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Using trigonometry and the Pythagorean theorem, the length of the hypotenuse AC is found to be 3.5 cm, and using the Pythagorean theorem again, the length of BC is found to be approximately 1.75 cm.

Using trigonometry and the given angle and side length information, we can solve for the length of side BC (x).

We know that

sin(A) = opposite/hypotenuse

sin(30) = AC/7

AC = 7 × sin(30)

AC = 3.5 cm

Using the Pythagorean theorem, we have

BC² = AC² - AB²

BC² = (3.5)² - (7)² sin²(30)

BC² = 3.0625

BC = √3.0625

BC = 1.75 cm (rounded to two decimal places)

Therefore, the value of x is approximately 1.75 cm.

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The exact value of sin 2θ is 12/37

Trigonometry deals with the relationship between the ratios of the sides of a right-angled triangle with its angles.

We have:

cot θ = 6

Thus, tan θ = 1/6

Using the given information, we can sketch the location of the angle θ in the quadrant (See the attached image).

Thus, we can calculate the value of the hypotenuse using the Pythagoras theorem. That is:

hypotenuse = √((-6)² + (-1)²) = √37

sin θ = -1/√37

cos θ = -6/√37

Using trig. identity:

sin 2θ = 2sinθ·cosθ

sin 2θ = 2 * (-1/√37) * (-6/√37)

sin 2θ = 12/37

Therefore, the exact value of sin 2θ is 12/37

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Complete Question

If cot θ = 6,and θ lies in quadrant 3, find the exact value of sin 2θ

The test statistic z ≈ -2.59 falls into the rejection region (z < -1.96).

Therefore, we reject the null hypothesis (H₀) in favor of the alternative hypothesis (H₁).

a) Null and alternative hypothesis:

The null hypothesis (H₀) states that there is no significant difference between the claimed weekly production volume and the actual production volume.

The alternative hypothesis (H₁) states that there is a significant difference between the claimed weekly production volume and the actual production volume.

H₀: μ = 370 units (the claimed weekly production volume is true)

H₁: μ ≠ 370 units (the claimed weekly production volume is not true)

b) Critical value:

Since we're using a two-tailed test at α = 0.05 significance level, we'll look for the critical value (z-score) that corresponds to the 2.5% in each tail (5% total) of the standard normal distribution.

The critical value for a two-tailed test at α = 0.05 is ±1.96. The rejection region consists of the areas where the z-score is less than -1.96 or greater than 1.96.

c) Test statistic:

To calculate the test statistic, we will use the following formula:

z = (X - μ) / (σ / √n)

z = (355 - 370) / (19 / √30) = -15 / (19 / √30) ≈ -2.59

d) Conclusion:

The test statistic z ≈ -2.59 falls into the rejection region (z < -1.96).

Therefore, we reject the null hypothesis (H₀) in favor of the alternative hypothesis (H₁).

This means that there is significant evidence to suggest that the claimed weekly production volume of 370 units is not true.

The Vice President's suspicion about the statement appears to be correct, and further investigation should be conducted to determine the actual production volume.

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Answer:

2

Step-by-step explanation:

The value of cube root of 8, ∛8, is 2.

Answer:

2

Step-by-step explanation:

cube root 8 means:

a number x such that x * x * x = 8

the answer is 2, since 2 * 2 * 2 = 8

Answer:

x = 16.2

Step-by-step explanation:

Using trig functions:

cos theta = adjacent side/ hypotenuse

cos 72 = 5/x

Solving for x

x = 5 /cos 72

x=16.18033

Rounding to the nearest tenth

x = 16.2

Answer: x = 16.2 units

Step-by-step explanation:

     We are given an angle (72°), the adjacent side (5), and the hypotenuse (x). We will use the cosine function to solve.

cos(θ) = [tex]\frac{adjacent}{hypotenuse }[/tex]

cos(72°) = [tex]\frac{5}{x }[/tex]

xcos(72°) = 5

x = [tex]\frac{5}{cos(72\°)}[/tex]

x = 16.1803398 ≈ 16.2

Thus, 9 percent of the total dance tickets available is found to be purchased in advance.

Although the usage of percent and percentage differs slightly, they both signify the same thing. It is customary to use percent or the symbol (%) along with a numerical value. One tenth of something is one percent.

Given data:

Total dance tickets = 200

Advanced purchased tickets = 18

Let x be the percentage of advance booked tickets.

Then,

x% of 200 = 18

x*200 / 100 = 18

2x = 18

x = 18/2

x = 9%

Thus, 9 percent of the total dance tickets available is found to be purchased in advance.

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The equivalent value of the expression is y = + 3/5 and y = -3/5

Given data ,

Let the expression be represented as A

Now , the value of A is

y = ±3/5

On simplifying the equation , we get

y = +3/5

And, y = -3/5

Now , the decimal values of y are

y = ±0.6

Hence , the expression is y = ±0.6

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The statement "1+1 is the hardest problem in the world" is generally meant to be taken as a joke or a humorous exaggeration.

The phrase may be used ironically to emphasize the difficulty of a seemingly simple task or to highlight the importance of attention to detail. For example, a complex mathematical proof may require multiple steps and involve intricate calculations, but the simplest mistake, such as an error in basic arithmetic, could render the entire proof invalid.

In this context, the phrase "1+1 is the hardest problem in the world" could be used to underscore the importance of checking and double-checking even the most basic assumptions and calculations in complex problem-solving.

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The sample space describing all possible outcomes is {1. 2. 3. 4. 5. 6}

From the question, we have the following parameters that can be used in our computation:

A number cube with faces labeled from 1 to 6 will be rolled once.

This means that

Sample space = {1. 2. 3. 4. 5. 6}

Using the above as a guide, we have the following:

The outcomes for the event of rolling the number 1 ,3 , or 4. is

Outcome = {1, 3, 4} where we have

P(1) = 1/6

P(3) = 1/6

P(4) = 1/6

Altogether, we have

P(1, 3, or 4) = 1/6 + 1/6 + 1/6

P(1, 3, or 4) = 3/6

P(1, 3, or 4) = 1/2

Hence, the probability is 1/2

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Complete question

A number cube with faces labeled from 1 to 6 will be rolled once. The number rolled will be recorded as the outcome.

Give the sample space describing all possible outcomes.

Then give all of the outcomes for the event of rolling the number 1 ,3 , or 4.

If there is more than one element in the set, separate them with commas.

The value of x in each expressions are:

1) 6

2) 1

3) -3

4) -21

5) -4

We have,

The expressions are:

1)

[tex]3^x = 9^3[/tex]

And,

9³ = (3²)³ = [tex]3^6[/tex]

So,

[tex]3^x = 3^6[/tex]

And,

x = 6

2)

[tex]4^{x + 1}[/tex] = 16

16 = 4²

So,

x + 1 = 2

x = 2 - 1

x = 1

3)

[tex](1/3)^x[/tex] = 27

27 = 3³

So,

[tex]3^{-1x}[/tex] = 3³

-x = 3

x = -3

4)

[tex]5^{3x}[/tex] = [tex]25^{x - 1}[/tex] =

[tex]5^{3x}[/tex] = [tex]5^{2x - 21}[/tex]

3x = 2x - 21

3x - 2x = -21

x = -21

5)

[tex]2^{-x}[/tex] = 16

16 = [tex]2^4[/tex]

So,

-x = 4

x = -4

Thus,

The value of x in each expression are:

1) 6

2) 1

3) -3

4) -21

5) -4

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The total money raised is (15/7) * $88,000 = $188,571.43.

a) Let x be the total amount raised.

Charity A received (1/3)(x - A), Charity C received (1/4)(x - C). We have A = (1/3)(B+C) and C = (1/4)(A+B).

Solving these equations, we find B's share to be 7/15 of the total amount.

p) Since Charity B received $88,000, which is 7/15 of the total amount, the total money raised is (15/7) * $88,000 = $188,571.43.

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An equation for this linear function that describes how height varies with age is y = 1/5(x) + 2.

The predicted height at 9 is 3.8 feet.

The predicted height at 31 is 8.2 feet.

In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical expression:

y - y₁ = m(x - x₁)

Where:

Conversion:

1 feet = 12 inches

24 inches = 24/12 = 2 feet.

Next, we would determine the slope of this line;

Slope (m) = (y₂ - y₁)/(x₂ - x₁)

Slope (m) = (4 - 2)/(10 - 0)

Slope (m) = 2/10

Slope (m) = 1/5

At data point (0, 2) and a slope of 1/5, a linear equation for this line can be calculated by using the point-slope form as follows:

y - y₁ = m(x - x₁)

y - 2 = 1/5(x - 0)  

y = 1/5(x) + 2

When x = 9, the predicted height is given by;

y = 1/5(9) + 2

y = 3.8 feet.

When x = 31, the predicted height is given by;

y = 1/5(31) + 2

y = 8.2 feet.

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The sum of all the number is found to be 630 which is not given in the options.

Let the middle number be x. Then the five numbers smaller than x have an average of 44, so their sum is 544 = 220. Similarly, the five numbers larger than x have an average of 68, so their sum is 568 = 340.

The total of all the numbers is 9x since x is the median and average of all the numbers. The total of all the numbers equals 9x since we know that x is the average of all the numbers.

Therefore, we have,

9x = 220 + x + 340

Simplifying and solving for x, we get,

8x = 560

x = 70

Therefore, the sum of all the numbers is 9x = 970 = 630.

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Complete question - Nine numbers are written in ascending order, the middle number is the average of all the numbers, the average of the five largest numbers is 68, and the average of the five smallest is 44. What is the sum of all the numbers? ?

a) 112

b) 504

c) 144

d) 560

d) 122

The coordinates of each point after the quadrilateral MNPQ is translated 2 units right and 5 units down are:

M' = (x1 + 2, y1 - 5)

N' = (x2 + 2, y2 - 5)

P' = (x3 + 2, y3 - 5)

Q' = (x4 + 2, y4 - 5)

To calculate the coordinates, we shall assume that the coordinates of the points of the quadrilateral MNPQ are:

M = (x1, y1)

N = (x2, y2)

P = (x3, y3)

Q = (x4, y4)

Next, we translate the quadrilateral 2 units right and 5 units down by adding 2 to the x-coordinate and subtracting 5 from the y-coordinate of each point.

The new coordinates of the points after the translation will be:

M' = (x1 + 2, y1 - 5)

N' = (x2 + 2, y2 - 5)

P' = (x3 + 2, y3 - 5)

Q' = (x4 + 2, y4 - 5)

Therefore, the coordinates of each point after the quadrilateral is translated 2 units right and 5 units down are:

M' = (x1 + 2, y1 - 5)

N' = (x2 + 2, y2 - 5)

P' = (x3 + 2, y3 - 5)

Q' = (x4 + 2, y4 - 5)

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Question completion:

Although part of your question is missing, you might be referring to the below question:

What are the coordinates of each point after quadrilateral MNPQ is translated 2 units right and 5 units down?

Answer:

[tex](f \circ g)(\text{x}) = \frac{13}{13-\text{x}}[/tex]

Domain: [tex](-\infty,0) \cup (0,13) \cup (13,\infty)[/tex]

=================================================

Explanation:

Let's find the function composition.

The notation [tex](f \circ g)(\text{x})[/tex] is the same as [tex]f(g(\text{x}))[/tex]

[tex]f(\text{x}) = \frac{\text{x}}{\text{x}-1}\\\\\\f(g(\text{x})) = \frac{g(\text{x})}{g(\text{x})-1}\\\\\\f(g(\text{x})) = g(\text{x}) \div \Big( g(\text{x}) - 1\Big)\\\\\\[/tex]

Then,

[tex]f(g(\text{x})) = \frac{13}{\text{x}} \div \left(\frac{13}{\text{x}}-1}\right)\\\\\\f(g(\text{x})) = \frac{13}{\text{x}} \div \left(\frac{13}{\text{x}}-\frac{\text{x}}{\text{x}}\right)\\\\\\f(g(\text{x})) = \frac{13}{\text{x}} \div \frac{13-\text{x}}{\text{x}}\\\\\\f(g(\text{x})) = \frac{13}{\text{x}} * \frac{\text{x}}{13-\text{x}}\\\\\\f(g(\text{x})) = \frac{13}{13-\text{x}}\\\\\\[/tex]

-----------------

Now let's find the domain.

If we plugged x = 0 into g(x), then we get a division by zero error.

This means we must exclude this value from the domain.

For similar reasoning, we must exclude x = 13 because we get a division by zero error in [tex]f(g(\text{x})) = \frac{13}{13-\text{x}}[/tex]

We could have any other real number to be plugged in for x.

Here's what the domain looks like in interval notation.

[tex](-\infty,0) \cup (0,13) \cup (13,\infty)[/tex]

We effectively poke holes at 0 and 13 on the number line.

The amount of oil in the lantern at 12 noon is 64.33 ounces

The time can be represented with x and the amount with y

Note that

x = number of hours from 12 noon

So, we have the following ordered pairs

(x, y) = (0, y) (2, 63), (5, 61)

Using the slope formula, we have

(y - 63)/(0 - 2) = (61 - 63)/(5 - 2)

So, we have

(y - 63)/-2 = -2/3

This gives

y - 63 = 4/3

Add 63 to both sides

y = 63 + 4/3

Evaluate

y = 64.33

Hence, the amount in the lantern at 12 noon is 64.33 ounces

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Complete question

A camper lights an oil lantern at 12 noon and lets it burn continuously. Once the lantern is lit, the lantern burns oil at a constant rate each hour. At 2 p.m., the amount of oil left in the lantern is 63 ounces. At 5 p.m., the amount of oil left in the lantern is 61 ounces.

Based on the average rate of oil burning per hour, how much oil, in ounces, was in the lantern at 12 noon?

Answer:ina can row 6mph in still water and 2 mph in current

Step-by-step explanation:

Answer: A) Let's first calculate how far the van would have traveled in the half hour before the car started. The van's speed is 50 km/h, which means in half an hour it would have traveled 50/2 = 25 km.

Now let's consider the time it takes for the car to catch up to the van. We can represent this using the formula:

distance = rate × time

Let's call the time it takes for the car to catch up "t". We know that during this time, the van is also traveling. In fact, it has been traveling for t + 0.5 hours (the half hour before the car started plus the time it takes for the car to catch up). So the distance the van has traveled is:

distance van = 50 × (t + 0.5)

The distance the car has traveled is:

distance car = 60t

When the car catches up to the van, they will have traveled the same distance. So we can set the two distances equal to each other:

50(t + 0.5) = 60t

Simplifying this equation:

50t + 25 = 60t

Subtracting 50t from both sides:

25 = 10t

So t = 2.5 hours.

But we're not done yet! We need to add the 0.5 hours that the van traveled before the car started to get the total time it took for the car to catch up:

t + 0.5 = 2.5 + 0.5 = 3 hours

So the car catches up to the van 3 hours after the van started, or at 12:15 pm.

B) We can use the formula:

distance = rate × time

to find the distance between the two towns. We know the car traveled for 6 hours (from 9:45 am to 3:45 pm) and its speed was 60 km/h. So the distance it traveled is:

distance car = 60 × 6 = 360 km

We also know that the van traveled for 6.5 hours (from 9:15 am to 3:45 pm) and its speed was 50 km/h. So the distance it traveled is:

distance van = 50 × 6.5 = 325 km

The distance between the two towns is the difference between these two distances:

distance = distance car - distance van = 360 - 325 = 35 km

So the distance between the two towns is 35 km.

We can start by finding the rate at which the printer is printing photos. Since it takes 3 minutes to print 6 photos, we can calculate the rate as 6 photos / 3 minutes = 2 photos/minute. This means that for every minute that passes, the printer prints 2 photos.

Now that we know the rate, we can use it to find out how long it will take to print 16 photos. Since the rate is 2 photos/minute, we can set up an equation to solve for y (time) when x (photos) is 16: 2 = 16/y. Solving for y, we get y = 16/2 = 8.

Therefore, when x (photos) is 16, y (time) is 8 minutes. This means that it will take the printer 8 minutes to print 16 photos.

If y (time) is 5 minutes, we can use the rate we calculated earlier to find out how many photos the printer will print in that time. Since the rate is 2 photos/minute, we can multiply it by the time to find out how many photos will be printed: 2 photos/minute * 5 minutes = 10 photos.

Therefore, if y (time) is 5 minutes, the printer will print 10 photos.

If y (time) is 7 minutes, we can use the rate we calculated earlier to find out how many photos the printer will print in that time. Since the rate is 2 photos/minute, we can multiply it by the time to find out how many photos will be printed: 2 photos/minute * 7 minutes = 14 photos.

Therefore, if y (time) is 7 minutes, the printer will print 14 photos.

The probability that none of the four adults is very confident in the nutritional information on the restaurant menus is 0.208 or approximately 20.8%. 

Ready to approach this issue by utilizing the hypergeometric conveyance, since we are examining without substitution from a limited populace of two sorts (those who are exceptionally sure and those who are not exceptionally sure).

Let X be the number of grown-ups in a sample of 4 who are not exceptionally sure about the wholesome data on eatery menus.

At that point, X takes after a hypergeometric dissemination with parameters N = 1000, n = 4, and K = 850 (since 850 grown-ups are not exceptionally certain).

The likelihood that none of the 4 grown-ups are exceptionally certain can be communicated as:

P(X = 4) = (K select 4) / (N select 4)

where (K select 4) speaks to the number of ways to select 4 grown-ups from the population of 850 who are not exceptionally sure,

and (N select 4) speaks to the entire number of ways to select 4 grown-ups from the populace of 1000.

Utilizing the equation for binomial coefficients, ready to disentangle this expression as:

P(X = 4) = [(850*849*848*847) / (4*3*2*1)] / [(1000*999*998*997) / (4*3*2*1)]

= 0.208

Hence, the likelihood that none of the four grown-ups are exceptionally certain within the wholesome data on eatery menus is 0.208 or approximately 20.8%. 

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Answer:

Step-by-step explanation:

4x - 5 = 7

4x = 7 + 5

4x = 12

x = 12/4

x = 3

The correct statements are as follows:-

There is a cluster from 3 to 4.

There is a gap between 1 and 3.

There is a peak at 4.

The data is skewed left.

We have,

Scatter plots are graphs that show how two variables in a data collection relate to one another. On a two-dimensional plane or in a Cartesian system, it represents data points.

The right statements for the dot plots are as follows:-

There is a cluster from 3 to 4.

There is a gap between 1 and 3.

There is a peak at 4.

The data is skewed left.

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The projection of (v + w) onto u is (-8 √(2)i + 6 √(2)j).

Now, let's consider the given vectors u = - i + j, v = 8i - 2j, and w = - 4j. The question asks us to find the projection of vector v + w onto the vector u, denoted as proj_u(v + w). To find this projection, we need to use the dot product between the two vectors.

First, we need to calculate v + w, which is (8i - 2j) + (-4j) = 8i - 6j. Next, we calculate the dot product of u and (v + w):

u · (v + w) = (-i + j) · (8i - 6j)

= -8i + 6j - 8i + 6j

= -16i + 12j

The dot product measures the similarity between two vectors, and in this case, it gives us the component of (v + w) that is parallel to u. To find the projection of (v + w) onto u, we need to divide this component by the magnitude of u:

proj_u(v + w) = (u · (v + w)) / ||u||

= (-16i + 12j) / √(2)

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Answer:

5)

The inverse of the function f(x) = x^7 can be found by following these steps:

Step 1: Replace f(x) with y. The equation becomes y = x^7.

Step 2: Interchange x and y in the equation, so it becomes x = y^7.

Step 3: Solve the equation for y by taking the seventh root of both sides. This yields y = x^(1/7).

Therefore, the inverse function of f(x) = x^7 is g(x) = x^(1/7), which maps any given value of x to its seventh root.

It's important to note that the domain and range of the inverse function are the opposite of those of the original function. The domain of the inverse function is all real numbers, while the range is only positive real numbers. The domain of the original function is all real numbers, while the range is also all real numbers.

6)

To find the inverse of the function f(x) = (-2/5)x^3, we can follow these steps:

Step 1: Replace f(x) with y. The equation becomes y = (-2/5)x^3.

Step 2: Solve the equation for x in terms of y.

Multiply both sides by -5/2:

(-5/2) y = x^3

Take the cube root of both sides:

x = [(-5/2) y]^(1/3)

Step 3: Replace x with f^-1(y) to obtain the inverse function.

f^-1(y) = [(-5/2) y]^(1/3)

Therefore, the inverse function of f(x) = (-2/5)x^3 is f^-1(y) = [(-5/2) y]^(1/3).

It is important to note that the domain and range of the inverse function are the opposite of those of the original function. The domain of the inverse function is all real numbers, while the range is also all real numbers. The domain of the original function is all real numbers, while the range is only negative real numbers if x is negative and only positive real numbers if x is positive.

Answer:

Step-by-step explanation:

Answer:

Step-by-step explanation:

In November 1998, former professional wrestler Jesse “The Body” Ventura was elected governor of Minnesota. Up until right before the election, most polls showed he had little chance of winning. There were several contributing factors to the polls not reflecting the actual intent of the electorate:

Ventura was running on a third-party ticket and most polling methods are better suited to a two-candidate race.

Many respondents to polls may have been embarrassed to tell pollsters that they were planning to vote for a professional wrestler.

The mere fact that the polls showed Ventura had little chance of winning might have prompted some people to vote for him in protest to send a message to the major-party candidates.

But one of the major contributing factors was that Ventura recruited a substantial amount of support from young people, particularly college students, who had never voted before and who registered specifically to vote in the gubernatorial election. The polls did not deem these young people likely voters (since in most cases young people have a lower rate of voter registration and a turnout rate for elections) and so the polling samples were subject to sampling bias: they omitted a portion of the electorate that was weighted in favor of the winning candidate.

SAMPLING BIAS

A sampling method is biased if every member of the population doesn’t have equal likelihood of being in the sample.

So even identifying the population can be a difficult job, but once we have identified the population, how do we choose an appropriate sample? Remember, although we would prefer to survey all members of the population, this is usually impractical unless the population is very small, so we choose a sample. There are many ways to sample a population, but there is one goal we need to keep in mind: we would like the sample to be representative of the population.

Returning to our hypothetical job as a political pollster, we would not anticipate very accurate results if we drew all of our samples from among the customers at a Starbucks, nor would we expect that a sample drawn entirely from the membership list of the local Elks club would provide a useful picture of district-wide support for our candidate.

One way to ensure that the sample has a reasonable chance of mirroring the population is to employ randomness. The most basic random method is simple random sampling.

Answer:

Ned has 20 white marbles.

Step-by-step explanation:

Let's call the number of red marbles "r".

We know that Ned has twice as many white marbles as red marbles, so the number of white marbles would be 2r.

We also know that he has five more blue marbles than white marbles, so the number of blue marbles would be 2r + 5.

The total number of marbles in Ned's collection is 55, so:

r + 2r + (2r + 5) = 55

Simplifying this equation, we get:

5r + 5 = 55

Subtracting 5 from both sides:

5r = 50

Dividing by 5:

r = 10

So Ned has 10 red marbles.

We can use this to find the number of white marbles:

2r = 2(10) = 20

So Ned has 20 white marbles.