5. [3 points) Consider the vectors v1 = |2| v2= |3|, v3=|1| . |2| |1| |h| . |-4| |-4| |4| |Determine all values of the parameter h such that the vectors Vi, V2 and v3 are linearly dependent. You should justify your answer.

The running time of QuickSort in this case is O(n log(n)), which is the same as the running time when QuickSort splits the list exactly in half.

When QuickSort splits the list into one sublist of size an and another sublist of size (1-a)n, the recurrence relation for the running time of QuickSort becomes T(n) = T(an) + T((1-a)n) + O(n). This is because the two sublists have different sizes and therefore take different amounts of time to sort.

To solve this recurrence relation, we can use the recursion tree method. The recursion tree for this recurrence relation looks like this:

```
       T(n)
      /     \
  T(an)    T((1-a)n)
 /   \      /     \
T(a^2n) T(a(1-a)n) T(a(1-a)n) T((1-a)^2n)
...
```

At each level of the recursion tree, the size of the subproblems decreases by a factor of a or (1-a), and the number of subproblems doubles. The work done at each level is O(n), since the partitioning step takes O(n) time.

The recursion tree has log_{1/a}(n) levels, since the size of the subproblems decreases by a factor of a at each level. Therefore, the total work done by QuickSort is O(n log_{1/a}(n)) = O(n log(n)), since log_{1/a}(n) = log(n)/log(1/a) = log(n)/(-log(a)) = -log(n)/log(a) = O(log(n)).

So the running time of QuickSort in this case is O(n log(n)), which is the same as the running time when QuickSort splits the list exactly in half.

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The equation that shows the volume of the rectangular prism as a product of its edge lengths is (5/3)(1/3)(2/3) = 10/27 in cube. Option 2 is the correct answer.

The quantity of space within a three-dimensional object is its volume. The fundamentals of such forms are described in our page on three-dimensional shapes. Calculating volume is probably not something you will do as frequently as calculating area in the real world.

Even so, it may still be significant. Knowing how to calculate volume will help you determine things like how much room you have for packing when moving house, how much room you need for an office, or how much jam you can put into a jar. It can also be helpful for deciphering what the media means when they discuss a dam's capacity or a river's flow.

The volume of the rectangular prims is given by the formula:

V = lwh

Substituting the values we have:

V = (5/3)(1/3)(2/3) = 10/27

Hence, the equation that shows the volume of the rectangular prism as a product of its edge lengths is (5/3)(1/3)(2/3) = 10/27 in cube. Option 2 is the correct answer.

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1) The ratios 1/3 , 7/21 are in proportion.

2) The ratios 2/5 , 40/16 are not in proportion.

3) the ratios 48/9 , 16/3 are in proportion.

We know that two ratios are said to be in proportion when both the ratios are equivalent.

Consider 1/3 , 7/21

Here 7/21 = (7 × 1)/(7 × 3)

                = 1/3

This means 1/3 = 7/21

Thus, the ratios 1/3 , 7/21 are in proportion.

Consider 48/9, 16/3

Here 48/9 = (16 × 3)/(3 × 3)

                = 16/3

This means 48/9 = 16/3

Thus, the ratios 48/9, 16/3 are in proportion.

Now consider 2/5 , 40/16

Here 40/16 = (8 × 5)/(8 × 2)

                = 5/2

This means 2/5 ≠ 5/2

Thus, the ratios 2/5 , 40/16 are not in proportion.

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Step-by-step explanation:

The formula for the standard error of a proportion is:

SE = sqrt((p*(1-p))/n)

where p is the sample proportion and n is the sample size.

To find p, we need to first calculate the proportion of people who smoke regularly:

p = 71/635 = 0.1118

Now we can plug in the values for p and n:

SE = sqrt((0.1118*(1-0.1118))/635) = 0.0194

Therefore, the standard error for the sample proportion is 0.0194.

Check the picture below.

[tex]\begin{array}{llll} f(x)~from\\\\ x_1 ~~ to ~~ x_2 \end{array}~\hfill slope = m \implies \cfrac{ \stackrel{rise}{f(x_2) - f(x_1)}}{ \underset{run}{x_2 - x_1}}\impliedby \begin{array}{llll} average~rate\\ of~change \end{array} \\\\[-0.35em] ~\dotfill\\\\ \begin{cases} x_1=0\\ x_2=2 \end{cases}\implies \cfrac{f(2)-f(0)}{2 - 0}\implies \cfrac{18-12}{2}\implies \cfrac{6}{2}\implies \text{\LARGE 3}[/tex]

Using synthetic division, we can find that h(-1) = -5.

To find h(-1) using synthetic division, we can use the following steps:

Write the coefficients of the polynomial in descending order of the exponents: 36, 36, 0, 7, 12, 39
Write the value of -1 to the left of the coefficients: -1 | 36 36 0 7 12 39
Bring down the first coefficient: -1 | 36 36 0 7 12 39
         36
Multiply the first coefficient by -1 and write the result under the second coefficient: -1 | 36 36 0 7 12 39
         36 -36
Add the second coefficient and the result: -1 | 36 36 0 7 12 39
         36   0
Repeat steps 4 and 5 for the remaining coefficients: -1 | 36 36 0 7 12 39
         36   0  0 -7 -5
The last number in the bottom row is the remainder, which is the value of h(-1): h(-1) = -5

Therefore, using synthetic division, we can find that h(-1) = -5.

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Answer:

Step-by-step explanation:

SHORT ANSWER: perimeter = 4 * (6x+8) = 24x + 32

So, depending on which expression represents the length of the side of the square, the perimeter would be either 40x or 24x + 32.

LONG ANSWER WITH STEP-BY STEP:

If one side of the square has a length of 10x, then all sides have a length of 10x. Alternatively, if one side has a length of 6x+8, then all sides have a length of 6x+8. Therefore, we can write:

perimeter = 4 * side length

Substituting in the given expressions for the side length, we get:

perimeter = 4 * (10x) = 40x

or

perimeter = 4 * (6x+8) = 24x + 32

So, depending on which expression represents the length of the side of the square, the perimeter would be either 40x or 24x + 32.

Answer: 30

Step-by-step explanation:

Use Pythagorean's Theorem, a² + b² = c²

18² + 24² = c²

324 + 576 = c²

900 = c²

√900 = c

c = 30

The missing side is 30.

Hope this helps!

Answer:

FDG is 28 degrees, and x = 4

Step-by-step explanation:

We can see for angle EDG that there is a little box at the angle. This means that it is a right angle, which means it is 90 degrees. This, in turn, means that the two angles formed there (angle EDF, and angle FDG) will add up to that number, 90 degrees because they are within the initial angle.

Let us create an equation using our angles. We know one is 62 degrees and the other is 7x degrees.

62 +7x = 90

7x = 28

x =4

This means that the unknown value is 4, and since the angle was 7x, it is really 7(4) = 28 degrees, and we know the other angle is 62 degrees.

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The division of polynomials "(4x^(3)-5x^(2)+8x-8)/(x^(2)-3x)" gives the expression "4x+7+(29x-8)/(x^(2)-3x)".

To perform the indicated division, we will use polynomial long division.

First, we will divide the first term of the dividend, 4x^3, by the first term of the divisor, x^2, to get 4x. This will be the first term of our quotient.

Next, we will multiply 4x by the divisor, x^2-3x, to get 4x^3-12x^2. We will then subtract this from the dividend to get 7x^2+8x-8.

We will then repeat this process by dividing the first term of the new dividend, 7x^2, by the first term of the divisor, x^2, to get 7. This will be the second term of our quotient.

We will then multiply 7 by the divisor, x^2-3x, to get 7x^2-21x. We will subtract this from the new dividend to get 29x-8.

Since the degree of the new dividend, 29x-8, is lower than the degree of the divisor, x^2-3x, we are done with the division and 29x-8 will be our remainder.

Therefore, the final answer is:

(4x^(3)-5x^(2)+8x-8)/(x^(2)-3x) = 4x+7+(29x-8)/(x^(2)-3x)

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when x=24, y = y(3)/(2).

Given that y is directly proportional to the cube root of (x+3), we can write this relationship as:

y = k * cube root (x+3)

Where k is the constant of proportionality. We can use the given values of x and y to find k:

y(2)/(3 ) = k * cube root (5+3)

y(2)/(3 ) = k * cube root (8)

y(2)/(3 ) = k * 2

k = y(2)/(3 ) / 2

Now we can use this value of k to find y when x=24:

y = k * cube root (24+3)

y = (y(2)/(3 ) / 2) * cube root (27)

y = (y(2)/(3 ) / 2) * 3

y = y(2)/(3 ) * (3/2)

y = y(3)/(2)

Therefore, when x=24, y = y(3)/(2).

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Thhe standard error of the sample mean is 0.208333.  The probability that the sample mean is within 0.5 units of the population mean is 0.9918. We need to take a sample size of at least 7 to be 98% confident that the sample mean is within one-half of the population mean.

(i) The standard error of the sample mean is calculated by dividing the standard deviation of the population by the square root of the sample size. In this case, the standard error of the sample mean is 1.25 / √36 = 1.25 / 6 = 0.208333.

(ii) To find the probability that the sample mean is within 0.5 units of the population mean, we need to use the standard normal distribution. We can find the z-score for 0.5 units away from the mean by dividing 0.5 by the standard error of the sample mean, which is 0.5 / 0.208333 = 2.4.

Using a standard normal table, we can find the probability that the sample mean is within 2.4 standard deviations of the population mean, which is 0.9918.

(iii) To be 98% confident that the sample mean is within one-half of the population mean, we need to find the sample size that corresponds to a z-score of 2.33 (the z-score for a 98% confidence interval). We can use the formula for the standard error of the sample mean to solve for the sample size:

0.5 = 1.25 / √n

√n = 1.25 / 0.5

n = (1.25 / 0.5)²

n = 6.25

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The vectors [[4],[0],[0]],[[9],[3],[-6]],[[6],[6],[-18]] are linearly independent.

To determine if the vectors are linearly independent, we can use the determinant of the matrix formed by the vectors. If the determinant is not equal to 0, then the vectors are linearly independent.

The matrix formed by the vectors is:
\begin{bmatrix}
4 & 9 & 6\\
0 & 3 & 6\\
0 & -6 & -18
\end{bmatrix}

The determinant of this matrix is:
\begin{vmatrix}
4 & 9 & 6\\
0 & 3 & 6\\
0 & -6 & -18
\end{vmatrix} = 4\begin{vmatrix}
3 & 6\\
-6 & -18
\end{vmatrix} - 9\begin{vmatrix}
0 & 6\\
0 & -18
\end{vmatrix} + 6\begin{vmatrix}
0 & 3\\
0 & -6
\end{vmatrix} = 4(-54-(-36)) - 9(0-0) + 6(0-0) = 4(-18) = -72

Since the determinant is not equal to 0, the vectors are linearly independent. Therefore, the vectors [[4],[0],[0]],[[9],[3],[-6]],[[6],[6],[-18]] are linearly independent.

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Option C. (x+2)/(x-3)*(4)/(x+2) is the equivalent rational expression to (4)/(x-3).

The rational expression that is equivalent to this expression (4)/(x-3) is option C. (x+2)/(x-3)*(4)/(x+2).
We can simplify the rational expression (x+2)/(x-3)*(4)/(x+2) by canceling out the common factor (x+2) from the numerator and denominator. This will give us the equivalent rational expression:

(x+2)/(x-3)*(4)/(x+2) = (4)/(x-3)

Therefore, option C. (x+2)/(x-3)*(4)/(x+2) is the equivalent rational expression to (4)/(x-3).

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Answer:

The height of the kite from the ground nearest to the tenth will be 443.6 feet.

Step-by-step explanation:

Answer:

Step-by-step explanation:

To calculate your new income after an increase of 8.2%, you can use the following formula:

New income = Old income + (Percentage increase * Old income)

Plugging in the values given in the problem, we get:

New income = 420,600,000 + (8.2% * 420,600,000)

New income = 420,600,000 + (0.082 * 420,600,000)

New income = 420,600,000 + 34,524,120

New income = 455,124,120

Therefore, your new income after an increase of 8.2% would be 455,124,120.

Answer: I could be wrong but its either going to be 13.5 or 13.

Step-by-step explanation:

Answer:

13.5 goals

Step-by-step explanation:

We can use a ratio to solve

3 goals                   x

---------------  = -------------

2 games            9 games

Using cross products

3 * 9 = 2x

27 = 2x

Divide each side by 2

27/2 = x

13.5 goals

A professional golfer to make about 40.7 holes over 5 rounds of golf with the new club, while an amateur golfer would only make about 1.6.

First, let's define some variables to represent the probabilities of making a hole for a professional golfer and an amateur golfer:

Let p be the probability that a professional golfer makes a hole with the new club.

Let q be the probability that an amateur golfer makes a hole with the new club.

According to the company's claims, we know that:

p = 1.2q (since the professional golfer makes a hole 120% of the time, which is 1.2 times the probability of the amateur golfer making a hole)

Next, we need to determine the probability of each golfer making a hole during one round of golf, which consists of 18 holes. Let's assume that each hole is independent of the others, meaning that the outcome of one hole does not affect the outcome of another. In that case, the probability of making at least one hole in a round can be calculated using the complement rule:

The probability that a professional golfer makes at least one hole in a round is 1 minus the probability that the golfer misses every hole: [tex]1 - (1-p)^{18} .[/tex]

The probability that an amateur golfer makes at least one hole in a round is[tex]1 - (1-q)^{18} .[/tex]

Now, let's use these probabilities to calculate the expected number of holes each golfer will make in 5 rounds of golf:

The expected number of holes made by a professional golfer in 5 rounds is 5 times the expected number of holes made in one round, which is [tex](1 - (1-p)^{18} )\times18.[/tex]

The expected number of holes made by an amateur golfer in 5 rounds is 5 times the expected number of holes made in one round, which is [tex](1 - (1-q)^{18} )\times18.[/tex]

We can simplify these expressions using the relationship between p and q:

The expected number of holes made by a professional golfer in 5 rounds is [tex]518(1 - (1-1.2q)^{18} ).[/tex]

The expected number of holes made by an amateur golfer in 5 rounds is [tex]518(1 - (1-q)^{18} ).[/tex]

We can now evaluate these expressions using the values of p and q:

[tex]p = 1.2q, so q = p/1.2[/tex]

Substituting this into the expressions above, we get:

The expected number of holes made by a professional golfer in 5 rounds is[tex]518(1 - (1-1.2(p/1.2))^{18} ) = 518(1 - (1-p)^{18} ).[/tex]

The expected number of holes made by an amateur golfer in 5 rounds is [tex]518(1 - (1-p/1.2)^{18} ).[/tex]

Finally, we can evaluate these expressions using the given probabilities:

The expected number of holes made by a professional golfer in 5 rounds is[tex]518(1 - (1-1.2q)^{18} ) = 518(1 - (1-1.2(0.1))^{18} ) = 40.7.[/tex]

The expected number of holes made by an amateur golfer in 5 rounds is [tex]518(1 - (1-q)^{18} ) = 518(1 - (1-0.1/1.2)^{18} ) = 1.6.[/tex]

So according to these calculations, we would expect a professional golfer to make about 40.7 holes over 5 rounds of golf with the new club, while an amateur golfer would only make about 1.6

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Answer:

We can use the given information to find the ant's crawling speed and then use that to calculate how far it can crawl in a given amount of time.

The ant crawls 12 feet in 10 minutes, so its crawling speed is:

12 feet / 10 minutes = 1.2 feet/minute

To find how far the ant can crawl in 20 minutes, we can multiply its crawling speed by the time:

Distance in 20 minutes = crawling speed x time

Distance in 20 minutes = 1.2 feet/minute x 20 minutes

Distance in 20 minutes = 24 feet

Therefore, the ant can crawl 24 feet in 20 minutes.

To find how far the ant can crawl in 30 minutes, we can use the same formula:

Distance in 30 minutes = crawling speed x time

Distance in 30 minutes = 1.2 feet/minute x 30 minutes

Distance in 30 minutes = 36 feet

Therefore, the ant can crawl 36 feet in 30 minutes.

Answer:

20 mins=24 feet

30 mins=36 feet

Step-by-step explanation:

if 12 feet = 10 mins and 10 times 2 = 20 then 12 times 2 =24

same for 30 mins.

10 times 3= 30 then 12 times 3= 36

The dot plot allows us to know the exact data we are graphing.

In a dot plot, we have the different possible values on the horizontal axis, and dots above each of these values that count how many times that value has appeared in an experiment.

Instead, in a whisker plot or a box plot we have a representation in a kinda of "rectangle with whiskers".

This type of plot is usefull to find the quartiles of the data set, but not to know exactly how many of each data points we have, like in the dot plot.

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300  kilowatt-hours per month does one have to return to the grid to break even.

To find out how many kilowatt-hours per month one has to return to the grid to break even, we need to set up an equation and solve for x, where x is the number of kilowatt-hours returned to the grid.

The equation would be:

24 = 0.08x

To solve for x, we need to isolate the variable on one side of the equation. We can do this by dividing both sides of the equation by 0.08:

24/0.08 = x

x = 300

Therefore, one would have to return 300 kilowatt-hours per month to the grid to break even.

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The partial derivatives of f with respect to u and v at (u,v) = (-2, -2) are ∂f/∂u(-2,-2) = 224 and ∂f/∂v(-2,-2) = -1056.

To evaluate the partial derivatives ∂f/∂u and ∂f/∂v at (u,v) = (-2, -2) using the Chain Rule, we need to find the derivatives of f with respect to x, y, and z, and then multiply them by the derivatives of x, y, and z with respect to u and v.

First, let's find the derivatives of f with respect to x, y, and z:
∂f/∂x = 3x^2
∂f/∂y = z^2
∂f/∂z = 2yz

Next, let's find the derivatives of x, y, and z with respect to u and v:
∂x/∂u = 2u
∂x/∂v = 1
∂y/∂u = 1
∂y/∂v = 2v
∂z/∂u = 4v
∂z/∂v = 4u

Now, we can use the Chain Rule to find the partial derivatives of f with respect to u and v:
∂f/∂u = (3x^2)(2u) + (z^2)(1) + (2yz)(4v)
∂f/∂v = (3x^2)(1) + (z^2)(2v) + (2yz)(4u)

Finally, we can plug in the values of u and v to find the partial derivatives at (u,v) = (-2, -2):
∂f/∂u(-2,-2) = (3((-2)^2 + (-2))^2)(2(-2)) + ((4(-2)(-2))^2)(1) + (2((-2)+(-2)^2)(4(-2)))
= (3(0)^2)(-4) + (16^2)(1) + (2(2)(-8))
= 0 + 256 - 32
= 224

∂f/∂v(-2,-2) = (3((-2)^2 + (-2))^2)(1) + ((4(-2)(-2))^2)(2(-2)) + (2((-2)+(-2)^2)(4(-2)))
= (3(0)^2)(1) + (16^2)(-4) + (2(2)(-8))
= 0 - 1024 - 32
= -1056

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These two numbers are -4, 2. The smallest possible product is -8.

To find a pair of numbers that satisfy the given conditions, we can use algebra. Let x be the first number and y be the second number. According to the problem, one number is 8 less than twice a second number. This can be written as:
x = 2y - 8
We need to find the product of these two numbers, which is x*y. Substituting the value of x from the equation above, we get:
x*y = (2y - 8)*y
= 2y^2 - 8y
To find the smallest possible product, we need to minimize this expression. We can do this by finding the vertex of the parabola represented by this equation. The vertex of a parabola in the form ax^2 + bx + c is given by (-b/2a, f(-b/2a)). In this case, a = 2, b = -8, and c = 0. So, the vertex is:
(-b/2a, f(-b/2a)) = (-(-8)/(2*2), f(-(-8)/(2*2)))
= (2, f(2))
Substituting y = 2 into the equation for the product, we get:
x*y = 2(2)^2 - 8(2)
= 8 - 16
= -8
So, the smallest possible product is -8. To find the pair of numbers that give this product, we can substitute y = 2 into the equation for x:
x = 2y - 8
= 2(2) - 8
= -4
Therefore, the pair of numbers are -4 and 2.

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To determine whether the given vectors in $\mathbb{R}^3$ form a basis for $\mathbb{R}^3$, we need to check if they are linearly independent and span $\mathbb{R}^3$.

To check for linear independence, we set up the augmented matrix:

[

0

5

7

0

0

2

2

0

2

9

11

0

]

⎣

⎡

​

0

0

2

​

5

2

9

​

7

2

11

​

0

0

0

⎦

⎤

We reduce this to echelon form:

[

1

0

2

0

0

1

1

0

0

0

0

0

]

⎣

⎡

​

1

0

0

​

0

1

0

2

1

0

​

0

0

0

​

⎦

⎤

​

Since there is a row of zeros, the rank of the matrix is less than 3, which means the vectors are linearly dependent.

Therefore, the given vectors do not form a basis for $\mathbb{R}^3$.

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Using the numbers 1 to 9 (one time each), the boxes can be filled in the following way to make the equation true. 2:2 = 3×3:9 = 4×4 =16.

An arithmetic contrast among two numbers is known as a percentage. Two sets of provided numbers are considered to be approximately equal with respect to one another in conformity with the rules of proportion if they increase or decrease by the same ratio.

We must create the equation to prove the equality by utilizing each of the numerals 1 through 9 once. Any of the numbers between 1 and 9 are equivalent to 1 and 2.

Let the ratio equal 1.

The comparable ratios are therefore 1:1, 2:2, and 3:3.

2:2 = 3×3:9 = 4×4 =16

Therefore, using the numbers 1 to 9 (one time each), the boxes can be filled in the following way to make the equation true. 2:2 = 3×3:9 = 4×4 =16.

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Answer:

sin(B)/b = sin(A)/a

sin(B)/12 = sin(58)/a

a = 12(sin(58)/sin(B))

Now we can use the Law of Cosines to find the remaining sides of the triangle:

a^2 = b^2 + c^2 - 2bc*cos(A)

a^2 = 12^2 + c^2 - 2(12)(c)*cos(58)

c^2 - 24c*cos(58) + 144 - a^2 = 0

Using the quadratic formula, we get:

c = (24*cos(58) ± sqrt((24*cos(58))^2 - 4(1)(144 - a^2)))/2(1)

c = 12*cos(58) ± sqrt(144*cos(58)^2 - 4(144 - a^2))

c = 12*cos(58) ± sqrt(576*cos(58)^2 - 4a^2)

c = 12*cos(58) ± sqrt(576*(1 - sin(58)^2) - 4a^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 4a^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 4(12(sin(58)/sin(B)))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/sin(B))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(180 - A - B)))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(180 - 58 - B)))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - B)))^2)

Now we can substitute the value we found for a into the equation for c to get:

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - B)))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(a/b))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(12/a))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(12/(12(sin(58)/sin(B)))))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(sin(58)/sin(B))))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(sin(58)/(12*sin(58)/a))))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(a/12))))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - arcsin(1/12)*a))))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(122 - 4.98)*a))))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(sin(117.02)*a))))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(sin(58)/(0.97*a))))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(1.03*a/sin(58))))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(1.03*(12*sin(58)/sin(B))/sin(58))))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(1.03*(12/sin(B)))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(1.03*(12/sin(180 - A - B)))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(1.03*(12/sin(180 - 58 - B)))))^2)

c = 12*cos(58) ± sqrt(576 - 576*sin(58)^2 - 576(1.03*(12/sin(122 - B)))))^2)

Now we can solve for c using the two possible values of B:

B = arcsin(b*sin(A)/a)

B = arcsin(12*sin(58)/a)

B = arcsin(12*sin(58)/(12*sin(58)/sin(B)))

B = arcsin(sin(B))

B = 58

or

B = 180 - arcsin(b*sin(A)/a)

B = 180 - arcsin(12*sin

The final answer is 16 - 1 = 15 linear extensions of the poset P.

The poset P has 8 linear extensions. We can determine the number of linear extensions by counting the number of topological sorts of the poset. A topological sort of a poset is a linear ordering of its vertices that respects the partial ordering. There are two ways to obtain a topological sort of the poset P:

1. Start with vertex A, then choose either B or F, then choose either C or G, then choose either D or H, and finally choose E. This gives us a total of 2 x 2 x 2 x 1 = 8 topological sorts.

2. Start with vertex F, then choose either A or G, then choose either B or H, then choose either C or D, and finally choose E. This also gives us a total of 2 x 2 x 2 x 1 = 8 topological sorts.

Therefore, the total number of linear extensions of the poset P is 8 + 8 = 16.

However, we need to subtract the number of topological sorts that are counted twice. These are the topological sorts that start with A, then choose F, then choose G, then choose H, and finally choose E. There are 1 x 1 x 1 x 1 x 1 = 1 of these topological sorts.

So the final answer is 16 - 1 = 15 linear extensions of the poset P.

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The two equivalent expressions are the second one and the last:

(56x + 24)/8 and 3 + 7x

We want to find an equivalent expression to 56x + 24 divided by 8, so we want to simplify the expression:

(56x + 24)/8   (that is the second expression)

We can distribute that division so we get:

(56x + 24)/8 = (56x)/8 + 24/8

Now we can simplify these two quotients so we get:

(56x)/8 + 24/8 = 7x + 3  (that is the last expression).

Then the two equivalent expressions are:

(56x + 24)/8 and 3 + 7x

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The approximate value of x to the nearest tenth of a degree is 36.9 degrees.

To find x to the nearest tenth of a degree, we can use trigonometric ratios for right triangles. First, we need to determine which trigonometric ratio to use based on the given information.

If we are given the opposite side and the adjacent side of the right triangle, we can use the tangent ratio:

tan(x) = opposite/adjacent

If we are given the opposite side and the hypotenuse, we can use the sine ratio:

sin(x) = opposite/hypotenuse

If we are given the adjacent side and the hypotenuse, we can use the cosine ratio:

cos(x) = adjacent/hypotenuse

Once we have determined the appropriate trigonometric ratio, we can plug in the given values and solve for x. To find x to the nearest tenth of a degree, we can use a calculator to find the approximate value of x and then round to the nearest tenth.

For example, if we are given a right triangle with an opposite side of 3 and an adjacent side of 4, we can use the tangent ratio to find x:

tan(x) = 3/4

x = tan^-1(3/4)

Using a calculator, we find that x is approximately 36.87 degrees. To the nearest tenth of a degree, x is approximately 36.9 degrees.

Therefore, the approximate value of x to the nearest tenth of a degree is 36.9 degrees.

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The expression is (+(13)/(18))+(-(1)/(3)). To simplify this expression, we need to combine like terms. First, we can combine the fractions with different denominators. The LCD (Least Common Denominator) for this expression is 18. Therefore, we need to convert each fraction to an equivalent fraction with a denominator of 18.

For the first fraction, (+(13)/(18)), 13/18 can be reduced to 1/2. To get this, multiply both the numerator and denominator by 2, resulting in 13/18 = 2/4 = 1/2.

For the second fraction, +(-(1)/(3)), 1/3 can be reduced to 6/18. To get this, multiply both the numerator and denominator by 6, resulting in 1/3 = 6/18.

Now that all the fractions have a common denominator, we can add the two fractions together. 1/2 + 6/18 = 8/18. Therefore, the simplified form of the expression is 8/18.

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