HELP ME PLZZ I AM GIVING 30 POINTS FOR THIS!!

The total area of the first floor of the tiny house is 121 square feet

The total area of the first floor of the tiny house can be expressed as the sum of the areas of each room. The area of a rectangle is calculated by multiplying the length by the width. Therefore, we can write:

Total area = Area of kitchen + Area of bathroom

The area of the kitchen is given by the product of the length and the width of the kitchen, which is 11 feet and 7 feet, respectively. Therefore, the area of the kitchen can be written as:

Area of kitchen = 11 x 7 = 77 square feet

Similarly, the area of the bathroom is given by the product of the length and the width of the bathroom, which is 11 feet and 4 feet, respectively. Therefore, the area of the bathroom can be written as:

Area of bathroom = 11 x 4 = 44 square feet

Substituting these expressions into the equation for the total area, we get:

Total area = Area of kitchen + Area of bathroom

= 77 + 44

= 121 square feet

Therefore, the total area of the first floor of the tiny house is 121 square feet, which can also be expressed as the sum of the areas of the kitchen and bathroom.

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The correct number sentence to find the number of black bugs would be:

A) 14 + 4 = (black)

Given that, there are 14 bugs crawling up the stairs.

We need to choose the number that can be used to determine how many of the bugs are black while just four are green.

The number sentence states that there are 14 bugs in total and 4 of them are green.

Since we want to find the number of black bugs, we need to add the number of green bugs (4) to the number of black bugs.

By using the number sentence 14 + 4 = (black), we can determine the value of "black" by performing the addition.

Hence the correct number sentence to find the number of black bugs would be:

A) 14 + 4 = (black)'

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Answer:

6.5

Step-by-step explanation:

This answer can be concluded by plugging in all of the variables into the problem

3r+s-t/2    ---->   3(4)+(3)-(2)   /   2

                          12+3-2 = 13/2 = 6.5

Answer:

6.5

Step-by-step explanation:

(3r+s-t)/2

Replace the variables with the terms.

[3(4)+3-2]/2

Then combine.

(12+3-2)/2

(15-2)/2

13/2

6.5

-hope it helps

To maximize the volume of an open-top container made from a 13-inch by 48-inch piece of plastic, you need to determine the optimal size of the squares to be cut out from each corner. Let 'x' be the side length of the square removed from each corner. After cutting, the dimensions of the container will be:

- Length: 48 - 2x
- Width: 13 - 2x
- Height: x

The volume of the container can be calculated using the formula: V = L * W * H.  the dimensions, we get:

V(x) = (48 - 2x)(13 - 2x)(x)

To find the maximum volume, we need to identify the value of 'x' that maximizes V(x). This can be achieved using calculus, by finding the critical points where the derivative of the function V(x) is zero or undefined.

Differentiating V(x) with respect to x and setting the derivative equal to zero, we can solve for the optimal value of 'x'. After performing these calculations, we find that the optimal size of the square to be cut out from each corner is approximately 1.52 inches. By removing 1.52-inch squares from each corner and folding up the flaps, the open-top container will have the maximum volume.

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Dividing 9x³ + 3x² + 10x + 3 by 3x + 2. will yield a quotient of 3x² - x + 4 and a remainder of -5 that is (3x² - x + 4) -5/(3x + 2).

Applying the long division method will require us to; divide, multiply, subtract, bring down the next number and repeat the process to end at zero or arrive at a remainder.

We shall divide the 9x³ + 3x² + 10x + 3 by 3x + 2. as follows;

9x³ divided by 3x equals 3x²

3x + 2. multiplied by 3x² equals 9x³ + 6x²

subtract 9x³ + 6x² from 9x³ + 3x² + 10x + 3 will result to -3x² + 10x + 3

-3x² divided by 3x equals -x

3x + 2 multiplied by -x equals -3x² - 2x

subtract -3x² - 2x from -3x² + 10x + 3 will result to 12x + 3

12x divided by 3x equals 4

3x + 2 multiplied by 4 equals 12x + 8

subtract 12x + 8 from 12x + 3 will result to a remainder of -5

Therefore by the long division method, 9x³ + 3x² + 10x + 3 divided by 3x + 2 gives a quotient 3x² - x + 4 and a remainder of -5 and can be written as (3x² - x + 4) -5/(3x + 2).

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The triangle PAT is an isosceles triangle because the side lengths PA and AT are congruent

The points are given as:

P = (1,6)

A = (4,5)

T = (5,2)

Calculate the distance between both points as follows:

\(d = \sqrt{(x_2 -x_1)^2 + (y_2 -y_1)^2}\)

So, we have:

\(PA = \sqrt{(1 -4)^2 + (6-5)^2}\)

\(PA = \sqrt{10}\)

\(PT = \sqrt{(1 -5)^2 + (6-2)^2}\)

\(PT = \sqrt{32}\)

\(AT = \sqrt{(4 -5)^2 + (5-2)^2}\)

\(AT = \sqrt{10}\)

Because the side lengths PA and AT are congruent, i.e. √10, then the triangle is an isosceles triangle

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Answer:

r = - 2.5

\(r=-2\frac{1}{2}\)

Step-by-step explanation:

21r - 7 - r + 5 = - 52

20r - 2 = - 52

20r - 2 + 2 = - 52 + 2

20r = - 50

20r ÷ 20 = - 50 ÷ 20

r = - 2.5

The Maclaurin series of \(xe^{8x}=\frac{\sum^\infty_0(8^n * x^n)}{n!}\)

What is the Maclaurin series?

The Maclaurin series is a special case of the Taylor series expansion, where the expansion is centered around x = 0. It represents a function as an infinite sum of terms involving powers of x. The Maclaurin series of a function f(x) is given by:

\(f(x) = f(0) + f'(0)x +\frac{ (f''(0)x^2}{2!} + ]\frac{(f'''(0)x^3)}{3! }+ ...\)

To find the Maclaurin series of the function f(x) = \(xe^{8x}\), we can start with the general formula for the Maclaurin series expansion:

\(f(x) = \frac{\sum^\infty_0(f^n(0) * x^n) }{ n!}\)

where\(f^n(0)\) represents the nth derivative of f(x) evaluated at x = 0.

Let's determine the appropriate series for the function \(f(x) = xe^{8x}\) from the given options:

a) \(\frac{\sum^\infty_0(8^n * x^{n+1})}{n!}\)

b) \(\frac{\sum^\infty_0(x^n )} {8^n*n!}\)

c)\(\sum^\infty_0(8^n * x^n)/n!\)

d)\(\frac{\sum^\infty_0(x^n )} {n!}\)

Comparing the given options with the general formula, we can see that option (c) matches the required form:

f(x) = \(=\frac{\sum^\infty_0(8^n * x^n)}{n!}\)

Therefore, the Maclaurin series of \(f(x) = xe^{8x}\) can be written as:

f(x) = \(=\frac{\sum^\infty_0(8^n * x^n)}{n!}\)

Option (c) is the correct series to represent the Maclaurin series of \(xe^{8x}.\)

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In conclusion for a. If the system is homogeneous , every solution is trivial true. b. If there exists a trivial solution, the system is homogeneous false. c. If there exists a nontrivial solution, there is no trivial solution false.

How to solve?

a. If the system is homogeneous, every solution is trivial.

This statement is true. A homogeneous system of linear equations has the form Ax = 0, where A is the coefficient matrix and x is the vector of variables. The trivial solution is always x = 0, which satisfies the equation. Any other solution would require a nonzero x vector, but then Ax would be nonzero, contradicting the fact that it equals zero in a homogeneous system.

b. If there exists a trivial solution, the system is homogeneous.

This statement is false. A system of linear equations can have a trivial solution (i.e., all variables are equal to zero) without being homogeneous. For example, the system

x + y = 0

2x + 2y = 0

has a trivial solution (x = 0, y = 0) but is not homogeneous.

c. If there exists a nontrivial solution, there is no trivial solution.

This statement is false. A homogeneous system of linear equations can have both trivial and nontrivial solutions. For example, the system

x + y = 0

2x + 2y = 0

has both a trivial solution (x = 0, y = 0) and a nontrivial solution (x = 1, y = -1).

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To determine the equation of a polynomial of the third degree with the given criteria, we know that the polynomial will have roots at x = 3 and x = -4. Therefore, the factors of the polynomial are (x - 3) and (x + 4).

Since the polynomial has a third degree, we need to introduce another factor of (x - a), where 'a' is a constant.

The equation of the polynomial is then:
f(x) = k * (x - 3) * (x + 4) * (x - a)

To find the value of 'a' and 'k,' we use the fact that f(6) = 8:
8 = k * (6 - 3) * (6 + 4) * (6 - a)
8 = k * 3 * 10 * (6 - a)
8 = 90k * (6 - a)
k * (6 - a) = 8/90
k * (6 - a) = 4/45

From this equation, we can solve for 'a' and 'k' to obtain the complete equation of the polynomial.

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Answer:

150 pennies were lost altogether

Step-by-step explanation:

Let the initial number of pennies owned by Dante be x pennies

Let the initial number of pennies owned by Mia be y pennies

Mathematically ;

x + y = 350 •••••••(i)

So after losing half, dante will have x/2 pennies left.

Mia lost 1/3 so she will have 2/3y left

So after all the losses, they both had equal amount of pennies

This means that;

x/2 = 2y/3

Cross multiply;

3x = 4y •••••••(ii)

Let’s solve both equations simultaneously;

From i , x = 350-y

Substitute this into equation ii

3(350-y) = 4y

1050-3y = 4y

7y = 1050

y = 1050/7

y = 150

since x = 350-y

x = 350-150 = 200

Now Dante loss x/2 = 200/2 = 100

Mia lost 1/3y = 1/3 * 150 = 50

Total pennies lost = 100 + 50 = 150

Using a standard normal distribution table or calculator, we can find the probability associated with these z-scores. The probability that their score will be between 21 and 26 is approximately 0.387.

For the first question, we are given that the average weight of children with ELBW is 815 grams with a standard deviation of 100 grams. We want to find the probability that an SRS of 45 children with ELBW weighs less than 825 grams.

To find this probability, we can use the normal distribution. Since we have the mean and standard deviation, we can standardize the variable by calculating the z-score.

The z-score is calculated as (825 - 815) / (100 / sqrt(45)) = 0.474.

Using a standard normal distribution table or calculator, we can find the probability associated with this z-score. The probability that an SRS of 45 children with ELBW weighs less than 825 grams is approximately 0.678.

For the second question, we are told that the length of human pregnancies is approximately normally distributed with a mean of 266 days and a standard deviation of 16 days. We want to find the probability that the average pregnancy length for 7 randomly chosen women exceeds 270 days.

To find this probability, we can use the central limit theorem. The distribution of sample means will be approximately normally distributed with the same mean as the population and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

The standard deviation of the sample mean is 16 / sqrt(7) = 6.048.

We can now standardize the variable by calculating the z-score. The z-score is (270 - 266) / 6.048 = 0.662.

Using a standard normal distribution table or calculator, we can find the probability associated with this z-score. The probability that the average pregnancy length for 7 randomly chosen women exceeds 270 days is approximately 0.253.

For the third question, we are given that scores on the MCAT are approximately normally distributed with a mean of 25 and a standard deviation of 6.5. We want to find the probability that a sample of 10 students will have an average score between 21 and 27.

To find this probability, we can use the central limit theorem. The distribution of sample means will be approximately normally distributed with the same mean as the population and a standard deviation equal to the population standard deviation divided by the square root of the sample size.

The standard deviation of the sample mean is 6.5 / sqrt(10) = 2.053.

We can now standardize the variable by calculating the z-scores for the lower and upper limits.

For the lower limit, the z-score is (21 - 25) / 2.053 = -1.949.
For the upper limit, the z-score is (27 - 25) / 2.053 = 0.975.

Using a standard normal distribution table or calculator, we can find the probability associated with these z-scores. The probability that this sample will have an average score between 21 and 27 is approximately 0.778.

For the fourth question, we are given that scores on the MCAT are approximately normally distributed with a mean of 25 and a standard deviation of 6.5. We want to find the probability that the score of one WSSU student will be between 21 and 26.

To find this probability, we can calculate the z-scores for the lower and upper limits.

For the lower limit, the z-score is (21 - 25) / 6.5 = -0.615.
For the upper limit, the z-score is (26 - 25) / 6.5 = 0.154.

Using a standard normal distribution table or calculator, we can find the probability associated with these z-scores. The probability that their score will be between 21 and 26 is approximately 0.387.

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Answer:

b 24:8

Step-by-step explanation:

please mark me as brainliest

Answer:

b

Step-by-step explanation:

33:11 is the same as 3:1, and 24:8 is the same as 3:1.

1) P(ANB) = 0.1

2) We cannot determine P(AUB) using the given information.

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event

To find P(ANB), we can use the formula: P(ANB) = P(A) + P(B) - P(AUB).

Plugging in the given values, we get:

P(ANB) = 0.3 + 0.4 - 0.6

= 0.1

Therefore, P(ANB) is 0.1.

2)Since A and B are not mutually exclusive events, we cannot use the formula P(AUB) = P(A) + P(B) - P(ANB) to find P(AUB). Instead, we need to use the formula: P(AUB) = P(A) + P(B) - P(ANB) only if A and B are mutually exclusive events.

Therefore, we cannot determine P(AUB) using the given information.

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Answer:

Step-by-step explanation:

well it  decides  on your parents but yeah tai is right pg 13 means 13 years old and.

TAI IS CORRECT BECUASE PG-13: Parents Strongly Cautioned, Some Material May Be Inappropriate for Children Under 13. This rating is a stronger caution for parents that content included may not be appropriate for children under 13 (pre-teen ages). This may include stronger language, extended violence or sexual situations and drug-use, and sex

Answer:

-3/5

Step-by-step explanation:

from 1st point to second, you go down 3, right 5

The person has walked a total of 28 miles. To calculate total distance walked by the person, we add up the distances in each direction

5 miles East + 6 miles Southeast + 6 miles South + 7 miles Southwest + 4 miles East

When we add these distances together, we get:

5 + 6 + 6 + 7 + 4 = 28

Therefore, the person has walked a total of 28 miles.

In more detail, let's break down the distances walked in each direction:

- 5 miles East: This means the person walked 5 miles in the East direction.

- 6 miles Southeast: This means the person walked 6 miles in the direction that is both South and East.

- 6 miles South: This means the person walked 6 miles in the South direction.

- 7 miles Southwest: This means the person walked 7 miles in the direction that is both South and West.

- 4 miles East: This means the person walked 4 miles in the East direction.

By adding up these distances, we find that the person has walked a total of 28 miles.

#A person starts walking from home and walks: 5 miles East 6 miles Southeast 6 miles South 7 miles Southwest 4 miles East This person has walked a total of ---miles

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Answer:

31kg will cost $ 182.9

Step-by-step explanation:

If 18kg = $106.20

than 31kg = ?

= 31kg/18kg × $106.20

= 3,292.2 /18

=$182.9

Answer:

6x+2y=-78

Step-by-step explanation:

8x-3y=-70

-2x+5y=-8

6x+2y=-78

Answer:

(-11, -6).

Step-by-step explanation:

8x-3y=-70

-2x+5y=-8     Multiply this equation by 4:

-8x+20y=-32

Now add this last equation to the first equation:

17y = -102   - we see that we have eliminated x.

y = -102/17

= -6.

Now plug y=-6 in the first equation to get x:

8x-3(-6)=-70

8x =-70 -18

8x =-88

x = -11.

The compound inequality is x<-4 and x<2.

What is a compound inequality?

A compound inequality in mathematics is a clause that joins two claims with the conjunction "and" or "or." Both of the statements in a compound sentence are true simultaneously if the conjunction "and" is used to connect the statements. If "or" is used to connect the statements, the compound phrase is true if at least one of the statements is true.

We are given that -4 is greater than x

This means that  -4>x

Also, it is given that x is less than 2

Therefore, x<2

Hence, the compound inequality is x<-4 and x<2.

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Answer:

There a'int none

Step-by-step explanation:1/10= one out of tenths and same with the other one

To solve this problem, we can use the formula for the circumference of a circle:

C = 2πr

where C is the circumference and r is the radius.

We are given that the diameter of the circle is 8.6 cm, so the radius is half of this:

r = 8.6 cm / 2 = 4.3 cm

Substituting this value of r into the formula for the circumference, we get:

C = 2π(4.3 cm) = 8.6π cm

Rounding this to the nearest hundredth gives:

C ≈ 26.93 cm

Therefore, the circumference of the circle is approximately 26.93 cm.

To construct a truth table for the given logical expression using De Morgan's Law, we'll break it down step by step and apply the law to simplify the expression.

Let's start with the given expression:

Xyz + X(Y Z)' + X'(Y + Z) + (Xyz)' (X + Y')(X' + Z')(Y' + Z')

Step 1: Apply De Morgan's Law to the term (Xyz)'

(Xyz)' becomes X' + y' + z'

After applying De Morgan's Law, the expression becomes:

Xyz + X(Y Z)' + X'(Y + Z) + (X' + y' + z')(X + Y')(X' + Z')(Y' + Z')

Step 2: Expand the expression by distributing terms:

Xyz + XY'Z' + XYZ' + X'Y + X'Z + X'Y' + X'Z' + y'z' + x'y'z' + x'z'y' + x'z'z' + xy'z' + xyz' + xyz'

Now we have the expanded expression. To construct the truth table, we'll create columns for the variables X, Y, Z, and the corresponding output column based on the expression.

The truth table will have 2^3 = 8 rows to account for all possible combinations of X, Y, and Z.

Here's the complete truth table:

```

| X | Y | Z | Output |

|---|---|---|--------|

| 0 | 0 | 0 |   0    |

| 0 | 0 | 1 |   0    |

| 0 | 1 | 0 |   0    |

| 0 | 1 | 1 |   1    |

| 1 | 0 | 0 |   0    |

| 1 | 0 | 1 |   0    |

| 1 | 1 | 0 |   1    |

| 1 | 1 | 1 |   1    |

```

In the "Output" column, we evaluate the given expression for each combination of X, Y, and Z. For example, when X = 0, Y = 0, and Z = 0, the output is 0. We repeat this process for all possible combinations to fill out the truth table.

Note: The logical operators used in the expression are:

- '+' represents the logical OR operation.

- ' ' represents the logical AND operation.

- ' ' represents the logical NOT operation.

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Answer: b

Step-by-step explanation:

Steps are shown in the attached document.

The feature that will be the same as the original is:

The perimeter is the same.

The coordinate of C' is (3, 4).

The slope of A'C' is 1/4.

We have,

To rotate a point 180 degrees clockwise around another point, you can follow these steps:

- Calculate the displacement vector from the center of rotation to the point you want to rotate.

- Reverse the direction of the displacement vector.

- Apply the reversed displacement vector to the center of rotation.

- Let's apply these steps to each vertex of triangle ABC to find the coordinates of A', B', and C'.

So,

- Coordinate of A' (rotated point of A around (3, 4)):

Displacement vector: (A' - Center of rotation) = (A - Center of rotation) = (-5, 2) - (3, 4) = (-8, -2).

Reverse the direction of the displacement vector:

Reversed displacement vector: (-8, -2) * (-1) = (8, 2).

Apply the reversed displacement vector to the center of rotation:

Coordinate of A': (3, 4) + (8, 2) = (11, 6).

- Coordinate of B' (rotated point of B around (3, 4)):

Displacement vector: (B' - Center of rotation) = (B - Center of rotation) = (-2, 5) - (3, 4) = (-5, 1).

Reverse the direction of the displacement vector:

Reversed displacement vector: (-5, 1) * (-1) = (5, -1).

Apply the reversed displacement vector to the center of rotation:

Coordinate of B': (3, 4) + (5, -1) = (8, 3).

- Coordinate of C' (rotated point of C around (3, 4)):

Displacement vector: (C' - Center of rotation) = (C - Center of rotation) = (3, 4) - (3, 4) = (0, 0).

Reverse the direction of the displacement vector:

Reversed displacement vector: (0, 0) * (-1) = (0, 0).

Apply the reversed displacement vector to the center of rotation:

Coordinate of C': (3, 4) + (0, 0) = (3, 4).

So,

The coordinate of A' is (11, 6).

The coordinate of B' is (8, 3).

The coordinate of C' is (3, 4).

To find the perimeter and area of a triangle, we can use the coordinates of its vertices.

Let's start by finding the perimeter and area of triangle ABC.

Triangle ABC:

A = (-5, 2)

B = (-2, 5)

C = (3, 4)

The perimeter of triangle ABC:

The perimeter of a triangle is the sum of the lengths of its sides. We can use the distance formula to calculate the lengths of each side and then sum them up.

Length of side AB:

\(d_{AB} = \sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)\)

= √((-2 - (-5))² + (5 - 2)²)

= √(3² + 3²)

= √(18)

= 3√2

Length of side BC:

\(d_{BC} = \sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)\)

= √((3 - (-2))² + (4 - 5)²)

= √(5² + 1²)

= √(26)

Length of side CA:

\(d_{CA}= \sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)\)

= √((-5 - 3)² + (2 - 4)²)

= √((-8)² + (-2)²)

= √(64 + 4)

= √(68)

= 2√17

The perimeter of triangle ABC:

\(P_{ABC} = d_{AB} + d_{BC} + d_{CA}\)

= 3√2 + √26 + 2√17

Area of triangle ABC:

The area of a triangle can be calculated using the coordinates of its vertices with the Shoelace formula.

Area of triangle ABC:

A_ABC = 1/2 x |(x1 x y2 + x2 x y3 + x3 x y1) - (y1 x x2 + y2 x x3 + y3 x x1)|

= 1/2 x |((-5 x 5) + (-2 x 4) + (3 x 2)) - ((2 x -2) + (5 x 3) + (4 x -5))|

= 1/2 x |(-25 - 8 + 6) - (-4 + 15 - 20)|

= 1/2 x |-27 - (-9)|

= 1/2 x |-27 + 9|

= 1/2 x |-18|

= 9

Now let's find the perimeter and area of triangle A'B'C', which is the rotated triangle of ABC.

Triangle A'B'C':

A' = (11, 6)

B' = (8, 3)

C' = (3, 4)

The perimeter of triangle A'B'C':

Using the same approach as before, we calculate the lengths of the sides:

Length of side A'B':

d_A'B' = √((x2 - x1)^2 + (y2 - y1)^2)

= √((8 - 11)^2 + (3 - 6)^2)

= √((-3)^2 + (-3)^2)

= √(18)

= 3√2

Length of side B'C':

d_B'C' = √((x2 - x1)^2 + (y2 - y1)^2)

= √((3 - 8)^2 + (4 - 3)^2)

= √((-5)^2 + 1^2)

= √(26)

Length of side C'A':

d_C'A' = √((x2 - x1)^2 + (y2 - y1)^2)

= √((3 - 11)^2 + (4 - 6)^2)

= √((-8)^2 + (-2)^2)

= √(68)

= 2√17

The perimeter of triangle A'B'C':

P_A'B'C' = d_A'B' + d_B'C' + d_C'A'

= 3√2 + √26 + 2√17

Area of triangle A'B'C':

Using the same Shoelace formula as before:

Area of triangle A'B'C':

A_A'B'C' = 1/2 x |(x1 x y2 + x2 x y3 + x3 x y1) - (y1 x x2 + y2 x x3 + y3 x x1)|

= 1/2 x |((11 x 3) + (8 x 4) + (3 x 6)) - ((6 x 8) + (3 x 3) + (4 x 11))|

= 1/2 x |(33 + 32 + 18) - (48 + 9 + 44)|

= 1/2 x |(83) - (101)|

= 1/2 x |-18|

= 9

Now,

The perimeter of triangle ABC is 3√2 + √26 + 2√17, and the area of triangle ABC is 9.

The perimeter of triangle A'B'C' is 3√2 + √26 + 2√17, and the area of triangle A'B'C' is 9.

The slope of A'C' can be calculated using the coordinates of A' and C'.

The slope of a line can be calculated using the formula:

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

For A'(11, 6) and C'(3, 4), the slope of A'C' is:

slope = (4 - 6) / (3 - 11)

= -2 / -8

= 1/4

Thus,

The feature that will be the same as the original is:

The perimeter is the same.

The coordinate of C' is (3, 4).

The slope of A'C' is 1/4.

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"Answer: 11x - 8y = -9

Step-by-step explanation:

For this exercise you need to remember:

1) The multiplication of signs:

(+) (+) = +

(-) (-) = +

(-) (+) = -

(+) (-) = -

2) By definition, like terms contain the same variables with the same exponent.

Then know this, and given the equations:

8x - 4y = -4 and -3x + 4y = 5

You must subtract the like terms, which means that you must subtract the numerical coefficients of each term.

Then, you get the result is:

8x - 4y = -4

-3x + 4y = 5

--------------------------------------------------

8x - (-3x) - 4y - 4y = - 4y -5

8x + 3x - 8y  = -9

11x - 8y = -9"

The expressions for Y(t) and X(t) obtained by applying inverse Laplace transforms to the given equations are :

For Y(t):

Y(t) = 2 + 2e^(-t) + 1/4 + 1/4 * sin(2t)

For X(t):

X(t) = 1/12 + e^(-0.5t) - e^(-4t) - e^(-3t)

To find Y(t) using Laplace transforms for the equation Y(s) = 2(s + 1) / (s(s^2 + 4)), we need to apply the inverse Laplace transform to the given expression.

Decompose the fraction using partial fraction decomposition:

1/(s(s^2 + 4)) = A/s + (Bs + C)/(s^2 + 4)

Multiplying through by s(s^2 + 4), we get:

1 = A(s^2 + 4) + (Bs + C)s

Expanding the equation, we have:

1 = As^2 + 4A + Bs^2 + Cs

Equating the coefficients of like powers of s, we get the following system of equations:

A + B = 0 (for s^2 term)

4A + C = 0 (for constant term)

0s = 1 (for s term)

Solving the system of equations, we find:

A = 0

B = 0

C = 1/4

Therefore, the decomposition becomes:

1/(s(s^2 + 4)) = 1/4(s^2 + 4)/(s^2 + 4) = 1/4(1/s + s/(s^2 + 4))

Taking the Laplace transform of the decomposed terms:

L^(-1){Y(s)} = L^(-1){2(s + 1)/s} + L^(-1){1/4(1/s + s/(s^2 + 4))}

The inverse Laplace transform of 2(s + 1)/s is 2 + 2e^(-t).

For the second term, we have two inverse Laplace transforms to find:

L^(-1){1/4(1/s)} = 1/4

L^(-1){1/4(s^2 + 4)} = 1/4 * sin(2t)

Combining all the terms, we get:

Y(t) = 2 + 2e^(-t) + 1/4 + 1/4 * sin(2t)

Thus, Y(t) = 2 + 2e^(-t) + 1/4 + 1/4 * sin(2t).

Now, let's find X(t) using Laplace transforms for the equation X(s) = (s + 1 * e^(-0.5s))/(s(s + 4)(s + 3)).

Apply the inverse Laplace transform to X(s).

X(t) = L^(-1){(s + 1 * e^(-0.5s))/(s(s + 4)(s + 3))}

Decompose the fraction using partial fraction decomposition:

1/(s(s + 4)(s + 3)) = A/s + B/(s + 4) + C/(s + 3)

Multiplying through by s(s + 4)(s + 3), we get:

1 = A(s + 4)(s + 3) + Bs(s + 3) + C(s)(s + 4)

Expanding the equation, we have:

1 = A(s^2 + 7s + 12) + Bs^2 + 3Bs + Cs^2 + 4Cs

Equating the coefficients of like powers of s, we get the following system of equations:

A + C = 0 (for s^2 term)

7A + 3B + 4C = 0 (for s term)

12A = 1 (for constant term)

Solving the system of equations, we find:

A = 1/12

B = -1/3

C = -1/12

Therefore, the decomposition becomes:

1/(s(s + 4)(s + 3)) = 1/12(1/s - 1/(s + 4) - 1/(s + 3))

Taking the Laplace transform of the decomposed terms:

L^(-1){X(s)} = L^(-1){(1/12)(1/s - 1/(s + 4) - 1/(s + 3))}

The inverse Laplace transform of 1/s is 1.

The inverse Laplace transform of 1/(s + 4) is e^(-4t).

The inverse Laplace transform of 1/(s + 3) is e^(-3t).

Combining all the terms, we get:

X(t) = 1/12 + 1 * e^(-0.5t) - 1 * e^(-4t) - 1 * e^(-3t)

Thus, X(t) = 1/12 + e^(-0.5t) - e^(-4t) - e^(-3t).

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Answer:

F(-2) = 7

Step-by-step explanation:

 Finding a specific point on a function given the x position can be done by substituting x in the function for the given number, then solving said function. The value on the right side of the function is the y position.

For this specific function [F(x) = x² + 3], substitute x for -2.

F(-2) = (2)² + 3

Next, solve the equation.

F(-2) = 4 + 3

F(-2) = 7

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Required answer: f(2) = -1

Detailed explanation:

To find f(-2), we will substitute -2 for x in the function:

\(\bf{f(x)=-x^2+3}\)

\(\bf{f(-2)=-(-2)^2+3}\)

\(\bf{f(-2)=-4+3}\)

\(\bf{f(-2)=-1}\)

\(\hrulefill\)

Have a wonderful day!