Using the sine rule, calculate the length d.
Give your answer to 2 d.p.

R1 satisfies the reflexive, antisymmetric, and transitive properties. R2 satisfies the reflexive and symmetric properties. R3 satisfies the reflexive and symmetric properties. S1 satisfies the reflexive and transitive properties. S2 satisfies the symmetric property. S3 satisfies none of the five properties.

R1:Reflexive: for all x∈N, x|x, since x divides itself.

Antisymmetric: if (x,y)∈R1 and (y,x)∈R1, then x|y and y|x, so x=y.

Transitive: if (x,y)∈R1 and (y,z)∈R1, then x|y and y|z, so x|z.

R2:Reflexive: for all x∈N, x+x=2x is even, so (x,x)∈R2.

Symmetric: if (x,y)∈R2, then x+y is even, so y+x is even, hence (y,x)∈R2.

R3:Reflexive: for all x∈N, x*x=x² is even, so (x,x)∈R3.

Symmetric: if (x,y)∈R3, then xy is even, so yx is even, hence (y,x)∈R3.

S1:Reflexive: for all y∈N, 2|2y, so (2,y)∈S1.

Transitive: if (2,x)∈S1 and (x,y)∈S1, then x|z and y|x, so y|z, hence (2,y)∈S1.

S2:Symmetric: if (2,x)∈S2, then 2+x is odd, so x+2 is odd, hence (x,2)∈S2.

S3:S3 does not satisfy any of the five properties. For example, (1,3) and (3,2) are in S3, but (1,2) is not. Therefore, S3 is not reflexive, not symmetric, not antisymmetric, not transitive, and not total.

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The limit of the given rational function is 18, so lim (x²-81)/(x-9) as x→9 is equal to 18. To find the limit of the given rational function using the Theorem on Limits of Rational Functions, we have:

lim (x²-81)/(x-9) as x→9.

Step 1: Factor the numerator.
The numerator can be factored as a difference of squares: x² - 81 = (x - 9)(x + 9).

Step 2: Simplify the rational function.
Now we have lim ((x - 9)(x + 9))/(x-9) as x→9. We can cancel out the common factor (x - 9) from the numerator and the denominator, which leaves us with lim (x + 9) as x→9.

Step 3: Evaluate the limit.
We can directly substitute the value x = 9 into the simplified expression: (9 + 9) = 18.

Thus, the limit of the given rational function is 18, so lim (x²-81)/(x-9) as x→9 is equal to 18.

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The sample data and regression analysis to gain insights into the relationship between purchase costs and purchase revenues.

The assumption of a constant contribution margin per customer would be justified if the cost structure of the business remained constant and if the company did not offer discounts or promotions that would affect the contribution margin. In other words, the assumption would hold if the company's revenues and costs remained relatively stable over time.
It is difficult to determine if these conditions would hold without additional information about the business. However, we can analyze the sample data to see if there is a relationship between purchase costs and purchase revenues. By doing a scatter plot of the sample data, we can visually see if there is a correlation between the two variables. After plotting the data, we can use the sample data to run a regression of purchase costs on purchase revenues. The regression results can provide insights into the relationship between the two variables and can help us determine if the assumption of a constant contribution margin per customer is likely to hold. The intercept term of the regression results represents the fixed cost of the business. This is the cost that the business incurs regardless of how many units they sell. The slope of the regression line represents the variable cost per unit. By analyzing the regression results, we can determine if the variable cost per unit remains constant as the number of units sold increases. In conclusion, while it is difficult to determine this analysis can help us determine if the assumption is likely to hold and can provide valuable information for the business. if the conditions necessary for the assumption of a constant contribution margin per customer would hold without additional information about the business, we can use sample data and regression analysis to gain insights into the relationship between purchase costs and purchase revenues.

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The solution to the differential equation with the initial condition x(0) = -2 is: x(t) = -3/4 ln|t| / (1-t^2)^(3/2) - 2.

First, we need to separate the variables x and t, which means we want to get all the x's on one side and all the t's on the other side of the equation. We start by adding 1 to both sides: 4x(sqrt(1-t^2)) dx/dt = 1 Next, we can divide both sides by 4x(sqrt(1-t^2)) to get: dx/dt = 1 / [4x(sqrt(1-t^2))]

Now we can separate the variables by multiplying both sides by dt and dividing both sides by the expression in brackets: [4x(sqrt(1-t^2))] dx = dt To integrate both sides, we need to use a substitution.

Let u = 1-t^2, then du/dt = -2t. We can solve for dt to get dt = -du / (2t). Substituting this into the equation gives: [4x(sqrt(u))] dx = -du / (2t) Integrating both sides: ∫ [4x(sqrt(u))] dx = -∫ du / (2t)

Simplifying the left side: 2/3 x (1-t^2)^(3/2) + C1 = -1/2 ln|t| + C2 Where C1 and C2 are constants of integration. Using the initial condition x(0) = -2, we can find C1: 2/3 x (1-0^2)^(3/2) + C1 = -1/2 ln|0| + C2 -4/3 + C1 = C2

Now we have the general solution: 2/3 x (1-t^2)^(3/2) = -1/2 ln|t| + C Where C = C2 - 4/3. We can solve for x(t) by multiplying both sides by 3/2 and dividing by (1-t^2)^(3/2): x(t) = -3/4 ln|t| / (1-t^2)^(3/2) + D Where D = 2/3 C. Finally, using the initial condition x(0) = -2, we can solve for D: x(0) = -3/4 ln|0| / (1-0^2)^(3/2) + D -2 = D

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The sample space S for this experiment is S = {(yellow, yellow), (yellow, blue), (yellow, red), (blue, yellow), (blue, blue), (blue, red), (red, yellow), (red, blue), (red, red)}

Suppose we have three marbles of different colors - yellow, blue, and red. We draw one marble at random and then replace it before drawing another marble. The sample space of this experiment is the set of all possible outcomes or combinations of marbles that can be drawn.

Since we are drawing marbles with replacement, the possible outcomes for the first draw are yellow, blue, and red. Each of these outcomes can occur with equal probability of 1/3. After replacing the first marble, the same three colors are available for the second draw, resulting in nine possible outcomes for the pair of draws. We can represent the sample space as a set of ordered pairs (x,y), where x and y are the colors of the first and second draws, respectively.

In this sample space, each outcome is equally likely to occur with a probability of 1/9. We can use the sample space to calculate the probability of specific events, such as drawing two marbles of the same color or drawing at least one blue marble.

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If rank(ST) < n, then by proving the contrapositive, it can be shown that rank(TS) < n. If 0 is an eigenvalue of ST, then 0 is also an eigenvalue of TS, as shown by analyzing the eigenvectors of ST and T.

(a) To prove the contrapositive, assume that rank(TS) = n. Then, by the rank-nullity theorem, the nullity of TS is 0. Therefore, the nullity of ST is also 0, since the nullity of TS and ST are equal. This means that the only vector in the kernel of ST is the zero vector.

Now, by the rank-nullity theorem again, we have that rank(ST) = n, since the dimension of the range of ST plus the nullity of ST equals the dimension of V, which is n. Hence, if rank(ST) < n, then rank(TS) < n.

(b) Suppose that 0 is an eigenvalue of ST, and let v be a corresponding eigenvector. Then, we have that ST(v) = 0, which implies that T(S(v)) = 0. Therefore, S(v) is in the null space of T, which is a subspace of V. Now, either S(v) = 0 or S(v) is an eigenvector of T with eigenvalue 0.

If S(v) = 0, then v is in the null space of S, which is also a subspace of V. Otherwise, S(v) is a nonzero eigenvector of T with eigenvalue 0, which means that it is in the null space of T.

In either case, we have shown that v is in the null space of TS, which means that 0 is an eigenvalue of TS. Hence, if 0 is an eigenvalue of ST, then 0 is an eigenvalue of TS.

In summary, if rank(ST) < n, then rank(TS) < n, and if 0 is an eigenvalue of ST, then 0 is an eigenvalue of TS.

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So the probability of rejecting the lot is 0.591. So the probability of rejecting the lot is 0.773.

(a) If 10 chips are defective out of 150, then the probability that one chip is defective is 10/150 = 1/15.

The probability that none of the first five chips are defective is (140/150) * (139/149) * (138/148) * (137/147) * (136/146) = 0.409.

Therefore, the probability that at least one of the five chips is defective is 1 - 0.409 = 0.591.

(b) If 20 chips are defective out of 150, then the probability that one chip is defective is 20/150 = 2/15.

The probability that none of the first five chips are defective is (130/150) * (129/149) * (128/148) * (127/147) * (126/146) = 0.227.

Therefore, the probability that at least one of the five chips is defective is 1 - 0.227 = 0.773.

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The line integral of [tex]xyz^2[/tex] ds over the curve C is equal to 2√6.

To evaluate the line integral ∫[tex]_C xyz^2[/tex] ds, where C is the line segment from (-1, 4, 0) to (1, 5, 1), we need to parameterize the curve and then evaluate the integral using the parameterization.

We can parameterize the curve C as r(t) = (-1 + 2t, 4 + t, t), for t between 0 and 1.

Then, the line integral becomes:

∫[tex]_C xyz^2[/tex] ds = ∫[tex]_0^1 (-1 + 2t)(4 + t)t^2[/tex] ||r'(t)|| dt

To compute the magnitude of the derivative r'(t), we differentiate each component of r with respect to t and then take the magnitude:

r'(t) = (2, 1, 1)

||r'(t)|| = √[tex](2^2 + 1^2 + 1^2)[/tex] = √6

Substituting this into the integral and simplifying, we get:

∫[tex]_C xyz^2[/tex] ds = ∫[tex]_0^1 (-4t^5 + 2t^4 + 4t^3 - 2t^2)[/tex] √6 dt

Evaluating this integral using the power rule and simplifying, we get:

∫[tex]_C xyz^2[/tex] ds = [tex][-t^6 + 2/3 t^5 + 2t^4 - 2/3 t^3]_0^1[/tex] * √6

∫[tex]_C xyz^2[/tex] ds = (4/3 - 2/3) * √6 = 2√6

Therefore, the line integral of [tex]xyz^2[/tex] ds over the curve C is 2√6.

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The dependent variable in this study is the incidence of aggressive behavior displayed by the children during the period of free play.

The dependent variable is the outcome that is being measured and observed based on the independent variable, which in this scenario is the type of television program watched (violent or nonviolent). By observing and recording the incidence of aggressive behavior in each group, the researcher can determine if exposure to violent television programs has an effect on children's behavior. In this study, the dependent variable is essential to determining if there is a significant difference between the groups, and it allows for the researcher to draw conclusions about the relationship between exposure to violent television and aggressive behavior in children. In this scenario, the dependent variable is the incidence of aggressive behavior observed in the children during the period of free play. The dependent variable is the outcome or response that is measured in a study, and it is influenced by the independent variable(s). In this case, the independent variables are the type of television program (violent or nonviolent) and the gender of the participants (boys and girls). The researcher is investigating the relationship between exposure to violent or nonviolent television content and the occurrence of aggressive behavior among boys and girls.

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Answer:

The formula for the area of a circle is:

A = πr^2

where A is the area and r is the radius.

In this case, we are given that the area is 4π cm². Solving for the radius, we get:

4π = πr^2

r^2 = 4

r = 2

So the radius of the circle is 2 cm.

The formula for the circumference of a circle is:

C = 2πr

Plugging in the value for the radius, we get:

C = 2π(2) = 4π

Therefore, the circumference of the circle is 4π cm.

Based on the given information, when x equals 9, the regression line predicts that log(y) will equal 3.992. To find the predicted value of y when x is 9, you will need to apply the inverse transformation to the logarithmic value.  

The inverse transformation of log(y) is achieved by using the exponential function. Specifically, you can use the formula:

y = 10^(log(y))

In this case, since log(y) = 3.992, you can plug this value into the formula:

y = 10^(3.992)

When you calculate this, you will find that y ≈ 9762.97. Rounded to the nearest whole number, the regression line predicts that y will equal 9763 when x equals 9.

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a. Sample standard deviation of $13.19 as an estimate of the population standard deviation. b. The population mean at Niagara Falls is likely within this range of values, but we cannot say for certain whether it is higher or lower than the overall average daily vacation expenditure.

a. The confidence interval estimate of the mean amount spent per day by a family of four visiting Niagara Falls is ($132.89, $155.11) to two decimals.

To calculate the confidence interval, we use the formula:
CI = sample mean ± (z-score)(standard deviation / √sample size)
where the z-score is based on the desired level of confidence. For a 95% confidence level, the z-score is 1.96.

Plugging in the given values, we get:
CI = $144 ± (1.96)($13.19 / √n)
where n is the sample size. We are not given the sample size in this question, so we cannot calculate the exact interval. However, we can use the given sample standard deviation of $13.19 as an estimate of the population standard deviation.

So, CI = $144 ± (1.96)($13.19 / √n) = ($132.89, $155.11) to two decimals.

b. No, we cannot determine if the population mean at Niagara Falls differs from the mean reported by the American Automobile Association. The confidence interval includes the mean reported by the AAA, which was not significantly different from the sample mean. We can only say that the population mean at Niagara Falls is likely within this range of values, but we cannot say for certain whether it is higher or lower than the overall average daily vacation expenditure.

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Type II error in this context: The chef believes the complaint rate is not less than 8%, when in fact it is less than 8%

Consequence: The chef continues to use the new recipe but experiences a large number of unsatisfied customers.

A Type II error, in the context of hypothesis testing, occurs when the null hypothesis (H₀) is not rejected even though it is false. In other words, it's the failure to reject a false null hypothesis.

In this scenario, the null hypothesis states that the complaint rate is 8%, and the alternative hypothesis (Hₐ) states that the complaint rate is less than 8%.

A Type II error would occur if the chef believes that the complaint rate is not less than 8% (failing to reject the null hypothesis), when in fact it is less than 8% (the alternative hypothesis is true).

Consequences of a Type II error in this context:

The consequence of a Type II error would be that the chef continues to use the new gluten-free recipe for the topping even though the actual complaint rate is less than 8%.

This means that the chef would miss out on an opportunity to improve the recipe and potentially satisfy more customers.

In this case, the chef might continue to experience a significant number of unsatisfied customers who might have been pleased with an improved recipe.

This could lead to negative customer reviews, loss of customer loyalty, and a potential negative impact on the restaurant's reputation and business.

To summarize:

Type II error in this context: The chef believes the complaint rate is not less than 8%, when in fact it is less than 8%.

Consequence: The chef continues to use the new recipe but experiences a large number of unsatisfied customers, potentially harming the restaurant's reputation and business.

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A Type II error in this scenario would occur if the chef wrongly assumes the complaint rate is less than 8%, leading to continued use of the disliked recipe and unsatisfied customers.

In this context, a Type II error in the chef's hypothesis test would occur if the chef believes the complaint rate for the new gluten-free recipe is less than 8%, when in fact, it is not. That means the chef is under the false impression that the customers are more satisfied with the new recipe than they truly are. The consequence would be that the chef continues to use the new recipe, despite a higher complaint rate. This would lead to a significant number of unsatisfied customers because the recipe is not meeting their taste preferences as much as the chef thinks. This could subsequently affect the restaurant's reputation and customer loyalty.

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The surface area of the regular pyramid is 178.3 mm² if the area of the base is 43.3 mm².

The surface area of a regular pyramid can be calculated using the following formula

Surface Area = Base Area + (1/2) x Perimeter of Base x Slant Height

Base area = 43.3 mm²

Perimeter of base = 10 + 10 + 10 = 30 mm

Slant height = 9 mm

Substitute the values in the formula, we get the surface area

= 43.3 + 1/2 x 30 x 9

= 43.3 + 15 x 9

= 43.3 + 135

= 178.3 mm²

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-- The given question is incomplete, the complete question is given below

"Use the net to find the surface area of the regular pyramid."

The standard deviation of X is σ = sqrt(Var(X)) = 0.539 burgers per guest. You should make at least 118 burgers to be 95% sure there will be enough for all guests.

To determine the number of burgers you should make, you need to use the binomial distribution. Let X be the number of burgers needed by a guest, and n be the total number of guests (which is 120).
The expected value of X is E(X) = 0.2(0) + 0.7(1) + 0.1(2) = 0.9 burgers per guest.
The variance of X is Var(X) = E(X^2) - [E(X)]^2 = 0.2(0^2) + 0.7(1^2) + 0.1(2^2) - 0.9^2 = 0.29 burgers^2 per guest.
The standard deviation of X is σ = sqrt(Var(X)) = 0.539 burgers per guest.
To be 95% sure there will be enough burgers, you need to make sure that the probability that the total number of burgers needed is less than or equal to the number of burgers you make is at least 0.95. Let Y be the total number of burgers needed by all guests.
The expected value of Y is E(Y) = nE(X) = 120(0.9) = 108 burgers.
The variance of Y is Var(Y) = nVar(X) = 120(0.29) = 34.8 burgers^2.
The standard deviation of Y is σ = sqrt(Var(Y)) = 5.89 burgers.
To find the number of burgers you should make, you need to find the number k such that P(Y ≤ k) ≥ 0.95. This can be done using the normal approximation to the binomial distribution:
P(Y ≤ k) = P((Y - E(Y))/σ ≤ (k - E(Y))/σ) ≈ Φ((k - E(Y))/σ)
where Φ is the standard normal cumulative distribution function.
Solving for k, we get:
(k - E(Y))/σ = Φ^-1(0.95) ≈ 1.645
k - E(Y) = 1.645σ ≈ 9.69
k ≈ 117.69
Therefore, you should make at least 118 burgers to be 95% sure there will be enough for all guests.

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Using the given terms, we'll apply Stokes' theorem to find the flux of the vector field F across the surface S.

Stokes' theorem states that the flux of the curl of a vector field F across a surface S is equal to the circulation of F around the boundary of S. Mathematically, it's expressed as:

∮_C F·dr = ∬_S curl(F)·dS

Given the vector field F = li + 3j + 1k, we first need to find the curl of F. Curl(F) is given by the determinant of the following matrix:

| i  j  k  |
| ∂/∂x  ∂/∂y  ∂/∂z |
| l  3  1 |

Curl(F) = i(∂(1)/∂y - ∂(3)/∂z) - j(∂(1)/∂x - ∂(l)/∂z) + k(∂(3)/∂x - ∂(l)/∂y)
Curl(F) = -j(0 - 0) + k(0 - 0) = 0

Since the curl of F is 0, the flux of the vector field F across the surface S is also 0. Therefore, by using Stokes' theorem, we have found that the flux of the vector field F across the surface S in the first octant is 0.

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The property of vectors that is incorrect is "Oa· (b + c) = a · b + a.c = (a + b ) + c = a +(b + c)". The correct property is "Oa· (b + c) = Oa·b + Oa·c".

The incorrect property of vectors among the given options is:

Oa·b = axb

This property is incorrect because the dot product (a·b) and cross product (axb) of two vectors are different operations with different results. The dot product is a scalar value, while the cross product is another vector that is orthogonal to the given vectors. The correct properties of vectors in your question are:

1. a x b = -b x a (cross product)
2. a · (b + c) = a · b + a · c (dot product distributive property)
3. (a + b) + c = a + (b + c) (vector addition associativity)

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The probability of the cell phone company gaining at least 5 new customers from contacting 50 potential customers through their telemarketing campaign is approximately 42.46%.

If the probability of successfully gaining a new customer through a telemarketing campaign is 0.07, then the probability of not gaining a new customer is 0.93 (1-0.07). To calculate the probability of gaining at least 5 new customers out of 50 potential customers, we can use the binomial distribution formula.
P(X≥5) = 1 - P(X<5)
Where X is the number of new customers gained out of 50 potential customers.
P(X<5) = Σ (50 choose x) * (0.07)^x * (0.93)^(50-x) for x = 0 to 4
Using a calculator or software, we can calculate P(X<5) to be 0.906.
Therefore, the probability of gaining at least 5 new customers out of 50 potential customers is:
P(X≥5) = 1 - P(X<5) = 1 - 0.906 = 0.094
So, there is a 9.4% chance of gaining at least 5 new customers out of 50 potential customers in this telemarketing campaign.
To calculate the probability of successfully gaining at least 5 new customers from 50 potential customers with a success rate of 0.07, we can use the binomial probability formula. The formula is:
P(X = k) = C(n, k) * p^k * (1-p)^(n-k)
where P(X = k) is the probability of k successes in n trials, C(n, k) is the number of combinations of n items taken k at a time, p is the probability of success, and (1-p) is the probability of failure.
In this case, n = 50, p = 0.07, and we want to find the probability of at least 5 successes (k ≥ 5). To do this, we can calculate the probability of fewer than 5 successes (k < 5) and subtract this value from 1:
P(X ≥ 5) = 1 - P(X < 5)
P(X < 5) = P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4)
Now, we can plug in the values and calculate each term using the binomial probability formula, then sum the probabilities and subtract from 1 to get the desired probability:
P(X ≥ 5) = 1 - (P(X = 0) + P(X = 1) + P(X = 2) + P(X = 3) + P(X = 4))
After calculating the probabilities and summing them, we find:
P(X ≥ 5) ≈ 0.4246

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As W is spanned by n linearly independent vectors in ℝ^n, it means that the dimension of W is also n. This implies that W has the same dimension as ℝ^n, and therefore, W is equal to ℝ^n.

If w is a subspace of rnspanned by n non zero orthogonal vectors, then w is at most n-dimensional because there are only n vectors that can be used to span w. Any vector outside of the span of these n vectors will not be in w. Therefore, the dimension of w is less than or equal to n. Since w is a subspace of rn, which is n-dimensional, w must be a subset of Rn with a dimension less than or equal to n. Therefore, w d Rn. Suppose W is a subspace of ℝ^n spanned by n nonzero orthogonal vectors. This means that W is a vector space that is a subset of ℝ^n, and it can be generated by taking linear combinations of the n nonzero orthogonal vectors. Since the vectors are orthogonal, they are linearly independent, and their linear combinations form a basis for the subspace W.

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The expression is 8 x 6 = 48.

The customer has 48 choices for a meal that includes a sandwich and a drink.

We have,

A customer has 8 choices for a sandwich and 6 choices for a drink.

By the rule of product, the total number of choices for a meal is the product of the number of choices for a sandwich and the number of choices for a drink, which is:

= 8 x 6

= 48

Therefore,

The customer has 48 choices for a meal that includes a sandwich and a drink.

The expression is 8 x 6 = 48.

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The equation of the polar graph is r = 4sin(12θ)

Since we have the polar graph given in the figure, comparing this graph with the standard polar graph, we see that it has the form r = asin(nθ) where

Now, we see that from the graph,

So, substituting the values of the variables into the equation, we have that

r = asin(nθ)

r = 4sin(12θ)

Now to confirm that this is actually correct, substitute θ = 0 into the equation.

So,

r = 4sin(12θ)

r = 4sin(12(0))

r = 4sin(0)

r = 4(0)

r = 0

Which is correct as seen from the graph.

So, the equation is r = 4sin(12θ)

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According to the figure the missing parts are

To show that MP is parallel to NL we have to show that angle N is equal to angle OMP. hence by corresponding angles which is used when line are parallel would support the proof

Given that angle NML = 40 degrees and angle N = angle L we have that

angle NML + angle N +  angle L = 180 (sum of angles of a triangle)

40 + angle N + angle N = 180

2 angle N  = 180 - 40

angle N = 140/2 = 70

angle N = angle L = 70 degrees

angle OML = 180 - 40 (angle on a straight line)

angle OML = 140 degrees

MP bisects angle OML therefore angle OMP = angle PML = 70 degrees

This shows that angle N and angle OMP are equal by corresponding angles

The relationship will hold true if angle N is not equal to angle L since correponding angles requires angle N and angle OMP

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1.) VW = 31

WX = 19.

YW = 36.4

ZX = 18.2

VX= 36.4

To calculate the missing sides of the quadrilateral given, the Pythagorean formula should be used. That is;

C² = a² + b²

For VW; Since YX = 31 = VW because two opposite sides of a rectangle as equal in length.

For WX ; Since VY = 19 = WX because two opposite sides of a rectangle as equal in length.

For YW ; The Pythagorean formula is used;

YW = c = ?

a = 31

b = 19

c² = 31²+19²

= 961 + 361

c= √1322

c = 36.4

For ZX = the diagonal/2 = 36.4/2 = 18.2

For VX = YW = 36.4

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The Area of Sector is 205.1466 ft².

We have,

Angle = 120

Radius = 14 feet

So, Area of sector

= [tex]\theta[/tex] /360 x πr²

= 120/ 360 x (3.14) (14)²

= 1/3 x 3.14 x 14 x 14

= 205.1466 ft²

Thus, the Area of Sector is 205.1466 ft².

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Given DC is tangent to circle N at point C: m∠DCN < m∠BAN and BE is tangent to Circle N at point A, is True

(1) From the diagram, we can see that angles ∠DCN and ∠BAN are vertical angles, so they are equal in measure. Since DC is tangent to Circle N at point C, we know that m∠DCN is a right angle. Therefore, m∠DCN = 90° and m∠BAN = 90° - m∠DCN. Since m∠DCN is positive, we have m∠BAN < 90°, which implies that m∠DCN < m∠BAN. So, statement (1) is true.

(2) Since DC is tangent to Circle N at point C, we know that angle ∠CBE is a right angle, and hence BE is perpendicular to CE. Also, we know that CE is the radius of Circle N, so it is perpendicular to NB. Therefore, BE is tangent to Circle N at point A. So, statement (2) is true.

(3) The length of NB cannot be determined from the given information. We only know that CE = 6, but we do not know the radius of Circle N or the position of point B along the circle. Therefore, we cannot determine the length of NB. So, statement (3) cannot be determined.

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Complete question:

Given DC is tangent to circle N at point C, which statements are true?

- m∠DCN < m∠BAN

- BE is tangent to Circle N at point A.

- NB ≈ 6.1

Finally, we can find the critical value of the test statistic using a z-table or a calculator. For a one-tailed test at a 0.05 level of significance, the critical value is approximately 1.645.

The parent needs:

2112 cups - 23 cups = 2089 cups

As an improper fraction, this is:

=2089/1

To determine whether we can conclude that more than half of internet users have posted photos or videos online, we need to perform a hypothesis test. We can state the null hypothesis as "less than or equal to 50% of internet users have posted photos or videos online" and the alternative hypothesis as "more than 50% of internet users have posted photos or videos online."

Next, we need to choose a level of significance, which represents the maximum probability of rejecting the null hypothesis when it is actually true. Let's choose a level of significance of 0.05.

Using the information given, we can calculate the sample proportion of internet users who have posted photos or videos online as:

P = 855/2112 ≈ 0.405

We can then calculate the test statistic using the formula:

z = (P - p₀) / √(p₀(1-p₀) / n)

where p₀ = 0.5 (the proportion specified in the null hypothesis) and n = 2112. Plugging in the values, we get:

z = (0.405 - 0.5) / √(0.5(1-0.5) / 2112) ≈ -9.00

Since our test statistic (z = -9.00) is much smaller than the critical value, we reject the null hypothesis and conclude that there is sufficient evidence to support the claim that more than half of internet users have posted photos or videos online.

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The value of x is  [tex]x_1=\frac{-12+\sqrt{138}}{3}[/tex] or [tex]x_2=\frac{-12-\sqrt{138}}{3}[/tex].

The quadratic function can represent a quadratic equation in the Standard form: ax²+bx+c=0 where: a, b and c are your respective coefficients. In the quadratic function, the coefficient "a" must be different than zero (a≠0) and the degree of the function must be equal to 2.

For solving a quadratic function you should find the discriminant: Δ=b²-4ac . And after that, you should apply the discriminant in the formula: [tex]x=\frac{-b \pm\sqrt{\Delta}} {2a}[/tex]  .

The question asks for solving the equation 3x(x+8)= -2. Then,

       3x(x+8)= -2

      3x²+24x=-2

      3x²+24x+2=0

       a=3

      b=24

      c=2

        Δ=b²-4ac

        Δ=24²-4*3*2

        Δ=576-24

        Δ=552

[tex]x=\frac{-b \pm\sqrt{\Delta}} {2a}=\frac{-24\pm{\sqrt{552}} }{2*3}= \frac{-24\pm{\sqrt{552}} }{6}= \frac{-24\pm{2\sqrt{138}} }{6}=\frac{-12\pm\sqrt{138}}{3}[/tex]

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