Ways to use coordinates to prove quadrilaterals are parallelograms, rectangles, rhombi, or squares:

Using a linear approximation (or differentials) we estimate 3√28 is 3.11111.

To estimate the cube root of 28 using linear approximation, we first need to find a nearby perfect cube. In this case, the closest perfect cube is 27, since [tex]3^3[/tex] = 27.
Let's use the function f(x) = [tex]x^{(1/3)}[/tex]. We want to approximate f(28) using linear approximation. To do this, we will find the tangent line to f(x) at x=27, which will give us a good approximation for f(28).
First, find the derivative of f(x):
f'(x) = [tex](1/3)x^{(-2/3)}[/tex]
Now, evaluate the derivative at x=27:
f'(27) = [tex](1/3)(27)^{(-2/3)}[/tex] = [tex](1/3)(3^{(-2)})[/tex] = 1/9
Next, find the value of f(27):
f(27) = [tex]27^{(1/3)}[/tex] = 3
Now, we can find the equation of the tangent line at x=27:
y = f(27) + f'(27)(x - 27)
y = 3 + (1/9)(x - 27)
Finally, approximate f(28) by plugging x=28 into the tangent line equation:
y ≈ 3 + (1/9)(28 - 27)
y ≈ 3 + (1/9)(1)
y ≈ 3 + 1/9
y ≈ 3.11111
So, using linear approximation, we estimate that the cube root of 28 is approximately 3.11111, rounded to five decimal places.

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Approximately 97 cans must be sampled to achieve a margin of error of 0.01 ounce.

To estimate the number of cans that must be sampled for the margin of error to be equal to 0.01 ounce, we'll use the formula for the sample size in a confidence interval:

[tex]n = \left(\frac{Z \cdot \sigma}{E}\right)^2[/tex]

where n is the sample size, Z is the Z-score for the desired confidence level (95% in this case), σ is the standard deviation (0.05 ounce), and E is the margin of error (0.01 ounce).

For a 95% confidence interval, the Z-score is approximately 1.96. Plugging the values into the formula, we get:

n = [tex]\(\left(\frac{{1.96 \cdot 0.05}}{{0.01}}\right)^2\)[/tex]
n ≈ (9.8)²
n ≈ 96.04

Since we can't have a fraction of a can, we'll round up to the nearest whole number. Therefore, a sample size of approximately 97 cans is needed for the margin of error to be equal to 0.01 ounce with a 95% confidence interval.

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Answer: 17 is a prime number.

Explanation:

17 is a prime number because it is a natural number greater than 1 that has no positive divisors other than 1 and itself. In other words, 17 can only be divided evenly by 1 and 17. It is not possible to find any other whole number that can divide into 17 without a remainder. This property is what makes 17 a prime number.

Answer:

17

Step-by-step explanation:

17 is the only # in the set that is only divisible by 1 and itself, make it prime.

On the other hand,

10 is divisible by:

1, 10

2, 5.

12 is divisible by:

1, 12

2, 6

3, 4

So only 17 is prime.

the probability that a randomly chosen point is in the black region is: 17/36 ≈ 0.472 or about 47.2%

To find the probability that a randomly chosen point is in the black region, we need to divide the number of black squares by the total number of squares in the grid.

Counting the number of black squares, we see that there are 17.

Counting the total number of squares in the grid, we see that there are 36.

Therefore, the probability that a randomly chosen point is in the black region is: 17/36 ≈ 0.472 or about 47.2%

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495 different committees can be formed by selecting 4 members from a list of 12 people.

In this case, we have 12 people to choose from and we want to form a committee of 4 people. The number of different committees that can be formed is represented by the combination formula:

C(n, r) = n! / [r! * (n - r)!]

Where n = total number of people (12), r = number of people in the committee (4), and ! represents the factorial function.

Applying the formula:
C(12, 4) = 12! / (4! * (12 - 4)!)

C(12, 4) = 12! / (4! * 8!)

C(12, 4) = (12 * 11 * 10 * 9 * 8!)/(4 * 3 * 2 * 1 * 8!)

The 8! in the numerator and denominator cancels out:

C(12, 4) = (12 * 11 * 10 * 9)/(4 * 3 * 2 * 1)

C(12, 4) = 11880/24

C(12, 4) = 495

So, 495 different committees can be formed by selecting 4 members from a list of 12 people.

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The length and the width of the bar are 11.06 feet and 5.83 feet respectively

Let the width of the chocolate bar be x

From the question, the length of the chocolate bar is 0.6 feet shorter than twice its width. Therefore, the length (l) can be expressed as:

l = 2x - 0.6

The base area of the chocolate bar is given as 63.80 square feet. We can use this information to write an equation for the area of the chocolate bar in terms of its length and width:

area = length * width

63.80 = (2x - 0.6) * x

Expanding the right-hand side, we get:

63.80 = 2x² - 0.6x

Subtracting 63.80 from both sides, we get:

2x² - 0.6x - 63.80 = 0

Dividing both sides by 2, we get:

x² - 0.3x - 31.90 = 0

We can now solve this quadratic equation using the quadratic formula:

x = (-b ± sqrt(b² - 4ac)) / 2a

where a = 1, b = -0.3, and c = -31.90

Substituting in the values, we get:

x = (-(-0.3) ± sqrt((-0.3)^2 - 4(1)(-31.90))) / 2(1)

x = (0.3 ± sqrt(0.09 + 127.60)) / 2

x = (0.3 ± 11.36) / 2

We can now solve for the two possible values of x:

x = (0.3 + 11.36) / 2 = 5.83 or x = (0.3 - 11.36) / 2 = -5.53

Since the width of the chocolate bar cannot be negative, therefore, the width of the chocolate bar is 5.83 feet.

We can now use the equation for the length of the chocolate bar to find its length:

l = 2x - 0.6

l = 2(5.83) - 0.6

l = 11.06

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Therefore, the vector form of the equation y = x² + 8x + 18 is y = (x + 4)² + 20. The vertex of the parabola is at the point (-4, 20).

To convert the quadratic equation y = x² + 8x + 18 from standard form to vertex form, we need to complete the square by adding and subtracting a constant term. Here's the step-by-step explanation:

Factor the coefficient of x²: The coefficient of x² is 1, so we don't need to factor it.

Group the x terms: Rewrite the quadratic equation as y = (x² + 8x) + 18.

Complete the square: To complete the square, we need to add and subtract a constant term that will make the expression inside the parentheses a perfect square trinomial. The constant we need to add is half of the coefficient of x, squared: (8/2)² = 16.

y = (x² + 8x + 16 - 16) + 2 + 18 // add and subtract 16

y = (x + 4)² + 20

Simplify: Now we can simplify the expression by combining the constants 2 and 18 to get 20.

y = (x + 4)² + 20

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The sides of the quadrilateral are

FG = 11

GH = 22

HJ = 33

FJ = 44

When the quadrilateral are similar according to the ratio 7 : 11 and the similar sides are:

FG : AB

GH : BC

HJ : CD

FJ : AD

Since FJ : AD the ratio is 11 : 7, therefore where FJ = 11 then AD = 7 = x

FG : AB

AB = 2x = 2 * 7 = 14

For FG

11 ⇔ 7

FG ⇔ 14

cross multiplying

7 * FG = 14 * 11

FG = 22

For GH

11 ⇔ 7

GH ⇔ BC = 3 * 7 = 21

cross multiplying

7 * GH = 21 * 11

GH = 33

For HJ

11 ⇔ 7

HJ ⇔ CD = 4 * 7 = 28

cross multiplying

7 * GH = 28 * 11

GH = 44

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The manager of the city recreation center aims to estimate the percentage of city residents who support the proposal to build a new dog park. To achieve this, the manager will gather data by selecting a random sample of city residents, which helps ensure an unbiased representation of the population's opinion on the matter.

The manager of the city recreation center can use statistical methods to estimate the percentage of city residents who support the proposal for a new dog park. By selecting a random sample of city residents, the manager can get an idea of what the overall population thinks about the proposal. However, it's important to note that the accuracy of the estimate will depend on the size of the sample and how representative it is of the overall population. Therefore, the manager should make sure to choose a large enough and diverse enough sample to ensure that the estimate is as accurate as possible.

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The probability that the rat presses the red button exactly 2 times is 1/9, which corresponds to option C in the multiple-choice list.

To calculate the probability of the rat pressing the red button exactly 2 times, we need to consider the sample space of possible outcomes. Since there are 3 buttons and the rat presses them twice, there are 3 x 3 = 9 possible outcomes.
Now, let's identify the favorable outcomes: The only outcome that satisfies the condition of pressing the red button exactly 2 times is when the rat presses the red button first and then presses the red button again (RR).
There is only 1 favorable outcome, so the probability of pressing the red button exactly 2 times is:
P(RR) = Number of favorable outcomes / Total number of possible outcomes
P(RR) = 1/9

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Given that the hits to a web site occur at a rate of 10 per minute between 7:00 P.M. and 10:00 P.M., we can say that the web site experiences a Poisson process. The value of land for this Poisson process is 260.

The random variable X represents the number of hits to the web site between 7:56 P.M. and 8:22 P.M.

To find the values of λ, we need to use the formula for the Poisson distribution:

P(X = k) = (λ^k * e^-λ) / k!

where λ is the average number of hits per unit time.

Since we are interested in the hits between 7:56 P.M. and 8:22 P.M., which is a duration of 26 minutes, we need to adjust the rate of hits accordingly. Therefore, λ for this duration would be:

λ = 10 hits/min * 26 min = 260 hits

So, the values of λ for this Poisson process would be λ = 260.

To determine the values of λ (lambda) for this Poisson process, we need to first calculate the time interval between 7:56 P.M. and 8:22 P.M. and then find the expected number of hits to the website during that time.

Step 1: Calculate the time interval
The time interval between 7:56 P.M. and 8:22 P.M. is 26 minutes.

Step 2: Determine the expected number of hits
The website receives hits at a rate of 10 per minute. To find the expected number of hits during the 26-minute interval, multiply the rate by the length of the interval:
10 hits/minute * 26 minutes = 260 hits

Step 3: State the value of λ for the Poisson process
The random variable X represents the number of hits to the website during this interval. Since the expected number of hits is 260, the value of λ for this Poisson process is 260.

Your answer: The value of λ for this Poisson process is 260.

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At maturity, using the U.S. rule, the interest calculated from the last partial payment is subtracted from beginning balance added to adjusted balance.

Here, we have,

The payment of money above the principal amount (the amount borrowed) at a predetermined rate to a lender or depositor by a borrower or deposit-taking financial institution is referred to as paying interest in the fields of finance and economics.

It is different from a fee that the borrower may pay to the lender or another entity.

It also differs from a dividend, which is money given to shareholders (owners) by a company from its profit or reserve, but not at a set rate predetermined beforehand, but rather on a pro rata basis as a portion of the rewards received by risk-taking businesspeople when revenue is earned that exceeds all costs.

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Answer:

39.25 mm

Step-by-step explanation:

It will take approximately 6.96 years for the fish population to reach 2700. The initial fish population (P0) is 400, and after the first year, the population triples, making it 1200. The carrying capacity (K) is 5400. We can use these values to find the constant k.

a. To determine the constant k, we can use the initial conditions given in the problem. In the first year, the number of fish tripled, which means the population increased by a factor of 3. Therefore, we can set P(1) = 3(400) = 1200. We also know that the carrying capacity is 5400, so we can set K = r(1-Pmax/P0) where r is the intrinsic growth rate and P0 is the initial population. Substituting in the values we get K = 0.372.

Now we can solve the logistic equation:

dP/dt = kP(1-P/5400)

Separating variables and integrating, we get:

ln|P/ (5400-P)| = 0.372t + C

where C is the constant of integration. To find C, we can use the initial condition that P(0) = 400:

ln|400/5000| = C

C = -4.605

Substituting C back into the equation, we get:

ln|P/ (5400-P)| = 0.372t - 4.605

Taking the exponential of both sides:

|P/ (5400-P)| = e^(0.372t-4.605)

Multiplying both sides by the denominator and simplifying:

P = 5400e^(0.372t-4.605) / (1 + e^(0.372t-4.605))

b. We want to find the time t when the population reaches 2700. So we can set P = 2700 and solve for t:

2700 = 5400e^(0.372t-4.605) / (1 + e^(0.372t-4.605))

Multiplying both sides by the denominator and simplifying:

2700 + 2700e^(0.372t-4.605) = 5400e^(0.372t-4.605)

Dividing both sides by 2700 and simplifying:

1 + e^(0.372t-4.605) = 2

Taking the natural logarithm of both sides:

0.372t - 4.605 = ln(1)

0.372t = 4.605

t = 12.4 years

Therefore, it will take approximately 12.4 years for the population to reach 2700.

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When the eigenvalues of a matrix are equal, it means that the matrix has repeated eigenvalues. This can occur for matrices of any size, but is most commonly encountered for 2x2 or 3x3 matrices.

When a matrix has repeated eigenvalues, it can exhibit some special behaviors that are not observed for matrices with distinct eigenvalues. In particular, the matrix may not be diagonalizable, meaning that it cannot be transformed into a diagonal matrix using a change of basis. Instead, it may have only one eigenvector associated with the repeated eigenvalue, or it may have a full set of linearly independent generalized eigenvectors.

In the case of a 2x2 matrix A with repeated eigenvalues λ, the matrix may be expressed in Jordan canonical form as:

A = P * J * P^-1

where J is a matrix with λ on the diagonal and 1 on the super-diagonal, and P is the matrix whose columns are the eigenvectors and generalized eigenvectors of A. In this case, the solutions of the system x' = Ax may have a non-trivial component along the generalized eigenvector, which can lead to more complicated behavior than for matrices with distinct eigenvalues.

Overall, the presence of repeated eigenvalues in a matrix can indicate that the matrix has some special structure or symmetry, and may require additional techniques beyond diagonalization to fully understand its behavior.

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Answer: 11/9 is the slope

11/9

I suggest just googling "slope calculator" if possible because that's a fast way to do these without doing the math

The coordinate of point w is (3.75, 0.25).

Option D is the correct answer.

We have,

From the graph, we can see that,

The coordinate on the x-coordinate at point w is 3.75.

The coordinate on the y-coordinate at point w is 0.25.

Thus,

The coordinate of point w is (3.75, 0.25).

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Answer:

Magnitude: 23.4 kph

Direction: 50.2° below horizontal.

Step-by-step explanation:

15² + 18² = s²

s² = 549

s = 23.4

Θ = tan^-1 18/15

Θ = 50.2°

The velocity vector is the sum of the two vectors which is a vector that starts at the boat and goes until the lower tip of the vertical arrow.

Its magnitude is 23.4 kph, and its direction is 50.2° below horizontal.

Answer:  boats vector is 23.45 kph, 50.2° south of east

Step-by-step explanation:

A  vector has both direction and magnitude.

The vector for the boat is the shortest route from the boat to that end point which is a diagonal.

In order to find magnitude you use pythagorean

c²=a²+b²

m=15²+18²

m=√549

magnitude=23.45 kph

direction you would use tan∅

tan∅=opp/adj=18/15

∅= [tex]tan^{-1} \frac{18}{15}[/tex]

∅=50.2

So the boats vector is 23.45 kph, 50.2° south of east

True.

The determinant  of a matrix A is defined as the scaling factor of the linear transformation determined by A.

The determinant  of a matrix A is defined as the scaling factor of the linear transformation determined by A. Geometrically, the absolute value of the determinant of a matrix A can be interpreted as the factor by which the linear transformation determined by A changes the volume of a unit cube.

Consider the columns of A as vectors in n-dimensional space. The parallelepiped determined by the columns of A is the n-dimensional generalization of a parallelogram in two dimensions or a parallelepiped in three dimensions. The volume of this parallelepiped can be computed as the absolute value of the determinant of A.

To see why this is true, consider the special case of a 2x2 matrix A:

```
A = [[a, c],
[b, d]]
```

The columns of A are the vectors `[a, b]` and `[c, d]`. The parallelogram determined by these vectors has area equal to the absolute value of the determinant of t of

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Step-by-step explanation:

There are  5 p 5 = 120 ways to line up    ( = 5!)

   only ONE will be shortest to tallest

1/120

it is possible that options (A) and (D) are both true, but without seeing the graph, it is difficult to say for sure.

If the graph shows a straight line with a constant slope, then the angular speed of the object is constant. If the graph shows a horizontal line, then the angular speed of the object is zero, which means the object is not rotating.

If the graph shows a straight line with a constant slope, then the angular acceleration of the object is zero because the angular speed is not changing. If the graph shows a straight line with a constant non-zero slope, then the angular acceleration of the object is constant because the angular speed is changing at a constant rate.

If the graph shows a horizontal line, then the angular position of the object is constant because the object is not rotating.

The net torque applied to an object is equal to the rate of change of its angular momentum. If the graph shows a straight line with a constant slope, then the angular momentum is changing at a constant rate, which means the net torque applied to the object is constant.

Based on the above information, it is possible that options (A) and (D) are both true, but without seeing the graph, it is difficult to say for sure.

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1.5 gallons are equivalent to 24 cups because 1 gallon = 4 quarts or 16 cups.

Equivalent values are values that are equal or equivalent, notwithstanding the unit of expression or measurement.

For instance, we know that 1 gallon equals 4 quarts, and 4 quarts are equivalent to 16 cups.

The total number of cups = 24

1 gallon = 4 quart

1 quart = 4 cups

1 gallon = 16 cups

24 cups = 1.5 gallons (24 ÷ 16).

Thus, since we can state categorically that 1 gallon is equivalent to 16 cups, 24 cups will be equivalent to 1.5 gallons.

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One key assumption of correlation analysis is that the variables have a linear relationship. This means that the relationship between the two variables can be described by a straight line. In other words, as one variable increases, the other variable increases or decreases in a consistent manner.

When conducting correlation analysis, it is important to ensure that the relationship between the variables is indeed linear. This can be done by examining a scatterplot of the data, which allows you to visualize the relationship between the two variables. If the relationship appears to be curvilinear, non-linear, or circular, then a different type of analysis may be more appropriate.

Overall, the assumption of a linear relationship is important for ensuring the accuracy and validity of the correlation analysis. It allows for a clear and interpretable understanding of the relationship between the two variables and can inform further statistical analysis or decision-making.

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Answer:

the height of the triangular pyramid is 2.25 inches.

Step-by-step explanation:

To find the height of the triangular pyramid, we can use the formula for the volume of a pyramid:

Volume = (1/3) × base area × height

where the base area is the area of the triangle and the height is the height of the pyramid.

We know that the volume of the wood is 18 cubic inches, so we can substitute the given values into the formula:

18 = (1/3) × (4 × 6) × h

Multiplying both sides by 3, we get:

54 = 24 × h

Dividing both sides by 24, we get:

h = 2.25

Therefore, the height of the triangular pyramid is 2.25 inches.

The dimensions of the actual painting are 32 inches by 40 inches

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

Scale = 4:1.

The dimensions of the scaled painting are 8 inches by 10 inches.

So, we have

Actual = 4 * 8 inches by 4 * 10 inches

Evaluate

Actual = 32 inches by 40 inches

Hence, the dimension is 32 inches by 40 inches

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Since the calculated t-value of 4.38 is greater than the critical value of -1.796, we reject the null hypothesis and conclude that there is evidence to suggest that the mean number of paintings on the walls of houses owned by teachers is larger than the mean number of paintings on the walls of all houses.

Let μ be the population mean number of paintings on the walls of all houses, and let μt be the population mean number of paintings on the walls of houses owned by teachers. The null hypothesis is that there is no difference between the mean number of paintings on the walls of all houses and the mean number of paintings on the walls of houses owned by teachers:

H0: μ = μt

The alternative hypothesis is that the mean number of paintings on the walls of houses owned by teachers is larger than the mean number of paintings on the walls of all houses:

Ha: μ < μt

We can use a one-sample t-test to compare the sample mean number of paintings on the walls of houses owned by teachers to the population mean of 15 paintings. Let X be the sample mean number of paintings on the walls of houses owned by teachers and s be the sample standard deviation. We are given the following data from the survey:

n = 12

X = 20

s = 5.48

The test statistic is:

t = (X - μ) / (s / √(n))

Using the null hypothesis μ = 15, we get:

t = (20 - 15) / (5.48 / √(12))

= 4.38

The degrees of freedom for this test are n - 1 = 11. Using a t-distribution table with 11 degrees of freedom and a significance level of 0.05, we find a critical value of -1.796.

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1. There are 3 decision nodes in the tree.
2. The best course of action for the hospital is option B - seek approval first, and proceed with Plan B if approved, and Plan A if denied.
3. The answer is E) B & C only. The decision tree shows conditional probabilities, but TreePlan is not maximizing the EMV. The probability of approval for Plan B does not depend on when approval is sought.
4. The EMV for choosing Plan B from the start is -137.5.

1. Decision nodes are points in a decision tree where a decision must be made. To determine the number of decision nodes in the tree, count the number of places where a choice must be made between different options.

2. To determine the best course of action, you would analyze the decision tree by considering the expected monetary value (EMV) of each path. Choose the path with the highest EMV.

3. Examine your decision tree and determine which conclusions apply. Some things to consider are:
  A) Conditional probabilities: Are there probabilities given based on the outcomes of previous decisions?
  B) "Rolling back" the decision tree: Is the tree being analyzed by working backward from the final outcomes to determine the best course of action?
  C) Probability of approval: Does the chance of getting approval for Plan B change based on when approval is sought?

4. To find the EMV for choosing Plan B from the start, calculate the probability-weighted average of the possible monetary outcomes for that path. Multiply the probability of each outcome by its monetary value, then sum those products.

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The value of the expression 200% of of $25,000 is $50,000

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

200% of $25,000

The above expression is a percentage expression and mathematically, can be expressed as

Value = 200% of $25,000

Express "of" as products

So, we have

Value = 200% of $25,000

Evaluate the products in the expression

So, we have the following

Value = $50,000

Hence, the solution of the expression 200% of of $25,000 is $50,000

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Based on the available data, children under age 18 comprise approximately 19.0% of the homeless population.

This statistic highlights the significance of addressing homelessness, as it impacts a significant portion of young individuals who are in their crucial developmental years.The term "population" is frequently used to describe the total number of people living in a particular location. To estimate the number of the resident population within a certain territory, governments conduct censuses. The phrase has particular use in the domains of ecology and genetics and is also used to refer to plants, animals, and microbes.

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The final volume of the wooden post after the hole has been drilled is 778.41 cm³.

The volume of a right rectangular prism is given by the formula V = l × w × h, where l, w, and h represent the length, width, and height of the prism, respectively. In this case, the wooden post has dimensions of 5 cm by 9 cm by 24 cm, so its original volume is:

V1 = 5 cm × 9 cm × 24 cm = 1080 cm³

Now, we need to find the volume of the cylindrical hole. The formula for the volume of a cylinder is V = πr²h, where r is the radius of the cylinder and h is its height. In this case, the hole has a radius of 2 cm and goes through the entire length of the wooden post, so its volume is:

V2 = π × 2 cm² × 24 cm ≈ 301.59 cm³

To find the final volume of the wooden post, we need to subtract the volume of the hole from the original volume:

Vfinal = V1 - V2 = 1080 cm³ - 301.59 cm³ = 778.41 cm³

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