A normal population has a mean of $74 and standard deviation of $15. You select random samples of nine. a. Apply the central limit theorem to describe the sampling distribution of the sample mean with n= 9. With the small sample size, what condition is necessary to apply the central limit theorem? Applying the central limit theorem requires the population distribution to be normal.

The position, s(t), of the particle at time t if initially s(0) = 3 is (2) s(t) = 10 - 7 cos(t) - 6 sin(t).

To find the position, s(t), of the particle at time t, we need to integrate the velocity function, v(t), with respect to time:

s(t) = ∫ v(t) dt

Since the velocity function is v(t) = 7 sin(t) - 6 cos(t), we have:

s(t) = ∫ (7 sin(t) - 6 cos(t)) dt

Integrating each term separately, we get:

s(t) = -7 cos(t) - 6 sin(t) + C

where C is the constant of integration.

To find the value of C, we use the initial condition s(0) = 3:

s(0) = -7 cos(0) - 6 sin(0) + C = -7 + C = 3

C = 10, and the position function is:

s(t) = -7 cos(t) - 6 sin(t) + 10

Rewriting this equation in the form of answer choices, we get:

s(t) = 10 - 7 cos(t) - 6 sin(t)

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The position, s(t), of the particle at time t, given the initial condition s(0) = 3 and the velocity v(t) = 7sin(t) - 6cos(t), is s(t) = 9 - 7sin(t) - 6cos(t).

To find the position, we integrate the velocity function with respect to time. Integrating the velocity function v(t) = 7sin(t) - 6cos(t) gives us the position function s(t).

The indefinite integral of sin(t) is -cos(t), and the indefinite integral of cos(t) is sin(t). When integrating, we also take into account the initial condition s(0) = 3 to determine the constant term.

Integrating the velocity function, we get:

s(t) = -7cos(t) - 6sin(t) + C

To determine the constant term C, we use the initial condition s(0) = 3:

3 = -7cos(0) - 6sin(0) + C

3 = -7(1) - 6(0) + C

3 = -7 + C

C = 10

Substituting the value of C back into the position function, we obtain:

s(t) = 9 - 7sin(t) - 6cos(t)

Therefore, the position of the particle at time t, with the initial condition s(0) = 3, is given by s(t) = 9 - 7sin(t) - 6cos(t).

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The cross section would be a circular sphere and a cylinder

A cylinder is defined as a shape that has there dimensional surface that is made up of two circles and a curved area.

The two flat circular bases are congruent to each other and It does not have any vertex.

A circular sphere is defined as a round object found in a space which is equally a three dimensional object.

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The inequality to describe the relationship between the number of floors (f) and the maximum number of offices (o) is:

f * o ≤ 195.

Let's assume that the number of floors in the building is represented by the variable "f" and the maximum number of offices on each floor is represented by the variable "o". To write an inequality describing the relationship between the number of floors and the maximum number of offices, we can use the following inequality:

f * o ≤ 195

In this inequality, the product of "f" and "o" represents the total number of offices in the building. We multiply the number of floors by the maximum number of offices per floor to obtain the total number of offices. The inequality states that the total number of offices must be less than or equal to 195.

This inequality ensures that the building does not exceed the maximum limit of 195 offices. It allows for flexibility in the distribution of offices across the floors, as long as the total number of offices does not exceed the given limit.

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Answer:

There are 9 Nickels & 12 Dimes.

Step-by-step explanation:

Let N = the number of nickels

Let D = the number of dimes

Each nickel is worth 0.05

Each dime is worth 0.10

So let's write some equations.

0.05N + 0.10D = 1.65

N + D = 21

We can solve by substitution.

Rewrite that 2nd equation.

N = 21-D

Now substitute that ^^^ equation into the first equation.

0.05(21-D) + 0.10D = 1.65

1.05-0.05D + 0.10D = 1.65

Combine like terms.

0.05D = 0.6

Divide both sides by 0.05

D = 12

There are 12 Dimes.

N + D = 21

N + 12 = 21

N = 21-12

N = 9

There are 9 Nickels & 12 Dimes.

Check that our answer is correct.

9(.05) + 12(.10) = 0.45 + 1.20 = 1.65

a. The employee will need to save $964,000 for retirement to have enough to live off 80% of their current annual gross wage and withdraw 4% the first year.

b. The account balance after 40 years would be approximately $1,173,919.74.

c. The difference between the employee's goal and the actual retirement account balance is -$209,919.74. The employee will not meet their retirement goal with the current contribution amount and interest rate.

a. Target Annual Income = 80% of Current Annual Gross Wage

= 80% of $48,200

= $48,200 * 0.8

= $38,560

Total Retirement Savings = Target Annual Income / Withdrawal Rate

= $38,560 / 0.04

= $964,000

Therefore, the employee will need to save $964,000 for retirement to have enough to live off 80% of their current annual gross wage and withdraw 4% the first year.

b. Account Balance = Monthly Contribution * (((1 + Monthly Interest Rate)^(Number of Months) - 1) / Monthly Interest Rate)

Convert the annual interest rate to a monthly interest rate:

Monthly Interest Rate = (1 + Annual Interest Rate)^(1/12) - 1

= (1 + 0.055)^(1/12) - 1

= 0.004433

Number of Months = Number of Years * 12

= 40 * 12

= 480

Calculate the account balance:

Account Balance = $400 * (((1 + 0.004433)^480 - 1) / 0.004433)

Using a calculator, the account balance after 40 years would be approximately $1,173,919.74 (rounded to the nearest cent).

c. The difference between the employee's retirement goal and the actual retirement account balance can be calculated by subtracting the account balance from the target amount:

Difference = Target Retirement Savings - Account Balance

= $964,000 - $1,173,919.74

= -$209,919.74

The result is negative, indicating that the actual retirement account balance falls short of the employee's goal by approximately $209,919.74.

Based on these calculations, the employee will not meet their retirement goal with the current contribution amount and interest rate.

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The odds that are in favour of guessing the right answer would be = 1:5. That is option A.

The given multiple choice questions has only 5 possible answers.

This means that when both the correct and wrong answers are added together, the total should be = 5.

That is;

4:1 = 4+1 = 5

1:4 = 1+4 = 5

3:2 = 3+2 = 5

Therefore, 1:5 = 1+5 = 6 which can't be a possible answer as it's more than the total of the multiple choice questions.

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The independent variable in this situation where fuel efficiency of a car decreases as tire pressure decreases is the Tire pressure.

Option B is correct.

The independent variable in a research study or experiment is described as  what the researcher is changing in the study or experiment

In the scenario above,  the tire pressure is being intentionally varied to observe its effect on the fuel efficiency of the car.

Researchers can study the cause-and-effect relationship between variables and gain a better understanding of how different factors influence the outcome of interest, in this case, the fuel efficiency of the car through the identification and manipulation the independent variable.

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The Lotka-Volterra predator-prey model is a set of differential equations used to describe the interactions between two species, a predator and its prey, in a given ecosystem. The model assumes that the population sizes of both species are influenced by factors such as predation, reproduction, and carrying capacity.

a) In the Lotka-Volterra predator-prey model, the terms have the following biological meanings:
- R(t) represents the number of rabbits (prey) at time t.
- F(t) represents the number of foxes (predators) at time t.
- a is the rabbit's reproduction rate.
- b is the predation rate at which rabbits are consumed by foxes.
- c is the natural death rate of foxes.
- d is the rate at which foxes consume rabbits and convert them into new foxes.

b) To make the equations dimensionless, we introduce dimensionless variables x and y, and a constant µ:

x = R/a, y = F/c, µ = ac/bd

The equations become:

x' = x(1 - y)
y' = µy(x - 1)

c) A conserved quantity in terms of the dimensionless variables can be found using the following function:

H(x, y) = - ln(x) + y - ln(y) + µx

The conserved quantity is dH/dt = 0.

d) The model predicts cycles in the populations of both species for almost all initial conditions, as indicated by the oscillatory behavior of the solutions in the phase plane. The trajectories in the phase plane are closed curves around a fixed point (x*, y*), showing that both species' populations will experience periodic fluctuations.

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Rounded to the nearest whole number, the container is approximately 52% full.

To find the percentage that the shipping container is full, we need to compare the volume of the shipped goods to the total volume of the container.

Given dimensions:

Length = 35 ft

Width = 8 ft

Height = 9 ft 9 in

We need to convert the height to feet by dividing the inches by 12:

Height = 9 ft + (9/12) ft = 9.75 ft

Total volume of the container:

Volume = Length × Width × Height

Volume = 35 ft × 8 ft × 9.75 ft

Volume = 2730 ft³

Volume of the shipped goods:

Given as 1420 ft³

To find the percentage filled, we divide the volume of the shipped goods by the total volume of the container and multiply by 100:

Percentage filled = (Volume of shipped goods / Total volume of container) × 100

Percentage filled = (1420 ft³ / 2730 ft³) × 100

Percentage filled ≈ 52.0%

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Option B, 0.005, represents the least risky stock based on the coefficient of variation.

In terms of the coefficients of variation provided, the value of 0.005 (Option B) represents the least risky stock. The coefficient of variation is a statistical measure used to assess the relative risk of an investment by comparing the standard deviation to the mean. A lower coefficient of variation indicates less variability and, therefore, less risk. Option B's coefficient of variation of 0.005 suggests a very small standard deviation in relation to the mean, implying a stable and predictable stock performance.

The coefficient of variation provides valuable insights into the risk associated with different investment options. By comparing the standard deviation to the mean, it allows investors to gauge the level of variability in returns. In the given options, the coefficient of variation value of 0.005 (Option B) suggests the least risky stock.

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To find the length of the diagonal of square C, we can use the Pythagorean theorem which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Since square C has equal sides, we only need to find the length of one side and then multiply it by the square root of 2 to get the length of the diagonal.

Using the coordinates given, we can find the length of one side by subtracting the x-coordinate of one vertex from the x-coordinate of the other vertex (11 - 1 = 10). We then multiply this by the conversion factor given in the problem (1 square unit = 25 feet) to get the length in feet (10 * 25 = 250). Finally, we multiply this by the square root of 2 to get the length of the diagonal (250 * sqrt(2) ≈ 353.55 feet).

Therefore, if square C has an area of 2500 square units and each unit is equal to 25 feet, then a square with a diagonal length of approximately 353.55 feet would be an accurate representation of square C.

The total distance traveled by David as obtained from the line plot is 158 miles.

The total distance traveled by David is obtained from the line plot showing the distance traveled by David each day for two weeks.

From the line plot:

The total distance = 10 × 2 + 21/2 × 1 + 11 × 3 + 23/2 × 4 + 12 × 3 + 25/2 × 1

The total distance = 20 + 21/2 + 33 + 46 + 36 + 25/2

The total distance = 135 + 46 / 2

The total distance =  135 + 23

The total distance = 158 miles.

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The exact length of the curve described by the parametric equations x = 8 + 3t², y = 3 + 2t³, for 0 ≤ t ≤ 2, is 2√5 - 2.

To find the exact length of the curve described by the parametric equations, we can use the arc length formula for parametric curves:

L = ∫[a,b] √[(dx/dt)² + (dy/dt)²] dt

Given the parametric equations x = 8 + 3t² and y = 3 + 2t³, we need to find dx/dt and dy/dt and then evaluate the integral over the given range 0 ≤ t ≤ 2.

First, let's find dx/dt:

dx/dt = d/dt (8 + 3t²)

       = 6t

Next, let's find dy/dt:

dy/dt = d/dt (3 + 2t³)

       = 6t²

Now, let's substitute these derivatives into the arc length formula and evaluate the integral:

L = ∫[0,2] √[(6t)² + (6t²)²] dt

  = ∫[0,2] √(36t² + 36t⁴) dt

  = ∫[0,2] √(36t²(1 + t²)) dt

  = ∫[0,2] 6t√(1 + t²) dt

To evaluate this integral, we can use a substitution. Let u = 1 + t², then du = 2t dt. Substituting these values, we get:

L = ∫[0,2] 6t√(1 + t²) dt

  = ∫[1,5] 3√u du

Integrating with respect to u:

L = [2√u] | [1,5]

  = 2√5 - 2√1

  = 2√5 - 2

Therefore, the exact length of the curve described by the parametric equations x = 8 + 3t², y = 3 + 2t³, for 0 ≤ t ≤ 2, is 2√5 - 2.

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A cross section parallel to the base is a rectangle measuring 15 inches by 8 inches. A cross section perpendicular to the base through the midpoints of the 8-inch sides is a rectangle measuring 6 inches by 15 inches. The correct options are 1, 3, and 4.

A cross section parallel to the base is a rectangle measuring 15 inches by 8 inches. This option is correct. If a cross section is taken parallel to the base of the right rectangular prism, it will result in a rectangle with the same dimensions as the base, which is 15 inches by 8 inches.

A cross section perpendicular to the base through the midpoints of the 8-inch sides is a rectangle measuring 6 inches by 15 inches. This option is correct. If a cross section is taken perpendicular to the base through the midpoints of the 8-inch sides, it will result in a rectangle with dimensions of 6 inches by 15 inches.

A cross section perpendicular to the base through the midpoints of the 8-inch sides is a rectangle measuring 4 inches by 15 inches. This option is incorrect. The dimensions mentioned here are not accurate for a cross section taken perpendicular to the base through the midpoints of the 8-inch sides.

Thus, the correct options are 1, 3, and 4.

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The sample space as Feechi makes three attempts at a basket in a basketball game is BBB, BMB, MBM, MMM).Option A

Here, we have,

to determine Feechi sample space:

The sample space represents all possible outcomes of Feechi's three attempts, where each attempt can either result in a basket (B) or a miss (M).

Option A lists the following four outcomes: BBB, BMB, MBM, and MMM.

Each outcome is a sequence of three letters, where B represents a basket and M represents a miss.

Feechi makes three attempts at a basket in a basketball game,

so, we get,

Therefore, the correct answer is (BBB, BMB, MBM, MMM).

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Answer:

I don't know all of them but:

Question 3 is x=17. Because angles on a straight line sum 180 degrees.

(8x-15)+(3x+8)=180

x= 17

Question 5 is 78 degrees. Because the angle at the center is double the angle at the circumference.

Answer:

1225

Step-by-step explanation

The sum of positive integers less than 50 can be found using the formula for the sum of an arithmetic sequence. An arithmetic sequence is a sequence of numbers in which each term is obtained by adding a fixed value (called the common difference) to the previous term.

In this case, the first term is 1, the common difference is 1, and we want to find the sum of the first 49 terms (since we are looking for the sum of positive integers less than 50).

The formula for the sum of an arithmetic sequence is:

S = n/2 * (a + l)

where S is the sum, n is the number of terms, a is the first term, and l is the last term.

We can find the last term by subtracting the common difference (1) from 50, since we want the last term to be less than 50. So:

l = 50 - 1 = 49

Using these values, we can plug into the formula:

S = 49/2 * (1 + 49)

= 24.5 * 50

= 1225

Therefore, the sum of positive integers less than 50 is 1+2+3+...+48+49 = 1225.

The focus of the parabola y = -132x² is located at (0, -1/528) and the equation of the directrix is y = 1/528.

In the general equation of a parabola, y = ax², the focus is located at (0, 1/(4a)), and the directrix is given by the equation y = -1/(4a). In this case, the coefficient of x² is -132, so we substitute this value into the formulas.

To find the coordinates of the focus, we set a = -132 in the focus formula: (0, 1/(4(-132))) = (0, -1/528). Therefore, the focus of the parabola is located at (0, -1/528).

For the equation of the directrix, we substitute a = -132 into the directrix formula: y = -1/(4(-132)) = -1/528. Hence, the equation of the directrix is y = 1/528.

conclusion, the focus of the parabola y = -132x² is located at (0, -1/528), and the equation of the directrix is y = 1/528.

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That the sampling distribution of the quantity (n-1)s^2 / sigma^2 is a chi-square distribution.

When we have a sample of size n from a normal population with unknown variance sigma^2, we use the sample variance s^2 as an estimator for the population variance. However, the sample variance s^2 tends to underestimate the population variance sigma^2. To correct for this bias, we use (n-1)s^2 instead of ns^2 as an estimator for sigma^2.

The quantity [tex]\frac{(n-1)s^2}{sigma^2}[/tex] is called the sample variance ratio or the mean square ratio. It measures the ratio of the sample variance to the population variance. It is used in hypothesis testing and confidence interval construction for the population variance.

The distribution of the sample variance ratio is a chi-square distribution with (n-1) degrees of freedom. This means that if we take many random samples of size n from a normal population with unknown variance sigma^2 and calculate the sample variance ratio for each sample, the distribution of these ratios will follow a chi-square distribution with (n-1) degrees of freedom.

Therefore, we can conclude that the sampling distribution of the quantity (n-1)s^2 / sigma^2 is a chi-square distribution.

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Thus,  the sampling distribution of (n-1)s^2 / sigma^2 is a chi-square distribution with n-1 degrees of freedom, assuming a normal population distribution.

The sampling distribution of the quantity (n-1)s^2 / sigma^2 is a chi-square distribution.

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32% of students in the sample do not use Fac/ebook.

The percentage of the students not using Faceb/ook is solved using the formula

= (number of students not using fa/cebook) / (total number of students) * 100

The number of students who do not use Fa/cebook is the sum of the values in the "Does not use Fac/ebook" column, which is 4 + 67 = 71.

The total number of students in the sample is the sum of the values in the "TOTAL" row, which is 219.

hence we have that

(71 / 219) × 100 ≈ 32.42

Rounding to the nearest percent  approximately 32% of students in the sample do not use Fa/cebook.

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By calculations, the width of the walkway should be 5 feet

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

Dimension = 12 feet by 8 feet

Area of the walkway = 396 feet

The missing diagram is attached

This means that

Area = (12 + 2x) * (8 + 2x)

Recall that

Area of the walkway = 396 feet

So, we have

(12 + 2x) * (8 + 2x) = 396

When solved using a graphing tool, we have

x = 5

Hence, the width of the walkway should be 5 feet

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Answer:

17.4

Step-by-step explanation:

The Hundredths place is above four so it has to be the next number up

Answer:17.40

Step-by-step explanation:

17.37 rounded to the nearest tenth is, 17.40, because when rounding, you see if the number is 5 or up ( that means you round it up.)

The Pythagorean Theorem for vectors states that for any two orthogonal vectors u and v, ||u+v||^2 = ||u||^2 + ||v||^2.


Step 1: To verify the Pythagorean Theorem, we first need to compute the dot product of u and v:

u.v = (-1)(-3) + (2)(0) + (3)(-1) = 3

Since u.v is not equal to zero, u and v are not orthogonal.

Step 2: Next, we need to compute the magnitudes of u and v:

||u||^2 = (-1)^2 + (2)^2 + (3)^2 = 14

||v||^2 = (-3)^2 + (0)^2 + (-1)^2 = 10

Step 3: Now, we can compute u + v and its magnitude:

u + v = (-1-3, 2+0, 3-1) = (-4, 2, 2)

||u + v||^2 = (-4)^2 + (2)^2 + (2)^2 = 24

Finally, we can apply the Pythagorean Theorem for vectors:

||u+v||^2 = ||u||^2 + ||v||^2

24 = 14 + 10

Therefore, the Pythagorean Theorem is verified for the vectors u and v.

The Pythagorean Theorem for vectors is a useful tool in determining whether two vectors are orthogonal or not. In this case, we found that u and v are not orthogonal, but the theorem was still applicable in verifying the relationship between their magnitudes and the magnitude of their sum.

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The average atomic mass of the element in the data table is given as follows:

28.1 amu.

The mean of a data-set is given by the sum of all observations in the data-set divided by the cardinality of the data-set, which represents the number of observations in the data-set.

For the weighed mean, we calculate the mean as the sum of each observation multiplied by it's weight.

Hence the average atomic mass of the element in the data table is given as follows:

0.922297 x 27.977 + 0.046832 x 28.976 + 0.030872 x 29.974 = 28.1 amu.

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The monthly payment if you put 10% down for a 30 year loan with a fixed rate of 7.947 is Option A

Using the formula for calculating the monthly mortgage payment:

P = PV / (1 - (1 + r)^(-n))

Where:

P = Monthly payment

PV = Loan amount (purchase price - down payment)

r = Monthly interest rate (annual interest rate divided by 12)

n = Total number of monthly payments (30 years = 30 * 12 = 360)

First, calculate the loan amount (PV):

PV = $555,750 - (10% of $555,750)

PV = $555,750 - $55,575

PV = $500,175

Next, calculate the monthly interest rate (r):

r = 7.947% / 12

r = 0.66225%

Finally, calculate the monthly payment (P):

P = $500,175 / (1 - (1 + 0.0066225)^(-360))

The monthly payment is approximately $3,740.19.

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Answer:

the answer is 8472373n3

The change of coordinates matrix that transforms the coordinates in the basis (1, 1) to the coordinates in the basis (1 - t, 2t) is:

[ 1 1 ]

[-1 2 ]

To find the change of coordinates matrix, we need to determine how the basis vectors in one coordinate system are represented in terms of the basis vectors in the other coordinate system. In this case, we want to find the matrix that transforms the coordinates in the basis (1, 1) to the coordinates in the basis (1 - t, 2t).

Let's denote the change of coordinates matrix as C, and the basis vectors of the original coordinate system (1, 1) as v1 and v2, and the basis vectors of the new coordinate system (1 - t, 2t) as u1 and u2.

To find C, we express the basis vectors u1 and u2 in terms of the original basis vectors v1 and v2. We can write this relationship as:

u1 = av1 + bv2

u2 = cv1 + dv2

To find the coefficients a, b, c, and d, we solve the system of equations formed by equating the components of u1 and u2 to their corresponding components in terms of v1 and v2.

From the given information, we have:

(1 - t) = a(1) + b(1)

2t = c(1) + d(1)

Simplifying these equations, we get:

1 - t = a + b

2t = c + d

Solving these equations, we find a = 1, b = -1, c = 1, and d = 2. Therefore, the change of coordinates matrix C is:

[ 1 1 ]

[-1 2 ]

This matrix C can be used to transform coordinates in the basis (1, 1) to the coordinates in the basis (1 - t, 2t). To transform a vector from one coordinate system to another, we multiply the vector by the change of coordinates matrix C.

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To parametrize the surface of the first-octant portion of the cone between the planes z = 0 and z = 3, we can use cylindrical coordinates.

Let's denote the cylindrical coordinates as (r, θ, z), where r represents the distance from the z-axis, θ represents the azimuthal angle in the xy-plane, and z represents the height.

The equation of the cone in cylindrical coordinates can be written as:

z = √(r^2)/2

To restrict the cone to the first octant, we can set the ranges for the coordinates as follows:

0 ≤ r ≤ √(6)

0 ≤ θ ≤ π/2

0 ≤ z ≤ 3

Now, we can express the surface parametrically as:

x = r * cos(θ)

y = r * sin(θ)

z = √(r^2)/2

This parametrization satisfies the equation of the cone in the given range of coordinates. The parameter r varies from 0 to √(6), θ varies from 0 to π/2, and z varies from 0 to 3, covering the first-octant portion of the cone between the planes z = 0 and z = 3.

Therefore, the parametrization of the surface is:

(r * cos(θ), r * sin(θ), √(r^2)/2)

where 0 ≤ r ≤ √(6), 0 ≤ θ ≤ π/2, and 0 ≤ z ≤ 3.

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Answer:

A. V = (4/3)π(2^3) = 32π/3 cm^3

B. V = (2/3)π(5^3) = 250π/3 cm^3

C. V = π(10^2)(7) = 700π cm^3

D. V = (1/3)π(4^2)(2) = 32π/3 cm^3

Containers A and D hold the same amount of batter.

Answer:

[tex]x =[/tex] 1 3/5

Step-by-step explanation:

Isolate the variable by dividing each side by factors that don't contain the variable.

Exact form:

[tex]x = 8/5[/tex]

Decimal Form:

[tex]x = 1.6[/tex]

Mixed Number Form:

[tex]x =[/tex] 1 3/5

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