A survey among freshmen at a certain university received that the number of hours spent studying the week before final exams was normally distributed with mean 25 and standard deviation 7. Round your answers to nearest hundredth. (e.g. 0.345 would be entered as 0.35) (a) Find the 98th percentile of the number of hours studying.

The one-way ANOVA test would be the best choice to compare the average number of adjustments made by service representatives at five different locations in Texas.

To compare the average number of adjustments made by service representatives at five different locations in Texas, the best choice of statistical test would be the one-way ANOVA (Analysis of Variance) test.

The one-way ANOVA test is used to compare the means of three or more independent groups to determine whether there is a statistically significant difference between them.

Five different groups (locations) that we want to compare.

The one-way ANOVA test allows us to test the null hypothesis that all the groups have the same population mean, against the alternative hypothesis that at least one group has a different population mean than the others.

If the p-value is less than the significance level (usually set at 0.05), we can reject the null hypothesis and conclude that there is a statistically significant difference between the means of the groups.

The one-way ANOVA test, we can determine whether there is a significant difference in the average number of adjustments made by service representatives across the five different locations in Texas.

If a significant difference is found, we can then conduct post-hoc tests to determine which specific locations have different means.

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Answer: 45

Step-by-step explanation: 3 is multiplied by 9 and it makes 27 so you do the same to the 5 which would make it 45 and that's how you find the answer

The value of θ for the given trigonometric equation are 37° and -63°.

Given trigonometric equation is,

4tan²θ + 5tanθ = 6

We have to find the value of θ.

The value of θ is in the range [0, 360).

This is in the form of a quadratic equation.

4tan²θ + 5tanθ - 6 = 0

Discriminant = 25 - (4 × 4 × -6) = 121

tan θ = (-5 ± √121) / 8

tan θ = (-5 ± 11) / 8

tan θ = 6/8 = 3/4 and tan θ = -16/8 = -2

The value of θ are tan⁻¹ (3/4) and tan⁻¹(-2).

The value of θ are approximately 37° and -63°.

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The amount of alcohol remaining after 4 hours would be approximately 64 mg/dL.

We can use the formula for exponential decay to model the amount of alcohol remaining in the blood after t hours:

[tex]Q(t) = Q_o* e^{(-kt)[/tex]

where Q₀ is the initial amount of alcohol in the blood, k is the decay constant, and t is the time elapsed.

We know that Q₀ = 200 mg/dL, and Q(2) = 128 mg/dL. We can use this information to solve for k:

[tex]128 = 200 * e^{(-k*2)[/tex]

[tex]e^{(2k)} = 200/128[/tex]

2k ≈ ln(1.5625)

k ≈ -0.345

Now we can use this value of k to find Q(4):

[tex]Q(4) = 200 * e^{(-0.345*4)[/tex]

Q(4) ≈ 64

Therefore, the amount of alcohol is 64 mg/dL.

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The second part of the hike was 2 1/2 miles longer than the first part. Therefore, the correct answer is option D) 1/2.

To find out how much longer the second part of the hike was, we need to subtract the distance covered in the first part from the total distance of the hike.

The distance covered in the first part of the hike is 1 3/4 miles. The distance covered in the second part of the hike is 2 1/2 miles. To add these two distances, we need to convert the fractions to a common denominator

1 3/4 = 7/4

2 1/2 = 5/2

Now we can add the distances

7/4 + 5/2 = 35/20 + 50/20

= 85/20

Simplifying, we get

85/20 = 4 1/4 miles

Therefore, the total length of the hike is 4 1/4 miles.

To find out how much longer the second part of the hike was, we need to subtract the distance covered in the first part (1 3/4 miles) from the total distance of the hike (4 1/4 miles)

4 1/4 - 1 3/4 = 2 1/2

So, the second part of the hike was 2 1/2 miles longer than the first part.

Therefore, the answer is option D) is 1/2.

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Maricella has 4 quarters and 31 nickels in her bag.

Let's use the given terms and set up a system of equations to solve this problem:
Let n = number of nickels
Let q = number of quarters
We know two things:
There are 35 coins in total, so n + q = 35.
The total value is less than $2.75, so 0.05n + 0.25q < 2.75
Now let's solve the system of equations:
Solve the first equation for n:
n = 35 - q.

Substitute this expression for n into the second equation:
0.05(35 - q) + 0.25q < 2.75
Distribute the 0.05 to the terms inside the parentheses:
1.75 - 0.05q + 0.25q < 2.75
Combine like terms:
0.20q < 1
Divide by 0.20:
q < 5
Since q must be a whole number (as it represents the number of quarters), the highest possible value for q is 4.

Now we need to find the number of nickels.
Step 6: Substitute q = 4 back into the equation for n:
n = 35 - q
n = 35 - 4
n = 31.

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Based on your requirements, the appropriate test to determine whether DHA is a better treatment than control for treating eczema, considering that SCORAD is not normally distributed, is the Mann-Whitney U test (also known as the Wilcoxon rank-sum test).

The Mann-Whitney U test is a non-parametric statistical test that compares the distribution of two independent samples. In this case, the samples would be the DHA group and the control group.

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Answer:

your share is 7/5 of a pizza.

Step-by-step explanation:

If there are 2 1/3 pizzas left and you get 3/5 of it, then your share would be:

(2 + 1/3) x 3/5

First, we need to convert the mixed number to an improper fraction:

2 + 1/3 = 7/3

Now we can multiply:

(7/3) x (3/5) = 21/15

We can simplify this fraction by dividing both the numerator and denominator by 3:

21/15 = 7/5

Therefore, your share is 7/5 of a pizza.

The necessary rate of change of the camera's angle as a function of time so that it stays focused on the rocket is approximately 0.38068 rad/s.

To solve this problem, we need to use the concept of similar triangles. Let's draw a diagram:

```
             /|
            / |
           /  |
          /   |
     3000 ft |
         \  |
          \ |
           \|
            O
           / \
          /   \
         /     \
        /       \
       /         \
      /           \
     /             \
    /               \
   /_________________\
          19000 ft
```

In this diagram, O represents the launch pad, and the rocket is at a height of 3000 ft. The camera is located at a distance of 19000 ft from the launch pad. Let's call the angle between the camera's line of sight and the ground α. We want to find dα/dt, the rate of change of α with respect to time.

Now, let's consider the two triangles OAB and OCD:

```
     A        C
     |\      /|
     | \α  / |
     |  \  /  |
  3000|   \/   |h
     |   /\   |
     |  /  \  |
     | / β \ |
     |/____\|
     B      D
```

In triangle OAB, we have:

tan(α + β) = h / 19000

In triangle OCD, we have:

tan(β) = h / x

where x is the distance from the camera to the rocket. We want to find dα/dt, which we can do by differentiating the equation for tan(α + β) with respect to time:

sec^2(α + β) (dα/dt + dβ/dt) = dh/dt / 19000

We can solve for dβ/dt using the equation for tan(β):

dβ/dt = x / h^2 (dh/dt)

Now, we can substitute this into the equation for dα/dt:

dα/dt = [dh/dt / 19000 - x / h^2 (dh/dt)] / sec^2(α + β)

We know that dh/dt = 400 ft/s, and we can find h using the Pythagorean theorem:

h^2 = 19000^2 - (3000 - vt)^2

where v is the velocity of the rocket. Substituting these values into the equation for dα/dt, we get:

dα/dt = [400 / 19000 - x / (19000^2 - (3000 - vt)^2) (400)] / sec^2(α + β)

We still need to find x and β. From the diagram, we can see that:

x = vt

and

tan(β) = 3000 / x

Solving for β and substituting into the equation for dα/dt, we get:

dα/dt = [400 / 19000 - vt / (19000^2 - (3000 - vt)^2) (400)] / sec^2(α + arctan(3000 / vt))

Now, we just need to simplify this expression and leave our answer as an exact number.

To find the necessary rate of change of the camera's angle as a function of time so that it stays focused on the rocket, we can use the tangent function and differentiate it with respect to time. Let θ be the angle between the ground and the camera's line of sight, and y be the rocket's height above the launch pad. The tangent function can be expressed as:

tan(θ) = y/19000

Now, differentiate both sides with respect to time (t):

sec^2(θ) * (dθ/dt) = dy/dt

Given that the rocket is 3000 ft above the launch pad (y = 3000 ft) and its velocity is 400 ft/s (dy/dt = 400 ft/s), we can find the angle θ using the tangent function:

tan(θ) = 3000/19000

θ ≈ 0.15708 radians

Next, find the secant squared of θ:

sec^2(θ) = 1.02485

Now, we can find the rate of change of the camera's angle (dθ/dt) by substituting the given values into the differentiated equation:

1.02485 * (dθ/dt) = 400

dθ/dt = 390.27/1024.85

dθ/dt = 0.38068 rad/s

So, the necessary rate of change of the camera's angle as a function of time so that it stays focused on the rocket is approximately 0.38068 rad/s.

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The approximate chi-square value for this study is 10.84.

In a brand awareness study involving 35 males and 35 females, 25 males and 15 females identified the brand correctly.

To find the chi-square value, we need to create a contingency table and apply the chi-square formula. Here's a quick breakdown of the process:

1. Create a contingency table:
       Males    Females
Correct    25          15
Incorrect  10          20

2. Calculate row and column totals:
       Males    Females    Total
Correct    25          15          40
Incorrect  10          20          30
Total         35          35          70

3. Find the expected frequencies for each cell by multiplying row and column totals and dividing by the overall total (for example, for the 'Correct Males' cell, multiply 40*35 and divide by 70):
       Males    Females
Correct    20          20
Incorrect  15          15

4. Apply the chi-square formula: Χ² = Σ[(Observed-Expected)²/Expected]
Χ² = [(25-20)²/20] + [(15-20)²/20] + [(10-15)²/15] + [(20-15)²/15]
Χ² ≈ 5 + 2.5 + 1.67 + 1.67
Χ² ≈ 10.84

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Samantha is baking cookies and the recipe calls for 7/8 cups of sugar to make 1 cookie, How much sugar is needed to make three cookies? and value of 3 x 7/8 is 21/8.

Samantha is baking cookies and the recipe calls for 7/8 cups of sugar to make 1 cookie, How much sugar is needed to make three cookies

This we get by multiplying 7/8 with 3

3×7/8

21/8

So 21/8 cups of sugar is needed to make three cookies

Hence, the value of multiplication 3×7/8 is 21/8.

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The correct answer is A, true. When the correlation coefficient is positive, it indicates that there is a positive relationship between two variables.

This means that as one variable increases, the other variable also tends to increase. Therefore, above-average values of one variable are associated with above-average values of the other. However, it is important to note that a positive correlation does not necessarily mean that there is a strong relationship between the two variables or that one variable causes the other. It simply means that there is a tendency for the two variables to vary in the same direction.

In statistical analysis, understanding the correlation coefficient and the relationship between variables can help to make better decisions and predictions.

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Goran earned a total of 516 points for the 6 tests.

To find the total points Goran earned for the 6 tests, you need to use the mean score and the number of tests.

Mean score: 86 points.

Number of tests: 6
Formula to calculate the total points:
Mean score = (Total points) / (Number of tests)
Now, rearrange the formula to find the total points:
Total points = Mean score × Number of tests
Plug in the values:
Total points = 86 × 6
Calculate the result:
Total points = 516.

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To calculate the interval for a high degree of confidence that at least 95% of all UHPC specimens adhered to steel will have work of adhesion values between the limits of the interval, we need to use a confidence interval formula. Assuming a normal distribution, we can use the formula:

1. Obtain the mean (µ) and standard deviation (σ) of the work of adhesion values from the sample data.
2. Determine the desired confidence level (95% in this case), which corresponds to a Z-score of 1.96 (from a standard normal distribution table).
3. Calculate the standard error (SE) by dividing the standard deviation (σ) by the square root of the sample size (n): SE = σ / √n.
4. Calculate the margin of error (ME) by multiplying the Z-score by the standard error: ME = 1.96 * SE.
5. Find the confidence interval by subtracting and adding the margin of error from the mean: (µ - ME, µ + ME).

Round your answers to two decimal places. This interval provides a 95% confidence that the true work of adhesion values for at least 95% of all UHPC specimens adhered to steel lie between these limits.

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The volume of the right pyramid with a square base and each edge 4 inches long is 32/3√3 cubic inches. We used the formula V = (1/3)Bh, where B is the area of the base and h is the height of the pyramid.

We found the area of the base to be 16 square inches and the height of the pyramid to be 2√3 inches, using the Pythagorean theorem.

To find the volume of the right pyramid, we first need to understand what a right pyramid is. A right pyramid is a pyramid where the apex is directly above the center of the base and the lateral edges are perpendicular to the base. In this case, we have a right pyramid with a square base, which means that the base of the pyramid is a square and the apex is directly above the center of the square.

To find the volume of this pyramid, we can use the formula V = (1/3)Bh, where B is the area of the base and h is the height of the pyramid. In this case, the base is a square with side length 4 inches, so the area of the base is 4 x 4 = 16 square inches.

To find the height of the pyramid, we can use the Pythagorean theorem, since we know that each edge of the pyramid is 4 inches long. The height is the distance from the apex of the pyramid to the center of the square base. Using the Pythagorean theorem, we can find that the height of the pyramid is √(4^2 - 2^2) = √12 = 2√3 inches.

Now we can plug in the values we've found into the formula V = (1/3)Bh. We get V = (1/3)(16)(2√3) = 32/3√3 cubic inches. This is the volume of the pyramid in cubic inches.

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If you arrive at the railway station at 3 o'clock in the afternoon, you can expect to wait an average of 20 minutes for a limousine to leave for the airport, with a standard deviation of 20 minutes.

To find the expected value and standard deviation of your waiting time at the railway station until a limousine leaves for the airport, we need to use the exponential distribution formula.
First, we know that the expected value of the interdeparture times is 20 minutes. This means that on average, a limousine will depart from the railway station every 20 minutes.
Next, we need to find the probability that a limousine will depart within a certain amount of time after you arrive at the railway station. To do this, we can use the cumulative distribution function (CDF) of the exponential distribution.
The CDF of the exponential distribution is given by:
F(x) = 1 - e^(-λx)
where λ is the rate parameter, which is equal to 1/20 (since the expected value is 20 minutes).
So if you arrive at the railway station at 3 o'clock in the afternoon, your waiting time T until a limousine leaves for the airport is given by:
T = X - 3:00
where X is the time at which the limousine departs from the railway station.
To find the expected value of T, we can use the formula for the mean of the exponential distribution:
E(T) = 1/λ = 20 minutes
So on average, you can expect to wait 20 minutes until a limousine leaves for the airport.
To find the standard deviation of T, we can use the formula for the standard deviation of the exponential distribution:
SD(T) = 1/λ = 20 minutes
So the standard deviation of your waiting time is also 20 minutes.

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The lower confidence bound for the percentage of all such homes that have electrical/environmental problems is 12.8% (rounded to one decimal place). This means we can be 99% confident that the true percentage of all homes with electrical/environmental problems is at least 12.8%.

To calculate the lower confidence bound using a confidence level of 99%, we need to use the formula:
Lower Bound = Sample Proportion - Z-score * Square Root[(Sample Proportion * (1 - Sample Proportion)) / Sample Size]
Here, we need to know the sample proportion, which is the percentage of homes that have electrical/environmental problems. Let's assume that the sample size is 500 and 85 homes out of those have electrical/environmental problems. Then the sample proportion would be:
Sample Proportion = 85/500 = 0.17
Next, we need to find the Z-score for a 99% confidence level. From the Z-tables, we can find that the Z-score for a 99% confidence level is 2.576.
Putting these values in the formula, we get:
Lower Bound = 0.17 - 2.576 * Square Root[(0.17 * (1 - 0.17)) / 500] = 0.128

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The probability that a person who tested positive has the disease is about 0.394 or 39.4%.

To solve this problem, we can use Bayes' theorem, which relates conditional probabilities. Let's define:

A: the event that a person has the disease

B: the event that a person tests positive

We want to calculate the conditional probability P(A|B), which is the probability that a person has the disease, given that they tested positive. Bayes' theorem says:

[tex]$P(A|B) = \frac{P(B|A) \times P(A)}{P(B)}$[/tex]

We know that P(A) = 0.01 since 1% of people have the disease. We also know that the test does not give a negative result if the person has the disease, so P(B|A) = 1. Finally, we need to calculate P(B), the probability that a person tests positive, regardless of whether they have the disease or not.

To calculate P(B), we can use the law of total probability, which says that:

[tex]$P(B) = P(B|A) \times P(A) + P(B|\neg A) \times P(\neg A)$[/tex]

We know that P(B|not A) = 0.015, since 1.5% of people who do not have the disease test positive. We also know that P(not A) = 0.99, since the complement of having the disease is not having the disease. Therefore:

[tex]$P(B) = 1 \times 0.01 + 0.015 \times 0.99 = 0.02535$[/tex]

Now we can substitute these values into Bayes' theorem:

[tex]$P(A|B) = \frac{1 \times 0.01}{0.02535} = 0.394$[/tex]

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The equation of the ellipse that has vertices at (-2,-3), (-2,5), (-4,1), and (0, 1) can be represented as x² + 4y² + 4x - 8y + 8 = 16.

Length of the major axis = distance between the points (-2, -3) and (-2, 5).

                                        = √[(-2 - -2)² + (5 - -3)²]

                                        = 8

2a = 8

a = 4

Length of the minor axis = distance between the points (-4,1), and (0, 1)

                                         = √[(0 - -4)² + (1 - 1)²]

                                         = 4

2b = 4

b = 2

Center = ((-2-2)/2, (-3+5)/2) = (-2, 1)

Equation of the elllipse is ,

[(x - -2)² / 4²] + [(y - 1)² / 2²] = 1

[(x + 2)² / 16] + [(y - 1)² / 4] = 1

Simplifying,

[(x + 2)² / 16] + [4(y - 1)² / 16] = 1

x² + 4x + 4 + 4y² - 8y + 4 = 16

x² + 4y² + 4x - 8y + 8 = 16

hence the required equation is x² + 4y² + 4x - 8y + 8 = 16.

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The type of bias that is most likely to affect the survey results in this scenario is selection bias. This is because the council representatives plan to conduct the survey outside of a large concert in the summer and ask selected adults if they would support an increase in the ticket tax to pay for the new concert hall. This means that the sample of people who will be surveyed will not be representative of the entire population of the city.

For example, the people who attend concerts may not be representative of the entire population of the city, as they may be more likely to be interested in music and cultural events, and therefore more likely to support the proposal for a new concert hall. Additionally, the people who choose to participate in the survey may not be representative of the people who attend concerts, as they may have stronger opinions on the issue or may have a personal interest in the outcome.

Therefore, the results of the survey may be skewed and not truly reflect the opinions of the entire population of the city. To ensure more accurate results, the survey should be conducted in a way that ensures a random and representative sample of the population is surveyed.

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The total surface area of the triangular prism is 845cm².

Surface area refers to the total area of all the faces or surfaces of a three-dimensional object. It is the measure of the exposed area of an object that can be seen or touched.

The surface area of a triangular prism can be calculated by adding the areas of the six faces that make up the prism.

The surface area of a triangular prism.

SA = bh + (S₁ + S₂ + S₃)H

SA = 5×9 + ( 12 + 13 + 15 )× 20

SA = 45 + 800

SA = 845 cm²

Therefore, the surface area is 845 cm².

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The complete question is given below.

Find the total surface area of this triangular prism base=5cm 9cm line in the middle 12cm width=20cm 2 lengths on triangle 13cm and 15cm

The probability is[tex]$\boxed{\frac{1}{3}}$.[/tex]

To find the probability that the product of two numbers chosen from {3, 4, 5, 6} is a multiple of 9, we first need to determine the total number of possible pairs of numbers that can be chosen without replacement from this set.

There are [tex]$\binom{4}{2} = 6$[/tex] ways to choose two numbers from the set. These pairs are:

(3, 4), (3, 5), (3, 6), (4, 5), (4, 6), (5, 6)

Next, we need to determine which of these pairs have a product that is a multiple of 9. A number is a multiple of 9 if and only if it is divisible by 9, so the product of two numbers is a multiple of 9 if and only if at least one of the numbers is a multiple of 3.

From the set {3, 4, 5, 6}, only the numbers 3 and 6 are multiples of 3. Therefore, the pairs with a product that is a multiple of 9 are:

(3, 6), (6, 3)

Note that we have listed both (3, 6) and (6, 3) because the order in which the numbers are chosen does not matter.

Therefore, the probability that the product of two numbers chosen from {3, 4, 5, 6} is a multiple of 9 is:

[tex]\frac{number of pairs with product divisible by 9 }{total number of pairs} = \frac{2}{6} =\frac{1}{3}[/tex]

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If you hear a 1000 Hz tone from a radio that is 20 feet away from you and listen for 5 seconds, approximately 5000 pressure maxima will pass by your ear.

The speed of sound in air at room temperature is approximately 343 meters per second or 1125 feet per second.

The wavelength of a 1000 Hz tone in air can be calculated using the formula:

wavelength = speed of sound / frequency

wavelength = 1125 ft/s / 1000 Hz = 1.125 ft = 13.5 inches

This means that one complete cycle of the wave has a length of 13.5 inches.

If the radio is 20 feet away, then the sound wave will travel a distance of 20 feet from the radio to your ear.

During the 5 seconds you listen to the tone, the wavefronts will be reaching your ear continuously. The number of pressure maxima that pass by your ear will depend on the frequency of the tone and the distance traveled by the wavefront during those 5 seconds.

The distance traveled by the wavefront can be calculated using the formula:

distance = speed of sound x time

distance = 1125 ft/s x 5 s = 5625 ft

The number of pressure maxima that pass by your ear can be calculated by dividing the distance traveled by the wavelength:

number of maxima = distance / wavelength

number of maxima = 5625 ft / 1.125 ft = 5000

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The volume of the solid of revolution is (23/14)π cubic units.

To find the volume of the solid of revolution obtained by rotating the region in the xy-plane bounded by[tex]y = x^3[/tex] + 1, x = 1, y = 1 about the y-axis, we need to use the formula:

V = [tex]∫[a, b] π (f(x))^2 dx[/tex]

where a = 0, b = 1, and f(x) = [tex]x^3 + 1.[/tex]

So, we have:

V = [tex]∫[0, 1] π (x^3 + 1)^2 dx[/tex]

= π[tex]∫[0, 1] (x^6 + 2x^3 + 1) dx[/tex]

= π [tex][1/7 x^7 + 1/2 x^4 + x] [0, 1][/tex]

= π (1/7 + 1/2 + 1)

= π (9/14 + 2/2)

= π (23/14)

Therefore, the volume of the solid of revolution is (23/14)π cubic units.

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full question: Find the volume of the solid of revolution obtained by rotating the region in the xy-plane bounded by y=x^3+1, x=1, y=1 about the y-axis.

The ΔADE ≅ ΔCDF using the SAS congruence theorem.

To Show that two sides and one included angle in one triangle correspond to two sides and one included angle in the second triangle to demonstrate the SAS congruence theorem's conclusion that two triangles are congruent.

We have, the image showing the two triangles

Statement                                  Reason                                                

1. DE ≅ DF; AD ≅ DC                1. Given

2. ∠ADE ≅ ∠CDF                      2. Vertical angles theorem  

2. ΔADE ≅ ΔCDF                      3. SAS congruence theorem

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The answer is that the form of fiber art that involves placing two sets of parallel fibers at right angles and interlacing one set with the other is called weaving.

Weaving is the process of creating fabric by interlacing two sets of threads, known as the warp and weft, at right angles. The warp threads are stretched vertically on a loom, while the weft threads are woven horizontally across the warp. This results in a stable and durable fabric that can be used for a variety of purposes. Weaving has been used for centuries in cultures around the world to create clothing, textiles, and other functional and decorative items.

In weaving, the two sets of fibers are called the warp and the weft. The warp fibers run vertically and are held in tension on a loom. The weft fibers run horizontally and are woven over and under the warp fibers. This process of interlacing the warp and weft fibers creates a strong and flexible fabric, and it allows for the creation of various patterns and designs. Weaving is used in many different cultures and can be seen in a variety of fiber art forms, such as tapestries, rugs, and clothing.

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Sarah left work at 5:20 p.m. on Friday. She work d) 5 1/3 hrs after noon.

To calculate the hours, we know that noon is 12:00 PM, so we have to find the difference between 12 PM and 5:20 PM. We know that clock resets at 12, so she works 5 hours 20 mins after noon.

We can write 5 hours 20 mins as

5 20/60

= 5 1/3 because in 1 hour there are 60 minutes.

Hence, the answer is 5 1/3.

Equal hours, also known as equinoctial hours, were defined as 124 of a day measured from noon to noon; small seasonal changes in this unit were subsequently smoothed out by making it 124 of a solar day.

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Ivory needs to make an additional 17 successful free throws without a miss to attain a success rate of 75%.

Let's start by finding Ivory's current free-throw success rate:

[tex]success rate = \frac{number of successful free throws}{total numbers of free throw} =\frac{7}{15}[/tex]

We want to find the number of additional successful free throws Ivory must make to attain a success rate of $75%$, which can be written as:

[tex]success rate = \frac{numbers of successful free throws}{total numbers of free throws} = \frac{3}{4}[/tex]

Let's call the number of additional successful free throws that Ivory needs to make "x". Then, we can write an equation based on the above expressions for success rate:

[tex]\frac{7 + x}{15 + x } =\frac{3}{4}[/tex]

To solve for x, we can cross-multiply:

4( 7 + x) = 3 (15 + x)

28 + 4x = 45 + 3x

x= 17

Therefore, Ivory needs to make an additional 17 successful free throws without a miss to attain a success rate of 75%.

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The shortest piece is 5 feet long.

Let's use the terms given and set up the equation:
Let x represent the shortest piece.

According to the problem:
- The longest piece is 4 feet longer than the shortest piece, so it is x + 4 feet.
- The remaining piece is 2 feet shorter than the longest piece, so it is (x + 4) - 2 = x + 2 feet.
- The total length of the board is 21 feet.
Now we can set up the equation:
x (shortest piece) + (x + 4) (longest piece) + (x + 2) (remaining piece) = 21 feet.
Combining the terms, we get:
x + x + 4 + x + 2 = 21.
Simplifying, we get:
3x + 6 = 21.
Now, we need to solve for x (the shortest piece):
Subtract 6 from both sides: 3x = 15
Divide both sides by 3: x = 5.

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