ILL GIVE BRAINLEIST THIS WAS DUE YESTERDAY!! 5. Use the following information to answer the questions.
.
A survey asked 75 people if they wanted a later school day start time.
.
45 people were students, and the rest were teachers.
.
50 people voted yes for the later start.
• 30 students voted yes for the later start.
.
a) Use this information to complete the frequency table. (5 points: 1 point for
each cell that was not given above)
Students
Teachers
Total
Vote YES for
later start
Vote NO for later
start
Total
b) Use the completed table from Part a. What percentage of the people surveyed
were teachers? (2 points)

The probability of obtaining a sum less than or equal to 4 is:  1/12

An experiment consists of tossing two ordinary dice and adding their numbers together. To determine the probability of obtaining a sum of 8, we need to first count the number of ways we can get a sum of 8. We can do this by listing all the possible combinations of dice rolls that add up to 8:

2+6, 3+5, 4+4, 5+3, 6+2

So there are 5 ways to get a sum of 8.

Next, we need to determine the total number of possible outcomes for this experiment. Each die has 6 sides, so there are 6 x 6 = 36 possible outcomes.

Therefore, the probability of obtaining a sum of 8 is:

Number of ways to get a sum of 8 / Total number of possible outcomes = 5/36

Now let's determine the probability of obtaining a sum less than or equal to 4. We can use the same method as before:

1+1, 1+2, 2+1

So there are 3 ways to get a sum less than or equal to 4.

The probability of obtaining a sum less than or equal to 4 is:

Number of ways to get a sum less than or equal to 4 / Total number of possible outcomes = 3/36 = 1/12

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Answer:

B

Step-by-step explanation:

I put the equations into math-way and it solved the system of equations. X=10 and Y=14.

10 three-point questions and 14 five-point questions

Geometric mean (G(n)): It is defined as the square root of the product of all the positive factors of n. In other words, G(n) = √(f1 * f2 * f3 * ... * fn), where fi represents the positive factors of n.

Arithmetic mean (A(n)): It is defined as the sum of all the positive factors of n divided by the total number of factors. In other words, A(n) = (f1 + f2 + f3 + ... + fn) / k, where fi represents the positive factors of n and k represents the total number of factors.

To determine when G(n) is an integer, we need to find values of n for which all the factors of n can be paired such that each pair multiplies to an integer.

For example, if n has four factors (f1, f2, f3, f4), and we can pair them as (f1 * f4) and (f2 * f3), such that both products are integers, then G(n) would be an integer.

This condition can be satisfied when n is a perfect square, as each factor will have an even count and can be paired.

To find values of n for which A(n) is an integer, we need to determine when the sum of all the factors of n is divisible by the total number of factors (k).

This condition can be satisfied when n has an odd number of factors, as the sum of factors will always be an integer and can be divided evenly by k.

To determine when A(n) is equal to 6 or 124, we need to find values of n for which the sum of all the factors of n is equal to 6k or 124k, where k is a positive integer.

This condition can be satisfied when n is a multiple of 6 or 124, respectively.

During the process of solving these problems, interesting conjectures may arise, such as the conjecture that G(n) is an integer if and only if n is a perfect square, and that A(n) is an integer if and only if n has an odd number of factors.

These conjectures can be proved using mathematical reasoning and properties of factors and means.

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c) The interest due on January 11th is $66 and the balance of the loan after the January 11th payment is $4,263.

d) The final payment that Peter must make on the due date (interest and principal) is $4,285.

The interest due, balances, and final payments are based on compound interest.

The compound interest system charges interest on both the accumulated interest and principal (balance).

April 15th to October 12th = 180 days

October 13th to January 11th = 90 days

January 12th to February 28th = 47 days

Total number of days for the loan = 317 days

Days in the year = 365 days

Principal = $10,000

Loan period = 317 days

Interest rate = 4%

Balance on October 12th = $10,197 ($10,000 + $10,000 x 4% x 180/365)

Payment on October 12th = $3,500

Balance from October 13th = $6,697 ($10,197 - $3,500)

Balance on January 11th = $6,763 ($6,697 + $66)

Interest = $66 ($6,697 x 4% x 90/365)

Payment on January 11th = $2,500

Balance from January 12th = $4,263 ($6,763 - $2,500)

Final Payment on February 28th = $4,285 ($4,263 + $4,263 x 4% x 47/365)

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We would need to collect at least 268 service times to return a 95% confidence interval whose width is at most 20 seconds.

(a) We can use the formula for a confidence interval for the mean of a normal distribution with known standard deviation:

CI = X ± z*(σ/√n)

where X is the sample mean, σ is the population standard deviation (in this case, the sample standard deviation is used as an estimate of the population standard deviation since it is known), n is the sample size, and z is the critical value from the standard normal distribution for the desired level of confidence.

For a 95% confidence interval, the critical value is z = 1.96. Plugging in the values, we get:

CI = 8 ± 1.96*(7/√100) = 8 ± 1.372

Therefore, a 95% confidence interval for the mean service time is (6.63, 9.37) minutes.

(b) To find the sample size required to return a 95% confidence interval whose width is at most 20 seconds, we can use the formula for the margin of error:

ME = z*(σ/√n)

where ME is the maximum allowed margin of error (which is 1/3 minute or 0.33 minutes in this case).

Solving for n, we get:

n = (z*σ/ME)^2

For a 95% confidence interval, the critical value is z = 1.96. Plugging in the values, we get:

n = (1.96*7/0.33)^2 ≈ 267.17

Therefore, we would need to collect at least 268 service times to return a 95% confidence interval whose width is at most 20 seconds.

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Answer:

Step-by-step explanation:

If 60% of a number is 80, find 6% of that number.

The autonomous differential equation dp/dt = p(a + bp) has a solution that is increasing if -a/b < P < 0 (option c). This is because the rate of change of P (dp/dt) is positive when -a/b < P < 0, leading to an increasing solution.

The given differential equation is autonomous, which means it does not explicitly depend on time 't'. We can find the equilibrium solutions by setting dp/dt = 0. So, we have p(a+bp) = 0, which gives p = 0 and p = -a/b as equilibrium solutions.

Now, we can analyze the behavior of the solution by considering the sign of dp/dt for different values of p.

For p < -a/b, we have a+bp < 0, which implies dp/dt < 0. So, the solution is decreasing in this region.

For -a/b < p < 0, we have a+bp > 0, which implies dp/dt > 0. So, the solution is increasing in this region.

For p > 0, we have a+bp > 0, which implies dp/dt > 0. So, the solution is increasing in this region.

Therefore, the correct answer is (c) increasing if -a/b < p < 0.

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Answer: see attached file

Step-by-step explanation:

Answer:

36.9 degrees

Step-by-step explanation:

Cos(x) = adjacent/hypoteneuse

cos(x) = 12/15

cos(x) = 4/5

x = cos^-1(4/5)

= 36.869898 degrees

= 36.9 degrees

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Answer:

19

Step-by-step explanation:

f'(c) = (f(b) - f(a))/(b - a)

In this case, we have a = 1 and b = 5, so we can write:

f'(c) = (f(5) - f(1))/(5 - 1)

Solving for f(5), we get:

f(5) = f(1) + f'(c)(5 - 1)

Since we know that f '(x) ≥ 1 for 1 ≤ x ≤ 5, we have:

f(5) = f(1) + f'(c)(5 - 1) ≥ 15 + 1(5 - 1) = 19

Therefore, the smallest value that f(5) can possibly be is 19.

*IG:whis.sama_ent

Answer:

  x ≈ 2.467

Step-by-step explanation:

You want the solution to 5^(3x -2) = 7^(x +2).

Logarithms turn an exponential problem into a linear problem. Taking logs, we have ...

  (3x -2)·log(5) = (x +2)·log(7)

  x(3·log(5) -log(7)) = 2(log(7) +log(5)) . . . . . separate variables and constants

  x = log(35²)/log(5³/7) = log(1225)/log(125/7) . . . . divide by x-coefficient

  x ≈ 2.46693

__

Additional comment

A graphing calculator can solve this nicely as the x-intercept of the function f(x) = 5^(3x-2) -7^(x+2). Newton's method iteration is easily performed to refine the solution to calculator precision.

The sample size must be at least 12.

The formula for the standard error of the mean is:

[tex]SE = \sigma / \sqrt(n)[/tex]

where SE is the standard error, σ is the population standard deviation, and n is the sample size.

In this case, we are given that the population mean is 100 and the population variance is 25. Therefore, the population standard deviation is:

[tex]\sigma = \sqrt(\sigma^2) = \sqrt(25) = 5[/tex]

We want the standard error of the mean to be 1.5, so we can set up the following equation:

[tex]1.5 = 5 / \sqrt(n)[/tex]

Solving for n, we get:

[tex]\sqrt(n) = 5 / 1.5[/tex]

[tex]\sqrt(n) = 3.33[/tex]

[tex]n = (3.33)^2[/tex]

n = 11.0889

Since we need a whole number of samples, we can round up to the next integer and say that the sample size must be at least 12.

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The polygon is a parallelogram and rectangle

The polygon is a parallelogram , quadrilateral and a rectangle

The sum of angles of a parallelogram is 360°

The four types are parallelograms, squares, rectangles, and rhombuses

Opposite sides are parallel

Opposite sides are congruent

Opposite angles are congruent.

Same-Side interior angles (consecutive angles) are supplementary

Each diagonal of a parallelogram separates it into two congruent triangles

The diagonals of a parallelogram bisect each other

Given data ,

The polygon is represented as OPQR

Now , the number of sides of the polygon = 4

So , it is a quadrilateral

Now , the measure of sides of the quadrilateral are

OP = 20 units

PQ = 40 units

QR = 20 units

RO = 40 units

So, it has 2 congruent sides and they are parallel in shape

So, it is a parallelogram

Now, the 2 opposite pairs of sides of the parallelogram are equal

So, it is a rectangle

Hence, the polygon is a parallelogram and rectangle

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For some function f(x), suppose that for a certain interval [a, b], we have:
∫(f(x))dx from a to b = 1
And for another interval [c, d], we have:
∫(f(x))dx from c to d = 10

In mathematics, a (real) interval is a set of real numbers that includes all the real numbers between two numbers in the set. For example, the set x of numbers satisfying 0 ≤ x ≤ 1 is the range containing 0, 1, and every number in between. Other examples of ranges are the set of numbers such as 0 < x < 1, the set of all real numbers {R}, the set of negative numbers, positive real numbers, free space, and a singular (similar sets).

Real numbers play an important role together because they are the simplest numbers whose "length" (or "measure" or "size") is easy to define. The concept of measure can be extended to more complex real numbers, giving rise to the Boral measure and eventually the Lebesgue measure.

To find the values of other integrals involving this function, you would need to either use additional information about the function or be provided with the specific integral expressions and interval limits.
For some function f(x), suppose that for a certain interval [a, b], we have:
∫(f(x))dx from a to b = 1
And for another interval [c, d], we have:
∫(f(x))dx from c to d = 10


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The volume of the solid obtained by rotating the region bounded by the curves y = x², y = 0, and x = 1 about the y-axis is π/3 cubic units.

To find the volume of the solid obtained by rotating the region bounded by the curves y = x², y = 0, and x = 1 about the y-axis, we can use the disk method.

The idea behind the disk method is to slice the solid into thin disks perpendicular to the axis of rotation and sum up their volumes. The volume of each disk is the product of its cross-sectional area and its thickness.

In this case, we are rotating about the y-axis, so the cross-sectional area of each disk will be a circle with radius x and area πx². The thickness of each disk will be dx, which represents an infinitesimal slice of the x-axis.

Thus, the volume of each disk is given by:

dV = πx² dx

To find the total volume of the solid, we need to integrate this expression over the range of x from 0 to 1:

V = ∫₀¹ πx² dx

Integrating this expression gives:

V = π/3

Therefore, the volume of the solid obtained by rotating the region bounded by the curves y = x², y = 0, and x = 1 about the y-axis is π/3 cubic units.

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The standard deviation of the random variable is approximately 3.4082.

To compute the mean and standard deviation of the given discrete probability distribution, we need to use the following formulas:

Mean (μ) = ∑ [xi * P(xi)]

Variance (σ^2) = ∑ [(xi - μ)^2 * P(xi)]

Standard deviation (σ) = sqrt(σ^2)

where xi represents each possible value of the random variable and P(xi) represents the probability of each value.

Using the given probability distribution, we can compute the mean as:

Mean (μ) = (-5 * 0.14) + (-4 * 0.03) + (0 * 0.17) + (3 * 0.30) + (4 * 0.23) + (5 * 0.16) = 1.39

Therefore, the mean of the random variable is 1.39.

To compute the variance, we first need to compute the squared deviation of each value from the mean. Using the formula for variance, we get:

Variance (σ^2) = (-5 - 1.39)^2 * 0.14 + (-4 - 1.39)^2 * 0.03 + (0 - 1.39)^2 * 0.17 + (3 - 1.39)^2 * 0.30 + (4 - 1.39)^2 * 0.23 + (5 - 1.39)^2 * 0.16 = 11.6109

Finally, we can compute the standard deviation by taking the square root of the variance:

Standard deviation (σ) = sqrt(11.6109) = 3.4082

Therefore, the standard deviation of the random variable is approximately 3.4082.

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Answer:

We can evaluate this integral using trigonometric substitution. Let x = 8 sin(θ). Then dx = 8 cos(θ) dθ. Substituting gives us:

```

∫√1−64x2dx = ∫√1−64(8sin(θ))^2(8cos(θ))dθ = ∫√1−64sin^2(θ)cos(θ)dθ

```

We can now use the identity sin^2(θ) + cos^2(θ) = 1 to simplify this integral:

```

∫√1−64sin^2(θ)cos(θ)dθ = ∫√cos^2(θ)cos(θ)dθ = ∫8cos^3(θ)dθ

```

We can now integrate using the power rule:

```

∫8cos^3(θ)dθ = 8cos^4(θ)/4 + C = 2cos^4(θ) + C

```

To reverse the substitution, we need to solve for θ in terms of x. We have:

```

x = 8sin(θ)

```

```

sin(θ) = x/8

```

```

θ = sin^-1(x/8)

```

Substituting gives us:

```

2cos^4(θ) + C = 2cos^4(sin^-1(x/8)) + C

```

```

= 2(1 - sin^2(sin^-1(x/8)))^2 + C

```

```

= 2(1 - (x/8)^2)^2 + C

```

```

= 2(1 - x^2/64)^2 + C

```

Therefore, the integral is equal to:

```

∫√1−64x2dx = 2(1 - x^2/64)^2 + C

```

Step-by-step explanation:

The integral ∫√(1-64x²)dx is equal to (1/128)(asin(8x) + 8x√(1-64x²) + C), where C is the constant of integration.

To solve this integral, we use trigonometric substitution. Let x = (1/8)sin(θ), so dx = (1/8)cos(θ)dθ. The integral becomes ∫√(1-64((1/8)sin(θ))²)(1/8)cos(θ)dθ = ∫(1/8)cos²(θ)dθ.

Now, apply the power-reduction formula: cos²(θ) = (1+cos(2θ))/2. The integral becomes ∫(1/16)(1+cos(2θ))dθ. Integrate with respect to θ: (1/16)(θ+(1/2)sin(2θ)) + C. Convert back to x using θ = asin(8x): (1/128)(asin(8x) + 8x√(1-64x²)) + C.

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a) There are 3ⁿ ternary digit strings with exactly n digits.

b) There is only 1 string with n digits and n 2's.

c) There are n ternary digit strings with n digits and n-1 2's.



a) For each digit in a ternary digit string, there are 3 possible values (0, 1, or 2). With n digits, you have 3 choices for each digit, giving 3ⁿ total possible strings.

b) If a string has n digits and all are 2's, there's only one possible string, which is '222...2' (with n 2's).

c) If a string has n digits and n-1 of them are 2's, there's one remaining digit that can be 0 or 1. There are n positions this non-2 digit can be in, resulting in n possible strings.

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A function that would match the graph shown is f(x) = -5(x + 4)(x - 1)

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

The graph

The zeros of the graph are

x = -4; multiplicity 2

x = 1; multiplicity 1

So, we have

f(x) = a(x + 4)^2(x - 1)

The function intersects with the y-axis at y = 80

So, we have

a(0 + 4)^2(0 - 1) = 80

Evaluate

x = -5

So, we have

f(x) = -5(x + 4)(x - 1)

Hence, the equation is f(x) = -5(x + 4)(x - 1)

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The four possible times that Car B could take to complete one lap are 2 seconds, 3 seconds, 5 seconds, and 15 seconds.

150 is represented as a factor of 2, 3, 5 and 5. As 150/t is considered as an integer, t should be a divisor of 150. Hence, we can consider the potential values for t which are:

- t=2 seconds (because 150/2 equals to 75, making it an integer)

- t=3 seconds (as 150/3 equates to 50, also forming an integer)

- t=5 seconds (since 150 divided by 5 produces 30 which is another integer)

- t=15 seconds (seeing that when dividing 150 with 15 gives us 10 which is likewise an integer).

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a) The 20th percentile is 21.4.

b) The 40th percentile is 31.

c) The 70th percentile is 66.4.

To determine the percentile, we need to first arrange the data in order from smallest to largest

11, 24, 28, 33, 37, 46, 64, 72

a) To find the 20th percentile, we need to first determine the rank of this percentile.

The formula for rank is given by:

Rank = (percentile/100) x (number of observations + 1)

So for the 20th percentile, we have:

Rank = (20/100) x (8+1) = 1.8

This tells us that the 20th percentile lies between the 1st and 2nd observations. To find the actual value, we can use linear interpolation

Value = 11 + 0.8 x (24 - 11) = 11 + 0.8 x 13 = 21.4

Therefore, the 20th percentile is 21.4.

b) To find the 40th percentile, we use the same formula:

Rank = (40/100) x (8+1) = 3.6

This tells us that the 40th percentile lies between the 3rd and 4th observations. Using linear interpolation:

Value = 28 + 0.6 x (33 - 28) = 28 + 0.6 x 5 = 31

Therefore, the 40th percentile is 31.

c) To find the 70th percentile, we use the same formula:

Rank = (70/100) x (8+1) = 6.3

This tells us that the 70th percentile lies between the 6th and 7th observations. Using linear interpolation:

Value = 64 + 0.3 x (72 - 64) = 64 + 0.3 x 8 = 66.4

Therefore, the 70th percentile is 66.4.

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a) The elasticity of the demand function q = 800 - x is -x / (800 - x)².

b) At x = 3, the elasticity of the demand function q = 800 - x is approximately -0.0000465.

(a) To find the elasticity of the demand function q = 800 - x, we first need to calculate the derivative of q with respect to x:

dq/dx = -1

Next, we can use the formula for elasticity:

E = (dq/dx) * (x/q)

Substituting the values of dq/dx and q, we get:

E = (-1) * (x/(800-x))

Simplifying this expression, we get:

E = -x / (800 - x)²

(b) To find the elasticity at x = 3, we substitute x = 3 into the expression we derived for E:

E = -(3) / (800 - 3)² = -0.0000465

Therefore, the elasticity at x = 3 is approximately -0.0000465.

Note that since the elasticity is negative, this indicates that the demand is inelastic, meaning that a change in price will have a relatively small effect on the quantity demanded.

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Since f(n) is defined using a primitive recursive function (exponentiation) and follows a recursive structure, we can conclude that f(n) is primitive recursive.

To show that the function f(n) is primitive recursive, we need to demonstrate that it can be defined using basic primitive recursive functions (zero, successor, and projection functions) and can be composed or recursed using only primitive recursive function schemes.

Given the definition of f(n), we can write it as:
- f(0) = 0
- f(1) = 1
- f(2) = 2²
- f(3) = (3³)³
- ...

We can observe that f(n) is defined as a stack of exponentiation operations with the base and the exponent both being n. We can use the following recursive formula to define f(n):

- f(0) = 0


We know that exponentiation is primitive recursive, as it can be defined using multiplication, which is also primitive recursive. We can define exponentiation recursively as:

- exp(a, 0) = 1
- exp(a, b) = a * exp(a, b-1) for b > 0

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The 95% confidence interval for this estimate is (2.0, 6.5).

The 95% confidence interval for this estimate is (1.01, 1.04).

The confidence interval is (0.85, 9.8).

The confidence interval is (0.92, 1.08).

The confidence interval is (0.05, 1.05).

a) The rate of disease is 3.5 times higher in the exposed group compared to the nonexposed group. The 95% confidence interval for this estimate is (2.0, 6.5).

b) The rate of disease is 1.02 times higher in the exposed group compared to the nonexposed group. The 95% confidence interval for this estimate is (1.01, 1.04).

c) The rate of disease is 6.0 times higher in the exposed group compared to the nonexposed group. However, the 95% confidence interval for this estimate is wide and includes 1, indicating that this estimate may not be statistically significant. The confidence interval is (0.85, 9.8).

d) The rate of disease is 0.97 times lower in the exposed group compared to the nonexposed group. The 95% confidence interval for this estimate includes 1, indicating that this estimate may not be statistically significant. The confidence interval is (0.92, 1.08).

e) The rate of disease is 0.15 times lower in the exposed group compared to the nonexposed group. The 95% confidence interval for this estimate includes 1, indicating that this estimate may not be statistically significant. The confidence interval is (0.05, 1.05).

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The test statistic is 1.177, and the p-value is approximately 0.120, which is greater than the significance level of 0.05, indicating that there is not enough evidence to conclude that the proportion of defective circuit boards is greater than 3%.

To test the hypothesis that more than 3% of circuit boards are defective, we can use a one-tailed test with the following null and alternative hypotheses:

[tex]H_0[/tex]: p ≤ 0.03 (proportion of defective boards is less than or equal to 3%)

[tex]H_a[/tex]: p > 0.03 (proportion of defective boards is greater than 3%)

where p is the true proportion of defective boards in the population.

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

z = (p-cap - p0) / √(p0(1-p0)/n)

where p is the sample proportion of defective boards, p0 is the hypothesized proportion (0.03), and n is the sample size.

In this case, we have p-cap = 0.04, p0 = 0.03, and n = 100, so the test statistic is:

z = (0.04 - 0.03) / √(0.03(1-0.03)/100) = 1.177

To determine the p-value associated with this test statistic, we can use a standard normal distribution table or a calculator to find the probability of observing a z-value of 1.177 or greater under the null hypothesis. This probability is approximately 0.120, which is the area to the right of z = 1.177 on the standard normal distribution curve.

Since this p-value is greater than the common significance level of 0.05, we fail to reject the null hypothesis.

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The sum of the resulting limits of integration is:

a + b + 0 + Jo = a + b + Jo

Assuming the limits of integration for the first integral are 0 to Jo for y and some limits for x.

The limits for the second integral are also 0 to Jo for y and the same limits for x, we can combine the integrals as follows:

3∫∫ f(x,y) dy dx + 3∫∫ f(x,y) dy dx

= 3∫∫ f(x,y) + f(x,y) dy dx (by combining the two integrals)

= 6∫∫ f(x,y) dy dx

Now, to switch the order of integration, we need to express the limits of integration of y in terms of x. Let's assume the limits of integration for x are a to b:

6∫∫ f(x,y) dy dx = 6∫[a,b]∫[0,Jo] f(x,y) dy dx

We can integrate with respect to y first, then with respect to x, so:

6∫[a,b]∫[0,Jo] f(x,y) dy dx = 6∫[a,b] (∫[0,Jo] f(x,y) dy) dx

The limits of integration for y are constant, so we can take them out of the inner integral:

6∫[a,b] (∫[0,Jo] f(x,y) dy) dx = 6∫[a,b] f(x,y) * Jo|0 dx

The limits of integration for x are a to b, and the limits for y are 0 to Jo, so the sum of the resulting limits of integration is:

a + b + 0 + Jo = a + b + Jo

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The mean of the variable given in the question is Option A. 41.6.

To find the mean of the variable, we need to multiply each range of weekly hours worked by its corresponding probability, then sum all of the results.

The calculations are as follows:

(23 * 0.08) + (36 * 0.10) + (43 * 0.74) + (54 * 0.08) = 41.6

Therefore, the mean of the variable is Option A. 41.6.

In this case, the probabilities for each range of weekly hours worked to represent the likelihood of an employee working within that range. For example, the probability of an employee working between 41-50 hours is 0.74, which is quite high compared to the other ranges. As a result, this range has a larger impact on the overall mean of the variable.

It is important to calculate the mean of a variable as it helps in understanding the central tendency of a distribution. In this case, the mean helps us to understand the average number of weekly hours worked by employees, which can be useful in making decisions related to employee scheduling, workload management, and compensation.

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Answer:

Asiah it’s ego the answer is D

Step-by-step explanation:

I got it right

L ms Brenda

Answer: The answer is Sample D

Step-by-step explanation:

264 is the value of x³ - 1/x³ in linear equation.

What is a linear equation in mathematics?

A linear equation is an algebraic equation of the form y=mx b. which contains only a constant and  first-order (linear) term, where m is the slope and b is the y-intercept.

                      Sometimes the above is called a "linear equation in two variables" where y and x are the variables. A linear equation can have more than one variable. If a linear equation has two variables,  it is called a bivariate linear equation, etc.

x - 1/x = 7

x³ - 1/x³ = ?

(x - 1/x )³ = 7³

x³  - 1/x³ - 3 * x * 1/x (x - 1/x) = 243

x³  - 1/x³ - 3 * 7 = 243

x³  - 1/x³ - 21 = 243

x³  - 1/x³ = 243 + 21

x³  - 1/x³ =  264

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(a)  The probability of rolling at least one 6 is approximately 0.665, or about 66.5%.

(b)  The probability of rolling 5 different numbers is approximately 0.0772, or about 7.72%.

(a) To calculate the probability of rolling at least one 6 in 6 rolls of a fair die, we can use the complement rule: the probability of the complement (rolling no 6s) is easier to calculate, and then we subtract that from 1.

The probability of rolling no 6s in a single roll is 5/6, so the probability of rolling no 6s in 6 rolls is (5/6)^6.

Therefore,  probability of rolling at least one 6 in 6 rolls is:

1 - (5/6)^6 ≈ 0.665

So the probability of rolling at least one 6 is approximately 0.665, or about 66.5%.

(b) To calculate the probability of rolling 5 different numbers in 6 rolls of a die, we can use the formula for combinations. There are 6 possible numbers that could be rolled first, 5 possible numbers that could be rolled second (since we want 5 different numbers), and so on down to 2 possible numbers that could be rolled fifth.

For the sixth roll, any of the 5 previous numbers would result in 5 different numbers, so there are 5 choices. Therefore, the total number of ways to roll 5 different numbers is:

6 × 5 × 4 × 3 × 2 × 5 = 3600

To find the probability, we divide this by the total number of possible outcomes for rolling a die 6 times, which is 6^6 = 46656. Therefore, the probability of rolling 5 different numbers in 6 rolls of a die is:

3600/46656 ≈ 0.0772

So the probability of rolling 5 different numbers is approximately 0.0772, or about 7.72%.

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