Find the value of x.

The selling price for the tickets is $209.

Here, we have

Given:

If the cost of tickets is 190 dollars, and the markup is 10 percent,

We have to find the selling price.

Markup refers to the amount that must be added to the cost price of a product or service in order to make a profit.

It is computed by multiplying the cost price by the markup percentage. To find out what the selling price would be, you just need to add the markup to the cost price.

The markup percentage is 10%.

10 percent of the cost of tickets ($190) is:

$190 x 10/100 = $19

Therefore, the markup is $19.

Now, add the markup to the cost of tickets to obtain the selling price:

Selling price = Cost price + Markup= $190 + $19= $209

Therefore, the selling price for the tickets is $209.

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The one-sided (right side) confidence interval expression for an 85% confidence interval for the population mean is:

[tex]x + 1.04σ/√n < μ\\[/tex]

For a one-sided (right side) confidence interval for the mean of a normal population, the general expression is:

[tex]x + zασ/√n < μ\\[/tex]

where x is the sample mean, zα is the z-score for the desired level of confidence (with area α to the right of it under the standard normal distribution), σ is the population standard deviation, and n is the sample size.

To find the value of a that results in an 85% confidence interval, we need to find the z-score that corresponds to the area to the right of it being 0.15 (since it's a one-sided right-tailed interval).

Using a standard normal distribution table or calculator, we find that the z-score corresponding to a right-tail area of 0.15 is approximately 1.04.

Therefore, the one-sided (right side) confidence interval expression for an 85% confidence interval for the population mean is:

[tex]x + 1.04σ/√n < μ[/tex]

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The percent yield of H2O is 31.01%.

Given: Amount of H2O obtained = 35.6 g

Amount of H2 given = 4.3 g

Amount of O2 given = unlimited

We need to find the percent yield.

Now, let's calculate the theoretical yield of H2O:

From the balanced chemical equation:

2H2 + O2 → 2H2O

We can see that 2 moles of H2 are required to react with 1 mole of O2 to form 2 moles of H2O.

Molar mass of H2 = 2 g/mol

Molar mass of O2 = 32 g/mol

Molar mass of H2O = 18 g/mol

Therefore, 2 moles of H2O will be formed by using:

2 x (2 g + 32 g) = 68 g of the reactants

So, the theoretical yield of H2O is 68 g.

From the question, we have obtained 35.6 g of H2O.

Therefore, the percent yield of H2O is:

Percent yield = (Actual yield/Theoretical yield) x 100

= (35.6/68) x 100= 52.35%

Therefore, the percent yield of H2O is 52.35%.

Given: Amount of H2O obtained = 23.64 g

Amount of H2 given = 6.14 g

Amount of O2 given = 24.0 g

We need to find the percent yield.

Now, let's calculate the theoretical yield of H2O:From the balanced chemical equation:

2H2 + O2 → 2H2O

We can see that 2 moles of H2 are required to react with 1 mole of O2 to form 2 moles of H2O.

Molar mass of H2 = 2 g/mol

Molar mass of O2 = 32 g/mol

Molar mass of H2O = 18 g/mol

Therefore, 2 moles of H2O will be formed by using:

2 x (6.14 g + 32 g) = 76.28 g of the reactants

So, the theoretical yield of H2O is 76.28 g.

From the question, we have obtained 23.64 g of H2O.

Therefore, the percent yield of H2O is:

Percent yield = (Actual yield/Theoretical yield) x 100

= (23.64/76.28) x 100= 31.01%

Therefore, the percent yield of H2O is 31.01%.

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Geometric summations and their variations often occur because of the nature of recursion. The sum of the series i=0 to n-1 (2^i) is 2^n - 1.

The sum of the geometric series i=0 to n-1 (2^i) can be expressed as:

2^n - 1

Therefore, the simple expression for the sum i=0 to n-1 (2^i) is 2^n - 1.

To derive this expression, we can use the formula for the sum of a geometric series:

S = a(1 - r^n) / (1 - r)

In this case, a = 2^0 = 1 (the first term in the series), r = 2 (the common ratio), and n = number of terms in the series (which is n in this case). Substituting these values into the formula, we get:

S = 2^0 * (1 - 2^n) / (1 - 2)

Simplifying, we get:

S = (1 - 2^n) / (-1)

S = 2^n - 1

Therefore, the sum of the series i=0 to n-1 (2^i) is 2^n - 1.

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The frequency of heterozygous individuals in the population of piranhas can be calculated using the Hardy-Weinberg equilibrium equation. The dominant allele 'A' occurs with a frequency of 0.8, Assuming that the recessive allele 'a' occurs with a frequency of 0.2 .

According to the Hardy-Weinberg equilibrium, the frequency of heterozygous individuals (Aa) can be determined using the formula 2 xp xq, where p represents the frequency of the dominant allele and q represents the frequency of the recessive allele. In this case, p = 0.8 and q = 0.2. By substituting the values into the equation, we can calculate the frequency of heterozygous individuals as follows: Frequency of heterozygous individuals = 2 x 0.8 x0.2 = 0.32. Therefore, the frequency of heterozygous individuals in the population of piranhas is 0.32, or 32% (rounded to two decimal places). This means that approximately 32% of the individuals in the population carry both the dominant and recessive alleles, while the remaining individuals are either homozygous dominant (AA) or homozygous recessive (aa).

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To find the pmf of (y1|u = u), where u is a nonnegative integer, we need to use the Poisson distribution. The Poisson distribution describes the probability of a given number of events occurring in a fixed interval of time or space, given that these events occur independently and at a constant average rate. The pmf of (y1|u = u) can be expressed as: P(y1=k|u=u) = (e^-u * u^k) / k! where k is the number of events that occur in the fixed interval, u is the average rate at which events occur, e is Euler's number (approximately equal to 2.71828), and k! is the factorial of k. Therefore, the named distribution for the pmf of (y1|u = u) is the Poisson distribution, with parameter u representing the average rate of events occurring in the fixed interval.

In probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of the number of events occurring in a given time period if the average of these events is known and in independent time since the last event.

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The angle of attack α at zero lift is equal to the zero-lift angle of attack α₀. To provide a specific value, we would need more information about the airfoil being used, such as its camber or profile.

Using thin airfoil theory, we can calculate the angle of attack α when the lift coefficient (Cl) is equal to zero. In thin airfoil theory, the lift coefficient is given by the formula:

Cl = 2π(α - α₀)

Where α₀ is the zero-lift angle of attack. To find α when Cl = 0, we can rearrange the formula:

0 = 2π(α - α₀)

Now, divide both sides by 2π:

0 = α - α₀

Finally, add α₀ to both sides:

α = α₀

So, the angle of attack α at zero lift is equal to the zero-lift angle of attack α₀. To provide a specific value, we would need more information about the airfoil being used, such as its camber or profile.

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The scores earned on the mathematics portion of the SAT, a college entrance exam, are approximately normally distributed with mean 516 and standard deviation 1 16. The scores that separate the middle 90% of test takers from the bottom and top 5% are 333.22 and 698.78, respectively.

Using the mean of 516 and standard deviation of 116, we can standardize the scores using the formula z = (x - μ) / σ, where x is the score, μ is the mean, and σ is the standard deviation.
For the 5th percentile, we want to find the score that 5% of test takers scored below. Using a standard normal distribution table or calculator, we find that the z-score corresponding to the 5th percentile is approximately -1.645.
-1.645 = (x - 516) / 116
Solving for x, we get:
x = -1.645 * 116 + 516 = 333.22
So the score separating the bottom 5% from the rest is approximately 333.22.
For the 95th percentile, we want to find the score that 95% of test takers scored below. Using the same method, we find that the z-score corresponding to the 95th percentile is approximately 1.645.
1.645 = (x - 516) / 116
Solving for x, we get:
x = 1.645 * 116 + 516 = 698.78
So the score separating the top 5% from the rest is approximately 698.78.
Therefore, the scores that separate the middle 90% of test takers from the bottom and top 5% are 333.22 and 698.78, respectively.

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The area of the parallelogram for the given vertices is equal to √110 square units.

To find the area of a parallelogram with vertices A(-1, 2, 4), B(0, 4, 8), C(1, 1, 5), and D(2, 3, 9),

we can use the cross product of two vectors formed by the sides of the parallelogram.

Let us define vectors AB and AC as follows,

AB

= B - A

= (0, 4, 8) - (-1, 2, 4)

= (1, 2, 4)

AC

= C - A

= (1, 1, 5) - (-1, 2, 4)

= (2, -1, 1)

Now, let us calculate the cross product of AB and AC.

AB × AC = (1, 2, 4) × (2, -1, 1)

To compute the cross product, we can use the determinant of a 3x3 matrix.

AB × AC

= (2× 4 - (-1) × 1, -(1 × 4 - 2 × 1), 1 × (-1) - 2 × 2)

= (9, 2, -5)

The magnitude of the cross product gives us the area of the parallelogram.

Let us calculate the magnitude,

|AB × AC|

= √(9² + 2² + (-5)²)

= √(81 + 4 + 25)

= √110

Therefore, the area of the parallelogram with vertices A(-1, 2, 4), B(0, 4, 8), C(1, 1, 5), and D(2, 3, 9) is √110 square units.

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Using the fundamental theorem of calculus, part 2, we have evaluated the integral ∫1−1(t3−t2)dt to be -1/6.

To use the fundamental theorem of calculus, part 2 to evaluate the integral ∫1−1(t3−t2)dt, we first need to find the antiderivative of the integrand. To do this, we can apply the power rule of calculus, which states that the antiderivative of x^n is (x^(n+1))/(n+1) + C, where C is the constant of integration. Using this rule, we can find the antiderivative of t^3 - t^2 as follows:
∫(t^3 - t^2)dt = ∫t^3 dt - ∫t^2 dt
= (t^4/4) - (t^3/3) + C
Now that we have found the antiderivative, we can use the fundamental theorem of calculus, part 2, which states that if F(x) is an antiderivative of f(x), then ∫a^b f(x)dx = F(b) - F(a). Applying this theorem to the integral ∫1−1(t3−t2)dt, we get:
∫1−1(t3−t2)dt = (1^4/4) - (1^3/3) - ((-1)^4/4) + ((-1)^3/3)
= (1/4) - (1/3) - (1/4) - (-1/3)
= -1/6
Therefore, using the fundamental theorem of calculus, part 2, we have evaluated the integral ∫1−1(t3−t2)dt to be -1/6.

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We have factored the matrix A as A = XDX^(-1), where D is the diagonal matrix and X is the invertible matrix.

To factor the matrix A = [[5, 6], [-2, -2]] into a product XDX^(-1), where D is diagonal, we need to find the diagonal matrix D and the invertible matrix X.

First, we find the eigenvalues of A by solving the characteristic equation:

|A - λI| = 0

|5-λ 6 |

|-2 -2-λ| = 0

Expanding the determinant, we get:

(5-λ)(-2-λ) - (6)(-2) = 0

(λ-3)(λ+4) = 0

Solving for λ, we find two eigenvalues: λ = 3 and λ = -4.

Next, we find the corresponding eigenvectors for each eigenvalue:

For λ = 3:

(A - 3I)v = 0

|5-3 6 |

|-2 -2-3| v = 0

|2 6 |

|-2 -5| v = 0

Row-reducing the augmented matrix, we get:

|1 3 | v = 0

|0 0 |

Solving the system of equations, we find that the eigenvector v1 = [3, -1].

For λ = -4:

(A + 4I)v = 0

|5+4 6 |

|-2 -2+4| v = 0

|9 6 |

|-2 2 | v = 0

Row-reducing the augmented matrix, we get:

|1 2 | v = 0

|0 0 |

Solving the system of equations, we find that the eigenvector v2 = [-2, 1].

Now, we can construct the diagonal matrix D using the eigenvalues:

D = |λ1 0 |

|0 λ2|

D = |3 0 |

|0 -4|

Finally, we can construct the matrix X using the eigenvectors:

X = [v1, v2]

X = |3 -2 |

|-1 1 |

To factor the matrix A, we have:

A = XDX^(-1)

A = |5 6 | = |3 -2 | |3 0 | |-2 2 |^(-1)

|-2 -2 | |-1 1 | |0 -4 |

Calculating the matrix product, we get:

A = |5 6 | = |3(3) + (-2)(0) 3(-2) + (-2)(0) | |-2(3) + 2(0) -2(-2) + 2(0) |

|-2 -2 | |-1(3) + 1(0) (-1)(-2) + 1(0) | |(-1)(3) + 1(-2) (-1)(-2) + 1(0) |

A = |5 6 | = |9 -6 | | -2 0 |

|-2 -2 | |-3 2 | | 2 -2 |

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Given: Circle D, AB is a tangent with point A as the point of tangency, and M∠CAB = 105°.

We need to calculate mCEA.

As we can see in the image attached below:[tex][tex][tex]\Delta[/tex][/tex][/tex]

Let us consider the below-given diagram:

[tex]\Delta[/tex]ABC is a right triangle as AB is tangent to circle D at A (a tangent to a circle is perpendicular to the radius of the circle through the point of tangency), therefore, ∠ABC = 90°.

So,

mBAC = 180° – 90°

= 90°.M

∠CAB = 105°

Now, as we know that,

m∠BAC + m∠CAB + m∠ABC = 180°

90° + 105° + m∠ABC = 180°

m∠ABC = 180° - 90° - 105°

m∠ABC = -15°

Therefore,

m∠CEA = m∠CAB - m∠BAC

m∠CEA = 105° - 90°

m∠CEA = 15°

Hence, the value of mCEA is 15 degrees.

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To determine the height of the tree using proportions, we can set up a ratio between the lengths of the shadows and the corresponding heights.

Let's assume:

Andy's height: 5.6 ft

Andy's shadow length: 4.2 ft

Tree's shadow length: 42.3 ft

Unknown tree height: x ft

The proportion can be set up as follows:

(Height of Andy) / (Length of Andy's shadow) = (Height of the tree) / (Length of the tree's shadow

Substituting the given values:

(5.6 ft) / (4.2 ft) = x ft / (42.3 ft)

To solve for x, we can cross-multiply:

(5.6 ft) * (42.3 ft) = (4.2 ft) * (x ft)

235.68 ft = 4.2 ft * x

Now, divide both sides of the equation by 4.2 ft to isolate x:

235.68 ft / 4.2 ft = x

x ≈ 56 ft

Therefore, the estimated height of the tree is approximately 56 feet.

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In the equation y = 6 - 3x, we can observe that the coefficient of x is -3. This coefficient represents the slope of the line. A positive slope indicates a line that rises as x increases, while a negative slope indicates a line that falls as x increases.

Since the slope is -3, it means that for every increase of 1 unit in the x-coordinate, the corresponding y-coordinate decreases by 3 units. This tells us that the line will move downward as we move from left to right along the x-axis.

We can also determine the direction by considering the signs of the coefficients. The coefficient of x is negative (-3), and there is no coefficient of y, which means it is implicitly 1. In this case, the negative coefficient of x implies that as x increases, y decreases, causing the line to slope downward.

So, to summarize, the line defined by y = 6 - 3x has a negative slope (-3), indicating that the line slopes downward as we move from left to right along the x-axis. Therefore, the statement "the line defined by y = 6 - 3x would slope up and to the right" is false. The line slopes down and to the right.

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The second stem-and-leaf combination of 8-5 indicates that the cost of chocolate bars is 85 cents. Similarly, the third stem-and-leaf combination of 8-5 indicates that the cost of chocolate bars is 85 cents. The fourth stem-and-leaf combination of 8-7 indicates that the cost of chocolate bars is 87 cents. The last stem-and-leaf combination of 8-9 indicates that the cost of chocolate bars is 89 cents.

Chocolate bars are on sale for the prices shown in the given stem-and-leaf plot. Cost of a Chocolate Bar (in cents) at Several Different Stores.

Stem Leaf

7 7

8 5 5 7 8 9

9 3 3 3

10 0 5

There are four stores at which the cost of chocolate bars is displayed. Their costs are indicated in cents, and they are categorized in the given stem-and-leaf plot. In a stem-and-leaf plot, the digits in the stem section correspond to the tens place of the data.

The digits in the leaf section correspond to the units place of the data.

To interpret the data, look for patterns in the leaves associated with each stem.

For example, the first stem-and-leaf combination of 7-7 indicates that the cost of chocolate bars is 77 cents.

The second stem-and-leaf combination of 8-5 indicates that the cost of chocolate bars is 85 cents.

Similarly, the third stem-and-leaf combination of 8-5 indicates that the cost of chocolate bars is 85 cents.

The fourth stem-and-leaf combination of 8-7 indicates that the cost of chocolate bars is 87 cents.

The last stem-and-leaf combination of 8-9 indicates that the cost of chocolate bars is 89 cents.

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The limit of the ratio is less than 1, the series converges. Therefore, the series [infinity] n25n − 1 (−6)n n = 1 converges.

To test the series for convergence or divergence, we can use the ratio test.
The ratio test states that if the limit of the absolute value of the ratio of consecutive terms in the series is less than 1, then the series converges. If the limit is greater than 1 or does not exist, then the series diverges.
Let's apply the ratio test to this series:
lim(n→∞) |(n+1)25(n+1) − 1 (−6)n+1| / |n25n − 1 (−6)n|
= lim(n→∞) |(n+1)25n(25/6) − (25/6)n − 1/25| / |n25n (−6/25)|
= lim(n→∞) |(n+1)/n * (25/6) * (1 − (1/(n+1)²))| / 6
= 25/6 * lim(n→∞) (1 − (1/(n+1)²)) / n
= 25/6 * lim(n→∞) (n^2 / (n+1)²) / n
= 25/6 * lim(n→∞) n / (n+1)²
= 0
Since the limit of the ratio is less than 1, the series converges. Therefore, the series [infinity] n25n − 1 (−6)n n = 1 converges.

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TRUE. The ANOVA table from a multiple regression includes the F statistic for assessing overall fit. In this case, the F statistic is 2.83. The F statistic is a ratio of two variances, the between-group variance and the within-group variance.

It is used to test the null hypothesis that all the regression coefficients are equal to zero, which implies that the model does not provide a better fit than the intercept-only model. If the F statistic is larger than the critical value at a chosen significance level, the null hypothesis is rejected, and it can be concluded that the model provides a better fit than the intercept-only model.The F statistic can also be used to compare the fit of two or more models. For example, if we fit two different regression models to the same data, we can compare their F statistics to see which model provides a better fit. However, it is important to note that the F statistic is not always the most appropriate measure of overall fit, and other measures such as adjusted R-squared or AIC may be more informative in some cases.Overall, the F statistic is a useful tool for assessing the overall fit of a multiple regression model and can be used to make comparisons between different models. In this case, the F statistic of 2.83 suggests that the model provides a better fit than the intercept-only model.

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Thus, option A is the correct answer choice which shows the quotient of the given division problem as a simplified fraction in 250 words.

To solve the given division problem and show the quotient as a simplified fraction, we need to follow the steps given below:

Step 1: We need to perform the division of 8/21 ÷ 6/7 by multiplying the dividend with the reciprocal of the divisor.8/21 ÷ 6/7 = 8/21 × 7/6Step 2: We simplify the obtained fraction by cancelling out the common factors.8/21 × 7/6= (2×2×2)/ (3×7) × (7/2×3) = 8/21 × 7/6 = 56/126

Step 3: We reduce the obtained fraction by dividing both the numerator and denominator by the highest common factor (HCF) of 56 and 126.HCF of 56 and 126 = 14

Therefore, the simplified fraction of the quotient is:56/126 = 4/9

Thus, option A is the correct answer choice which shows the quotient of the given division problem as a simplified fraction in 250 words.

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The composite function f(g(x)) consists of an inner function g and an outer function f. When doing a change of variables, the likely choice for a new variable u is: b) u = g(x)

The composite function f(g(x)) consists of an inner function g and an outer function f. When doing a change of variables, the likely choice for a new variable u is: b) u = g(x).
This is because when you choose u = g(x), you can substitute u into the outer function f, making it easier to work with and solve the problem.

A composite function, also known as a function composition, is a mathematical operation that involves combining two or more functions to create a new function.

Given two functions, f and g, the composite function f(g(x)) is formed by first evaluating the function g at x, and then using the result as the input to the function f.

In other words, the output of g becomes the input of f. This can be written as follows:

f(g(x)) = f( g( x ) )

The composite function can be thought of as a chaining of functions, where the output of one function becomes the input of the next function.

It is important to note that the order in which the functions are composed matters, and not all functions can be composed. The domain and range of the functions must also be compatible in order to form a composite function.

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Answer:

total number of votes = 6,492

Step-by-step explanation:

We are given that the ratio of yes to no votes is 7 to 5

This means
[tex]\dfrac{\text{ number of yes votes}}{\text{ number of no votes}}} = \dfrac{7}{5}[/tex]

Number of no votes = 2705

Therefore
[tex]\dfrac{\text{ number of yes votes}}{2705}} = \dfrac{7}{5}[/tex]

[tex]\text{number of yes votes = } 2705 \times \dfrac{7}{5}\\= 3787[/tex]

Total number of votes = 3787 + 2705 = 6,492

Use the fundamental theorem of calculus and the given information the value of f(7) is 15.

First, we know that f'(x) is continuous, which means we can use the fundamental theorem of calculus to find the antiderivative of f'(x), denoted as F(x):

F(x) = ∫ f'(x) dx

Since we know that ∫ 7 5 f'(x) dx = 15, we can use this to find the value of F(7) - F(5):

F(7) - F(5) = ∫ 7 5 f'(x) dx = 15

Next, we can use the fact that f(5) = 13 to find F(5):

F(5) = ∫ f'(x) dx = f(x) + C

f(5) + C = 13

where C is the constant of integration.

Now we can solve for C:

C = 13 - f(5)

Plugging this back into our equation for F(7) - F(5), we get:

F(7) - F(5) = ∫ 7 5 f'(x) dx = 15

F(7) - (f(5) + C) = 15

F(7) = 15 + f(5) + C

F(7) = 15 + 13 - f(5)

F(7) = 28 - f(5)

Finally, we can use the fact that F(7) = f(7) + C to solve for f(7):

f(7) + C = F(7)

f(7) + C = 28 - f(5)

f(7) = 28 - f(5) - C

Substituting C = 13 - f(5), we get:

f(7) = 28 - f(5) - (13 - f(5))

f(7) = 15

Therefore, the value of f(7) is 15.

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For a pattern of dimensions of a quilting square, the blue fabric part that is parallelogram will she need to make one square is equals to the 48 inch².

We have a pattern present in attached figure. It shows the dimensions of a quilting square. We have to determine the length of fabric needed make a complete square. From the figure, there is formed different shapes with different colours, Side of square, a = 12 in.

length of blue parallelogram part of square = 8 in.

So, base length red triangle in square = 12 in. - 8 in. = 4 in.

Height of red triangle, h = 6in.

Same dimensions for other red triangle.

Length of pink parallelogram = 3 in.

Area of square = side²

= 12² = 144 in.²

Now, In case of blue parallelogram, the ares of blue parallelogram, [tex]A = base × height [/tex]

so, Area of blue fabric parallelogram= 8 × 6 in.² = 48 in.²

Hence, required value is 48 in.²

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Complete question:

The above figure complete the question.

The pattern shows the dimensions of a quilting square that need to will use to make a quilt How much blue fabric will she need to make one square

The surface area of the solid in this problem is given as follows:

D. 189 cm².

The figure in the context of this problem is a composite figure, hence we obtain the area of the figure adding the areas of all the parts of the figure.

The figure for this problem is composed as follows:

The surface area of the triangles is given as follows:

4 x 1/2 x 7 x 10 = 140 cm².

The surface area of the square is given as follows:

7² = 49 cm².

Hence the total surface area is given as follows:

A = 140 + 49

A = 189 cm².

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The trailing moving average of span 5 is 6.92.

The trailing moving average of span 5 for the given time series data is as follows:

t-5: (7.6 + 6.2 + 5.4 + 5.4 + 10)/5 = 6.92

t=6: (6.2 + 5.4 + 5.4 + 10 + 7.6)/5 = 6.92

t=7: (5.4 + 5.4 + 10 + 7.6 + 6.2)/5 = 6.92

t=8: (5.4 + 10 + 7.6 + 6.2 + 5.4)/5 = 6.92

t=9: (10 + 7.6 + 6.2 + 5.4 + 5.4)/5 = 6.92

t=10: (7.6 + 6.2 + 5.4 + 5.4 + 10)/5 = 6.92

Therefore, the trailing moving average of span 5 for time periods 5 through 10 is 6.92.

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The angle of refraction is approximately 46.4°, which is close to but not exactly 47°.

When light passes from one medium to another, its path changes due to a phenomenon known as refraction. Snell's Law describes the relationship between the angle of incidence and the angle of refraction when light travels between two media with different indices of refraction. The law is given by:

n1 * sin(θ1) = n2 * sin(θ2)

Here, n1 and n2 are the indices of refraction of medium A and B, respectively, θ1 is the angle of incidence (36° in this case), and θ2 is the angle of refraction.

It is given that the index of refraction of medium A (n1) is 1.25 times that of medium B (n2). Therefore, n1 = 1.25 * n2.

Substituting this relationship into Snell's Law:

(1.25 * n2) * sin(36°) = n2 * sin(θ2)

Dividing both sides by n2:

1.25 * sin(36°) = sin(θ2)

To find the angle of refraction θ2, we can take the inverse sine (arcsin) of both sides:

θ2 = arcsin(1.25 * sin(36°))

Calculating the value:

θ2 ≈ 46.4°

The angle of refraction is approximately 46.4°, which is close to but not exactly 47°.

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The given sequence is 64, 112, 196, 343. We will look for a pattern in the given sequence.

Step 1: The first term is 64.

Step 2: The second term is 112, which is the first term multiplied by 1.75 (112 = 64 x 1.75).

Step 3: The third term is 196, which is the second term multiplied by 1.75 (196 = 112 x 1.75).

Step 4: The fourth term is 343, which is the third term multiplied by 1.75 (343 = 196 x 1.75).

Step 5: Hence, we can see that each term in the sequence is the previous term multiplied by 1.75.So, the recursive formula that can be used to describe the given sequence is: a₁ = 64; aₙ = aₙ₋₁ x 1.75, n ≥ 2.

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The value of the Fourier cosine series at x = 2 is -3/8.

a0 = -3/4 for 0 ≤ x < 2 and a0 = 1/4 for 2 ≤ x ≤ 4.

The value of the Fourier cosine series at x = -3 is -3/8.

To compute the Fourier cosine coefficients for the function f(x) = {0 - (4 - x) for 0 ≤ x < 2, 4 - x for 2 ≤ x ≤ 4}, we need to evaluate the following integrals:

a0 = (1/2L) ∫[0 to L] f(x) dx

an = (1/L) ∫[0 to L] f(x) cos(nπx/L) dx

where L is the period of the function, which is 4 in this case.

Let's calculate the coefficients:

a0 = (1/8) ∫[0 to 4] f(x) dx

For 0 ≤ x < 2:

a0 = (1/8) ∫[0 to 2] (0 - (4 - x)) dx

= (1/8) ∫[0 to 2] (x - 4) dx

= (1/8) [x^2/2 - 4x] [0 to 2]

= (1/8) [(2^2/2 - 4(2)) - (0^2/2 - 4(0))]

= (1/8) [2 - 8]

= (1/8) (-6)

= -3/4

For 2 ≤ x ≤ 4:

a0 = (1/8) ∫[2 to 4] (4 - x) dx

= (1/8) [4x - (x^2/2)] [2 to 4]

= (1/8) [(4(4) - (4^2/2)) - (4(2) - (2^2/2))]

= (1/8) [16 - 8 - 8 + 2]

= (1/8) [2]

= 1/4

Now, let's calculate the values of the Fourier cosine series at the given points:

x = 2:

The Fourier cosine series at x = 2 is given by a0/2 + ∑[n=1 to ∞] an cos(nπx/4).

For x = 2, we have:

a0/2 = (-3/4)/2 = -3/8

an cos(nπx/4) = 0 (since cos(nπx/4) becomes zero for all values of n)

x = -3:

The Fourier cosine series at x = -3 is given by a0/2 + ∑[n=1 to ∞] an cos(nπx/4).

For x = -3, we have:

a0/2 = (-3/4)/2 = -3/8

an cos(nπx/4) = 0 (since cos(nπx/4) becomes zero for all values of n)

x = 5:

The Fourier cosine series at x = 5 is given by a0/2 + ∑[n=1 to ∞] an cos(nπx/4).

For x = 5, we have:

a0/2 = (1/4)/2 = 1/8

an cos(nπx/4) = 0

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The side b has a length of 19.8 cm.

To find the value of side b in the scalene triangle, we can follow these steps:

Step 1: Understand the information given.

The perimeter of the triangle is 54.6 cm.

The base of the triangle, labeled 3a, is three times the length of the shortest side, a.

Side a measures 8.7 cm.

Step 2: Set up the equation.

The equation to find the value of b is: b = 54.6 - (3a + a).

Step 3: Substitute the given values.

Substitute a = 8.7 cm into the equation: b = 54.6 - (3 * 8.7 + 8.7).

Step 4: Simplify and calculate.

Calculate 3 * 8.7 = 26.1.

Calculate (3 * 8.7 + 8.7) = 34.8.

Substitute this value into the equation: b = 54.6 - 34.8.

Calculate b: b = 19.8 cm.

By substituting a = 8.7 cm into the equation, we determined that side b has a length of 19.8 cm.

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