what is the median of this data set 60,70,69,65,62,70,72

The three complex cube roots of z^3 are:

z_1 = 2^(1/3) [cos(π/9) + i sin(π/9)]

z_2 = 2^(1/3) [cos(5π/9) + i sin(5π/9)]

z_3 = 2^(1/3) [cos(7π/9) + i sin(7π/9)]

First, we can find the modulus of the complex number as |z^3| = |√3+i√5| = √(3+5) = 2. We can also find the argument of the complex number as arg(z^3) = arctan(√5/√3) = π/3 - arctan(√3/√5).

Now, we can express the complex number in polar form as z^3 = 2(cosθ + i sinθ), where θ = π/3 - arctan(√3/√5).

Using De Moivre's theorem, we can find the cube roots of z as:

z_1 = 2^(1/3) [cos(θ/3) + i sin(θ/3)]

z_2 = 2^(1/3) [cos((θ+2π)/3) + i sin((θ+2π)/3)]

z_3 = 2^(1/3) [cos((θ+4π)/3) + i sin((θ+4π)/3)]

Simplifying further, we get:

z_1 = 2^(1/3) [cos(π/9) + i sin(π/9)]

z_2 = 2^(1/3) [cos(5π/9) + i sin(5π/9)]

z_3 = 2^(1/3) [cos(7π/9) + i sin(7π/9)]

Therefore, the three complex cube roots of z^3 are:

z_1 = 2^(1/3) [cos(π/9) + i sin(π/9)]

z_2 = 2^(1/3) [cos(5π/9) + i sin(5π/9)]

z_3 = 2^(1/3) [cos(7π/9) + i sin(7π/9)]

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The graph of a Square ABCD with vertices A(-7,5) B(-4,7) C(-2,4) and D(-5,2) is represent upper square and after 90° counterclockwise rotation the lower square represents ABCD in above figure.

A quadrilateral is a polygon that has number of four sides. This also implies that a quadrilateral has exactly four vertices, and exactly four angles. We have to graph a square with vertices A(-7,5), B(-4,7) ,C(-2,4) and D(-5,2) 90 counterclockwise. Now, steps to draw the square :

In case of rotating a figure of 90 degrees counterclockwise, each point of the figure has to be changed from (x, y) to (-y, x) and graph the rotated figure. So, now the vertices of square be A(-7,5), B(-4,7) ,C(-2,4) and D(-5,2). Now, draw the square for these point, lower square in above figure. Both graphs of square ABCD present in above figure.

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Complete question:

The above figure complete the question.

graph and label 9 and 10 and their given rotation about the origin. Give the coordinates of the images.

Square ABCD with vertices A(-7,5) B(-4,7) C(-2,4) and D(-5,2) 90 counterclockwise.

Matching the algebraic expressions with their correct solutions gives:

8y - 6 - 4y + 3 → 4y - 3

y - 1 - 2 + 3y → 4y - 3

1 + y - 1 + 4y + 2 → 2 + 5y

4 + 5y - 3y - 4 + 3y + 2 → 2 + 5y

6 - 3y + 6y - 6 + 6y → 9y

Let us solve each of the algebraic expressions given:

1) 8y - 6 - 4y + 3

= 4y - 3

2) 6 - 3y + 6y - 6 + 6y

= 9y

3) y - 1 - 2 + 3y

= 4y - 3

4) 1 + 18y - 1 - 9y

= 9y

5) 1 + y - 1 + 4y + 2

= 5y + 2

6) 4 + 5y - 3y - 4 + 3y + 2

= 5y + 2

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To estimate a in a binomial distribution, you can use the maximum likelihood estimation (MLE) method. Here are the steps:

1. Define the terms:
  - a: The probability of success in a single trial
  - n: The number of observations (trials)
  - k: The number of successful events among the n trials

2. Write the binomial probability function:
  P(k events | a) = (nCk) * (a^k) * (1 - a)^(n - k)

3. Calculate the likelihood function, which is the product of the binomial probability functions for all observed data points (for k = 0, 1, 2, ..., n).

4. Differentiate the logarithm of the likelihood function with respect to a (using logarithmic properties to simplify the expression) to obtain the first-order condition.

5. Set the first-order condition equal to zero and solve for a, which will give you the maximum likelihood estimate of a.

By following these steps, you can estimate a in a binomial distribution based on the number of events among n observations using the maximum likelihood estimation method.

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As given below find the suitable option which gives you the answer for the question. "There are 11 questions in 5 pages. No credit will be given without sufficient work 1. Let Z be a standard normally distributed random variable."

1. Let Z be a standard, normally distributed random variable.
a. P(Z ≤ 2.32)
To find this probability, you need to use the standard normal distribution table (also known as the Z-table) to look up the value corresponding to Z = 2.32. The value you find in the table is the probability P(Z ≤ 2.32).
b. P(Z ≥ -1.56)
To find this probability, first look up the value corresponding to Z = -1.56 in the standard normal distribution table. This value represents P(Z ≤ -1.56). Since we want P(Z ≥ -1.56), we need to find the complement, which is 1 - P(Z ≤ -1.56).
c. P(-1.43 ≤ Z ≤ 2.47)
To find this probability, look up the values corresponding to Z = -1.43 and Z = 2.47 in the standard normal distribution table. The difference between these two values will give you the probability P(-1.43 ≤ Z ≤ 2.47).
d. Find z* so that P(-z* ≤ Z ≤ z*) = 0.99
To find the z* value, you need to look up the value in the standard normal distribution table that corresponds to the area of 0.995 (since 0.99 is the area between -z* and z*, and each tail contains 0.005). Once you find the value in the table, look at the corresponding Z value. This value will be your z*.

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That kx^2 > 0 for x > 0, so we have 1+(k+1)x+kx^2 > 1+(k+1)x. Therefore, (1+x)^(k+1) > 1+(k+1)x, and the result follows by mathematical induction.

(a) To study the variations of f(x) = r - ln(1+x), we need to find the derivative of f(x) and analyze its sign.

The derivative is f'(x) = -1/(1+x), which is negative for all x > 0.

Therefore, f(x) is a decreasing function on (0, ∞).

Also, lim x→0 f(x) = r > -∞, and lim x→∞ f(x) = -∞.

Therefore, f(x) has a maximum at x = 0, which is r.

(b) To study the variations of g(x) = (1 + x) ln(1 + x) - 2, we need to find the derivative of g(x) and analyze its sign.

The derivative is g'(x) = ln(1 + x), which is positive for all x > -1.

Therefore, g(x) is an increasing function on (-1, ∞). Also, lim x→-1+ g(x) = -∞, and lim x→∞ g(x) = ∞.

Therefore, g(x) has a minimum at some point in (-1, ∞).

(c) To conclude that for all positive integer n, we have (1+x)^n > 1+nx, we can use mathematical induction.

For n = 1, we have (1+x)^1 = 1+x > 1+1x. Assume that (1+x)^k > 1+kx for some positive integer k. Then, for n = k+1, we have (1+x)^(k+1) = (1+x)^k * (1+x) > (1+kx) * (1+x) = 1+(k+1)x+kx^2.

Note that kx^2 > 0 for x > 0, so we have 1+(k+1)x+kx^2 > 1+(k+1)x. Therefore, (1+x)^(k+1) > 1+(k+1)x, and the result follows by mathematical induction.

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start with 18 multiplied by 16 which is 288

then i believe other side length next to the 6 might be 2

so that would mean you do 6 multiplied by 12 and you subtract that fthe 288

so im pretty sure the answer is 276, but im not entirely sure

The option that shows trigonometric functions of the same angle is:

Option C: cosY = 8 / 17, cotY = 8 / 15, secY = 17 / 8

The three primary trigonometric ratios are:

sin x = opposite/hypotenuse

cos x = adjacent/hypotenuse

tan x = opposite/adjacent

Here,

cosY = 8/17, cotY = 8/15, secY = 17 / 8

We know that in trigonometric ratios that:

cosY = 1 / SecY

Thus:

8 / 17 = 1 / secY

secY = 17 / 8

Now, using pythagoras theorem, we have:

P = √[17² - 8²]

P = 15

Thus:

Cot Y = 8 / 15

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When X and Y be independent random variables with X = N(0,1) and Y = exp(1) then E([[X|(Y+1)-1)=0.

To find the expected value E([X|(Y+1)-1]), we first need to clarify the expression inside the brackets. Since Y+1-1 = Y, the expression becomes E([X|Y]). Now, let's proceed to find E(X|Y):

1. X and Y are independent random variables, with X following a normal distribution N(0, 1) and Y following an exponential distribution with a rate parameter of 1.

2. To find the expected value of X given Y, we can use the property of independent random variables:

E(X|Y) = E(X), since Y's value does not affect X's expected value X.

3. Given that X follows a normal distribution with a mean of 0 and a variance of 1, the expected value E(X) is equal to its mean, which is 0.

So, E([X|Y]) = E(X) = 0.

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The probability that a wedding costs less than $20,000 is approximately 0.3745.

The probability that a wedding costs between $20,000 and $31,000 is approximately 0.6188.

The minimum cost for a wedding to be included among the most expensive 5% of weddings is approximately $31,229.

(a) To find the probability that a wedding costs less than $20,000, we need to standardize the value of $20,000 by subtracting the mean and dividing by the standard deviation:

z = (20000 - 21858) / 5800 = -0.32

We can then use a standard normal distribution table or technology to find the corresponding probability:

P(z < -0.32) ≈ 0.3745

Therefore, the probability that a wedding costs less than $20,000 is approximately 0.3745.

(b) To find the probability that a wedding costs between $20,000 and $31,000, we need to standardize both values and find the area between the corresponding z-scores:

z1 = (20000 - 21858) / 5800 = -0.32

z2 = (31000 - 21858) / 5800 = 1.58

Using a standard normal distribution table or technology, we can find the probabilities:

P(-0.32 < z < 1.58) ≈ 0.6188

Therefore, the probability that a wedding costs between $20,000 and $31,000 is approximately 0.6188.

(c) To find the minimum cost for a wedding to be included among the most expensive 5% of weddings, we need to find the z-score that corresponds to the 95th percentile of the standard normal distribution. We can use a standard normal distribution table or technology to find this value:

z = invNorm(0.95) ≈ 1.645

We can then use the formula for standardizing a value to find the minimum cost:

z = (x - 21858) / 5800

Solving for x, we get:

x = z(5800) + 21858

x = 1.645(5800) + 21858

x ≈ 31229

Therefore, the minimum cost for a wedding to be included among the most expensive 5% of weddings is approximately $31,229.

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To find the probability that Bo, Colleen, Jeff, and Rohini win the first, second, third, and fourth prizes, respectively, in a drawing with 50 people, follow these steps:

1. Since winning more than one prize is allowed, the probability of Bo winning the first prize is 1/50.

2. Likewise, the probability of Colleen winning the second prize is also 1/50.

3. Similarly, the probability of Jeff winning the third prize is 1/50.

4. Finally, the probability of Rohini winning the fourth prize is 1/50.

5. Since these events are independent, we can multiply the probabilities together to find the overall probability of this specific  :

  Probability = (1/50) * (1/50) * (1/50) * (1/50)

6. Calculate the result:

  Probability ≈ 0.00000016

7. Round the answer to two decimal places:

  Probability ≈ 0.00

So, the probability that Bo, Colleen, Jeff, and Rohini win the first, second, third, and fourth prizes, respectively, in a drawing with 50 people is approximately 0.00.

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Answer: The ratio of A's lateral surface area to B's lateral surface area is 16:1.

Step-by-step explanation: Let B's radius be x and the height be y. Then, the radius of A will be 4x and the height will be 4y.

As we know, the formula for the lateral surface area of a cylinder is

2[tex]\pi[/tex]rh.

So, the lateral surface area of A is 2[tex]\pi[/tex](4x)(4y)= 32[tex]\pi[/tex]xy

lateral surface area of B is 2[tex]\pi[/tex](x)(y)= 2[tex]\pi[/tex]xy

Ratio,

Lateral surface area of A/ Lateral surface area of B = [tex]\frac{32\pi xy}{2\pi xy}[/tex]

=[tex]\frac{16}{1}[/tex]

=16:1

The image of the point after it is rotated 180° about the z-axis is A' = (1, -2, -2)

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

Point A = (-1, 2, 2)

The rule of rotation is given as rotated 180° about the z-axis

Mathematically, this rule can be expressed as

(x, y, z) = (-x, -y, -z)

Substitute the known values in the above equation, so, we have the following representation

A' = (1, -2, -2)

Hence. the image of the point after the rotation is A' = (1, -2, -2)

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The domain and range for the relation are

for the graph: domain is [-3 3] and range is [-1 3]

domain is [-2 4] and range is [-3 0]

The mathematics domain and range refer, respectively, to a function's input values as well as its output.

The set of possible input values that can be used for the function is called the domain or independant variable(s), while also comprising all necessary values for the calculation of appropriate results.

Conversely, the range or dependent variable(s) represents every conceivable result obtainable from specific sets of inputs within the domain. It essentially displays the function's abilities to produce an output value based on any given input it receives.

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The parametric equations for the line of intersection are:
x = 61 - 5t
y = 2t - 30
z = t

To find the parametric equations for the line of intersection of the given planes, we first need to solve the system of equations:

1. x + 2y + z = 1
2. 2x + 5y + 32 = 4

Step 1: Solve for x from equation 1:
x = 1 - 2y - z

Step 2: Substitute x in equation 2 with the expression found in step 1:
2(1 - 2y - z) + 5y + 32 = 4

Now we can use elimination to solve for one variable. Let's eliminate y by multiplying the first equation by 5 and subtracting it from the second equation:

Step 3: Simplify and solve for y:
2 - 4y - 2z + 5y + 32 = 4
y - 2z = -30

Step 4: Designate z as the parameter t:
z = t

Step 5: Substitute z with t in the expression for y:
y = 2t - 30

Step 6: Substitute z with t in the expression for x:
x = 1 - 2(2t - 30) - t
x = 1 - 4t + 60 - t
x = 61 - 5t

Now we have the parametric equations for the line of intersection:
x = 61 - 5t
y = 2t - 30
z = t

Note that we can choose any value of z for the parameter t, since z is unconstrained.

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Answer:

Sure, let's break down each part step by step.

Part A:

To calculate how much fabric is needed to make 3 aprons, we need to multiply the amount of fabric needed for one apron by 3.1 apron requires

1 1/2 yards of fabric for the front and 1/8 yards of fabric for the tie.1 1/2 yards + 1/8 yards = 15/8 yards (Adding fractions with a common denominator)

Now we can multiply the total fabric needed for one apron by 3 to get the fabric needed for 3 aprons:

3 * 15/8 yards = 45/8 yards (Multiplying by a whole number)

So, the total fabric needed to make 3 aprons is 45/8 yards.

Part B:

If Susan originally has 7 yards of fabric and she uses 45/8 yards to make 3 aprons, we can subtract the amount used from the original amount to find out how much fabric is left over.

7 yards - 45/8 yards = 56/8 yards - 45/8 yards (Subtracting fractions with a common denominator)

= 11/8 yards (Subtracting fractions)

So, after making the aprons, Susan will have 11/8 yards of fabric left over.

Part C:

To determine if Susan has enough fabric left to make another apron, we need to compare the amount of fabric left (11/8 yards) with the amount of fabric needed for one apron (1 1/2 yards + 1/8 yards = 15/8 yards).

Since 15/8 yards is greater than 11/8 yards, Susan does not have enough fabric left to make another apron. She is short by 4/8 yards (or 1/2 yard) of fabric.

Hope this helps! Let me know if you have any further questions.

Step-by-step explanation:

Answer: A. 10.2

Step-by-step explanation: For this problem we have to create a second right triangle to find the length. You can apply the pythagorean theorem which continues to 10^2+2^2=c^2 which would get us 104. Then find the root of 104 which is equal to 10.2

Therefore, the quotient of the difference between 1200 and 700 divided by 5 is 100.

The slope of the secant line connecting two points on the graph of a function, f, is determined using the difference quotient. Just to refresh your memory, a function is a line or curve where there is just one y value and one x value.  The slope of secant lines may be calculated using the difference quotient.

Almost identical to a tangent line, a secant line traverses at least two points on a function. The slope of a secant line serves as the basis for the difference quotient formula. A function's difference quotient, y = f(x),

The difference between 1200 and 700 is: 1200 - 700

= 500

To divide this by 5, we simply divide 500 by 5:

500 ÷ 5 = 100

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The height of the original sculpture in centimeters is 195 cm

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

Scale = 1 : 15

Scale height = 13 cm

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

13 cm : height = 1 : 15

Express the ratio as fraction

So, we have

height/13 cm = 15/1

Cross multiply

So, we have

height = 13 cm * 15/1

Evaluate

height = 195 cm/1

So, we have

height = 195 cm

Hence, the value of the actial height = 195 cm

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Answer:

Ive attached a picture

Step-by-step explanation:

The probability that the airline will lose at least 10 suitcases in a week is 0.1723 or about 17.23%.

Given information:

µ = 6.7 (mean)

σ = 3.5 (standard deviation)

We need to find the probability of losing at least 10 suitcases in a week. We can use the normal distribution formula to solve this problem:

P(X ≥ 10) = 1 - P(X < 10)

To use this formula, we need to standardize the variable X to the standard normal distribution with mean 0 and standard deviation 1. We can do this using the following formula:

Z = (X - µ) / σ

Substituting the given values, we get:

Z = (10 - 6.7) / 3.5

Z = 0.943

Now, we can use a standard normal distribution table or calculator to find the probability of Z being greater than or equal to 0.943. The table or calculator will give us the probability of Z being less than 0.943, which we can then subtract from 1 to get the desired probability.

Using a standard normal distribution table, we find that P(Z < 0.943) = 0.8277.

Therefore, P(X ≥ 10) = 1 - P(X < 10) = 1 - P(Z < 0.943) = 1 - 0.8277 = 0.1723.

So, the probability that the airline will lose at least 10 suitcases in a week is 0.1723 or about 17.23%.

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The value of the variable is 3

Proportion can be defined as a method of comparing numbers in mathematics such that one is made equal to another.

Note that a fraction is described as the part of a whole

From the information given, we have that;

×:8 = 9:24​

To determine the fraction, we divide the numerator by the denominator, we have;

x/8= 9/24

Now, cross multiply the values

24(x) =9(8)

multiply the values, we have;

24x = 72

Now, make 'x' the subject

Divide both sides by the coefficient

x = 3

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Answer:

3

Step-by-step explanation:

Answer:

To find the mean height of the sample, we need to sum up all the values and divide them by the total number of values.

Sum of values = 59+61+62+63+63+64+65+65+66+67+67+67+67+69+73 = 964

Total number of values = 15

Mean height = sum of values / total number of values = 964/15 = 64.2666... ≈ 65.2

Therefore, the mean height, in inches, of this sample is approximately 65.2.

The answer is B.

The minimum sample size that should be taken to be 90% confident that the sample mean is within 17 emails of the true population mean is 82.

To find the minimum sample size required to estimate a population mean with a given confidence level, we need to use the following formula:

n = (Z * σ / E)^2

where:
- n is the sample size
- Z is the Z-score corresponding to the desired confidence level (90% in this case)
- σ is the population standard deviation (94 emails)
- E is the margin of error (17 emails)

First, find the Z-score for a 90% confidence level. You can do this by checking a Z-table or using a calculator. For a 90% confidence level, the Z-score is approximately 1.645.

Now, plug the values into the formula:

n = (1.645 * 94 / 17)^2
n ≈ (153.63 / 17)^2
n ≈ 9.036^2
n ≈ 81.65

Since we cannot have a fraction of a person, round up to the nearest whole number. Therefore, the minimum sample size that should be taken to be 90% confident that the sample mean is within 17 emails of the true population mean is 82.

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The gardener used 61.5 gallons of gasoline in his lawn mowers in the one month.

Let's call the amount of gasoline used in the lawn mowers "x".

We know that the total amount of gasoline used is 61.5 gallons, so:

x + (the amount used for other things) = 61.5

We don't know how much was used for other things, but we do know that "of the total amount of gasoline" used, a certain percentage was used in the lawn mowers. Let's call that percentage "p".

"Of" means "times", so we can write:

p * 61.5 = x

Now we have two equations:

x + (the amount used for other things) = 61.5

p * 61.5 = x

We want to solve for x, so let's isolate it in the second equation:

p * 61.5 = x

x = p * 61.5

Now we can substitute that into the first equation:

p * 61.5 + (the amount used for other things) = 61.5

Simplifying:

p * 61.5 = 61.5 - (the amount used for other things)

p = (61.5 - the amount used for other things) / 61.5

We don't know the exact amount used for other things, but we do know that it's less than or equal to 61.5, so:

p = (61.5 - something) / 61.5

p = (61.5 - 0) / 61.5

p = 1

So all of the gasoline was used in the lawn mowers, and:

x = 1 * 61.5

x = 61.5

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