters are both less than the circumference of the circle.

To compare the perimeters of the two polygons, we need to know the number of sides of each polygon and the length of one side. A regular hexagon has six sides, and a regular octagon has eight sides. Since the circle is inscribed in both polygons, the sides of each polygon are tangents to the circle, forming right angles with the radii of the circle.

Thus, we can draw a right triangle with the radius of the circle as the hypotenuse, and the side of the hexagon (or octagon) as one leg. Using trigonometry, we can find the length of one side of the hexagon (or octagon) in terms of the radius of the circle.

After calculating the lengths of one side of each polygon, we can compare their perimeters. It turns out that the perimeter of the octagon is greater than the perimeter of the hexagon, since the octagon has more sides.

Therefore, statement B is true.

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Complete question is:

A regular hexagon and a regular octagon are both inscribed in the same circle. which of these statements is true?

A) the perimeter of the hexagon is less than the perimeter of the octagon, and each perimeter is less than the circumference of the circle.

B) the perimeter of the hexagon is greater than the perimeter of the octagon, and each perimeter is greater than the circumference of the circle.

The number of different combo meals possible in a restaurant that is serving a special dinner combo meal is 90.

We are given that the customers can choose from 5 drinks, 6 main dishes, and 3 desserts. We have to find that how many different combos are possible. It means that we have to do an arrangement for such a situation. Arrangement of things means to group them in a systematic order, in all the possible ways.

We know that the number of possible ways to arrange is n! where n is the number of objects. As we know that the dinner includes 5 drinks, 6 main types of dishes, and 3 types of desserts. The number of different combo meals possible can be found by simply multiplying all the meals. Thus,

n = 5 * 6 * 3

n = 90

Therefore, the number of different combo meals possible in a restaurant that is serving a special dinner combo meal that includes a drink, a main dish, and a dessert is 90.

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As a result, the angle formed by Luis's backpack and the ground is roughly 53 degrees (rounded to the nearest degree).

A function in mathematics is a relationship between a set of potential outputs and a number of potential inputs, with the feature that each input is associated to only one possible output. It is a principle or set of guidelines that allots a different output value towards each input value. Equations, graphs, and tables are frequently used to depict functions in order to explain how the output changes as the input does. They are employed to express relationships between various quantities, such as the length of time it takes for an automobile to drive a certain distance or the height of an object in relation to its weight. The concept of a function is crucial to many departments of science and math, and it is widely applied in areas like engineering,

given

We can use trigonometry to determine the angle that Luis's backpack creates with the ground.

Consider the backpack's handle to represent the hypotenuse of a right triangle, with the vertical leg being Luis's hand's distance from the ground (3 feet) and the horizontal leg being the backpack's height (3.5 feet).

The angle can be determined using the inverse tangent function (tan-1):

53.13 degrees at tan-1(3.5/3)

As a result, the angle formed by Luis's backpack and the ground is roughly 53 degrees (rounded to the nearest degree).

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Answer:

252 mi

Step-by-step explanation:

volume= L x W x H

9x 7 x 4 = 252 mi

The total number of problems he solved is 36 if after completion of 27 problems he has completed 75% of the assignments.

Percentage of completion = 75 % of the work

The percentage can be converted to decimal by dividing the percentage by 100.

Thus, the part that has been completed = 75% = 0.75 of the work

The number of questions done = 27

Let the total number of questions be x

Thus, 75% of x is given as 27

75% of x = 27

0.75 * x = 27

0.75x = 27

x = 27/0.75

x = 36

Thus, the total number of questions is 36.

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