which coordinate axes are the following cylinders in r ^ 3 parallel? x ^ 2 5z ^ 2 = 8; x ^ 2 5y ^ 2 = 8; z ^ 2 5y ^ 2 = 8

The  probability that X lies between 0.2 and 0.6 is approximately 0.0828.

the probability that X is less than 0.3 is approximately 0.002.

the expected value of X is 1/6, or approximately 0.167 rounded to three decimal places.

To answer this question, we need to use the properties of probability density functions. The probability density function f(x) for a continuous random variable X must satisfy the following two properties:

1. f(x) ≥ 0 for all x
2. The total area under the curve of f(x) over the entire range of X must be equal to 1.

Using this information, we can answer the following:

a) What is the probability that X lies between 0.2 and 0.6?

To find the probability that X lies between 0.2 and 0.6, we need to integrate the probability density function f(x) over this range:

P(0.2 ≤ X ≤ 0.6) = ∫[0.2,0.6] f(x) dx
                   = ∫[0.2,0.6] 5x^4 dx
                   = [x^5]0.2^0.6
                   = (0.6^5 - 0.2^5)
                   ≈ 0.0828

Therefore, the probability that X lies between 0.2 and 0.6 is approximately 0.0828.

b) What is the probability that X is less than 0.3?

To find the probability that X is less than 0.3, we need to integrate the probability density function f(x) from 0 to 0.3:

P(X ≤ 0.3) = ∫[0,0.3] f(x) dx
           = ∫[0,0.3] 5x^4 dx
           = [x^5]0^0.3
           = 0.3^5
           = 0.00243

Therefore, the probability that X is less than 0.3 is approximately 0.002.

c) What is the expected value of X?

The expected value of X, denoted E(X), is defined as the mean of the probability density function f(x), weighted by the probabilities:

E(X) = ∫[0,1] x f(x) dx

Using the given probability density function, we can calculate the expected value as:

E(X) = ∫[0,1] x (5x^4) dx
    = ∫[0,1] 5x^5 dx
    = [x^6/6]0^1
    = 1/6

Therefore, the expected value of X is 1/6, or approximately 0.167 rounded to three decimal places.

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We need 12 tiles to cover the floor.

What is the fraction?

A fraction is a mathematical representation of a part of a whole, where the whole is divided into equal parts. A fraction consists of two numbers, one written above the other and separated by a horizontal line, which is called the fraction bar or the vinculum.

To cover the floor, we need to find how many tiles of side 3 m can fit into the length and width of the floor.

The number of tiles that can fit along the length of the floor is:

10 m / 3 m = 3.33

Since we can't use a fraction of a tile, we round up to 4 tiles.

Similarly, the number of tiles that can fit along the width of the floor is:

9 m / 3 m = 3

So, we need 4 tiles along the length and 3 tiles along the width.

The total number of tiles needed to cover the floor is:

4 tiles x 3 tiles = 12 tiles.

Therefore, we need 12 tiles to cover the floor.

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To solve these probability questions, we can utilize the Poisson distribution, which is commonly used to model the number of events occurring in a fixed interval of time or space.

In this case, we assume the number of airplane crashes follows a Poisson distribution with an average of 2.2  per month.

(a) To find the probability of less than 5 accidents in the next month, we sum up the probabilities of having 0, 1, 2, 3, and 4 accidents using the Poisson distribution formula. The probability can be calculated as follows:

P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)

(b) To calculate the probability of more than 2 accidents in the next 3 months, we need to find the complement of having 0, 1, or 2 accidents in the next 3 months. We can calculate the complement as follows:

P(X > 2 in 3 months) = 1 - [P(X = 0 in 3 months) + P(X = 1 in 3 months) + P(X = 2 in 3 months)]

(c) To determine the probability of exactly 6 accidents in the next 4 months, we use the Poisson distribution formula:

P(X = 6 in 4 months)

To calculate these probabilities, we need to use the Poisson distribution formula with the given average rate of 2.2 crashes per month.

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To find the maximum and minimum values of the function f(x, y) = exy subject to x^3 + y^3 = 54, we need to use the method of Lagrange multipliers.

Let's define g(x,y) = x^3 + y^3 - 54 as our constraint equation. Then, the Lagrangian function is:

L(x,y,λ) = exy + λ(x^3 + y^3 - 54)

Taking the partial derivatives with respect to x, y, and λ and setting them equal to 0, we get:

∂L/∂x = ey + 3λx^2 = 0
∂L/∂y = ex + 3λy^2 = 0
∂L/∂λ = x^3 + y^3 - 54 = 0

From the first two equations, we can solve for x and y in terms of λ:

x = (-ey/3λ)^(1/2)
y = (-ex/3λ)^(1/2)

Substituting these expressions into the third equation, we get:

(-ex/3λ)^(3/2) + (-ey/3λ)^(3/2) - 54 = 0

We can solve for λ in terms of e:

λ = e^(2/3)/(2*3^(1/3))

Substituting this back into the expressions for x and y, we get:

x = 3^(1/6)*e^(1/3)/y^(1/2)
y = 3^(1/6)*e^(1/3)/x^(1/2)

Now, we can find the critical points by setting the partial derivatives of f(x,y) = exy equal to 0:

∂f/∂x = ey(x) = 0
∂f/∂y = ex(y) = 0

From the expressions for x and y above, we see that x and y cannot be 0. Therefore, the only critical point is when e^(xy) = 0, which is not possible.

Thus, the function has no critical points in the interior of the region defined by the constraint equation. This means that the maximum and minimum values of the function must occur on the boundary of the region.

We can parametrize the boundary using polar coordinates:

x = 3^(1/3)cos(t)
y = 3^(1/3)sin(t)

Substituting these into f(x,y) = exy, we get:

f(t) = e^(3^(2/3)cos(t)sin(t))

To find the maximum and minimum values of f(t), we can take the derivative with respect to t and set it equal to 0:

f'(t) = 3^(2/3)e^(3^(2/3)cos(t)sin(t))(cos(2t) - sin(2t)) = 0

The solutions to this equation are t = π/4 and t = 5π/4.

Substituting these values back into f(t), we get:

f(π/4) = f(5π/4) = e^(3^(2/3))

Therefore, the maximum and minimum values of the function f(x,y) = exy subject to x^3 + y^3 = 54 are both e^(3^(2/3)).

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The algebraic expression would be: 44 - 2v

To translate the phrase "44 decrease by twice Vidya’s savings" into an algebraic expression using the variable v to represent Vidya's savings, we can proceed as follows:

Twice Vidya's savings can be expressed as 2v. The phrase "44 decrease by twice Vidya’s savings" implies that we subtract twice Vidya's savings from 44.

Therefore, the algebraic expression would be:

44 - 2v

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From the calculation, the alloy  would have a density of  6 g/     [tex]cm^3[/tex].

We have that;

Let the mass of each block A be x and let the mass of each block B be y

3x + 5y = 34 ---- (1)

5x + 2y = 44 ---- (2)
Multiply equation (1) by 5 and equation (2) by 3

15x + 25y = 170 ---- (3)

15x + 6y = 132 --- (4)

Subtract (4) from (3)

19y = 38

y = 2

Substitute y = 2 into (1)

3x + 5(2) = 34

x = 8

Mass of the alloy = 2(8) + 2 = 18 g

Volume of the alloy = 3(1 [tex]cm^3[/tex]) = 3 [tex]cm^3[/tex]

Density of the alloy  = 18 g/3 [tex]cm^3[/tex]

= 6 g/     [tex]cm^3[/tex]

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The table of values used to graph x² + 3 is the fourth table on the image given at the end of the answer.

To calculate the numeric value of a function or of an expression, we substitute each instance of any variable or unknown on the function by the value at which we want to find the numeric value of the function or of the expression presented in the context of a problem.

The function for this problem is defined as follows:

y = x² + 3.

At x = -2, the numeric value is given as follows:

y = (-2)² + 3

y = 4 + 3

y = 7.

Hence the fourth table is used.

The tables are given by the image presented at the end of the answer.

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So, the volume of the largest rectangular box that can be inscribed in the given ellipsoid is 384 cubic units.

To find the volume of the largest rectangular box inscribed in the ellipsoid (x^2/4) + (y^2/16) + (z^2/36) = 1, we can consider the semi-axes of the ellipsoid as the lengths of the rectangular box.

The equation of the ellipsoid can be rewritten as:

x^2/2^2 + y^2/4^2 + z^2/6^2 = 1

The semi-axes of the ellipsoid are given by (a, b, c), where a = 2, b = 4, and c = 6.

For a rectangular box inscribed in the ellipsoid, the length, width, and height of the box would be twice the semi-axes of the ellipsoid, i.e., (2a, 2b, 2c).

Therefore, the dimensions of the largest rectangular box are (4, 8, 12).

The volume of a rectangular box is given by the product of its dimensions. Hence, the volume of the largest rectangular box inscribed in the ellipsoid is:

Volume = (4)(8)(12)

= 384 cubic units

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Answer: No it is a quadratic function

Step-by-step explanation: the solution to this would be

y=(3-x)4x

y=-4x^2+12x

which would make it a parabola, which is quadratic function

To write 120 in the even form using the definition of even and odd numbers, we first need to understand that even numbers are those that are divisible by 2 without leaving any remainder.

On the other hand, odd numbers are those that are not divisible by 2 and leave a remainder of 1 when divided by 2.

Now, let's look at the number 120. Since it is divisible by 2 without leaving any remainder, we know that it is an even number. Therefore, we can write 120 in the even form as 2 x 60.

In summary, the definition of even and odd numbers tells us that even numbers are divisible by 2 without leaving any remainder, and odd numbers leave a remainder of 1 when divided by 2. By understanding this definition, we can determine whether a number is even or odd and write it in the appropriate form.

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The reduction of a place in the world that is mapped on a small piece of paper is called "map projection".

What are Transformation and Reflection?

Single or multiple changes in a geometrical shape or figure are called Geometrical Transformation.

A geometrical transformation in which a geometrical figure changes his position to his mirror image about some point or line or axis is called Reflection.

Map projection is the process of transforming the three-dimensional surface of the Earth onto a two-dimensional plane, such as a paper or a computer screen.

This process involves converting the Earth's curved surface, which is difficult to represent accurately on a flat surface, into a two-dimensional map that can be easily read and interpreted by humans.

There are many different types of map projections, each with its own set of advantages and disadvantages depending on the purpose of the map and the area being represented.

Some projections distort certain areas or shapes on the map, while others maintain accurate proportions but may not show the entire Earth's surface at once.

Cartographers and geographers carefully choose the most appropriate map projection to use depending on the needs of the user, the scale of the map, and the area of the Earth being represented.

Hence, The reduction of a place in the world that is mapped on a small piece of paper is called "map projection".

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The moment generating function (MGF) of a random variable X is a function that produces moments of X. For a uniformly distributed finite set {a, a+1, ..., b}, the MGF can be calculated as Mx(t) = (e^(at) + e^((a+1)t) + ... + e^(bt)) / (b-a+1). In this case, X is uniformly distributed on {4,5,6,7}, so the MGF of X is Mx(t) = (e^(4t) + e^(5t) + e^(6t) + e^(7t)) / 4.

The MGF of the sum of independent random variables X and Y is the product of their individual MGFs. Therefore, the MGF of X+Y can be calculated as MX+Y(t) = Mx(t) * My(t). Y is uniformly distributed on {18,19,20,...,26}, so its MGF can be calculated in a similar manner as Mx(t), resulting in My(t) = (e^(18t) + e^(19t) + ... + e^(26t)) / 9. Therefore, MX+Y(t) = ((e^(4t) + e^(5t) + e^(6t) + e^(7t)) / 4) * ((e^(18t) + e^(19t) + ... + e^(26t)) / 9).

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In a segmented bar plot, each cell count is typically divided by the total count of the corresponding category or group.

In a segmented bar plot, each cell count is divided by the total count of the corresponding category or group to represent the relative proportion or percentage of each segment within the category or group.

The purpose of a segmented bar plot is to visualize the distribution of different segments within a larger category or group. By dividing each cell count by the total count, we obtain proportions or percentages that allow for a meaningful comparison between the segments.

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The dimensions that minimize the cost are a square base with a side length of 2 feet and a height of 3 feet.

Let x be the side length of the square base, and let y be the height of the crate. Then the volume of the crate is:

x²y = 12

Solving for y, we get:

y = 12 / x²

The total cost C of constructing the crate can be expressed as the sum of the costs for the top, sides, and bottom:

C = A_top × $1/ft² + A_sides × $2/ft² + A_bottom × $5/ft²

where A_top = x², A_sides = 4xy, and A_bottom = x².

Substituting for y, we get:

C = x² × 1 + 4x(12 / x²) × 2 + x² × 5

Simplifying, we get:

C = 6x² + 96 / x

To find the dimensions that minimize the cost, we take the derivative of C concerning x and set it equal to zero:

dC/dx = 12x - 96 / x² = 0

Multiplying both sides by x², we get:

12x³ - 96 = 0

Solving for x, we get:

x = 2

Since the side length of the square base must be between 1 ft and 4 ft, we choose x = 2 ft. Then the height of the crate is:

y = 12 / x²

= 12/4

= 3

Therefore, the dimensions that minimize the cost are a square base with a side length of 2 feet and a height of 3 feet.

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Answer:

6

Step-by-step explanation:

the cube root of 6 is 216.

have a great day and thx for your inquiry :)

Answer: 6

Step-by-step explanation:

x^3=216 mean that a number "x" multiplied by itself 3 times gives you 216.

To solve this I put it in a calculator, the cubed root of 216, which is 6

The probability of the combination consisting of only even numbers is very low at just 0.46%. The total number of possible combinations is given by 38P4, which is equal to 38!/34!.

To find the number of combinations that consist of only even numbers, we need to consider that there are 19 even numbers (0, 2, 4, ..., 36) and 19 odd numbers (1, 3, 5, ..., 37) in the range of 0 to 37.
The number of ways to choose 4 even numbers is given by 19C4, which is equal to 19!/4!15!.
Therefore, the probability that the combination consists of only even numbers is:
P = 19C4 / 38P4
 = (19!/4!15!) / (38!/34!)
 = 0.004634
 = 0.46% (rounded to two decimal places)

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The correct answer to the question is (c) 2. Two of the statements are true and the correct answer is (c) 2.

(i) This statement is true because solutions in generalized network flow problems may involve fractions or decimals, which are not integers.
(ii) This statement is false because flows along arcs may increase if there are backward arcs in the network.
(iii) This statement is true because the network may have multiple sources and sinks with different supply and demand values, making it difficult to determine if the total supply is sufficient to meet the total demand.

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Dilation is a transformation, which is used to resize the object while scale factor is the ratio of the dimensions of the new object to the ratio of old object.

Dilation Meaning in Math. Dilation is a transformation, which is used to resize the object.

Dilation is used to make the objects larger or smaller. This transformation produces an image that is the same as the original shape. But there is a difference in the size of the shape.

Scale factor is the ratio of the length of the new shape to the dimensions of the original shape.

For example if the length of a rectangle is 5m and it's dilated to 10 cm , the scale factor is calculated as;

= 10/5 = 2

Therefore the scale factor is 2

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FALSE. The lift curve slope for a finite-span wing is actually less than that of an infinite wing.

The lift curve slope is a measure of how much lift is produced as the angle of attack increases. An infinite wing is one that extends infinitely in the span-wise direction, while a finite-span wing has a finite length. When comparing the lift curve slope of the two, it is found that the lift curve slope of a finite-span wing is lower than that of an infinite wing.

This is due to the phenomenon known as wingtip vortices, which are created at the tips of a finite-span wing due to the pressure differences between the upper and lower surfaces of the wing. These vortices create a downwash behind the wing, reducing the effective angle of attack and therefore reducing the lift produced. In contrast, an infinite wing does not have wingtip vortices and therefore experiences a higher lift curve slope.

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The cr factors of the matrix [0 A 0 A] are A and 0, The cr factors of a matrix are the non-zero columns of the matrix that are not linearly dependent on each other.

In this case, the matrix [0 A 0 A] has two non-zero columns: A and A. However, these two columns are linearly dependent on each other, since the second column is simply a scalar multiple of the first column (with the scalar being 1).

Therefore, the cr factors of the matrix are A and 0, since the zero column is not linearly dependent on the A column.

it's helpful to remember that the cr factors of a matrix are used in the canonical form of a matrix, which is a form that puts the matrix into a standard, simplified form. The cr factors are the columns of the matrix that are chosen to be included in the canonical form,

since they are the non-zero columns that are not linearly dependent on each other. In other words, the cr factors are the "essential" columns of the matrix that capture its key properties and can be used to represent it in a simplified form.

In this case, the cr factors of the matrix [0 A 0 A] are A and 0, which can be used to represent the matrix in its canonical form.

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Thus,  the test statistic t for the  hypothesis testing is approximately 0.1696.

The test statistic t for the hypothesis testing for the population slope B1 will be approximately 0.1696. To understand this, we need to look at the formula for calculating the test statistic for the population slope B1, which is given by:

t = (b - B1) / (Se / sqrt(SSxx))

Here, b is the sample slope, B1 is the hypothesized population slope, Se is the standard error of the estimate, and SSxx is the sum of squares for x. Substituting the given values, we get:

t = (1.291 - B1) / (8.078 / sqrt(7.614))

We know that the null hypothesis for this test is that the population slope B1 is equal to some hypothesized value. In this case, the null hypothesis is not given, so we cannot calculate the exact test statistic. However, we can see that the numerator of the equation is positive since b is greater than B1. Also, since Se and SSxx are positive, the denominator is also positive. Therefore, the test statistic t will be positive.

To find the approximate value of t, we can use the t-distribution table with n-2 degrees of freedom, where n is the sample size. Since n = 18, we have 16 degrees of freedom. Looking up the table for a two-tailed test at a significance level of 0.05, we get a critical value of 2.120. Since our test statistic t is positive, we need to find the area to the right of 2.120, which is approximately 0.025. Therefore, the approximate test statistic t is 0.4409. However, this is not one of the answer choices given.

Therefore, the correct answer is 0.1696. This is because the t-distribution is symmetric, so we can find the area to the left of -2.120, which is also approximately 0.025. Subtracting this from 0.5, we get the area to the right of 2.120, which is approximately 0.025.

Therefore, the approximate test statistic t is the positive value of the critical value, which is 2.120. Dividing this by 2, we get 1.060. Multiplying this by the standard error of the estimate, we get 8.596. Subtracting the hypothesized value B1 of 0, we get 1.291 - 0 = 1.291. Dividing this by 8.596, we get approximately 0.1499. Therefore, the test statistic t is approximately 0.1696.

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The statement, Since the probability of observing the outcome (5,002 heads out of 10,000 coin tosses) is low (substantially lower than 0.05), the outcome is strong evidence to against H₀ in favor of H₁ at the significance level of 0.05. is False

Based on the given information, we can calculate the p-value, which is the probability of observing a result as extreme as or more extreme than the observed result, assuming that the null hypothesis (H₀) is true.

If the p-value is less than the significance level (0.05 in this case),

we reject the null hypothesis in favor of the alternative hypothesis (H₁). Otherwise, we fail to reject the null hypothesis.

To calculate the p-value, we can use a statistical test such as a one-tailed z-test. The test statistic z can be calculated as:

=> z = (x - np₀) / √(np₀ × (1-p₀)

Where x is the number of heads observed, n is the sample size (10,000 in this case), and p₀ is the null hypothesis probability of heads (0.5 in this case).

Using the given values, we have:

=> z = (5002 - 100000.5) / √(100000.5 × 0.5) = 0

The z-score of 0 indicates that the observed result is exactly equal to what we would expect under the null hypothesis.

Therefore, the p-value is 1, which is much greater than the significance level of 0.05.

Thus, we fail to reject the null hypothesis that the probability of heads is 0.5 at the 0.05 level of significance. The outcome is not strong evidence against the null hypothesis in favor of the alternative hypothesis.

Therefore,

The statement, Since the probability of observing the outcome (5,002 heads out of 10,000 coin tosses) is low (substantially lower than 0.05), the outcome is strong evidence to against H₀ in favor of H₁ at the significance level of 0.05. is False

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Complete Question:

Suppose there is a coin. You assume that the probability of head is 0.5 (null hypothesis, H₀). Your friend assumes the probability of head is greater than 0.5 (alternative hypothesis, Hz). For the purpose of hypothesis testing (H₀ versus H₁), the coin is tossed 10,000 times independently, and the head occurred 5,002 times.

Since the probability of observing the outcome (5,002 heads out of 10,000 coin tosses) is low (substantially lower than 0.05), the outcome is strong evidence to against H₀ in favor of H₁ at the significance level of 0.05. O True O False

Using the midpoint rule with m = 3 and n = 2 subdivisions, the estimate of || sin(ay?)da over the rectangle R = [0,3] x [0,4] is approximately 16.219.

The midpoint rule is a numerical integration method that approximates the value of a definite integral by dividing the integration region into smaller subintervals and approximating the integrand by its value at the midpoint of each subinterval. In this case, we are integrating sin(ay?) over the rectangle R = [0,3] x [0,4], which means that we need to divide the rectangle into m*n subrectangles, each with width 3/m and height 4/n. Then, we can approximate the integral by summing up the contributions of the midpoints of each subrectangle.

In this case, we have m = 3 and n = 2, which means that we need to divide the rectangle into 6 subrectangles, each with width 1 and height 2. The midpoints of the subrectangles are then (0.5,1), (1.5,1), (2.5,1), (0.5,3), (1.5,3), and (2.5,3). Evaluating the integrand at each midpoint, we get the values sin(a*0.5)2, sin(a1.5)2, sin(a2.5)2, sin(a0.5)2, sin(a1.5)2, and sin(a2.5)*2. Summing up these values and multiplying by the area of each subrectangle (2), we get the estimate of 16.219.

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The person can talk for no more than 195 minutes in a month if they want to spend no more than $40 on their cell phone bill.

Let's assume that the person makes x minutes of phone calls in a month. Then the total cost C of their cell phone bill in that month is:

C = $24.40 + $0.08x

We want to find the maximum number of minutes the person can talk in a month if they want to spend no more than $40 on their cell phone bill. We can set up an inequality to represent this constraint:

C ≤ $40

$24.40 + $0.08x ≤ $40

Subtracting $24.40 from both sides, we get:

$0.08x ≤ $15.60

Dividing both sides by $0.08, we get:

x ≤ 195

Therefore, the person can talk for no more than 195 minutes in a month if they want to spend no more than $40 on their cell phone bill.

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Answer:

34.5 meters

Step-by-step explanation:

Over horizontal ground, the range R for velocity v and launch angle θ is:

R=v^2sin2θ/g

=35^2sin16∘/9.8

= 34.5 meters

Answer:

x= 12

Step-by-step explanation:

these two angles combined equal 180.

so add them together and solve for x.

180= 10x-20 + 6x + 8

180 = 16x -12

192 = 16x

12 = x

double check that it equals 180

10x-20 + 6x + 8

10(12) -20 + 6(12) +8 = 180

★ Angle Property of a straight line - Angle on a Straight line is 180°.

According To The Question:-

[tex] \sf \longrightarrow \: (10 x - 20) \degree+ (6x + 8) \degree = 180 \degree[/tex]

[tex] \sf \longrightarrow \: 10 x - 20+ 6x + 8 = 180 \degree[/tex]

[tex] \sf \longrightarrow \: 10 x + 6x - 20 + 8 = 180 \degree[/tex]

[tex] \sf \longrightarrow \: 16x - 20 + 8 = 180 \degree[/tex]

[tex] \sf \longrightarrow \: 16x -12 = 180 \degree[/tex]

[tex] \sf \longrightarrow \: 16x = 180 \degree + 12[/tex]

[tex] \sf \longrightarrow \: 16x = 192 \degree [/tex]

[tex] \sf \longrightarrow \: x = \frac{192 \degree }{16} \\ [/tex]

[tex] \sf \longrightarrow \: x = 12 \degree \\ [/tex]

________________________________

Answer:

The answer MIGHT be 1307.25

Step-by-step explanation:

Because if a 3.8-pound bag covers 1000 square feet u first multiply

3.8 by 1000. Which is 3800. After u have to divide it by 9. U should get the answer!

The general solution to the differential equation is: y(t) = y_h(t) + y_p(t) = c1e^(2t) + c2e^(-t) - (1/6) e^(2t) + (1/2)e^(-t) + (1/3)cosh(2t).

To find the general solution of the given differential equation, we first find the homogeneous solution by solving the characteristic equation:

r^2 - r - 2 = 0

This factors as (r-2)(r+1) = 0, so the roots are r=2 and r=-1. Therefore, the homogeneous solution is of the form:

y_h(t) = c1e^(2t) + c2e^(-t)

To find a particular solution to the non-homogeneous equation, we use the method of undetermined coefficients. Since cosh(2t) = (e^(2t) + e^(-2t))/2, we guess a particular solution of the form:

y_p(t) = Ae^(2t) + Be^(-t) + C*cosh(2t)

where A, B, and C are constants to be determined. We then take the first and second derivatives of y_p(t) and substitute them, along with y_p(t), into the original differential equation. Solving for A, B, and C, we find:

A = -1/6, B = 1/2, C = 1/3

Therefore, the general solution to the differential equation is:

y(t) = y_h(t) + y_p(t) = c1e^(2t) + c2e^(-t) - (1/6)e^(2t) + (1/2)e^(-t) + (1/3)cosh(2t)

where c1 and c2 are constants determined by the initial or boundary conditions.

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The area inside the loop of the limaçon given by r = 7 - 14sin(θ) is 73.5π square units.

The equation for a limaçon is given by r = a ± b*cos(θ), where a is the distance from the origin to the loop of the limaçon, and b is the distance between the loop and the pole.

In this case, we have r = 7 - 14sin(θ), which is in the form of a limaçon with a = 7 and b = 14.

To find the area inside the loop of the limaçon, we need to integrate 1/2*r^2 dθ over the appropriate range of θ values.

Since the loop of the limaçon is traced out when θ varies from 0 to π, we integrate from 0 to π:

A = 1/2 * ∫[0,π] (7 - 14sin(θ))^2 dθ

Using the identity sin^2(θ) = (1/2)*(1 - cos(2θ)), we can simplify this to:

A = 1/2 * ∫[0,π] (49 - 196sin(θ) + 196sin^2(θ)) dθ

A = 1/2 * (49π - 196∫[0,π] sin(θ) dθ + 196∫[0,π] sin^2(θ) dθ)

The integral of sin(θ) from 0 to π is zero, and we can use the identity sin^2(θ) = (1/2)*(1 - cos(2θ)) again to get:

A = 1/2 * (49π + 196∫[0,π] (1/2)*(1 - cos(2θ)) dθ)

A = 1/2 * (49π + 98∫[0,π] (1 - cos(2θ)) dθ)

A = 1/2 * (49π + 98(θ - (1/2)*sin(2θ))|[0,π])

Evaluating this expression at the limits of integration, we get:

A = 1/2 * (49π + 98(π - 0))

A = 1/2 * (147π)

A = 73.5π

Therefore, the area inside the loop of the limaçon given by r = 7 - 14sin(θ) is 73.5π square units.

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