Simplify the expression to a polynomial in standard form (x^2+3x+3) (-2x^2-x+6)

Answer:

- 8a²

Step-by-step explanation:

using the rule of exponents

[tex]\frac{a^{m} }{a^{n} }[/tex] = [tex]a^{(m - n)}[/tex]

given

[tex]\frac{80a^9}{-10a^7}[/tex]

= [tex]\frac{80}{-10}[/tex] × [tex]a^{(9-7)}[/tex]

= - 8 × a²

= - 8a²

a. An equation to describe the total cost of RV rental:

Cost = (125 * d) + (0.32 * m) + 2500

b. Comparing the two costs will determine which option is cheaper.

c. For the more direct route: m ≤ 3500

For fewer overnight stays: d ≤ 14

These domains ensure that we don't exceed the original values for miles and days.

d. I recommend compromising by lessening the number of days of the rental. By reducing the rental period to 11 days, you can stay within the $5,000 budget while still allowing your little sister to take the two-week trip.

a. To write an equation for the total cost of RV rental, we can use the given information. The cost per day is $125, and there is an additional charge of 32 cents per mile. Let's denote the number of days as d and the number of miles as m. The equation for the total cost of RV rental can be written as:

Cost = (125 * d) + (0.32 * m) + 2500

b. To compare the costs of the two options, we need to calculate the total cost for each. Option 1 has 3500 miles and takes 11 days, while option 2 has 3000 miles and takes 14 days. We can substitute these values into the equation from part a to find the total costs for each option. Comparing the two costs will determine which option is cheaper.

c. To compromise and stay within a budget of $5,000, we can adjust either the number of miles or the number of days. For the more direct route, we can reduce the number of miles, and for fewer overnight stays, we can reduce the number of days. The domains for these compromises would be:

For the more direct route: m ≤ 3500

For fewer overnight stays: d ≤ 14

These domains ensure that we don't exceed the original values for miles and days.

d. To find the number of miles or days to eliminate in order to stay under the $5,000 budget, we can set up equations using the total cost equation from part a. Let's denote the reduced number of miles as m' and the reduced number of days as d'. We need to solve the following equation for each compromise:

(125 * d') + (0.32 * m') + 2500 ≤ 5000

By substituting the appropriate values into the equation and solving for m' or d', we can determine how many miles or days need to be eliminated.

Based on the given information, I recommend compromising by lessening the number of days of the rental. By reducing the rental period to 11 days, you can stay within the $5,000 budget while still allowing your little sister to take the two-week trip. This compromise ensures that you don't have to sacrifice too much scenic beauty or make drastic changes to the route.

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The correct value of volume of the pyramid is 48 cubic inches.

To find the volume of the solid oblique pyramid, we can use the formula V = (1/3) * Base Area * Height. The base of the pyramid is a triangle, and the height is given as 6 inches.The formula for the area of a triangle is (1/2) * base * height. In this case, the base length is 8 inches and the height is 6 inches. Base Area = (1/2) * 8 * 6 = 24 square inches

Now, we can calculate the volume of the pyramid:

V = (1/3) * Base Area * Height

V = (1/3) * 24 * 6

V = 48 cubic inches

Therefore, the volume of the pyramid is 48 cubic inches.

None of the provided options (a, b, c, d) match the calculated volume of 48 cubic inches. Please double-check the given options or provide the correct options for further comparison.

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The direct labor cost included in the Preble Company's flexible budget for March is $819,000.

To find the direct labor cost included in the company's flexible budget for March, we shall estimate the actual direct labor cost incurred during the period.

Given:

Actual production and sales =n26,600 units

Actual direct labor rate = $13.00 per hour

Actual direct labor hours worked = 63,000 hours

Direct labor cost = Actual direct labor rate × Actual direct labor hours worked

Direct labor cost = $13.00/hour × 63,000 hours

Direct labor cost = $819,000

Hence, the direct labor cost included in the company's flexible budget for March would be $819,000.

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HI Your answer is 42

I have calculated it you can trust me

Well you have marked right in the pic

PLEASE MARK AS BRAINLIEST

y₁ = 1/t is indeed a known solution of the given differential equation.

The second solution can be found using reduction of order or other methods specific to the equation.

Let's find the first and second derivatives of y₁ with respect to t:

y₁ = 1/t

First derivative:

y'₁ = d/dt (1/t) = -1/t²

Second derivative:

y''₁ = d/dt (-1/t²) = 2/t³

Now, let's substitute y₁, y'₁, and y''₁ into the differential equation:

-t²y'' + 3ty' + 5y = 0

Substituting the values:

-t²(2/t³) + 3t(-1/t²) + 5(1/t) = 0

Simplifying the expression:

-2/t + (-3/t) + 5/t = 0

(-2 - 3 + 5)/t = 0

0/t = 0

We can see that the expression simplifies to 0/t, which is equal to 0.

Therefore, y₁ = 1/t is indeed a known solution of the given differential equation.

To find the second solution, we can use the method of reduction of order. Let's assume the second solution is of the form y₂ = v(t)y₁, where v(t) is a function to be determined.

Substituting this into the differential equation, we have:

-t²(y₂'' + v'y₁' + v''y₁) + 3t(y₂' + vy₁') + 5y₂ = 0

Expanding and rearranging the terms, we get:

-t²(v''y₁ + v'y₁' + v'y₁ + vy₁'') + 3t(vy₁' - v'y₁) + 5vy₁ = 0

Simplifying further:

(-t²v''y₁ - 2t²v'y₁' + 3tvy₁' + 5vy₁) + (-t²v'y₁ + 3tvy₁ - 5v'y₁) = 0

Combining like terms:

-t²v''y₁ - 2t²v'y₁' - t²v'y₁ - t²v'y₁ + 3tvy₁' + 3tvy₁ + 5vy₁ - 5v'y₁ = 0

Simplifying:

-t²v''y₁ - 3t²v'y₁' + 6tvy₁' + (5v - 5v')y₁ = 0

Since y₁ = 1/t, we have:

-t²v''(1/t) - 3t²v'(1/t²) + 6tv(1/t²) + (5v - 5v')(1/t) = 0

Simplifying further:

-v'' - 3v' + 6v(1/t) + (5v - 5v')(1/t) = 0

Reducing the equation:

-v'' - 3v' + 6v/t + (5v/t - 5v'/t) = 0

-v'' - 3v' + (6v + 5v - 5v')/t = 0

-v'' - 3v' + (11v - 5v')/t = 0

To simplify the equation, we can multiply through by t:

-tv'' - 3tv' + 11v - 5v' = 0

Now, we have a differential equation in terms of v(t) only. To solve this equation, we can apply appropriate techniques such as separation of variables, integrating factors, or other methods depending on the specific form of the equation. Solving for v(t) will give us the second solution to the original differential equation -t²y" + 3ty' + 5y = 0.

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Answer:

AB = √3

Step-by-step explanation:

Since ABCD is a rectangle, all angles are 90°

∠CDA = 90°

⇒ ∠CDB + ∠BDA = 90

⇒ ∠BDA = 60

In ΔABD,

sin(∠BDA) = opposite/ hypotenuse = AB / BD

⇒ sin(60) = AB/2

⇒ AB = 2 sin(60)

⇒ AB = 2 (√3)/2

AB = √3

The data set has two outliers, namely the points (2, 225) and (8, 75).

Based on the given information about the scatterplot, we can observe that most of the points are grouped closely together and show a slight increase.

There are two points that lie outside of this cluster, specifically (2, 225) and (8, 75).

To determine if these points are outliers, we need to consider their deviation from the general pattern exhibited by the majority of the data points.

If these points deviate significantly from the overall trend, they can be considered outliers.

In this case, since (2, 225) and (8, 75) lie outside of the cluster of closely grouped points and do not follow the general pattern, they can be considered outliers.

These points are noticeably different from the majority of the data points and may have influenced the overall trend of the scatterplot.

The data set has two outliers, namely the points (2, 225) and (8, 75).

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Answer:

Step-by-step explanation:

To solve these problems, we can use the Poisson probability formula:

P(x; λ) = (e^(-λ) * λ^x) / x!

Where:

P(x; λ) is the probability of x events occurring

e is the base of the natural logarithm (approximately 2.71828)

λ is the average rate of events occurring in the given time period

x is the number of events

(a) Probability of exactly 4 orders arriving in 30 minutes:

The average rate of orders is given as 5 orders per hour. To find the average rate of orders in 30 minutes, we divide it by 2 (since 30 minutes is half an hour):

λ = 5 orders/hour / 2 = 2.5 orders/30 minutes

Using the Poisson probability formula:

P(x = 4; λ = 2.5) = (e^(-2.5) * 2.5^4) / 4!

Calculating this:

P(x = 4; λ = 2.5) ≈ (0.082 * 39.0625) / 24

P(x = 4; λ = 2.5) ≈ 3.22265625 / 24

P(x = 4; λ = 2.5) ≈ 0.134

Therefore, the probability that exactly 4 orders will arrive in 30 minutes is approximately 0.134, or 13.4%.

(b) Probability of at least 2 orders arriving in an hour:

To find the probability of at least 2 orders, we need to calculate the probabilities of having 0 and 1 order and subtract it from 1 (since it's the complement).

Using the Poisson probability formula:

P(x = 0; λ = 5) = (e^(-5) * 5^0) / 0! = e^(-5) ≈ 0.0067

P(x = 1; λ = 5) = (e^(-5) * 5^1) / 1! ≈ 0.0337

P(at least 2 orders) = 1 - P(x = 0) - P(x = 1) ≈ 1 - 0.0067 - 0.0337 ≈ 0.9596

Therefore, the probability of at least 2 orders arriving in an hour is approximately 0.9596, or 95.96%.

Answer:

The parabola has a vertex at (3, -4), has a p-value of -6 and it opens downwards.

Step-by-step explanation:

The given directrix of the parabola is y = 2, which is a horizontal line.

This means that the parabola is vertical, with a vertical axis of symmetry.

The focus of a parabola is a fixed point located inside the curve. The y-coordinate of the given focus is y = -10. As this is below the directrix, it means that the parabola opens downwards.

The standard form of a vertical parabola is:

[tex]\boxed{(x-h)^2=4p(y-k)}[/tex]

where:

As the focus is (3, -10), then:

[tex](h, k+p)=(3,-10)[/tex]

     [tex]\implies h = 3[/tex]

[tex]\implies k+p=-10[/tex]

As the directrix is y = 2, then:

[tex]k - p=2[/tex]

To find the value of k, sum the equations involved k and p to eliminate p:

[tex]\begin{array}{crcccr}&k &+& p& =& -10\\+&k& -& p& = &2\\\cline{2-6}&2k&&& =& -8\\\cline{2-6}\\\implies &k&&&=&-4\end{array}[/tex]

To find the value of p, substitute the found value of k into one of the equations:

[tex]-4-p=2[/tex]

        [tex]p=-4-2[/tex]

        [tex]p=-6[/tex]

Therefore, the values of h, k and p are:

The parabola has a vertex at (3, -4), has a p-value of -6 and it opens downwards.

The parabola has a vertex at (3, -4), has a p-value of -6 and it opens downwards.

In Mathematics, the standard form of the equation of the directrix lines for any parabola is given by this mathematical equation:

(x - h)² = 4p(y - k).

Where:

Since the directrix is horizontal, the axis of symmetry would be vertical. This ultimately implies that, we would have the following parameters;

directrix is y = 2

Focus, (h, k + p) = (3, -10)

Next, we would determine the value of k as follows;

k + p = -10     .......equation 1

k - p = 2     .......equation 2

By solving the equations simultaneously, we have:

2k = -8

k = -4

For the value of p, we have the following from equation 2:

k - p = 2

-4 - p = 2

p = -4 - 2

p = -6

In conclusion, we can logically deduce that the parabola opens downward because the p-value is negative.

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The box plot that best represents the numerical data is: A. A box plot using a number line from 3 to 12.25 with tick marks every one-fourth unit. The box extends from 6.25 to 9.25 on the number line. A line in the box is at 7.5. The lines outside the box end at 4.5 and 11. The graph is titled Shoe Sizes, and the line is labeled Size of Shoe.

In order to determine the five-number summary for the survey, we would arrange the data set in an ascending order:

4.5,6,6,6.5,7,7,7.5,8,8.5,9,9.5,10,11

Based on the information provided about the list of shoe sizes for a group of 13 people, we would use a graphical method (box plot) to determine the five-number summary for the given data set as follows:

Minimum (Min) = 4.5.

First quartile (Q₁) = 6.25.

Median (Med) = 7.5.

Third quartile (Q₃) = 9.25.

Maximum (Max) = 11.

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a) The factored expression for h(t) is -16t(t - 10), obtained by factoring out a common factor of -16 and a common factor of t from the original expression -16t^2 + 160t.

b) Both the original expression -16t^2 + 160t and the factored expression -16t(t - 10) yield the same result of 144 when evaluated at t = 1, confirming the correctness of the factoring.

a) To factor out a common factor with a negative coefficient from the expression h(t) = [tex]-16t^2 + 160t[/tex], we can rewrite it as:

h(t) = [tex]-16(t^2 - 10t)[/tex]

Now, let's focus on factoring the quadratic expression inside the parentheses. We can factor out a common factor of t:

h(t) = -16t(t - 10)

Therefore, the factored expression for h(t) is -16t(t - 10).

b) To check the factoring by evaluating both expressions for h(t) at t = 1, we substitute t = 1 into the original expression and the factored expression and compare the results.

Using the original expression:

h(1) = [tex]-16(1)^2 + 160(1)[/tex]

h(1) = -16 + 160

h(1) = 144

Using the factored expression:

h(1) = -16(1)(1 - 10)

h(1) = -16(1)(-9)

h(1) = 144

Both expressions yield the same result of 144 when evaluated at t = 1. Therefore, the factoring is correct.

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The value of `a5 = λ5 = 824`.Therefore, the value of `a5` is 824.

Given the following values; `λ1 = 4` and `λn = na(n-1) + 4`.

We are required to calculate the value of `λ5` which is `a5`.

Solution We are given that;`λ1 = 4` which can also be expressed as `a1 = 4`. We are also given that `λn = na(n-1) + 4`. For `n=2`, `λ2 = 2a1 + 4 = 2(4) + 4 = 12`.

For `n=3`, `λ3 = 3a2 + 4 = 3(12) + 4 = 40`. For `n=4`, `λ4 = 4a3 + 4 = 4(40) + 4 = 164`. For `n=5`, `λ5 = 5a4 + 4 = 5(164) + 4 = 824`.

Hence, the value of `a5 = λ5 = 824`.Therefore, the value of `a5` is 824.

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In mathematics, a set is a well-defined collection of distinct objects, called elements or members of the set. These objects can be anything: numbers, letters, people, or even other sets.

he concept of sets is fundamental in various branches of mathematics, including set theory, algebra, and statistics.There are different kinds of sets based on their properties:

Finite set: A set with a specific number of elements, which can be counted.Infinite set: A set with an endless number of elements.Empty set: A set with no elements. It is denoted by the symbol Ø or {}.

Singleton set: A set with only one element.Subset: A set whose elements are all contained within another set.Universal set: A set that includes all the possible elements of interest in a particular context.Operations on sets involve various ways of combining or manipulating sets:

Union: The union of two sets A and B is the set that contains all the elements from both sets. It is denoted by A ∪ B.Intersection: The intersection of two sets A and B is the set of elements that are common to both sets. It is denoted by A ∩ B.

Complement: The complement of a set A, denoted by A', is the set of all elements that are not in A but are in the universal set.Difference: The difference between two sets A and B is the set of elements that are in A but not in B. It is denoted by A - B.

Cartesian Product: The Cartesian product of two sets A and B is the set of all possible ordered pairs, where the first element is from set A and the second element is from set B. It is denoted by A × B.

For teaching the concept of the union of sets, you can use the following activity:

Activity: Venn Diagrams

Draw two overlapping circles on the board or use physical cut-out circles.Label one circle as Set A and the other as Set B.

Ask the students to suggest elements for each set and write them inside the circles.Discuss the elements that are common to both sets and write them in the overlapping region.Explain that the union of sets A and B represents all the elements in both sets.

Combine the elements from sets A and B, including the elements in the overlapping region, and write them in a new circle labeled as A ∪ B.Emphasize that the union includes all the distinct elements from both sets without repetition.

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Sampling based upon equal probability is called d. Simple random sampling. The correct answer is d. Simple random sampling.

Simple random sampling is a sampling technique where each individual in the population has an equal probability of being selected for the sample. It is based on the principle of equal probability, ensuring that every element has the same chance of being chosen. This method involves randomly selecting samples without any specific grouping or stratification.

Cluster sampling involves dividing the population into clusters or groups and randomly selecting entire clusters for inclusion in the sample. It does not guarantee equal probability for individual units within each cluster.

Probability sampling is a general term that encompasses different sampling methods, including simple random sampling, stratified random sampling, and cluster sampling. It refers to sampling techniques that rely on random selection and allow for the calculation of probabilities associated with sample estimates.

Stratified random sampling involves dividing the population into distinct strata based on certain characteristics and then selecting samples from each stratum in proportion to their representation in the population. It does not guarantee equal probability of selection for all individuals.

Systematic sampling involves selecting every kth individual from a population list after randomly selecting a starting point. It does not guarantee equal probability of selection for all individuals.

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The area of the segment is 9( π-2) units²

The area of a figure is the number of unit squares that cover the surface of a closed figure.

A segment is the area occupied by a chord and an arc. A segment can be a major segment or minor segment.

Area of segment = area of sector - area of triangle

area of sector = 90/360 × πr²

= 1/4 × π × 36

= 9π

area of triangle = 1/2bh

= 1/2 × 6²

= 18

area of segment = 9π -18

= 9( π -2) units²

therefore the area of the segment is 9(π-2) units²

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Answer:

In Grade 4, students are introduced to the concept of chance and the idea that different situations have different probabilities of occurring. They learn that for many situations, there are a finite number of different possible outcomes. However, at this stage, students are not expected to calculate the probability of events occurring. In Grade 5, students continue to build on their understanding of probability and may begin to learn more advanced concepts and techniques for calculating probabilities.

Step-by-step explanation:

Answer:

B. Lenghts of the diagonals

Step-by-step explanation:

The proposal is as follows

Part III  -  The distance between buoys A and B is 12.8 kilometers.

Part IV  -  The length of the other two triangle legs are 10.2 kilometers and 8.4 kilometers.

Part V  - The total length of the race course is 31.4 kilometers.

Part VIII  - The winner's speed during last year's race was 10.8 knots.

See the proposal attached.

Dear Sponsor,

I'm seeking sponsorship for the San Francisco sailboat race.

With a proven track record and the determination to win, your investment of $5,500 covers travel costs and potential hazards.

By associating your brand with a winning sailor, you'll gain significant exposure to thousands of spectators. Join me in this thrilling race for success.

Sincerely,

[Your Name]

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Answer:

  19 or 26

Step-by-step explanation:

You want the value of 3n² -8n -9, given that n(n -3) = 10.

We recognize that 10 = 5·2 and that these factors differ by 3. This means n(n -3) = 10 is equivalent to saying n ∈ {-2, 5}.

The value of 3n² -8n -9 will be one of ...

  (3n -8)n -9 = (3(-2) -8)(-2) -9 = (-14)(-2) -9 = 19 . . . . for n = -2

or

  (3(5) -8)(5) -9 = (7)(5) -9 = 26 . . . . . . . for n = 5

The expression 3n² -8n -9 will be either 19 or 26.

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The function Sam can use to figure his total earnings for the day, based on the number of people he serves, is E(n) = 42 + 5n.

To calculate Sam's total earnings for the day, we need to consider both his hourly wages and the tips he receives based on the number of people he serves. Let's break it down step by step.

First, we know that Sam earns $7 per hour as his wage. Since he works for 6 hours, his earnings from wages alone would be $7 multiplied by 6, which equals $42.

Next, Sam also earns about $5 in tips for each person he serves. We can represent the number of people Sam serves as "n". Therefore, his total tip earnings would be $5 multiplied by "n", which gives us 5n.

To calculate Sam's total earnings for the day, we add his earnings from wages and tips together. So the function representing his total earnings, "E", can be written as:

E(n) = 7(6) + 5n

Simplifying further, we get:

E(n) = 42 + 5n

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Answer:

25

Step-by-step explanation:

5+5+5+5+5=25

Answer:25

Step-by-step explanation:

Answer:

f(-1/2) = -1/6

Step-by-step explanation:

To find the value of f(x) when x = -1/2, we substitute -1/2 for x in the expression for f(x) and simplify:

f(x) = (-1/3)x - 1/3

f(-1/2) = (-1/3)(-1/2) - 1/3

= 1/6 - 1/3

= -1/6

So, f(-1/2) = -1/6.

Answer:

A: the probability that a 1 is received is 0.56.

B:  the probability that a 0 was sent given that a 1 is received is (2/25) * (1 - P(0 sent)).

Step-by-step explanation:

To solve this problem, we can use conditional probabilities and the concept of Bayes' theorem.

a. To find the probability that a 1 is received, we need to consider the two possibilities: either a 1 was sent and remained unchanged, or a 0 was sent and got flipped to a 1 by the random error.

Let's denote:

P(1 sent) = 0.6 (probability a 1 is sent)

P(0→1) = 0.2 (probability a 0 is flipped to 1)

P(1 received) = ?

P(1 received) = P(1 sent and unchanged) + P(0 sent and flipped to 1)

= P(1 sent) * (1 - P(0→1)) + P(0 sent) * P(0→1)

= 0.6 * (1 - 0.2) + 0.4 * 0.2

= 0.6 * 0.8 + 0.4 * 0.2

= 0.48 + 0.08

= 0.56

Therefore, the probability that a 1 is received is 0.56.

b. If a 1 is received, we want to find the probability that a 0 was sent. We can use Bayes' theorem to calculate this.

Let's denote:

P(0 sent) = ?

P(1 received) = 0.56

We know that P(0 sent) + P(1 sent) = 1 (since either a 0 or a 1 is sent).

Using Bayes' theorem:

P(0 sent | 1 received) = (P(1 received | 0 sent) * P(0 sent)) / P(1 received)

P(1 received | 0 sent) = P(0 sent and flipped to 1) = 0.4 * 0.2 = 0.08

P(0 sent | 1 received) = (0.08 * P(0 sent)) / 0.56

Since P(0 sent) + P(1 sent) = 1, we can substitute 1 - P(0 sent) for P(1 sent):

P(0 sent | 1 received) = (0.08 * (1 - P(0 sent))) / 0.56

Simplifying:

P(0 sent | 1 received) = 0.08 * (1 - P(0 sent)) / 0.56

= 0.08 * (1 - P(0 sent)) * (1 / 0.56)

= 0.08 * (1 - P(0 sent)) * (25/14)

= (2/25) * (1 - P(0 sent))

Therefore, the probability that a 0 was sent given that a 1 is received is (2/25) * (1 - P(0 sent)).

A message is coded into the binary symbols 0 and 1 and the message is sent over a communication channel. The probability a 0 is sent is 0.4 and the probability a 1 is sent is 0.6. The channel, however, has a random error that changes a 1 to a 0 with probability 0.2 and changes a 0 to a 1 with probability 0.1. (a) What is the probability a 0 is received? (b) If a 1 is received, what is the probability a 0 was sent?

The difference in size between the two guys is approximately -0.3108 inches, which implies that the first guy is bigger than the second guy.

To calculate the difference in size between two people, one measuring in inches and the other measuring in centimeters, we must first convert all measurements to a common unit.

Guy measures 3 inches + 1/4 inch. We can convert 1/4 inch to a decimal fraction by dividing 1 by 4, which gives us 0.25 inches. So your measurement in inches would be 3 + 0.25 = 3.25 inches.

The other guy measures 9.045 cm. To convert centimeters to inches, we use the following relationship: 1 cm = 0.3937 inches. Multiplying the measurement in centimeters by 0.3937, we get the measurement in inches: 9.045 cm * 0.3937 = 3.5608 inches (approximately).

Now we can calculate the size difference between them. We subtract the measurement of the second chavo (3.5608 inches) from the measurement of the first chavo (3.25 inches):

3.25 inches - 3.5608 inches = -0.3108 inches.

The resulting difference is -0.3108 inches. This means that the second chavo is smaller in size than the first. Since the difference is negative, it indicates that the first chavo is bigger than the second.

In summary, the difference in size between the two guys is approximately -0.3108 inches, which implies that the first guy is bigger than the second guy.

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Answer:

(-4,4)

Step-by-step explanation:

You rewrite the terms:

(x + 4)^2 => [x - (-4)]^2

(y - 4)^2 => [y - (4)]^2

so h = -4 and k = 4

so center of ellipse is (h,k) or (-4,4)

Answer:

Center = (-4, 4)

Step-by-step explanation:

The standard form of the equation of an ellipse with center (h, k) is:

[tex]\boxed{\dfrac{(x-h)^2}{a^2}+\dfrac{(y-k)^2}{b^2}=1}[/tex]

The given equation is:

[tex]\dfrac{(x+4)^2}{25}+\dfrac{(y-4)^2}{9}=1[/tex]

Comparing the given equation with the standard form, we can see that h = -4 and k = 4. Therefore, the center (h, k) of the ellipse is (-4, 4).

The vertex of point A' in the dilated triangle A'B'C' is (6, 4.5).

1. Start by calculating the distance between the vertices of the original triangle ABC:

  - Distance between A(4, 3) and B(3, -2):

    Δx = 3 - 4 = -1

    Δy = -2 - 3 = -5

    Distance = √((-[tex]1)^2[/tex] + (-[tex]5)^2[/tex]) = √26

  - Distance between B(3, -2) and C(-3, 1):

    Δx = -3 - 3 = -6

    Δy = 1 - (-2) = 3

    Distance = √((-6)² + 3²) = √45 = 3√5

  - Distance between C(-3, 1) and A(4, 3):

    Δx = 4 - (-3) = 7

    Δy = 3 - 1 = 2

    Distance = √(7² + 2²) = √53

2. Apply the scale factor of 1.5 to the distances calculated above:

  - Distance between A' and B' = 1.5 * √26

  - Distance between B' and C' = 1.5 * 3√5

  - Distance between C' and A' = 1.5 * √53

3. Determine the coordinates of A' by using the distance formula and the given coordinates of A(4, 3):

  - A' is located Δx units horizontally and Δy units vertically from A.

  - Δx = 1.5 * (-1) = -1.5

  - Δy = 1.5 * (-5) = -7.5

  - Coordinates of A':

    x-coordinate: 4 + (-1.5) = 2.5

    y-coordinate: 3 + (-7.5) = -4.5

4. Thus, the vertex of point A' in the dilated triangle A'B'C' is (2.5, -4.5).

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Step-by-step explanation:

Perimeter =  pi * diameter

  radius = 1  1/3  yard   then diameter =  2  2/3 yd

perimter = pi *  2 2/3   yds = 8.38 yds