Find the theoretical probability of the event occurring on a single roll of a number cube. P(multiple of 3) = A) 0
B) 1/3
C) 1/2
D) 2/3​

The probability of drawing at least one red card is 1 - 30/90 = 60/90, which can be reduced to 2/3.

The probability of drawing at least one red card can be found by calculating the probability of drawing two yellow cards and subtracting that from 1.

The probability of drawing a yellow card on the first draw is 6/10.
If a yellow card is drawn on the first draw, there are 5 yellow cards and 4 red cards left, so the probability of drawing another yellow card is 5/9.
The probability of drawing two yellow cards in a row is (6/10) * (5/9) = 30/90.

Therefore, the probability of drawing at least one red card is 1 - 30/90 = 60/90, which can be reduced to 2/3.

So the answer is: 2/3.

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Answer:

Selecting people for the survey based on those who say they watch television with their children would introduce bias into the sample. People who watch television with their children may have different preferences than those who do not, and this could skew the results of the survey. Additionally, this method would exclude people who do not have children, or who do not watch television with their children for other reasons. On the other hand, using a random process to select participants for the survey would ensure that the sample is representative of the population, and would help to minimize bias. Random sampling would give each person in the population an equal chance of being selected for the survey, which would help to ensure that the results of the survey are accurate and can be generalized to the broader population.

Answer:

if you survey only parents who are watching with children, they would have a bias towards animated comedy. it will not represent all of the viewers, and therefore will not be accurate. other viewers (who are not parents) would be less likely to watch cartoons.

Step-by-step explanation:

Emma is currently 21 years old.

Let's denote Emma's age as E and Claire's age as C. We are given the following information:

The age of a person can be counted differently in different cultures. This calculator is based on the most common age system. In this system, age increases on a person's birthday. For example, the age of a person who has lived for 3 years and 11 months is 3, and their age will increase to 4 on their next birthday one month later. Most western countries use this age system.

1. When Emma was born, Claire was E years old.
2. Currently, Claire is 42 years old (C = 42).

Since Emma is currently the same age as Claire was when Emma was born, we can say E = C - E.

Now let's solve for E:

E = 42 - E
2E = 42
E = 21

So, Emma is currently 21 years old.

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The plane P can be represented as the vector (8s, 7s + 8t, 7t), where s and t are real numbers.

Let's call the plane spanned by v1 and v2 as P.

A point that lies in P can be represented as a linear combination of v1 and v2.

Let's say the point we want to represent is (x, y, z). Then, we can write:

(x, y, z) = s * v1 + t * v2

Expanding this expression using vector addition and scalar multiplication, we get:

(x, y, z) = (8s, 7s, 0s) + (0t, 8t, 7t)

= (8s, 7s + 8t, 7t)

Therefore, any point that lies in the plane P can be represented as the vector (8s, 7s + 8t, 7t), where s and t are real numbers.

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The following parts can be answered by the concept of Standard deviation.

4. To test the scenario, we can set up the following hypotheses:

Null Hypothesis (H0): The proportion of positions where men are paid more than women is equal to 0.5 (equal pay).
Alternative Hypothesis (H1): The proportion of positions where men are paid more than women is greater than 0.5 (men are paid more).

5. To calculate the test statistic and p-value, we'll use a one-sample proportion test. In this case, the sample proportion (p-hat) is 19/21. The hypothesized proportion (p0) is 0.5. The sample size (n) is 21.

Test Statistic (z) = (p-hat - p0) / √((p0 × (1 - p0)) / n)
z = (19/21 - 0.5) / √((0.5 × (1 - 0.5)) / 21)
z ≈ 6.43

To find the p-value, we look at the upper tail of the standard normal distribution corresponding to the z-score of 6.43. Since this z-score is very large, the p-value is extremely small (approaching 0).

6. With such a small p-value (close to 0) and a significance level (α) of 0.05, we reject the null hypothesis in favor of the alternative hypothesis. This sample provides convincing evidence that men are paid more than women in the examined positions at the 0.05 significance level.

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The following parts can be answered by the concept of Standard deviation.

4. To test the scenario, we can set up the following hypotheses:

Null Hypothesis (H0): The proportion of positions where men are paid more than women is equal to 0.5 (equal pay).
Alternative Hypothesis (H1): The proportion of positions where men are paid more than women is greater than 0.5 (men are paid more).

5. To calculate the test statistic and p-value, we'll use a one-sample proportion test. In this case, the sample proportion (p-hat) is 19/21. The hypothesized proportion (p0) is 0.5. The sample size (n) is 21.

Test Statistic (z) = (p-hat - p0) / √((p0 × (1 - p0)) / n)
z = (19/21 - 0.5) / √((0.5 × (1 - 0.5)) / 21)
z ≈ 6.43

To find the p-value, we look at the upper tail of the standard normal distribution corresponding to the z-score of 6.43. Since this z-score is very large, the p-value is extremely small (approaching 0).

6. With such a small p-value (close to 0) and a significance level (α) of 0.05, we reject the null hypothesis in favor of the alternative hypothesis. This sample provides convincing evidence that men are paid more than women in the examined positions at the 0.05 significance level.

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The value of Arc length = ∫√(1 + (dy/dx)²) dx from 0 to 15/16

The arc length of y = ln(1-x²) on the interval 0 < x < 15/16 can be found using the arc length formula:


To find the arc length, follow these steps:

1. Differentiate y with respect to x: dy/dx = -2x/(1-x²)
2. Square dy/dx and add 1: (dy/dx)² + 1 = (4x² + 1 - x²) / (1-x²)²
3. Find the square root: √((4x² + 1 - x²) / (1-x²)²)
4. Integrate with respect to x from 0 to 15/16: ∫√((4x² + 1 - x²) / (1-x²)²) dx from 0 to 15/16

Unfortunately, the integral cannot be evaluated using elementary functions. You will need to use numerical methods, like Simpson's rule or a numerical integration calculator, to approximate the arc length.

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To determine if there are any outliers in the data, we can use the Interquartile Range (IQR) method. The IQR is the range between the first quartile (Q1) and the third quartile (Q3). Here's a step-by-step explanation:

Step 1: Calculate the IQR. IQR = Q3 - Q1 = 18.0 - 15.0 = 3.0

Step 2: Find the lower and upper bounds for outliers.

Lower Bound = Q1 - 1.5 * IQR = 15.0 - 1.5 * 3.0 = 15.0 - 4.5 = 10.5

Upper Bound = Q3 + 1.5 * IQR = 18.0 + 1.5 * 3.0 = 18.0 + 4.5 = 22.5

Step 3: Compare the bounds to the minimum and maximum values. The minimum value (13.0) is greater than the lower bound (10.5) and the maximum value (22.0) is less than the upper bound (22.5).

Since both the minimum and maximum values are within the bounds, we know that there are no outliers in the data.

So, the correct answer is: A) there are no outliers in the data.

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The magnitude of an earthquake 25 times as strong as a 4.2 earthquake would be 6.3 on the Richter scale. If a small earthquake measures 4.2 on the Richter scale, the magnitude of an earthquake 25 times as strong will be 6.3 on the Richter scale.

The Richter scale measures the magnitude of an earthquake, which is a logarithmic scale. This means that each increase of one on the Richter scale corresponds to a ten-fold increase in the amplitude of the earthquake waves.
If a small earthquake measures 4.2 on the Richter scale, an earthquake 25 times as strong would have an amplitude that is 25 times greater than the amplitude of the 4.2 earthquakes.
To find the magnitude of the stronger earthquake, we need to determine how many times greater the amplitude of the 25 times stronger earthquake is than the amplitude of the 4.2 earthquakes.
25 times stronger = 25 x 10 = 250 (because each increase of one on the Richter scale corresponds to a ten-fold increase in amplitude)
So the magnitude of the 25 times stronger earthquake can be calculated using the formula:
M2 = M1 + log(A2/A1)
Where M1 is the magnitude of the 4.2 earthquakes, A1 is its amplitude, A2 is the amplitude of the 25 times stronger earthquake, and M2 is the magnitude we want to find.
Substituting the values we have:
M1 = 4.2
A1 = 1 (by definition)
A2 = 250
M2 = 4.2 + log(250/1)
M2 = 6.3
Therefore, the magnitude of an earthquake 25 times as strong as a 4.2 earthquake would be 6.3 on the Richter scale.

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Let's say Dali's walking speed is x and his running speed is 3x.

In the morning, let's say the total distance is d. Therefore, Dali walks d/2 with speed x and runs d/2 with speed 3x. The time it takes for him to complete this distance is given by:

d/2x + d/2(3x) = 24

Simplifying this equation, we get:

d = 3x(8)

d = 24x

So, the total distance Dali covers in the morning is 24x.

After school, let's say the distance from school to home is d'. Therefore, Dali walks d'/2 with speed x and runs d'/2 with speed 3x. The time it takes for him to complete this distance is given by:

d'/2x + d'/2(3x) = ?

We don't know how much time it takes Dali to get home, so we'll leave the right-hand side of the equation blank for now.

Now, let's look at the ratios of Dali's walking and running speeds:

Walking speed : Running speed = x : 3x = 1 : 3

This means that for every 4 parts of the distance Dali covers, he walks one part and runs three parts. So, we can write:

d' = 4y, where y is the distance Dali walks

This also means that Dali spends one-fourth of the total time walking and three-fourths running. So, we can write:

d'/2x + d'/2(3x) = (1/4)t + (3/4)t, where t is the total time it takes Dali to get home

Simplifying this equation, we get:

2d' = 2t(x + 3x)

4y = 8tx

y = 2tx

Substituting this value of y in the equation d' = 4y, we get:

d' = 8tx

So, the total distance Dali covers in the afternoon is 8tx.

Now, we have two equations:

d = 24x

d' = 8tx

We need to simplify these equations further to find the value of t (the total time it takes Dali to get home).

From the first equation, we get:

x = d/24

Substituting this value of x in the second equation, we get:

d' = 8t(d/24)

d' = (1/3)dt

So, the total distance Dali covers in the afternoon is (1/3)dt.

Now, we can equate the two expressions we have for d':

d' = 8tx = (1/3)dt

Simplifying this equation, we get:

24x = t

Therefore, it takes Dali 24 minutes to get home.

Step-by-step explanation:

Remember for RIGHT triangles   sin Φ = opposite leg / hypotenuse

then  

sin x =  6 sqrt(105) / (12 sqrt (35))

arcsin (  6 sqrt(105) / (12 sqrt (35)) ) = x = 60 degrees

Answer:

2(3/8) + 4/8 + 5/8 + 7/8

= 6/8 + 4/8 + 5/8 + 7/8 = 22/8 = 2 6/8

B is correct.

A committee consists of 11 men and 12 women. In 304,920 ways can a subcommittee of 4 men and 6 women be chosen

To solve this problem, we will use the combination formula:

nCr = n! / (r! * (n-r)!)

where n is the total number of people (in this case, 23), r is the number of people we want to choose (4 men and 6 women), and ! means factorial (the product of all positive integers up to that number).

First, we need to find the number of ways we can choose 4 men from the 11 available. This is:

11C4 = 11! / (4! * 7!) = 330

Next, we need to find the number of ways we can choose 6 women from the 12 available. This is:

12C6 = 12! / (6! * 6!) = 924

To find the total number of ways we can choose a subcommittee of 4 men and 6 women, we need to multiply these two numbers:

330 * 924 = 304,920

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The relationship between arithmetic mean return, geometric mean return, and variability of returns is they all provide different insights into the performance of an investment.

The arithmetic mean return and geometric mean return are two commonly used measures of investment returns. The arithmetic mean return is calculated by adding up all the returns over a period of time and dividing by the number of returns. The geometric mean return, on the other hand, takes into account the compounding effect of returns over time. It is calculated by multiplying all the returns over a period of time and taking the nth root, where n is the number of returns.

The variability of returns refers to the degree of fluctuation in investment returns over a period of time. This variability can be measured using standard deviation or variance.

The relationship among arithmetic mean return, geometric mean return, and variability of returns is that they all provide different insights into the performance of an investment. The arithmetic mean return is a simple measure of the average return over a period of time, while the geometric mean return takes into account the effect of compounding. The variability of returns, measured by standard deviation or variance, provides an indication of the risk associated with an investment.

Investments with high variability of returns are generally considered riskier than those with low variability. When comparing two investments with the same arithmetic mean return, the one with a higher geometric mean return will generally be preferred because it indicates that the investment has been compounding at a higher rate. Overall, it is important to consider all three measures when evaluating the performance of an investment.

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The parameterization of the portion of the circular cylinder is :

x = 4 cos(t) cos(theta)
y = ±1/sin(theta)
z = 4 cos(t) sin(theta)
where t varies from 0 to pi/3 and theta varies from 0 to pi/6.

To find a parameterization of the portion of the circular cylinder y2 z2=16 between the planes x=2 and x=6, we can use cylindrical coordinates. Let r be the radius of the cylinder, and let theta be the angle of rotation around the z-axis. Then, we can parameterize the cylinder as:

x = r cos(theta)
y = y
z = r sin(theta)

Since we want the portion of the cylinder between x=2 and x=6, we can set x=r cos(theta) equal to these values:

2 = r cos(theta)
6 = r cos(theta)

Solving for r in each equation, we get:

r = 2/cos(theta)
r = 6/cos(theta)

Since the cylinder has radius sqrt(16) = 4, we know that r must be between 2 and 4. Therefore, we can set:

r = 4 cos(t)

where t is a parameter that varies from 0 to pi/3 (corresponding to theta varying from 0 to pi/6). Substituting this expression for r into the parameterization above, we get:

x = 4 cos(t) cos(theta)
y = y
z = 4 cos(t) sin(theta)

To find the range of y, we can look at the equation y2 z2=16 and substitute the expressions for y and z above:

y2 (4 cos(t) sin(theta))2 = 16
y2 sin2(theta) = 1
y = ±1/sin(theta)

Since theta varies from 0 to pi/6, sin(theta) varies from 0 to 1/2, so y varies from -2 to -∞ and from 2 to +∞. Therefore, we can write the parameterization as:

x = 4 cos(t) cos(theta)
y = ±1/sin(theta)
z = 4 cos(t) sin(theta)

where t varies from 0 to pi/3 and theta varies from 0 to pi/6.

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Answer:

The median is number 12!!!!

The probability is 4/9.

What is the probability?

The probability of an occurrence is a figure that represents how likely it is that the event will take place. In terms of percentage notation, it is expressed as a number between 0 and 1, or between 0% and 100%. The higher the likelihood, the more likely it is that the event will take place.

Here, we have

Given: A set of 3 cards, spelling the word ADD, are placed face down on the table.

we have to determine the probability of selecting two cards with D and D.

P(D) = Number of favorable outcomes to D / Total number of possible outcomes

Number of favorable outcomes to D = 2

Number of possible outcomes = 3

P(D) = 2/3

The probability of selecting two cards that have D and D with replacement is

P(D, D) = P(D) × P(D)

P(D, D) = 2/3 × 2/3

P(D, D) = 4/9

Hence, the probability is 4/9.

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The correct statements are a and b. Statement d is incorrect as the training data set is used to build the model, not to test its prediction accuracy.

The data should be partitioned into training and validation sets because we need two sets of data: one to build the model that depicts the relationship between the predictor variable(s) and the predicted variable, and another to validate the model's predictive accuracy. The training data set is used to build the model, and the validation data set is used to test the prediction accuracy of the model. In this process, the model (built using the training data set) is used to make predictions with the validation data - data that were not used to fit the model. In this way, we get an unbiased estimate of how well the model performs. Therefore, the correct statements are a and b. Statement d is incorrect as the training data set is used to build the model, not to test its prediction accuracy.
Your answer: a and b are the correct statements.

a. The training data set is used to build the model, and the validation data is used to test the prediction accuracy of the model.
b. In this process, the model (built using the training data set) is used to make predictions with the validation data - data that were not used to fit the model. In this way, we get an unbiased estimate of how well the model performs.
c. The data should be partitioned into training and validation sets because we need two sets of data: one to build the model that depicts the relationship between the predictor variables and the predicted variable, and another to validate the model's predictive accuracy.

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For desired completion of the project, specific proportions must be established for the School Spirit Cups.

These essential elements involve:

Height: The cups claiming 12.5 cm in stature.

Diameter: Generously constructed to accommodate a substantial amount of yield (e.g., 8-10 cm).

Volume: The vessels should leave no expectations unmet with at least 300 mL or more as needed.

Material: Structural durability is paramount and thus only food-grade plastic or glass should oblige here.

Design: It's important that the classic school symbolism and its affiliated tone be present through vibrant colors by way of logo trends.

Subsequently, these prerequisites followed ensures that the Teacher's request of a stack of 10 cups amounts to 125 cm tall is achieved successfully.

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If The company has three possible types of glue to choose from and the book can either be bound as a paperback or a hardback, then the company is considering 6 treatments. hence, the fourth option is correct.

Explanation:

Given that: The company has three possible types of glue to choose from and the book can either be bound as a paperback or a hardback.

To determine the number of treatments, follow these steps:

Step 1: The company is considering 6 treatments:

glue 1 + paperback, glue 1 + hardback, glue 2 + paperback, glue 2 + hardback, glue 3 + paperback, and glue 3 + hardback.
Thus, the company is considering 3 types of glue and 2 types of binding (paperback and hardback).

Step 2: To determine the total number of treatments, you would multiply the options for each factor. In this case, 3 types of glue * 2 types of binding = 6 treatments.

The company is considering 6 treatments. hence, the fourth option is correct.

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Tthe 99th percentile for the average number of residents within five miles of each store in a sample of 50 stores is: x = μ + z*SE = 17 + 2.326*0.4949 = 18.1662. Rounding to the nearest tenth, the answer is 18.2 thousand residents.

To find the 99th percentile for the average number of residents within five miles of each store in a sample of 50 stores, we need to use the z-score formula:

z = (X - μ) / (σ / √n)

In this case, the mean (μ) is 17,000 and the standard deviation (σ) is 3,500. The sample size (n) is 50 stores.

First, we need to find the z-value for the 99th percentile. Based on the z-table, the z-value for the 99th percentile is approximately 2.33.

Now we can use the z-score formula to find X, the 99th percentile for the average number of residents:

X = μ + z * (σ / √n)

X = 17,000 + 2.33 * (3,500 / √50)

X = 17,000 + 2.33 * (3,500 / 7.071)

X = 17,000 + 2.33 * 495.4

X = 17,000 + 1,153.4

X = 18,153.4

Since we need to provide the answer in thousands and round to the nearest tenth, the 99th percentile for the average number of residents within five miles of each store in a sample of 50 stores is approximately 18.2 thousand residents.

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The required answer is alpha = 2.31 rad/s^2

To find the angular acceleration of the propeller, we can use the formula:

angular acceleration (alpha) = (final angular speed - initial angular speed) / time
Angular acceleration is measured in units of angle per unit time squared (which in SI units is radians per second squared), and is usually represented by the symbol alpha (α). In two dimensions, angular acceleration is a pseudoscalar whose sign is taken to be positive if the angular speed increases counterclockwise or decreases clockwise, and is taken to be negative.

It should be noted that angular acceleration simply means the time rate for change of the angular velocity.
Plugging in the given values, we get:

alpha = (39 rad/s - 10 rad/s) / (2.0 revolutions * 2*pi rad/revolution)

Simplifying the denominator, we get:

alpha = (39 rad/s - 10 rad/s) / (12.57 rad)

Calculating the numerator, we get:

alpha = 29 rad/s / 12.57 rad

Simplifying, we get:

alpha = 2.31 rad/s^2
Therefore, the angular acceleration of the propeller is 2.31 rad/s^2, expressed using two significant figures.

To find the angular acceleration of the propeller, we can use the following formula:
angular acceleration (α) = (final angular speed - initial angular speed) / time
It should be noted that angular acceleration simply means the time rate for change of the angular velocity.

Given that the angular speed increases from 10 rad/s to 39 rad/s in 2.0 revolutions, we first need to convert revolutions to time. To do this, we can use the following relationship:

angular speed = angular distance / time

Let's solve for time:

time = angular distance / angular speed

We are given the initial angular speed (10 rad/s) and the angular distance (2.0 revolutions). We need to convert revolutions to radians:

angular distance = 2.0 revolutions * 2π radians/revolution ≈ 12.57 radians

Now we can find the time taken for the 2.0 revolutions:

time = 12.57 radians / 10 rad/s = 1.257 s
angular acceleration refers to the time rate of change of angular velocity. As there are two types of angular velocity, namely spin angular velocity and orbital angular velocity,

Now that we have the time, we can find the angular acceleration:

α = (39 rad/s - 10 rad/s) / 1.257 s ≈ 23.07 rad/s²

Expressing the answer using two significant figures:

α ≈ 23 rad/s²

So, the angular acceleration of the propeller is approximately 23 rad/s².

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The scale factor of the dilation is k = 1/3

Given data ,

Let the coordinates of the triangle KLM be

K ( -2 , 8 ) , L ( 7 , 2 ) and M ( -15 , -1 )

Let the coordinates of the triangle RST be

R ( 4 , 8 ) , S ( 7 , 6 ) and T ( 0 , 5 )

Scale factor = Distance from center of dilation to corresponding vertex of image triangle / Distance from center of dilation to corresponding vertex of pre-image triangle

Distance from center of dilation C to vertex R = √((xR - xC)^2 + (yR - yC)^2)

Distance from center of dilation C to vertex K = √((xK - xC)^2 + (yK - yC)^2)

Plugging in the given coordinates:

Distance from center of dilation C to vertex

R = √((4 - 7)² + (8 - 8)²) = √9 = 3

Distance from center of dilation C to vertex

K = √((-2 - 7)² + (8 - 8)²) = √81 = 9

Now, we can calculate the scale factor:

Scale factor = Distance from center of dilation to corresponding vertex of image triangle / Distance from center of dilation to corresponding vertex of pre-image triangle

Scale factor = 3 / 9 = 1/3

Hence , the scale factor of the dilation that maps triangle KLM to triangle RST with the center of dilation at point C(7, 8) is 1/3

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The sunfish gains 2/7 of it's mass every 0.5 days.

An exponential function has the definition presented as follows:

y = ab^x.

In which the parameters are given as follows:

The growth rate after t days is given as follows:

(81/49)

When the sunfish gains 2/7 of it's mass, the fraction change is given as follows:

1 + 2/7 = 7/7 + 2/7 = 9/7.

Hence the number of days is obtained as follows:

9/7 = (81/49)^x

(9/7)^2x = 9/7

2x = 1

x = 0.5.

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Answer: The sunfish gains 2/7 of it's mass every 0.5 days.

How to define an exponential function?

An exponential function has the definition presented as follows:

y = ab^x.

In which the parameters are given as follows:

a is the value of y when x = 0.

b is the rate of change.

The growth rate after t days is given as follows:

(81/49)

When the sunfish gains 2/7 of it's mass, the fraction change is given as follows:

1 + 2/7 = 7/7 + 2/7 = 9/7.

Hence the number of days is obtained as follows:

9/7 = (81/49)^x

(9/7)^2x = 9/7

2x = 1

x = 0.5.

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Answer:

4°C

Step-by-step explanation:

-8 + 12 = 4°C

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The correct answer of cos(θ) is option (b) -40/41.

To find cos(θ) when the terminal side of angle θ in standard position intersects the unit circle at point P(-40/41, -9/41), we can use the coordinates of P.

In a unit circle, the x-coordinate represents the value of cos(θ). Therefore, cos(θ) is given by the x-coordinate of the point P.

In this case, P has coordinates (-40/41, -9/41). The x-coordinate is -40/41.

So, cos(θ) = -40/41.

The correct answer is option (b) -40/41.

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A) The estimated cost per year for Jorden would be:

   $X - ($7,500 + $3,700)

B) Jo needs to contribute each year:

    65% of $Y = 0.65 * $Y

Estimating cost means to calculate the approximate amount of money required for a particular expense or project.

It may involve considering various factors, such as known costs, expected expenses, and potential variables that may affect the final cost.

To estimate Jorden's cost per year, we need to subtract the scholarships, grants, and work-study earnings from the total cost of attendance. Let's assume the total cost of attendance is $X.

Given:

Scholarships and grants = $7,500 per year

Work-study earnings = $3,700 per year

So, the estimated cost per year for Jorden would be:

$X - ($7,500 + $3,700)

B) To calculate Jo's contribution, we need to find 65% of the estimated cost per year. Let's assume the estimated cost per year is $Y.

Given:

Estimated cost per year = $Y

Family contribution percentage = 65%

So, Jo's contribution per year would be:

65% of $Y = 0.65 * $Y

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The Wronskian of the functions e and e(3x) is 2e(4x).

The Wronskian of the functions e^x and e^(3x) is found by computing the determinant of the matrix formed by their derivatives. In this case, the Wronskian is:

W(e^x, e^(3x)) = |(d/dx)e^x   (d/dx)e^(3x)|
                      |e^x         3e^(3x)|

Now, compute the determinant:

W(e^x, e^(3x)) = e^x(3e^(3x)) - e^(3x)e^x
W(e^x, e^(3x)) = 2e^(4x)

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The general solution to the differential equation y''' - 2y'' - 8y = 0 is [tex]y(t) = c1 e^{(4t)} + c2 e^{(-t)} + c3 t e^{(-t).[/tex]

To find the general solution of the given differential equation:

y''' - 2y'' - 8y = 0

We first find the characteristic equation by assuming a solution of the form:

y = [tex]e^{(rt)}[/tex]

where r is a constant to be determined.

Substituting this solution into the differential equation, we get:

[tex]r^3 e^{(rt)} - 2r^2 e^{(rt)} - 8e^{(rt)} = 0[/tex]

Dividing both sides by [tex]e^{(rt)[/tex], we get:

r³ - 2r² - 8 = 0

This is the characteristic equation, which we can solve for r using factoring or the quadratic formula. Factoring gives:

(r - 4)(r + 1)² = 0

So the roots are:

r = 4 (with multiplicity 1)

r = -1 (with multiplicity 2)

Therefore, the general solution to the differential equation is:

[tex]y(t) = c1 e^{(4t)} + c2 e^{(-t)} + c3 t e^{(-t).[/tex]

where c1, c2, and c3 are constants determined by the initial or boundary conditions of the problem.

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The power of ten is basic concept to solve the largest numbers. The number 1,23456789 x 10⁻⁶is not the same value as 1,23456789000000. So, option(d) is right one.

The powers of 10 refer to the numbers in which the base is 10 and the exponent is an integer. Exponent and power concept is used to simplify the smallest and largest number. For example, 10², 10³ shows different powers of 10. The formula for powers of 10 is 10ˣ , where x is an integer.

We have a number 1,23456789 x 10⁻⁶

As we know, 10⁻ˣ is written as '0 decimal point followed by (x -1) number of zeros followed by 1". We can write the power expansion, 10⁻⁶ = 0.000001. So, the above expression can be represent flin following forms,

Hence, we cannot write it in form of 1,23456789000000.

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Complete question:

1,23456789 x 10^{-6} is not the same value as

a) 1,23456789/1000000

b)1,23.456789

c) 1,23456789/10⁶

d) 1,23456789000000

The number of dimes students has is 12 dimes.

We are given that;

Ratio= 2:3

Cost= $1.60

This is a question that can be solved using a system of linear equations. Let x be the number of nickels and y be the number of dimes. Then we have:

0.05x+0.10y=1.60

for the total amount of money, and

yx​=32​

for the ratio of nickels to dimes. To solve this system, we can use the substitution method. First, we isolate x in the second equation:

x=32​y

Then we substitute this expression for x in the first equation:

0.05(32​y)+0.10y=1.60

Simplifying, we get:

301​y+101​y=1.60

Adding the fractions, we get:

304​y=1.60

Multiplying both sides by 30, we get:

4y=48

Dividing both sides by 4, we get:

y=12

The student has 12 dimes.

Therefore, by the given ratio the answer will 12 dimes.

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