As each person entered a theatre, the manager counted how many of the 105 people had popcorn and how many had a drink. She found that out of 84 people that had popcorn, only 10 did not have a drink. Six people walked in without popcorn or a drink. Construct a two-way table summarizing the results. ​

To use Lagrange multipliers, we need to set up the Lagrangian function, Therefore, the maximum value of f(x,y,2) = xyz subject to the constraint x+y+z-6=0 is f(2,2,2) = 8.

L(x,y,z,λ) = xyz + λ(x+y+z-6)
Then we need to find the critical points of L by setting its partial derivatives equal to zero:
∂L/∂x = yz + λ = 0
∂L/∂y = xz + λ = 0
∂L/∂z = xy + λ = 0
∂L/∂λ = x + y + z - 6 = 0
Solving this system of equations, we get:
x = y = z = 2
λ = -4
This critical point satisfies the constraint, since 2+2+2-6 = 0. To check whether it is a maximum or minimum, we need to use the second partial derivative test:
∂²L/∂x² = 0, ∂²L/∂y² = 0, ∂²L/∂z² = 0
∂²L/∂x∂y = z, ∂²L/∂x∂z = y, ∂²L/∂y∂z = x
The Hessian matrix is:
| 0  z  y |
| z  0  x |
| y  x  0 |
At the critical point (2,2,2), the Hessian matrix is:
| 0  2  2 |
| 2  0  2 |
| 2  2  0 |
The eigenvalues of this matrix are -4, -4, and 8. Since the eigenvalues are not all positive or all negative, we cannot conclude whether the critical point is a maximum or minimum.
Therefore, the maximum value of f(x,y,2) = xyz subject to the constraint x+y+z-6=0 is f(2,2,2) = 8.

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Step-by-step explanation:

1. To find the mean number of rebounds Jamie had per game, we need to add up all the rebounds she had and then divide by the total number of games played:

Mean = (8 + 12 + 8 + 11 + 15 + 6 + 9 + 8 + 10 + 13) / 10

Mean = 100 / 10

Mean = 10

Therefore, the mean number of rebounds Jamie had per game was 10.

2. To find the median number of rebounds Jamie had for her final 10 games, we need to first arrange the data in order from least to greatest:

6, 8, 8, 8, 9, 10, 11, 12, 13, 15

Since there are an even number of data points, the median will be the average of the two middle values, which are 9 and 10:

Median = (9 + 10) / 2

Median = 9.5

Therefore, the median number of rebounds Jamie had for her final 10 games was 9.5.

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A number cube has sides numbered 1 through 6. the probability of rolling a 2 is 1/6. The probability of not rolling a 2 on a number cube with sides numbered 1 through 6 is 5/6 (in simplest form). The correct answer is 5/6.

The probability of not rolling a 2 on a number cube with sides numbered 1 through 6 can be calculated as follows:
Determine the total number of outcomes, which is 6 (1, 2, 3, 4, 5, and 6).
Determine the number of favorable outcomes, which is 5 (1, 3, 4, 5, and 6), since you're looking for the probability of not rolling a 2.
Calculate the probability by dividing the number of favorable outcomes by the total number of outcomes.
The probability of not rolling a 2 on a number cube with sides numbered 1 through 6 is 5/6 (in simplest form).

So, the correct answer is 5/6.

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The equation of the line that best fits the given data is y = (5/6)x + 1/3

The equation of the line that best fits the given data can be found by using the slope-intercept form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept. To find the slope, we can use the formula:

m = (y2 - y1) / (x2 - x1)

Using the points (1, 1.5) and (4, 4), we get:

m = (4 - 1.5) / (4 - 1) = 2.5 / 3 = 5/6

Now we can use one of the given points to find the y-intercept. Let's use the point (2, 2):

y = mx + b
2 = (5/6)(2) + b
2 = 5/3 + b
b = 2 - 5/3
b = 1/3

Therefore, the equation of the line that best fits the given data is:

y = (5/6)x + 1/3

The best answer is C. Y = x + 1.

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The given problem involves evaluating the surface integral of the vector field F(X, y, 2) over the top half of a sphere x^2 + y^2 + z^2 = 1, oriented upwards, using the Divergence Theorem.

The Divergence Theorem states that the flux of a vector field F through a closed surface S is equal to the triple integral of the divergence of F over the region enclosed by S.

In this problem, the given vector field F(X, y, z) is F(X, y, 2) = 3z^2xi + (y^3 + tan(2)J + (3x^2z + 1y^2)k.

The surface S is the top half of the sphere x^2 + y^2 + z^2 = 1, oriented upwards. This means that z is positive on S, and the normal vector points in the positive z-direction.

To use the Divergence Theorem, we need to find the divergence of F. The divergence of F is given by div(F) = ∂Fx/∂x + ∂Fy/∂y + ∂Fz/∂z, where ∂Fx/∂x, ∂Fy/∂y, and ∂Fz/∂z are the partial derivatives of F with respect to x, y, and z, respectively.

Taking the partial derivatives of F with respect to x, y, and z, we get:

∂Fx/∂x = 6xz

∂Fy/∂y = 3y^2 + 2y

∂Fz/∂z = 0

So, the divergence of F is: div(F) = 6xz + 3y^2 + 2y

Now, we can apply the Divergence Theorem, which states that the surface integral of F over S is equal to the triple integral of the divergence of F over the region enclosed by S.

The triple integral of the divergence of F over the region enclosed by S can be written as: ∫∫∫ div(F) dV, where dV is the volume element.

Since the given problem asks for the surface integral of F over S, we only need to consider the part of the triple integral that involves the surface S.

The surface integral of F over S can be written as: ∫∫ F · dS, where dS is the outward-pointing normal vector on S and · represents the dot product.

The dot product F · dS can be expressed as: Fx * dSx + Fy * dSy + Fz * dSz, where Fx, Fy, and Fz are the components of F, and dSx, dSy, and dSz are the components of the outward-pointing normal vector on S.

Since the normal vector on S points in the positive z-direction, we have dSx = 0, dSy = 0, and dSz = 1.

Substituting the components of F and the components of dS into the expression for the dot product, we get: Fx * dSx + Fy * dSy + Fz * dSz = (3z^2x)(0) + (y^3 + tan(2)J + (3x^2z +

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Answer: 8x +29

Step-by-step explanation:

5+8(3+x)

5+8(x+3)

__________

5 + 8(x+3)

5+ 8x +25

_________

5+8x+ 24

29+8x

____

8x+29

The ratios of successive numbers in the Fibonacci sequence eventually get closer to a. 1.61

In mathematics, the Fibonacci sequence is a sequence in which each number is the sum of the two preceding ones. Numbers that are part of the Fibonacci sequence are known as Fibonacci numbers, commonly denoted Fn . The sequence commonly starts from 0 and 1, although some authors start the sequence from 1 and 1 or sometimes (as did Fibonacci) from 1 and 2.

The Fibonacci sequence is a series of numbers where each number is the sum of the two preceding ones, usually starting with 0 and 1 (0, 1, 1, 2, 3, 5, 8, 13, ...). When you take the ratio of successive numbers in the sequence (e.g., 5/3 or 8/5), it converges to approximately 1.618, also known as the Golden Ratio or Phi.

The closest option in your list is 1.61, which is option (a).

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It's essential to consider the depreciation rate when deciding whether to invest in a new car or keep your current one.

Cars typically depreciate, and the amount can vary depending on factors such as make, model, and age.

For this example, let's assume you're interested in purchasing a 3-year-old used Honda Accord. I found an advertisement for this vehicle online, but since I cannot attach a copy here, please search for a similar advertisement and include it with your work.

It's common for new cars to depreciate by approximately 20-30% in the first year, and around 10-15% each subsequent year. So, a 3-year-old car may have already experienced around 40-60% of its total depreciation.

After researching, the used 3-year-old Honda Accord is priced at $18,000. If you compare it to the price of a new Honda Accord, which starts around $25,000, you can see that there has been a considerable depreciation in value.

In conclusion, cars do depreciate, and the rate can vary depending on the vehicle's age and other factors. In this case, a 3-year-old Honda Accord has already experienced significant depreciation, making it a more affordable option compared to buying a brand new car.

Considering depreciation can help you make an informed decision when deciding between a new or used car.

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The probability of the length of a randomly selected cane being between 205cm and 210cm is approximately 0.645 (rounded to 3 decimal places).

To find the probability of the length of a randomly selected cane being between 205cm and 210cm, we need to calculate the z-scores for these values and then use the standard normal distribution.

The z-score formula is given by:

z = (x - μ) / σ,

where x is the observed value, μ is the mean, and σ is the standard deviation.

For 205cm:

z1 = (205 - 208.5) / 2.5 = -1.4

For 210cm:

z2 = (210 - 208.5) / 2.5 = 0.6

Now, we can use a standard normal distribution table or a calculator to find the probability between these two z-scores.

Using a standard normal distribution table or a calculator, we find that the probability associated with z1 = -1.4 is approximately 0.0808, and the probability associated with z2 = 0.6 is approximately 0.7257.

To find the probability between these two z-scores, we subtract the probability corresponding to z1 from the probability corresponding to z2:

P(205cm < length < 210cm) ≈ P(z1 < z < z2) ≈ P(z < 0.6) - P(z < -1.4) ≈ 0.7257 - 0.0808 ≈ 0.6449.

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Both Zola and Amir are correct in writing the area of the rectangle. They have simply used different ways of expressing the same value.

Zola and Amir have both written the area of a rectangle using different algebraic expressions.

Zola wrote the area of the rectangle as `2a + 3a + 4a`, which can be simplified using the distributive property of multiplication:

2a + 3a + 4a = (2 + 3 + 4)a

Therefore, Zola's expression simplifies to `(2 + 3 + 4)a`, which is the same as Amir's expression.

Amir wrote the area of the rectangle as `(2 + 3 + 4) a`, which can also be simplified:

(2 + 3 + 4) a = 9a

Therefore, Amir's expression simplifies to `9a`, which is the same as the sum of the terms in Zola's expression.

Therefore, both Zola and Amir are correct in writing the area of the rectangle. They have simply used different ways of expressing the same value.

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The height of the cylinder is 3.6 inches.

We know that the volume of a cylinder of radius R and height H is:

V = pi*R²*H

where pi = 3.14

We know that the radius is R = 5in and the volume is 284.7 inches cubed, replacing that in the formula above we will get:

284.7 in³= 3.14*(5 in)²*H

Solving that for H we will get:

H= (284.7 in³)/ 3.14*(5 in)²

H = 3.6 inches.

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In this scenario, the sampling units are four households, the variable of interest is the average food cost, and auxiliary information associated with the units is the number of people in each household and total food cost.

Sampling Units: The sampling units are the four households selected from the 25,000 households in the population.

They are as follows:

1. Household with 4 people that spent $150 on food

2. Household with 2 people that spent $100 on food

3. Household with 4 people that spent $200 on food

4. Household with 3 people that spent $140 on food

Variable of Interest: The variable of interest is the average cost on food per household for a week.

Auxiliary Information: The auxiliary information associated with the units includes the number of people in each household and the total amount spent on food for that week.

To estimate the average cost on food per household for a week, follow these steps:

1. Calculate the total cost on food for all four households: $150 + $100 + $200 + $140 = $590

2. Divide the total cost by the number of households: $590 / 4 = $147.50

So, the estimated average cost on food per household for a week is $147.50.

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The value of x that maximizes the concentration is 119 weeks, and the maximum concentration is approximately 7.19 ml.

Based on the provided equation, the concentration C(x) is a quadratic function of x with a negative coefficient for the quadratic term, which means that it has a maximum point.

(a) To find the value of x that maximizes the concentration, we can take the derivative of the concentration function with respect to x, set it equal to zero, and solve for x. The derivative of C(x) is:

C'(x) = 56x + 7

Setting C'(x) equal to zero, we get:

56x + 7 = 0

Solving for x, we get:

x = -7/56 = -0.125

However, x represents the number of weeks from now, which cannot be negative. Therefore, the maximum concentration occurs at the endpoint of the interval we are considering, which is x = 119 weeks.

To find the maximum concentration, we can substitute x = 119 into the concentration function:

C(119) = 28²(2119)+7 - 119²-2119+2 ≈ 7.19 ml

So, the value of x that maximizes the concentration is 119 weeks, and the maximum concentration is approximately 7.19 ml.

(b) For very large values of x, the quadratic term (-x²) dominates the concentration function, and the concentration appears to decrease without bound.

This is because the negative quadratic term becomes much larger than the linear term (2x) and the constant term (2), causing the concentration to become more and more negative as x increases.

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(a) When the input is (-7,-3), the rule (x,y) → (x-3,y+8) moves the point 3 units to the left and 8 units up.

So we can apply this rule to the input (-7,-3) as follows:

(-7,-3) → (-7-3,-3+8)

(-7,-3) → (-10,5)

Therefore, the output is (-10,5).

(b) When the input is (10,-5), the rule (x,y) → (x-3,y+8) moves the point 3 units to the left and 8 units up. So we can apply this rule to the input (10,-5) as follows:

(10,-5) → (10-3,-5+8)

(10,-5) → (7,3)

Therefore, the output is (7,3).

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The possible values of x that satisfy the equation |x+5| = c are x = c - 5 and x = -c - 5.

what is algebra?

Algebra is a branch of mathematics that deals with mathematical operations and symbols used to represent numbers and quantities in equations and formulas.

The value of x in the solution to this system of equations 5x - 4y = 27 and y = 2x + 3 is -13.

To find the value of x in this system of equations, we can use substitution method to find the its solution. Start by isolating x in one of the equations and then substituting that value into the other equation.

Let's start by isolating x in the second equation:

y = 2x + 3

Subtracting 3 from both sides:

y - 3 = 2x

Dividing both sides by 2:

(1/2)y - (3/2) = x

Now we can substitute this expression for x into the first equation:

5x - 4y = 27

5((1/2)y - (3/2)) - 4y = 27

Simplifying:

(5/2)y - 15/2 - 4y = 27

Combining like terms:

-(3/2)y = 69/2

Dividing by -(3/2):

y = -23

Now we can substitute this value of y back into the expression we found for x:

x = (1/2)y - (3/2)

x = (1/2)(-23) - (3/2)

x = -13

Therefore, the solution to this system of equations is x = -13, y = -23.

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To find the surface area of a spherical object, we need to know the radius of the sphere. However, in this case, only the circumference of the pincushion is given, which is not enough information to directly determine the radius.

The formula relating the circumference (c) and the radius (r) of a sphere is:

c = 2πr

To find the surface area (A) of the sphere, we can use the formula:

A = 4πr^2

Since we don't have the radius, we need to solve the circumference formula for the radius first:

c = 2πr

Divide both sides of the equation by 2π:

r = c / (2π)

Now we can substitute the value of c = 18 cm into the equation to find the radius:

r = 18 cm / (2π)

r ≈ 2.868 cm (approximately)

Now that we have the radius, we can calculate the surface area using the formula:

A = 4πr^2

A = 4π(2.868 cm)^2

A ≈ 103.05 cm² (approximately)

Therefore, the surface area of the pincushion is approximately 103.05 square centimeters.

To use the Limit Comparison Test, we need to find a series whose behavior is well-known and similar to the given series. Let's consider the series aₙ = n². We have:

limₙ→∞ (aₙ / (n² - 5naₙ - 6)) = limₙ→∞ (n² / n²) = 1

Since this limit is finite and positive, and aₙ is a convergent series (by the p-series test with p = 2), we can apply the Limit Comparison Test and conclude that the given series is convergent.
To determine if the series ∑(n² - 5n) from n=1 to infinity is convergent or divergent using the Limit Comparison Test, we need to find a comparable series and then calculate the limit of the ratio between the two series as n approaches infinity.

Let's compare the given series to a simpler series ∑n² (n=1 to infinity). Now, we'll find the limit of the ratio:

Limit (n→∞) [(n² - 5n) / n²]

As n approaches infinity, the -5n term becomes insignificant compared to the n² term. So, the limit becomes:

Limit (n→∞) [n² / n²] = 1

Since the limit is a finite, nonzero value (1 in this case), the given series and the comparison series will have the same convergence behavior. We know that the series ∑n² (n=1 to infinity) is a divergent series, as it is a p-series with p=2 (less than or equal to 1). Therefore, the given series ∑(n² - 5n) from n=1 to infinity is also divergent.

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Answer:

4

Step-by-step explanation:

14 student tickets times $12 = 168

168 + $17 = 185

210-185=25

6*4=$24

so they can buy 4 pies with 1 dollar left over

sorry if I am wrong

Answer:

Graph 2

Step-by-step explanation:

There is 1 from 1-5

There are 4 from 6-10

There are 5 from 11-15

There is 1 from 16-20

There are 4 from 21-25

Answer: 27 inches sq.

Step-by-step explanation:

3*3=9

2*9=18

18+9=27

The equation is y = x + 3 which shows the relationship of the two activities.

The equation to show the relationship between the number of hours Julie practices her violin each week (y) and the number of hours she studies (x) is:

y = x + 3

This means that the number of hours Julie practices her violin each week is equal to the number of hours she studies, plus three additional hours.

A linear equation is an equation that represents a straight line on a graph. It can be written in the form:

y = mx + b

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The minimum dimensions of the box will be; 7 cm × 6 cm × 6 cm

Since the dimension is described as the measurement of something in physical space such as length, width, or height.

Given that the there will be maximum dimension when the height of the cylinder and the radius of the hemisphere are aligned together.

Maximum height = 4 cm + 3 cm = 7 cm

Maximum diameter = 2 × 3 cm = 6 cm

Therefore, we can see that the minimum dimensions of the box are :

7 cm × 6 cm × 6 cm.

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The function has a local maximum at x = -√(2/7) with value -3√14 and a local minimum at x = √(2/7) with value 3√14.

To find the local maximum and minimum of the function f(x) = 2x + 7x⁻¹, we need to find the critical points of the function and then use the second derivative test to determine if they are local maxima or minima.

First, we find the derivative of f(x):

f'(x) = 2 - 7x⁻²

Setting f'(x) = 0, we get:

2 - 7x⁻² = 0

Solving for x, we get:

x = ±√(2/7)

Next, we compute the second derivative of f(x):

f''(x) = 14x⁻³

At x = ±√(2/7), we have:

f''(±√(2/7)) = ±∞

Since f''(±√(2/7)) has opposite signs at the critical points, ±√(2/7), we conclude that f(x) has a local maximum at x = -√(2/7) and a local minimum at x = √(2/7).

To find the values of the local maximum and minimum, we plug them into the original function:

f(-√(2/7)) = 2(-√(2/7)) + 7/(-√(2/7)) = -3√14

f(√(2/7)) = 2(√(2/7)) + 7/(√(2/7)) = 3√14

Therefore, the function has a local maximum at x = -√(2/7) with value -3√14 and a local minimum at x = √(2/7) with value 3√14.

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In this case, it's important for the librarian to make sure that the sample of patrons who were randomly selected is representative of the larger population of patrons in the children's section, and that any assumptions made in the statistical inference process are valid.

To infer the ages of all patrons in the children's section of the library, the librarian should use statistical inference techniques such as estimation or hypothesis testing.

If the librarian wants to estimate the average age of all patrons in the children's section, they can use a point estimate or an interval estimate. A point estimate would involve calculating the sample mean age of the patrons who were randomly selected and using that as an estimate for the population means age. An interval estimate would involve calculating a confidence interval around the sample mean, which would give a range of likely values for the population means.

Alternatively, if the librarian wants to test a hypothesis about the ages of patrons in the children's section, they can use a hypothesis test. For example, they could test whether the average age of patrons in the children's section is significantly different from a certain value (such as the national average age of children), or whether there is a significant difference in age between male and female patrons.

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Maryam answered 26 questions correctly and got 4 questions wrong on the test with 30 questions.

To find how many questions Maryam got wrong, we need to first determine how many questions she got right. Since she scored 86.7%, we can multiply the total number of questions by the percentage to get the number of questions she answered correctly.

86.7% of 30 questions is (86.7/100) * 30 = 26.01 questions.

Since Maryam cannot have answered a fractional number of questions correctly, we round down to the nearest whole number. Thus, she answered 26 questions correctly.

To find out how many questions she got wrong, we can simply subtract the number of questions she got right from the total number of questions. Therefore, Maryam got 30 - 26 = 4 questions wrong.

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The volume of blood flow through the artery in one minute is 600 milliliters.

We can integrate the given equation to get the volume of blood flow.

a) Integrating both sides of the equation with respect to time from 0 to 10 seconds, we get:

∫dᏙ = ∫(10 - 2cos(120t)) dt

ΔᏙ = [10t - (1/60)sin(120t)] from 0 to 10

ΔᏙ = [(10 x 10) - (1/60)sin(1200)] - [(10 x 0) - (1/60)sin(0)]

ΔᏙ ≈ 100 - 0

So, the volume of blood flow through the artery in 10 seconds is 100 milliliters.

b) To find the volume of blood flow through the artery in one minute, we need to integrate the given equation from 0 to 60 seconds:

∫dᏙ = ∫(10 - 2cos(120t)) dt

ΔᏙ = [10t - (1/60)sin(120t)] from 0 to 60

ΔᏙ = [(10 x 60) - (1/60)sin(7200)] - [(10 x 0) - (1/60)sin(0)]

ΔᏙ ≈ 600 - 0

So, the volume of blood flow through the artery in one minute is 600 milliliters.

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Answer:

There are 16 unit squares on the geoboard. Each unit square represents 5 meters, so the total area is 16*5 = 80 square meters.

Here is a diagram of the geoboard with the unit squares labeled:

[Image of a geoboard with 16 unit squares labeled]

I hope this helps! Let me know if you have any other questions.