what is 15t+8t-2t=16?
find the vaule of t

The correct rearrangement is: 5026 6634 6453 5243 6638 7026 7044 7377 5061 3500 5477 1009 6215 6376 2199. The Option C is correct.

Rearrangement means the action or process of changing the position, time or order of something or number in this context.

To rearrange the set of numbers, we can choose a method such as sorting them in ascending or descending order or grouping them in pairs.

The correct option is C. 5026 6634 6453 5243 6638 7026 7044 7377 5061 3500 5477 1009 6215 6376 2199. This option groups the numbers in pairs which is suitable for simulating two free throws in a row.

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The probability of one robot succeeding in a task less than or equal to 80 times out of 100 can be calculated using a binomial distribution formula. Assuming the probability of success for one robot is p, the probability of success for both robots is p^2. Using the binomial distribution formula, we can calculate the probability of success for each robot and then multiply them together to find the probability of both robots succeeding less than or equal to 80 times out of 100. The formula is P(X<=80) = sum of P(X=k) from k=0 to k=80, where X is the number of successes in 100 attempts.

To calculate the probability of both robots succeeding less than or equal to 80 times out of 100, we need to first find the probability of success for one robot. Let's assume the probability of success for one robot is p = 0.7. The probability of success for both robots is then p^2 = 0.7^2 = 0.49.

Next, we need to use the binomial distribution formula to calculate the probability of success for each robot. The formula is P(X=k) = (n choose k) * p^k * (1-p)^(n-k), where n is the number of attempts, k is the number of successes, and (n choose k) is the binomial coefficient.

Using this formula, we can calculate the probability of one robot succeeding less than or equal to 80 times out of 100. P(X<=80) = sum of P(X=k) from k=0 to k=80 = sum of [(100 choose k) * 0.7^k * 0.3^(100-k)] from k=0 to k=80.

We can use a calculator or a software program like Excel to calculate this sum. The result is 0.9899, which means the probability of one robot succeeding less than or equal to 80 times out of 100 is almost 99%.

To find the probability of both robots succeeding less than or equal to 80 times out of 100, we just need to multiply the probability of one robot succeeding by itself: 0.9899 * 0.9899 = 0.9799. So the probability of both robots succeeding less than or equal to 80 times out of 100 is about 98%.

The probability of both robots succeeding less than or equal to 80 times out of 100 can be calculated using the binomial distribution formula. Assuming the probability of success for one robot is p, the probability of success for both robots is p^2. Using the formula P(X<=80) = sum of P(X=k) from k=0 to k=80, we can calculate the probability of one robot succeeding less than or equal to 80 times out of 100. Multiplying this probability by itself gives us the probability of both robots succeeding less than or equal to 80 times out of 100. For the given values, the probability is about 98%.

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To find the fourth corner of the rectangular mural, we can use the fact that opposite sides of a rectangle are parallel and perpendicular. This means that if we draw a line between the first two points, we can find the direction of one side of the rectangle, and if we draw a line between the second and third points, we can find the direction of the adjacent side of the rectangle. The intersection of these two lines will give us the fourth corner of the rectangle.

First, let's find the direction of the side of the rectangle that connects the first two points:

Slope of line connecting (–1, –1) and (–1, 1) = (change in y) / (change in x) = (1 - (-1)) / (-1 - (-1)) = 2 / (-2) = -1

So the side of the rectangle that connects the first two points has a slope of -1. We also know that this line passes through the midpoint of the segment connecting these two points, which is ((-1 + (-1))/2, (-1 + 1)/2) = (-1, 0).

Using point-slope form, we can write the equation of this line as:

y - 0 = -1(x - (-1))

y = -x - 1

Next, let's find the direction of the side of the rectangle that connects the second and third points:

Slope of line connecting (–1, 1) and (4, 1) = (change in y) / (change in x) = (1 - 1) / (4 - (-1)) = 0 / 5 = 0

So the side of the rectangle that connects the second and third points has a slope of 0. We also know that this line passes through the midpoint of the segment connecting these two points, which is ((-1 + 4)/2, (1 + 1)/2) = (1.5, 1).

Using point-slope form, we can write the equation of this line as:

y - 1 = 0(x - 1.5)

y = 1

Now we have two equations for the sides of the rectangle:

y = -x - 1    (from the first two points)

y = 1         (from the second and third points)

To find the fourth corner of the rectangle, we need to find the point where these two lines intersect. We can do this by setting the two equations equal to each other:

-x - 1 = 1

-x = 2

x = -2

Now that we know that x = -2, we can substitute this value into either equation to find the corresponding value of y:

y = -(-2) - 1 = 1

Therefore, the fourth corner of the rectangular mural is located at (-2, 1) in the coordinate plane.

A z-statistic reports how many standard deviations an observed value is from the expected value, where the expected value is calculated using the population mean and standard deviation.

To calculate the z-statistic, use the following formula:

Where:

x = the observed value

μ = the population mean

σ = the population standard deviation

n = the sample size

The z-statistic tells us how many standard deviations an observed value is from the expected value, which is the population mean. A positive z-score indicates that the observed value is above the expected value, while a negative z-score indicates that the observed value is below the expected value. A z-score of 0 indicates that the observed value is equal to the expected value. By calculating the z-statistic, we can determine how unusual or significant an observed value is relative to the population.

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Complete Question:

a z-statistic reports how many sds an observed value is from the expected value, where the expected value is calculated using the___.

Answer:

180 degrees

Step-by-step explanation:

Hope this helps! Pls give brainliest!

Answer:

The annual percent increase in this model is 67%.

Step-by-step explanation:

The given function Y = 4.56(1.67)^x represents an exponential growth model, where x represents the number of years and Y is the value after x years. The base of the exponent, 1.67, represents the growth factor.

To find the annual percent increase, we can convert the growth factor to a percentage increase. Subtract 1 from the growth factor and multiply the result by 100:

(1.67 - 1) * 100 = 0.67 * 100 = 67%

So, the annual percent increase in this model is 67%.

By trigonometric functions, the length of side AB of the right triangle can be found by using any of the following expressions:

7 / cos A = 24 / cos B = 7 / sin B

In this problem we find the case of a right triangle, whose side AB must determine by means of trigonometric functions. Side AB can be determine by following expressions:

AB = 7 / cos A = 24 / sin A = 24 / cos B = 7 / sin B

Where A, B are angles of the right triangle.

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The term "distance" can refer to different types of measuring  depending on the context. In general, distance is a numerical value that indicates the amount of separation between two objects or points in space.

In mathematics, distance usually refers to the length of a line segment connecting two points in a Euclidean space. The distance can be measured using various formulas, such as the Pythagorean theorem, which computes the length of the hypotenuse of a right triangle formed by the two points and the origin.

In physics, distance can refer to the separation between two objects in three-dimensional space, often measured in meters or kilometers.

In computer science and data analysis, distance can refer to a measure of similarity or dissimilarity between two objects or data points. For example, the Euclidean distance, Manhattan distance, or cosine distance can be used to quantify the difference between two vectors or sets of features.

In general, the meaning of the distance value depends on the specific context and the type of measurement being used.

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A simple (1-for-1) substitution would provide a direct replacement of one element or variable with another element or variable.

This can be useful in simplifying equations or formulas by replacing complex expressions with simpler ones. However, it may not always be applicable or accurate in more complex situations.

A simple 1-for-1 substitution provides a straightforward replacement of one element with another in a given context. This can be applied in various scenarios, such as replacing letters in cryptography, swapping ingredients in a recipe, or substituting variables in mathematical equations. The primary purpose of this substitution is to maintain the overall structure while changing a specific component.

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The probability that the sample proportion will be within 10.05 of the population proportion is approximately 0.0139.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a number between 0 and 1, where 0 means the event is impossible and 1 means the event is certain to happen.

(a) To find the probability that the sample proportion will be within 10.03 of the population proportion, we need to first calculate the standard error of the sample proportion:

SE = sqrt(p*(1-p)/n) = sqrt(0.3*(1-0.3)/300) = 0.0308

Then, we can use the normal distribution to find the probability:

P(|p - 0.3| <= 0.1003) = P(-0.1003/0.0308 <= (p - 0.3)/0.0308 <= 0.1003/0.0308)

≈ P(-3.2565 <= Z <= 3.2752) = 2*P(Z <= 3.2752) - 1 ≈ 0.0146

where Z is the standard normal distribution.

Therefore, the probability that the sample proportion will be within 10.03 of the population proportion is approximately 0.0146.

(b) To find the probability that the sample proportion will be within 10.05 of the population proportion, we can follow the same steps as in part (a), but with a different margin of error:

SE = sqrt(0.3*(1-0.3)/300) = 0.0308

P(|p - 0.3| <= 0.1005) = P(-0.1005/0.0308 <= (p - 0.3)/0.0308 <= 0.1005/0.0308)

≈ P(-3.2649 <= Z <= 3.2836) = 2*P(Z <= 3.2836) - 1 ≈ 0.0139

where Z is the standard normal distribution.

Therefore, the probability that the sample proportion will be within 10.05 of the population proportion is approximately 0.0139.

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Using the unitary method, we were able to determine that Lin ran each lap in 1.2 minutes, or 72 seconds.

To use the unitary method, we first need to determine the ratio between the number of laps and the time it took to run them. We can do this by dividing the total time by the number of laps:

Ratio = Total time / Number of laps

Ratio = 12 minutes / 10 laps

Ratio = 1.2 minutes per lap

Now we have the ratio between the time and the number of laps. We can use this ratio to find the time it took to run one lap by dividing the ratio by the number of laps:

Time per lap = Ratio / Number of laps

Time per lap = 1.2 minutes per lap / 1 lap

Time per lap = 1.2 minutes

Therefore, it took Lin 1.2 minutes, or 72 seconds, to run one lap around the track.

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An equation can be used to solve for x in the following diagram:

x + (4x - 85) = 90

The correct answer is an option (A)

From the attached figure we can observe that the right angle is divided into two angles i.e., angle x degree and angle (4x - 85) degrees

We know that the two angles are called complementary angles when the sum of their is equal to 90 degrees.

We can observe that  angle x degree and angle (4x - 85) degrees are complementary angles.

This means that the sum of these angles must be 90 degrees.

x + (4x - 85) = 90

this is the required equation.

Thus, the correct answer is an option (A)

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Find the complete question below.

The correct option is (d) i.e. one of the basic assumptions is Each sample is assumed to be normal.

What is ANOVA test?

ANOVA stands for Analysis of Variance. It is a statistical test used to analyze the differences between two or more groups of data. ANOVA tests whether the means of the groups are significantly different from each other.

The correct option is (d).

Each population from which a sample is taken is assumed to be normal. This is one of the five basic assumptions that must be fulfilled in order to perform a one-way ANOVA test. The other assumptions include:

Homogeneity of variance: The population variances are assumed to be equal for all groups.

Independence: The samples are assumed to be independent of each other.

Random sampling: The samples are assumed to be selected at random from their respective populations.

Interval or ratio data: The data being analyzed is assumed to be measured on an interval or ratio scale.

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There is no outcome that satisfies this condition, while the product of the marginal probabilities is 4/16=1/4. Therefore, we conclude that x and y are not independent.

What is probability?

Probability is a measure of the likelihood of an event occurring. It is a numerical value between 0 and 1 (inclusive), where 0 represents an impossible event and 1 represents a certain event.

To show that x and y are not independent, we need to demonstrate that the joint probability distribution of x and y does not factor into the product of their marginal distributions.

The possible outcomes when two coins are flipped are:

HH

HT

TH

TT

Since the coins are fair, each of these outcomes has probability 1/4. We can define the random variables x and y as follows:

x is the number of heads, which can take on the values 0 or 1 or 2.

y is the number of tails, which can also take on the values 0 or 1 or 2.

The joint probability distribution of x and y can be represented by a 3x3 matrix:

[tex]\left[\begin{array}{ccc}1&0&0\\0&2&0\\0&0&1\end{array}\right][/tex]

The (i, j) entry of this matrix represents the probability that x=i and y=j. For example, the probability that x=1 and y=2 is 0, since there is only one outcome (HT) that satisfies this condition and its probability is 1/4.

The marginal distribution of x is obtained by summing the entries of each row:

x | 0 1 2

--|-----

p | 1 2 1

Similarly, the marginal distribution of y is obtained by summing the entries of each column:

y | 0 1 2

--|-----

p | 1 2 1

If x and y were independent, then the joint probability distribution would factor into the product of the marginal distributions:

[tex]\left[\begin{array}{ccc}p&p&p\\p&p&p\\p&p&p\end{array}\right][/tex]

However, this is not the case, as can be seen by comparing the joint probability distribution with the product of the marginal distributions. For example, the probability that x=1 and y=1 is 0, since there is no outcome that satisfies this condition, while the product of the marginal probabilities is 4/16=1/4.

Therefore, we conclude that x and y are not independent.

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A. Reject Upper H0. There is not sufficient evidence to warrant support of the claim that more than 20% of users develop nausea.

B. Fail to reject Upper H0. There is sufficient evidence to warrant support of the claim that more than 20% of users develop nausea.

C. Fail to reject Upper H0. There is not sufficient evidence to warrant support of the claim that more than 20% of users develop nausea.

D. Reject Upper H0. There is sufficient evidence to warrant support of the claim that more than 20% of users develop nausea.

The test statistic for this hypothesis test is the difference between the sample proportion and the hypothesized proportion. In this case, the hypothesis is that more than 20% of users develop nausea, and the sample proportion is the proportion of users who reported nausea in the sample.

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To test if there is a difference between the means of the two populations, we can perform a two-sample t-test. The null hypothesis is that there is no difference between the mean number of hours per week that a family with no children participates in recreational activities and a family with children participates in recreational activities.

Let's assume that the researcher has collected the following data:

Sample of families with no children: n1 = 30, sample mean = 4.5 hours per week, sample standard deviation = 1.2 hours per week.

Sample of families with children: n2 = 40, sample mean = 3.8 hours per week, sample standard deviation = 1.5 hours per week.

Using the critical value method, we need to calculate the t-statistic and compare it to the critical value from the t-distribution table with n1+n2-2 degrees of freedom and a significance level of α = 0.05.

The formula for the t-statistic is:

t = (x1 - x2) / sqrt(s1^2/n1 + s2^2/n2)

where x1 and x2 are the sample means, s1 and s2 are the sample standard deviations, and n1 and n2 are the sample sizes.

Plugging in the numbers, we get:

t = (4.5 - 3.8) / sqrt((1.2^2/30) + (1.5^2/40)) = 2.08

The degrees of freedom for the t-distribution is df = n1 + n2 - 2 = 68.

Using a t-distribution table, we find the critical value for a two-tailed test with α = 0.05 and df = 68 is ±1.997.

Since our calculated t-statistic of 2.08 is greater than the critical value of 1.997, we can reject the null hypothesis and conclude that there is a statistically significant difference between the mean number of hours per week that a family with no children participates in recreational activities and a family with children participates in recreational activities.

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The probability of selecting a white bead, replacing it, and then selecting a red bead is 0.15 and this is an independent event.

To fill in the blanks:

1)The probability of selecting a white bead, replacing it, and then selecting a red bead is:

P(white, then red) = P(white) * P(red | white) = (3/10) * (5/10) = 0.15

Here, P(white) is the probability of selecting a white bead on the first draw, which is 3/10

And P(red | white) is the probability of selecting a red bead on the second draw, given that a white bead was selected on the first draw.

Since the first bead is replaced before the second draw, the probability of selecting a red bead on the second draw is still 5/10.

2)This is an independent event.

This is because the first draw does not affect the probability of the second draw, since the bead is replaced.

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This right rectangular prism, which measures 6 by 6 by 15 cm, has a surface area of 432 square centimetres.

The sum of the areas of the six faces of a right rectangular prism gives the prism's surface area. The prism in this instance is 6 centimetres x 6 centimetres by 15 centimetres in size.

We must first determine the size of each face's area before adding them all up to determine the surface area. Each of the top and bottom faces measures 6 cm by 6 cm, giving them a combined area of 6 cm by 6 cm, or [tex]36 cm^2[/tex].

The front and back faces each have an area of [tex]90 cm^2[/tex] because they are each 6 cm by 15 cm in size.

Last but not least, the left and right faces have a combined area of 6 cm by 15 cm, or [tex]90 cm^2[/tex], each.

The total area of all six faces is as follows:

[tex]36 cm^2 +90 cm^2 +90 cm^2 +90 cm^2 = 432 cm^2[/tex].

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a.  P(X ≤ 2) = 0.77

b.  P(X = 1 or X = 2) = 0.60

c.  P(X > 0) = 0.83

d. The cumulative probability distribution function for X is given in the table.

(a) The probability of a randomly selected student holding two or fewer jobs during the past year is given by the sum of the probabilities of holding 0, 1, or 2 jobs:

P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2) = 0.17 + 0.35 + 0.25 = 0.77

Therefore, P(X ≤ 2) = 0.77.

(b) The probability of a randomly selected student holding either one or two jobs during the past year is given by the sum of the probabilities of holding 1 or 2 jobs:

P(X = 1 or X = 2) = P(X = 1) + P(X = 2) = 0.35 + 0.25 = 0.60

Therefore, P = 0.60.

(c) The probability of a randomly selected student holding at least one job during the past year is the complement of the probability of holding no jobs:

P(X > 0) = 1 - P(X = 0) = 1 - 0.17 = 0.83

Therefore, P(X > 0) = 0.83.

(d) The cumulative probability distribution function for X is the sum of the probabilities up to and including the value k:

P(X ≤ k) = ∑P(X = i) for i = 0 to k

Using the given probabilities, we can fill in the table:

k   0    1    2    3    4

P(X ≤ k) 0.17 0.52 0.77 0.93 1.00

Therefore, the cumulative probability distribution function for X is:

P(X ≤ k) = 0.17 for k = 0

P(X ≤ k) = 0.52 for k = 1

P(X ≤ k) = 0.77 for k = 2

P(X ≤ k) = 0.93 for k = 3

P(X ≤ k) = 1.00 for k = 4

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The polynomial representing the shaded area between two squares with given dimensions is 3x² + x - 6. Solving for x using the given area of 24 square units, x equals 3.

To find the polynomial that represents the area of the shaded region, we need to subtract the area of the inner square from the area of the outer square.

The area of the outer square is (3x-2) * (x+3) = 3x² + 7x - 6.

The area of the inner square is 6 * x = 6x.

So, the area of the shaded region is (3x² + 7x - 6) - 6x = 3x² + x - 6.

To solve for x given that the area of the shaded region is 24 square units, we can set the polynomial equal to 24 and solve for x

3x² + x - 6 = 24

3x² + x - 30 = 0

Using the quadratic formula, we get

x = (-1 ± √(1 - 4(3)(-30))) / (2(3))

x = (-1 ± 19) / 6

x = 3 or x = -10/3

Since x must be a positive value in this context, we choose x = 3.

Therefore, the polynomial that represents the area of the shaded region is 3x² + x - 6, and x = 3 satisfies the condition that the area of the shaded region is 24 square units.

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

" Write a polynomial to represent the area of the shaded region. Then solve for x given that the area of the shaded region is 24 square units.

3x2 + x - 6; x = 3

3x2 + 7x - 6; x = 2

3x2 + 7x - 6; x = 5

WILL GIVE BRAINLIEST WITH EXPLANATION"--

Answer:Any number that we can write as a decimal number, between 1.0 and 10.0, multiplied by a power of 10, is said to be in standard form. 1.23 × 108; If you observe carefully, 1.23 is a decimal number between 1.0 and 10.0 and so we have the standard form of 123,000,000 as 1.23 × 10 8.

Step-by-step explanation:

P.S i am emo

The total FICA tax amount on the annual salary of $38,480 is given by  option d. $2,943.72

Annual salary is equals to $38,480.

FICA (Federal Insurance Contributions Act) tax includes two separate taxes.

Social Security tax and Medicare tax.

The Social Security tax rate is 6.2% and the Medicare tax rate is 1.45%, making the total FICA tax rate 7.65%.

To determine the total annual FICA tax for an annual salary of $38,480,

we need to multiply the salary by the FICA tax rate,

Total FICA tax = Annual salary × FICA tax rate

⇒ Total FICA tax = 0.0765 x $38,480

⇒ Total FICA tax = $2,943.72

Therefore, the total annual FICA tax is equal to option d) $2,943.72.

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Answer:

a = 6

Step-by-step explanation:

using the sine ratio in the right triangle

sin30° = [tex]\frac{opposite}{hypotenuse}[/tex] = [tex]\frac{a}{12}[/tex] ( multiply both sides by 12 )

12 × sin30° = a

12 × 0.5 = a , then

a = 6

Answer:

The answer is D!!!

Step-by-step explanation:

x=-7/5,=8/5 or (x=-1 2/5,y= 1 3/5) D one -7/5 = 1 2/5. 8/5 = 1 3/5

D IS THE CORRECT ANSWER

The volume of the cylinder for the given radius and height is equal to 4973.8 square units

Radius of the cylinder = 12 units

height of the cylinder = 11 units

Value of π = 3. 14

Volume of the cylinder = πr²h

where 'r' is the radius of the cylinder.

And 'h' is the height of the cylinder.

Substitute the value we have in the formula we get,

Volume of the cylinder =  3.14 × ( 12 )² × 11

⇒Volume of the cylinder =  4973.76 square units

⇒Volume of the cylinder =  4973.8 square units (  nearest tenth  )

Therefore, the volume of the cylinder is equal to 4973.8 square units

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The volume of cylinder is  6458.98 cubic feet

We know that the formula for the volume of cylinder is:

V = π × r²  × h

where r is the radius of the cylinder

and h is the height of the cylinder

Here, r = 11 ft and h = 17 ft

Using above formula, the volume of the cylinder would be,

⇒ V = π × r²  × h

⇒ V = 3.14 × 11²  × 17

⇒ V = 3.14 × 121  × 17

⇒ V = 6458.98 cubic ft

this is the required volume of cyinder.

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The complete question is:

What is the volume of the cylinder that has a height of 17 feet and a radius of 11 feet? Use ​ ≈ 3.14 and round your answer to the nearest hundredth.

The probability of drawing a white ball given that the ball is odd-numbered is 1/3.

We have,

There are a total of 15 balls in the bag, out of which 6 are white and odd-numbered.

To find the probability of drawing a white ball given that it is odd-numbered, we need to use conditional probability.

Let A be the event that the ball is white and B be the event that the ball is odd-numbered.

Then, we need to find P(A|B), the probability of A given B.

We know that P(B), the probability of drawing an odd-numbered ball, is:

P(B) = number of odd-numbered balls / total number of balls

= 9 / 15

= 3 / 5

We also know that P(A and B), the probability of drawing a white odd-numbered ball, is:

P(A and B) = number of white odd-numbered balls / total number of balls

= 3 / 15

= 1 / 5

Using the formula for conditional probability, we have:

P(A|B) = P(A and B) / P(B)

= (1/5) / (3/5)

= 1/3

Therefore,

The probability of drawing a white ball given that the ball is odd-numbered is 1/3.

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Answer:

3 = -2m + 6

-2m = -3, so m = 3/2

y = (3/2)x + 6

Answer: y=6x+15

Step-by-step explanation:

Formula for point-slope (you were given a point and the slope)

[tex]y-y_{1} =m(x-x_{1} )[/tex]

where m is your slope

[tex](x_{1} ,y_{1} )[/tex] is your point

y-3=6(x-(-2))      plug in and simplify the negative in the parentheses

y-3=6(x+2)        distribute

y-3=6x+12         add 3 to both sides

y=6x+15            this is your answer in slope-intercept form

The coordinates of the centre of mass are:

[tex]x = \frac{M_x}{M} = \frac{r}{2}\\y = \frac{M_y}{M} = \frac{r}{2}\\z = \frac{M_z}{M} = \frac{r}{2}[/tex]

To find the centre of mass of a solid of uniform density, we need to calculate the triple integral of the position vector (x, y, z) over the volume of the solid, and divide by the total mass of the solid.

In this case, the solid is a hemispherical shell of radius r and uniform density, so we can use spherical coordinates to simplify the calculations.

0 ≤ θ ≤ π/2

0 ≤ φ ≤ 2π

The mass of the solid is proportional to its volume, so we can assume that the total mass is [tex]M = \frac{2\pi r^3}{3}[/tex] (the mass of a full sphere of radius r, divided by 2).

To calculate the triple integral for the centre of mass, we need to compute the following integrals:

[tex]M_x = \iiint x \rho \, dV\\M_y = \iiint y \rho \, dV\\M_z = \iiint z \rho \, dV[/tex]

We can simplify the integrals using spherical coordinates:

[tex]\int_0^R \int_0^{\frac{\pi}{2}} \int_0^{2\pi} (r \sin\theta \cos\phi) \rho r^2 \sin\theta \, d\phi \, d\theta \, dr[/tex]

[tex]\int_0^R \int_0^{\frac{\pi}{2}} \int_0^{2\pi} (r \sin{\theta} \sin{\phi}) \rho r^2 \sin{\theta} \, \mathrm{d}\phi \, \mathrm{d}\theta \, \mathrm{d}r[/tex]

[tex]\int_0^R \int_0^{\frac{\pi}{2}} \int_0^{2\pi} (r \cos \theta) \rho r^2 \sin \theta \,d\phi \,d\theta \,dr[/tex]

Since the density is uniform, we can factor it out of the integrals:

[tex]M_x = \rho \int_0^R \int_0^{\frac{\pi}{2}} \int_0^{2\pi} (r^3 \sin^2 \theta \cos \phi) \, d\phi \,d\theta \,dr M_y = \rho \int_0^R \int_0^{\frac{\pi}{2}} \int_0^{2\pi} (r^3 \sin^2 \theta \sin \phi) \,d\phi \,d\theta \,dr M_z = \rho \int_0^R \int_0^{\frac{\pi}{2}} \int_0^{2\pi} (r^3 \cos \theta \sin \theta) \,d\phi \,d\theta \,dr[/tex]

The integrals over φ and θ can be evaluated using the standard formulas for integrating trigonometric functions over a range of angles:

[tex]\int_0^{2\pi}\cos\phi\, d\phi = \int_0^{2\pi}\sin\phi\, d\phi = 0\\\int_0^{\frac{\pi}{2}}\cos\theta \sin\theta\,d\theta = \frac{1}{2}\\x = \frac{M_x}{M} = \frac{r}{2}\\y = \frac{M_y}{M} = \frac{r}{2}\\z = \frac{M_z}{M} = \frac{r}{2}[/tex]

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To determine how well Luke must score on exam B in order to do equivalently well as he did on exam A, we need to first standardize his score on exam A using z-scores.

A z-score represents the number of standard deviations a given data point is away from the mean. The formula for calculating z-scores is:

z = (x - μ) / σ

where x is the data point, μ is the mean, and σ is the standard deviation.

In this case, Luke's score on exam A has a z-score of:

z = (850 - 750) / 50 = 2

This means that his score on exam A is 2 standard deviations above the mean.

To do equivalently well on exam B, Luke needs to achieve a score that has the same z-score of 2. We can use the formula for z-scores again to determine what score he needs to achieve:

2 = (x - 38) / 5

Solving for x, we get:

x = 48

Therefore, Luke needs to score 48 on exam B in order to do equivalently well as he did on exam A.

It's important to note that we're assuming that the distributions of the two exams are both normal distributions. If this assumption is not valid, then our answer may not be accurate.

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