MAKE CONNECTIONS Use the quadratic function f(x)=x² and exponential function g (x)-2. Graph both functions on the same coordinate plane on a separate
sheet of paper Select statements to compare and contrast the domains and ranges of the two functions and any symmetry of the two graphs
A) The domain of f(x)=x² is all real numbers and the domain of g (x) - 2' is x 20.
B) The domains of the two functions are the same, all real numbers.
C) The quadratic function has a vertical line of symmetry. x=0, and is symmetric about that line
DD) The exponential function has no symmetry
DE) The range of the two functions are the same. y 2 0.
DF) Both functions have vertical line symmetry, x-0, and are symmetric about that line.
G) The range of f(x)=x²5 y 20 while the range of g (x) - 2 is y> 0.

A level of 0.05 is used, which means that there is a 5% chance of making a type I error.

A type I error occurs when we reject the null hypothesis, but it is actually true. This means that we have made a mistake in concluding that there is a significant difference between two groups or variables, when in fact there is not. This can happen due to factors such as sample size, random variability or bias.

For example, if a drug company tests a new medication and concludes that it is effective in treating a certain condition, but in reality it is not, this would be a type I error. This could lead to the medication being approved and prescribed to patients, which could potentially harm them and waste  resources.

In statistical analysis, a type I error is represented by the significance level, or alpha level, which is the probability of rejecting the null hypothesis when it is actually true. It is important to set a reasonable alpha level to minimize the risk of making a type I error. Generally, a level of 0.05 is used, which means that there is a 5% chance of making a type I error.
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The annual interest rate for the bank account is r = 3.7 %

Given data ,

A = P(1 + r/n)^(nt)

where A is the amount after t years, P is the principal (initial amount), r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the time in years.

a)

To find the principal, we can use the information from the first year:

$1739.50 = P(1 + r/n)^(n*5)

$1739.50 = P(1 + r/n)^5

We don't know the values of P, r, or n, but we can simplify the equation by dividing both sides by (1 + r/n)^5:

$1739.50 / (1 + r/n)^5 = P

Now we can use the information from the second year to solve for r:

$1803.40 = P(1 + r/n)^(n*6)

$1803.40 = ($1739.50 / (1 + r/n)^5) * (1 + r/n)^6

Simplifying, we get:

$1803.40 = $1739.50 * (1 + r/n)

Dividing both sides by $1739.50, we get:

1.036928 = 1 + r/n

Subtracting 1 from both sides, we get:

0.036928 = r/n

b)

To find the annual interest rate, we need to solve for r.

We know that r/n = 0.036928, so we can solve for r by multiplying both sides by n:

r = 0.036928n

$1739.50 = P(1 + r/n)^5

Substituting r = 0.036928n, we get:

$1739.50 = P(1 + 0.036928n/n)^5

Simplifying, we get:

$1739.50 = P(1.036928)^5

Dividing both sides by (1.036928)^5, we get:

P = $1500

Now we can substitute P = $1500 and r/n = 0.036928 into the equation r = 0.036928n and solve for r:

r = 0.036928n = 0.036928 * 1 = 0.036928

Hence , the annual interest rate is 3.7%

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The event that is not possible in the context of sock drawer with 5 black socks, 10 white socks, and 6 multicolored socks is (black socks, red socks). So, option(c) is right one.

We have a sock drawer, Number of blacks socks in drawer = 5

Number of white socks in drawer = 10

Number of multicolored socks in drawer = 2

Total number of socks in drawer = 17

Now, schaid will randomly pull out 2 socks without seeing in the drawer.

We have to determine the event that is not possible in this context.

a) If selected two socks are black and white, then we have sufficient that for this event and easily determine the probability, so it is an event.

b) If selected two socks are white and multicolor then we have sufficient details that for this event and easily determine the probability, so it is an event. Similarly, in case of option (d) is also an event in same reason.

But in case of optio(c), If selected two socks are black and multicolor, we have data of black socks but no idea about number red socks in drawer. That is no data related to this, so it not possible event in context. Hence, required answer is ( black socks, red socks).

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Complete question:

a sock drawer has 5 black socks, 10 white socks, and 6 multicolored socks. schaid will randomly pull out 2 socks without looking in the drawer. which answer describes an event that is not possible in this context?

a) ( black socks , white socks)

b)(white socks, multicolour socks)

c) ( black socks, red socks)

d) ( black socks , multicolour socks)

The occurrence (black socks, red socks) is not feasible in the context of a sock drawer containing 5 black socks, 10 white socks, and 6 multi-colored socks. So, choice (c) is the best one.

Probability is a way to gauge how likely something is to happen. Many things are difficult to forecast with absolute confidence.

We have a sock drawer, Number of blacks socks in drawer = 5

Number of white socks in drawer = 10

Number of multicolored socks in drawer = 2

Total number of socks in drawer = 17

Schaid will now take two socks out of the drawer at random.

The event that cannot occur in this situation must be identified.

a) If selected two socks are black and white, then we have sufficient that for this event and easily determine the probability, so it is an event.

b) If selected two socks are white and multicolor then we have sufficient details that for this event and easily determine the probability, so it is an event. Similarly, in case of option (d) is also an event in same reason.

But in case of option (c), If selected two socks are black and multicolor, we have data of black socks but no idea about number red socks in drawer. That is no data related to this, so it not possible event in context. Hence, required answer is ( black socks, red socks).

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Complete question:

a sock drawer has 5 black socks, 10 white socks, and 6 multicolored socks. schaid will randomly pull out 2 socks without looking in the drawer. which answer describes an event that is not possible in this context?

a) ( black socks , white socks)

b)(white socks, multicolour socks)

c) ( black socks, red socks)

d) ( black socks , multicolour socks)

The h'(3) = h'(x)|x=3 = 22. We can see from the calculation that the value of h'(3) depends on the slopes of both f(x) and g(x) at x=3, as well as their values at x=3.

To evaluate h'(3) for h(x) = f(x) * g(x), we will use the product rule. The product rule states that if h(x) = f(x) * g(x), then h'(x) = f'(x) * g(x) + f(x) * g'(x).
Given the graphs of f(x) and g(x), we need to find the values of f(3), g(3), f'(3), and g'(3).
1. Locate the point x = 3 on the graph of f(x) and determine the value of f(3).
2. Locate the point x = 3 on the graph of g(x) and determine the value of g(3).
3. Determine the slope of the tangent line at x = 3 on the graph of f(x), which represents f'(3).
4. Determine the slope of the tangent line at x = 3 on the graph of g(x), which represents g'(3).
Once you have the values of f(3), g(3), f'(3), and g'(3), plug them into the product rule formula:
h'(3) = f'(3) * g(3) + f(3) * g'(3)
After calculating, you'll get the value of h'(3). Please refer to the graphs of f(x) and g(x) to obtain the necessary values and complete the evaluation.

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B = {2.5, 3.5, 4.5} will make the statement false.

Option B is the correct answer.

We have,

A is the set of all integers.

And,

B ⊆ A

This means,

B is a subset of A where all elements in B is in A.

Now,

B = {1, 2, 3}.

This is true.

B = {2.5, 3.5, 4.5}

This is not true.

2.5, 3.5, and 4.5 are not integers.

They are real numbers.

B = {0}

This is true.

B = {-4, -1, 0, 5, 10}

This is true.

Thus,

B = {2.5, 3.5, 4.5} will make the statement false.

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The correct representation of the expression 10 * 10^-8 in scientific notation would be 1.0 x 10^-7

The expression 10 * 10^-8 is not a correct representation of scientific notation because it is not in the proper form. Scientific notation is a way of writing very large or very small numbers in a more compact and convenient form. It consists of two parts: a coefficient (which is always greater than or equal to 1 and less than 10) and a power of 10.

The correct representation of the expression 10 * 10^-8 in scientific notation would be:

1.0 x 10^-7

To put a number in scientific notation, we need to move the decimal point to the left or right until we have a number between 1 and 10. In this case, we can move the decimal point one place to the left to get 1.0, and we need to increase the power of 10 by one to compensate for the decimal point movement. Therefore, the correct exponent is -7.

In summary, the expression 10 * 10^-8 is not a correct representation of scientific notation because it does not have the proper coefficient (which should be between 1 and 10) and exponent (which should be a power of 10). The correct scientific notation representation for this expression is 1.0 x 10^-7.

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Answer:

C) reflection across y = 1

Step-by-step explanation:

We can see that the shape was reflected vertically, so that means that the line of reflection must be horizontal, or be in the form: y = constant.

This immediately narrows down the options to:

From here, we need to find what y-value is halfway between the image and the preimage. We can see that this y-value is 1.

In other words, the shortest distance from any point on the preimage to the line of reflection should be the same as the shortest distance from its corresponding point on the image to the line of reflection.

For example, the shortest distance from E to y = 1 is 2, and the shortest distance from E' to y = 1 is also 2.

Therefore, the correct answer is C) reflection across y = 1.

If the current price of breakfast burritos is lower than the equilibrium price, we would expect an increase in demand and a decrease in supply, leading to a shortage of breakfast burritos.

Producers may choose to raise the price of breakfast burritos to reach the equilibrium price.

Based on the given information, the current price of breakfast burritos is lower than the equilibrium price. We would expect an increase in demand and a shortage of supply.

                                               When the price is below equilibrium, consumers will demand more breakfast burritos, while suppliers might not produce enough to meet the demand, resulting in a shortage. Eventually, the market will adjust, and the price will rise to reach equilibrium, balancing supply and demand.

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The like terms in the following 12a, -4b, -2a, and 8ab are 12a and -2a.

The term "like terms" in algebra refers to terms that have the same variable increased to the same degree. Just the numerical variables can change in terms that are similar to algebra.

We can combine similar terms to make algebraic expressions simpler, making it much simpler to determine the expression's outcome. The term "unlike terms" are those in algebra that do not have identical literal coefficients and cannot be increased to the same power.

Terms that have variables and exponents that differ from one another are said to be unlike terms. For example, 3xy and 4xy are like terms because they have the same coefficient (xy), while 3a and 4ab are unlike terms because they have different coefficients (a) and (ab).

Given that the sequence is 12a, -4b, -2a, and 8ab. Here, the like terms are 12a and -2a, because they have the same coefficient (a).

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To solve for t2 in the equation t2/t1=(p2/p1)^(n-1/n), we can cross-multiply to get t2=(p2/p1)^(n-1/n) * t1.


In the given equation, t2/t1 represents the ratio of two temperatures, while (p2/p1)^(n-1/n) is a ratio of pressures raised to a power. The exponent of (n-1/n) in the equation represents the ratio of specific heats, which is a constant value for a given gas. Thus, we can rewrite the equation as t2/t1= (p2/p1)^k, where k = (n-1/n).

To solve for t2, we can multiply both sides of the equation by t1, which gives us t2= (p2/p1)^k * t1.


Therefore, the equation to solve for t2 is t2= (p2/p1)^(n-1/n) * t1.

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The function f(x) = arcsin([tex]x^2[/tex]- 2x) is increasing for x > 1. To determine where the function f(x) = arcsin([tex]x^2[/tex] - 2x) is increasing, we need to find the intervals where its derivative, f'(x), is positive.

First, find the derivative of the inner function, g(x) = [tex]x^2[/tex] - 2x:
g'(x) = 2x - 2
Now, find the derivative of the outer function, h(u) = arcsin(u), where u = g(x):
h'(u) = 1/√(1 - [tex]u^2[/tex])
Using the chain rule, the derivative of f(x) is the product of the derivatives of the inner and outer functions:
f'(x) = h'(g(x)) * g'(x) = (1/√(1 - [tex](x^2 - 2x)^2)[/tex]) * (2x - 2)
To find where f(x) is increasing, we need to determine where f'(x) > 0. Since the denominator of the fraction is always positive, we only need to focus on the numerator:
2x - 2 > 0
Solving for x, we get:
x > 1

The complete question is:-

Determine where f(x) = arcsin ([tex]x^{2}[/tex] − 2x) is increasing.

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An increase in the exchange rate of the U.S. dollar relative to a trading partner can result from higher interest rates and higher inflation in the U.S. The correct answer is option (e).

This is because higher interest rates attract foreign investors to the U.S., increasing demand for the U.S. dollar and causing its value to rise relative to other currencies.

Higher inflation in the U.S. also makes the dollar less attractive to foreign investors, causing its value to decline.

The other options listed may affect trade and economic activity, but do not directly impact the exchange rate.

Therefore, the correct option is (e) higher interest rates and higher inflation in the U.S.

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The vector sum [tex]\mathbf{u}+\mathbf{v}[/tex]  is approximately equal to [tex]3.9\left < \cos 26^{\circ},,\sin 26^{\circ}\right > , 3.9⟨cos26∘, sin26∘⟩[/tex] in magnitude and direction form.

To find the sum of two vectors, we add their corresponding components. In this case, [tex]\mathbf{u}+\mathbf{v} = 7\left < \cos 0^{\circ},,\sin 0^{\circ}\right > + 4\left < \cos 135^{\circ},,\sin 135^{\circ}\right >[/tex]. Evaluating the x- and y-components separately, we get:

[tex]\mathbf{u}+\mathbf{v} = (7\cos 0^{\circ} + 4\cos 135^{\circ}),\mathbf{i} + (7\sin 0^{\circ} + 4\sin 135^{\circ}),\mathbf{j}[/tex]

[tex]\mathbf{u}+\mathbf{v} = 3.9,\left < \cos 26^{\circ},,\sin 26^{\circ}\right >[/tex]

To write the answer in magnitude and direction form, we can use the ormula [tex]\left|\mathbf{u}+\mathbf{v}\right| = \sqrt{(u_1+v_1)^2 + (u_2+v_2)^2}[/tex] for the magnitude, and [tex]\theta = \tan^{-1}\left(\frac{u_2+v_2}{u_1+v_1}\right)[/tex] for the direction. Plugging in the values, we get:

[tex]\left|\mathbf{u}+\mathbf{v}\right| \approx 3.9\theta \approx 26^{\circ}[/tex]

Therefore, the vector sum [tex]\mathbf{u}+\mathbf{v} is approximately equal to 3.9\left < \cos 26^{\circ},,\sin 26^{\circ}\right > , 3.9⟨cos26∘, sin26∘⟩[/tex] in magnitude and direction form.

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The sign of the expression can be determined by examining the signs of the individual terms involved.

3/5: The sign of this term depends on the sign of the numerator (3) and the denominator (5). As both the numerator and denominator are positive, this term is positive.

(3/5) / (5/3): To simplify this expression, we can multiply the numerator and denominator of the second fraction by 3/5, resulting in (3/5) * (3/5) / 1. The numerator is positive as it is the product of two positive numbers, and the denominator is positive as well. Therefore, this term is positive.

Combining the two positive terms, the overall expression will have a positive sign.

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The answers to all parts sre whicn below.

1. Since there are two theaters each showing three different movies, the total number of possible movie choices is the product of the number of choices at each theater, which is 3 in this case. Therefore, the total number of possible movie choices is 3 x 3 = 9.

Answer: 9 choices

2. The tree diagram is most likely to go with the scenario of "two types of cake in two different shapes with two different types of frosting". The two boxes in the tree diagram represent the two types of cake, the five lines coming out of each box represent the two different shapes of cake and the two types of frosting, and the five boxes at the end of each line represent the different combinations of cake shapes and frosting types.

3. The correct organized list that represents the outcomes for choosing gloves (G) or mittens (M) in red (R), tan (T), or black (B) is:

GR, GT, GB, MR, MT, MB

4. HH, HT, TH, TT

5. There are 4 different cell phone models, each of which has 3 different plans. To find the total number of possible outcomes, we need to multiply the number of options for each choice. Using the multiplication principle, the total number of possible outcomes is:

4 (cell phone models) × 3 (plans for each model) = 12 outcomes

Therefore, the total number of possible outcomes is 12.

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The cost of six dresses is $145.74.

Let's assign variables to represent the costs of the watches and dresses. Let w be the cost of one watch, and d be the cost of one dress.

From the given information, we can set up the following equations:

3w = 5d (Three ladies watches cost as much as five dresses)

(5w + 3d) / 8 = $148.75 (Average cost of 5 watches and 3 dresses is $148.75)

From the first equation, we can solve for w in terms of d:

w = (5d) / 3

Substituting this expression for w into the second equation, we can solve for d:

(5[(5d) / 3] + 3d) / 8 = $148.75

Simplifying the equation:

(25d + 24d) / 24 = $148.75 * 8

49d = $1190

Dividing both sides by 49, we find:

d = $24.29

The cost of six dresses would be:

6 * $24.29 = $145.74

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The mean and the standard deviation for the average of the 36 measurements is given as follows:

The Central Limit Theorem defines the distribution of the sampling distribution of sample means with size n of a normally distributed variable with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex].

For the sampling distribution, the mean and the standard deviation are given as follows:

Considering the sample size of n = 36, the standard deviation is obtained as follows:

[tex]s = \frac{54}{\sqrt{36}} = 9[/tex]

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To find the distance d from y to the subspace of R4 spanned by v1 and v2, we can use the formula. The distance d from y to the subspace of R4 spanned by v1 and v2 is approximately 6.558.

d = ||y - proj_v(y)||

where proj_v(y) is the projection of y onto the subspace spanned by v1 and v2. We can find proj_v(y) as:

proj_v(y) = ((y · v1)/(v1 · v1)) v1 + ((y · v2)/(v2 · v2)) v2

where · represents the dot product. Plugging in the given values, we get:

proj_v(y) = ((-3)(3) + (5)(-4) + (0)(2) + (-4)(-1))/(3^2 + (-4)^2 + 2^2 + (-1)^2) [3 -4 2 -1] + ((-3)(-4) + (5)(1) + (0)(-2) + (-4)(-20))/((-4)^2 + 1^2 + (-2)^2 + (-20)^2) [-4 1 -2 -20]
         = (-26/30) [3 -4 2 -1] + (53/441) [-4 1 -2 -20]
         = [-34/35 77/210 -43/105 -617/441]

Then, we can calculate the distance d as:

d = ||y - proj_v(y)|| = ||[-3 5 0 -4] - [-34/35 77/210 -43/105 -617/441]|| = ||[77/35 -131/210 43/105 -1117/441]||
 = sqrt((77/35)^2 + (-131/210)^2 + (43/105)^2 + (-1117/441)^2)
 = 6.558

Therefore, the distance d from y to the subspace of R4 spanned by v1 and v2 is approximately 6.558.

To compute the distance d from y to the subspace of R4 spanned by v1 and v2, we first need to find the orthogonal projection of y onto the subspace. Let's denote the projection as p(y).

1. Find the coordinates of y with respect to the basis {v1, v2}. Solve the equation: c1 * v1 + c2 * v2 = y.
2. Calculate p(y) using the coordinates found in step 1: p(y) = c1 * v1 + c2 * v2.
3. Calculate the distance d using the formula d = ||y - p(y)||.

After following these steps, you will find the distance d from y to the subspace of R4 spanned by v1 and v2.

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The cost of Rudy's calls on the pay-as-you-go option, totaling 82 minutes and 15 seconds, is approximately R214.05.

1.1 The equation C = 180 + 1.60t represents the total monthly cost (C) in rands for calls made using the business contract for a given amount of time (t) in minutes.

The term 180 represents the fixed monthly subscription fee of R180 that Rudy has to pay regardless of the number of minutes he uses. The term 1.60t represents the cost of the peak-call minutes, with each minute costing R1.60.

Multiplying the cost per minute (1.60) by the number of minutes (t) gives the total cost for the peak-call minutes. Adding the fixed monthly subscription fee to the cost of the peak-call minutes gives the total monthly cost.

1.2 The equation for Rudy's total monthly cost (C) in rand, for calls using the pay-as-you-go option for a given amount of time (t) in minutes can be written as C = 2.60t. Since there is no monthly subscription fee for the pay-as-you-go option, the cost is solely based on the number of minutes (t) multiplied by the cost per minute (2.60).

1.3 To calculate the cost of Rudy's calls on the pay-as-you-go option if they add up to 82 minutes and 15 seconds, we convert the seconds into minutes. Since there are 60 seconds in a minute, 15 seconds is equivalent to 15/60 = 0.25 minutes. Therefore, the total time is 82 + 0.25 = 82.25 minutes. Substituting this value into the equation C = 2.60t, we find that the cost of Rudy's calls on the pay-as-you-go option is C = 2.60 * 82.25 = R214.05.

Thus, the cost of Rudy's calls on the pay-as-you-go option, totaling 82 minutes and 15 seconds, is approximately R214.05.

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To find the point of intersection (P) of the lines r1 and r2. First, let's express the parametric equations of the lines as follows:r1(t) = (2, 1 - 2t, 8 + t)
           r2(t) = (6.5 - 3t, 0.5 + 3t, 10.5 - 3t)

For these lines to intersect, the corresponding coordinates must be equal. Let's denote the parameter for r1 as t1 and for r2 as t2. This gives us the following system of equations:

1. 2 = 6.5 - 3t2
2. 1 - 2t1 = 0.5 + 3t2
3. 8 + t1 = 10.5 - 3t2

Solving equation 1 for t2, we get t2 = (6.5 - 2) / 3 = 1.5. Now, we can plug this value into equations 2 and 3:

1 - 2t1 = 0.5 + 3(1.5) => t1 = -2
8 + (-2) = 10.5 - 3(1.5) => t1 = -2

Since both equations result in t1 = -2, we have a consistent solution. Now, we can find the point of intersection P by plugging t1 and t2 into the parametric equations of r1 and r2:

P = r1(-2) = (2, 1 - 2(-2), 8 - 2) = (2, 5, 6)

Therefore, the point of intersection P is (2, 5, 6).

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Answer:

Step-by-step explanation:

The longer you study the more you can remember so if she studied for 3.50 hrs she is more likely to get a grade in the 90s

If wrong im sry

The bearings of B from A in the given diagrams are shown to be :

To be able to tell the bearing of B from A, you need to Identify and mark two points on a map or diagram of the area. Proceed to draw a straight line between these two points demarcating them clearly.

Employ a protractor in measuring the angle created between the aforementioned line and the north-south axis; this calculated measurement will be referred to as your "bearing."

Then, from the North - south line going up from A, you check the degree measure clockwise to B. Doing this, the bearings of B from A would be 50 °, 115° and 223 ° respectively.

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The volume of the given rectangular prism is 315[tex]cm^3[/tex]

We have the information from the question:

A right rectangular prism has a height of 17.5 cm.

The area of the base of the prism is 18 square cm.

To find the volume, in cubic cm of the right rectangular prism.

We know that the volume of a right rectangular prism is given by-

Volume = A × h

Volume = (18)(17.5)

Volume = 315[tex]cm^3[/tex]

So the volume of the given rectangular prism is 315[tex]cm^3[/tex]

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The answer would be C, (The measure of ZY is the greatest of the three angles.)

YX=3, ZY=5, ZX= 4

(YX)≤(ZX)≤(ZY)

or

3≤4≤5

Answer:

A)  The measure of ∠X is the greatest of the three angles.

Step-by-step explanation:

In a triangle:

Therefore, in triangle XYZ, where XY = 3, YZ = 5 and XZ = 4:

The measure of the all the angles cannot be the same since the three side lengths are different.

(a) A range vector b that satisfies b1 = 5 and b2 = 2 can be expressed as: b = [5; 2; 0] + x3[-2; -2; 1]

(b) The family of solutions of the system ax = b is :

x = [5; 2; 0] + x3[-2; -2; 1], where x3 is any real number.

(a) In exercise 6, part (b), we are given matrix A as:

A = [1 2 3; 4 5 6; 7 8 9]

To find a range vector b in which b1 = 5 and b2 = 2, we can set up a system of equations using the row echelon form of A:

[1 2 3 | b1]    [1  2  3 | 5]
[0 1 2 | b2] => [0 -3 -6 | -8]
[0 0 0 | 0]     [0  0  0 | 0]

From the row echelon form, we can see that the pivot variables are x1 and x2, while the free variable is x3. Using the pivot variables, we can express b1 and b2 in terms of x1 and x2:

x1 + 2x2 + 3x3 = 5
x2 + 2x3 = 2

Solving for x1 and x2 in terms of x3, we get:

x1 = 5 - 2x2 - 3x3
x2 = 2 - 2x3

Thus, a range vector b that satisfies b1 = 5 and b2 = 2 can be expressed as:

b = [5; 2; 0] + x3[-2; -2; 1]

(b) Using the b vector we found in part (a), we can solve the system ax = b using Theorem 1, part (iv):

If a system of linear equations ax = b has a nontrivial solution, then the solution set is a translate of the solution set of the homogeneous system ax = 0.

The homogeneous system is given by:

[1 2 3 | 0]
[0 1 2 | 0]
[0 0 0 | 0]

From the row echelon form, we can see that the pivot variables are x1 and x2, while the free variable is x3. Using the pivot variables, we can express x1 and x2 in terms of x3:

x1 = -2x2 - 3x3
x2 = -2x3

Thus, the general solution of the homogeneous system is:
xh = x3[-2; -2; 1]

To obtain the general solution of the system ax = b, we add the particular solution we found in part (a) to the general solution of the homogeneous system:
xp = [5; 2; 0] + x3[-2; -2; 1]

Thus, the family of solutions of the system ax = b is:
x = [5; 2; 0] + x3[-2; -2; 1], where x3 is any real number.

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[tex]\qquad \qquad \textit{direct proportional variation} \\\\ \textit{\underline{y} varies directly with \underline{x}}\qquad \qquad \stackrel{\textit{constant of variation}}{y=\stackrel{\downarrow }{k}x~\hfill } \\\\ \textit{\underline{x} varies directly with }\underline{z^5}\qquad \qquad \stackrel{\textit{constant of variation}}{x=\stackrel{\downarrow }{k}z^5~\hfill } \\\\[-0.35em] ~\dotfill[/tex]

[tex]\stackrel{\textit{"y" varies directly with "x"}}{y = k(x)}\hspace{5em}\textit{we also know that} \begin{cases} y=24\\ x=4 \end{cases} \\\\\\ 24=k(4)\implies \cfrac{24}{4}=k\implies 6=k\hspace{9em}\boxed{y=6x}[/tex]

Answer:  The probability that a random chosen test taker will score between 400 and 550 is approximately 0.5328.

Step-by-step explanation:

From question 33, we know that the mean score is 500 and the standard deviation is 100.

To obtain the probability that a random test taker will score between 400 and 550, we need to standardise these scores using the formula:

z = (x - μ) / σ

where x is the score we are interested in, μ is the mean score, and σ is the standard deviation.

For x = 400:

z = (400 - 500) / 100 = -1

For x = 550:

z = (550 - 500) / 100 = 0.5

Using a standard normal distribution table or calculator, we can find the probability of z being between -1 and 0.5:

P(-1 < z < 0.5) = P(z < 0.5) - P(z < -1)

= 0.6915 - 0.1587

= 0.5328

Therefore, the probability that a random chosen test taker will score between 400 and 550 is approximately 0.5328.

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A hamburger costs $4. We can substitute this value into one of the original equations to solve for the cost of a hot dog. Carter charges $3 for a hot dog.

Let's represent the cost of a hamburger with "h" and the cost of a hot dog with "d". We can then set up two equations based on the information given:

30h + 25d = 195 (Equation 1)

15h + 20d = 120 (Equation 2)

We can solve this system of equations using either substitution or elimination. Let's use the elimination method.

First, let's multiply Equation 1 by 4:

120h + 100d = 780 (Equation 1 multiplied by 4)

Next, let's multiply Equation 2 by 5:

75h + 100d = 600 (Equation 2 multiplied by 5)

Now we can subtract Equation 2 from Equation 1 to eliminate the "d" variable:

(120h + 100d) - (75h + 100d) = 780 - 600

45h = 180

h = 4

We now know that a hamburger costs $4. We can substitute this value into one of the original equations to solve for the cost of a hot dog. Let's use Equation 1:

30h + 25d = 195

30(4) + 25d = 195

120 + 25d = 195

25d = 75

d = 3

Therefore, Carter charges $3 for a hot dog.

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To show that AεCFG is decidable, we will describe an algorithm that can decide whether a given context-free grammar (CFG) G generates the empty string ε. The terms we will include in this explanation are CFG, decidable, and ε.

Algorithm to decide AεCFG:

1. Convert the given CFG G into Chomsky Normal Form (CNF). In CNF, each production is either of the form A -> BC or A -> a, where A, B, and C are non-terminal symbols and a is a terminal symbol. Converting to CNF ensures that we can easily analyze the structure of G.

2. Perform a depth-first search (DFS) on the CFG G starting from the start symbol S.

3. During the DFS, if we encounter a production of the form S -> ε, then G generates the empty string ε, and we can accept the input.

4. If we do not find a production of the form S -> ε after exploring all possible paths in the DFS, then G does not generate the empty string ε, and we can reject the input.

Since this algorithm can decide whether a given CFG G generates the empty string ε in a finite number of steps, we can conclude that AεCFG is decidable.

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