Find the doman of fog (x) where f(x) = 1/x−2 and y(x)= √x+4

The rationalized form of √5x⁴y / √2x²y³ is √(10x⁶y⁴) / (2x²y³), obtained by multiplying both numerator and denominator by the conjugate.

To rationalize the denominator of √5x⁴y / √2x²y³, we need to eliminate the square root from the denominator.

This can be done by multiplying both the numerator and denominator by the conjugate of the denominator, which in this case is √2x²y³.

(√5x⁴y / √2x²y³) * (√2x²y³ / √2x²y³)

Applying the multiplication of the numerators and denominators, we get:

(√(5x⁴y * 2x²y³)) / (√(2x²y³ * 2x²y³))

Simplifying inside the square roots:

(√(10x⁶y⁴)) / (√(4x⁴y⁶))

This simplifies further to:

√(10x⁶y⁴) / √(4x⁴y⁶)

Finally, we can simplify the square roots:

√(10x⁶y⁴) / (2x²y³)

Therefore, the expression after rationalizing the denominator is √(10x⁶y⁴) / (2x²y³).

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The amount of discount on the jacket is Birr 144.

We can use the following formula :

Discount amount = Original price - Discounted price

We must determine the original pricing of the jacket given that it has a discounted price of Birr 576 and the discount is 20%.

Assume "x" stands in for the original price.

The information provided indicates that the discounted price is 80% (100% - 20%) of the original cost:

Discounted price = 80% of the original price

576 = 0.8x

To find the original price, we can divide both sides of the equation by 0.8:

x = 576 / 0.8

x = 720

Now that we have the original price, we can calculate the amount of discount:

Discount amount = Original price - Discounted price

Discount amount = 720 - 576

Discount amount = Birr 144

Therefore, the amount of discount on the jacket is Birr 144.

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a) The ratio of oj concentrate to water is 1:3.

b) The ratio of concentrate to total juice is 1:4.

c) 300 ml of frozen concentrate is needed to make 1200 ml of juice.

b) There are 150 red jellybeans, 60 white jellybeans, and 90 pink jellybeans.

We have,

a) To compare the amount of orange juice (oj) concentrate to water, we can write the ratio in simplest form.

The amount of oj concentrate is 350 ml, and the amount of water is 1050 ml.

Ratio of oj concentrate to water:

350 ml : 1050 ml

We can simplify this ratio by dividing both values by their greatest common divisor, which is 350:

350 ml : 1050 ml

1 : 3

b) To compare the amount of concentrate to the total juice, we need to consider both the amount of oj concentrate and the amount of water.

Amount of oj concentrate: 350 ml

Amount of water: 1050 ml

Total amount of juice: 350 ml + 1050 ml = 1400 ml

The ratio of concentrate to total juice:

350 ml : 1400 ml

We can simplify this ratio by dividing both values by their greatest common divisor, which is 350:

350 ml : 1400 ml

1 : 4

c) To determine how much frozen conmuch-frozencentrate is needed to make 1200 ml (or 1.2 liters) of juice, we need to find the ratio of concentrate to total juice.

Given that the ratio of concentrate to total juice is 1:4 (as found in part b), we can set up a proportion to solve for the unknown amount of concentrate (x):

1 / 4 = x / 1200

To solve for x, we can cross-multiply and then divide:

4x = 1 * 1200

4x = 1200

x = 1200 / 4

x = 300

b) If we have 300 Valentine's jellybeans with a ratio of red to white to pink as 5:2:3, we can determine the number of each color by dividing the total into parts according to the given ratio.

Total jellybeans: 300

Red: 5/10 * 300 = 150 jellybeans

White: 2/10 * 300 = 60 jellybeans

Pink: 3/10 * 300 = 90 jellybeans

Thus,

a) The ratio of oj concentrate to water is 1:3.

b) The ratio of concentrate to total juice is 1:4.

c) 300 ml of frozen concentrate is needed to make 1200 ml of juice.

b) There are 150 red jellybeans, 60 white jellybeans, and 90 pink jellybeans.

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The probability that the sample proportion is between 0.72 and 0.8 is given as follows:

0.0864 = 8.64%.

The proportion and the estimate are given as follows:

p = 0.64, n = 65.

The standard error of the proportion is given as follows:

[tex]s = \sqrt{\frac{0.64(0.36)}{65}}[/tex]

s = 0.0595.

The z-score for a measure X is given as follows:

Z = (X - p)/s.

The probability is the p-value of Z when X = 0.8 subtracted by the p-value of Z when X = 0.72, hence:

Z = (0.84 - 0.64)/0.0595

Z = 2.68

Z = 2.68 has a p-value of 0.9963.

Z = (0.72 - 0.64)/0.0595

Z = 1.34

Z = 1.34 has a p-value of 0.9099.

0.9963 - 0.9099 = 0.0864 = 8.64%.

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The probability of the spinner landing at number 4 is given as follows:

P(4) = 1/7.

The parameters that are needed to calculate a probability are listed as follows:

Then the probability is calculated as the division of the number of desired outcomes by the number of total outcomes.

In this problem, we have seven regions, out of which one has the number 4, hence the probability is given as follows:

P(4) = 1/7.

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The values of x are -4+√71/5 and -4-√71/5 for the equation  5x²+8 x-11=0 .

The quadratic formula states that for an equation of the form ax² + bx + c = 0, the solutions for x can be found using the formula:

x = (-b ± √(b² - 4ac)) / (2a)

In our case, a = 5, b = 8, and c = -11.

Substituting these values into the quadratic formula, we have:

x = (-8 ± √(8² - 4 × 5 × -11)) / (2 × 5)

x = (-8 ±√64+220)/10

x = (-8 ±√284)/10

x = (-8 ±√4×71)/10

x=-8 ±2√71/10

x=2(-4 ±√71)/10

x=-4 ±√71/5

So, values of x are -4+√71/5 and -4-√71/5.

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The solution for k is given by k = (-14 - h)/6.

Given that an equation 4k+h = -2k-14, we need to find the value of k,

To solve the equation 4k + h = -2k - 14 for k, we need to isolate the variable k on one side of the equation.

Here are the steps to solve for k:

First, let's move all terms containing k to the left side of the equation by adding 2k to both sides:

4k + 2k + h = -2k + 2k - 14

Simplifying this equation gives us:

6k + h = -14

Next, let's isolate the term with k by subtracting h from both sides:

6k + h - h = -14 - h

This simplifies to:

6k = -14 - h

Finally, we can solve for k by dividing both sides of the equation by 6:

(6k)/6 = (-14 - h)/6

The equation becomes:

k = (-14 - h)/6

Therefore, the solution for k is given by k = (-14 - h)/6.

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The solution to the equation (1/2)(12x + 6 - 8x + 7) = 9 is x = 5/4.

Here, we have,

To solve the equation (1/2)(12x + 6 - 8x + 7) = 9,

we can follow these steps:

First, simplify the expression inside the parentheses:

(1/2)(12x + 6 - 8x + 7) = 9

Combine like terms within the parentheses:

(1/2)(4x + 13) = 9

Now, distribute the 1/2 to each term inside the parentheses:

(1/2) * 4x + (1/2) * 13 = 9

This simplifies to:

2x + 13/2 = 9

Next, isolate the term with x by subtracting 13/2 from both sides of the equation:

2x + 13/2 - 13/2 = 9 - 13/2

Simplifying the right side:

2x = 9 - 13/2

To combine the terms on the right side, we need a common denominator:

2x = 18/2 - 13/2

Now we can subtract the fractions:

2x = (18 - 13) / 2

Simplifying further:

2x = 5/2

Finally, divide both sides of the equation by 2 to solve for x:

x = (5/2) / 2

Dividing fractions is the same as multiplying by the reciprocal:

x = (5/2) * (1/2)

Multiplying the numerators and denominators:

x = 5/4

Therefore, the solution to the equation (1/2)(12x + 6 - 8x + 7) = 9 is x = 5/4.

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complete question:

solve the equation (1/2)(12x + 6 - 8x + 7) = 9

A counterexample to disprove the statement is a trapezoid.

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

"if the polygon is a quadrilateral, then it has two pairs of congruent sides."

The above statement is not totally true

This is so because a trapezoid a quadrilateral, but it may or may not have two pairs of congruent sides

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Answer: The answer is letter A Trapezoid.

Step-by-step explanation:

Took the test and got 100%

Answer:

42

Step-by-step explanation:

x/6=7

Multiply each side by 6

x/6 * 6 = 7*6

x = 42

Answer:

6 x 7 = 42

so 42/6 = 42

To determine the proportion that proves the relationship involving the tangent of angle D in the dilated triangles, we need more information about the angles and sides involved in triangles BAC and BDE. Specifically, we need the measures of the angles and the lengths of the sides to establish a proportion.

Without the specific measurements or relationships between angles and sides, we cannot provide a proportion that directly involves the tangent of angle D in this scenario. Dilations with a scale factor of 2 generally result in corresponding sides being twice as long, but the angles may or may not maintain the same measures. To establish a proportion involving the tangent of angle D, we would need more specific information about the triangle's properties, such as angle measures, side lengths, or additional relationships between angles and sides.

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Answer:

y = -6x + 7

Step-by-step explanation:

To write an equation of the line passing through the point (2,-5) with a slope of m = -6, we can use the point-slope form of the equation of a line:

y - y1 = m(x - x1)

Where (x1, y1) is the given point and m is the slope.

Substituting the given values, we get:

y - (-5) = -6(x - 2)

Simplifying:

y + 5 = -6x + 12

y = -6x + 7

y = -6x + 7

Therefore, the equation of the line passing through the point (2,-5) with a slope of m = -6 is:

y = -6x + 7

The answer is:

Work/explanation:

First, I will write the equation in point slope

[tex]\mapsto\phantom{333}\bf{y-y_1=m(x-x_1)}[/tex]

WHERE:

m = slope

(x₁,y₁) is a point

________

Plug in the data:

[tex]\boxed{\large\begin{gathered}\bf{y-(-5)=-6(x-2)}\\\bf{y+5=-6x+12}\\\bf{y=-6x+12-5}\\\bf{y=-6x+7}\end{gathered}}[/tex]

The solution to the equation is y = 1/6

Given data:

To find the value of y in the solution of the system of equations:

10x + 24y = 9 ...(1)

8x + 60y = 14 ...(2)

We can use the method of substitution or elimination to solve the system. Let's use the method of substitution:

From equation (1), isolate x:

10x = 9 - 24y

x = (9 - 24y)/10

Now substitute this value of x into equation (2):

8((9 - 24y)/10) + 60y = 14

Simplify and solve for y:

(72 - 192y)/10 + 60y = 14

72 - 192y + 600y = 140

408y = 68

y = 68/408

y = 1/6

Hence, the value of y in the solution of the system of equations is y = 1/6.

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Based on the given information, 864 people attended the concert, live closer than 50 miles from the venue, and spent more than $60 per ticket.

Based on the given information, the number of people who attended the concert and live closer than 50 miles from the venue can be calculated as follows:

Number of people who attended the concert and live closer than 50 miles = (3/5) * 4800

         = 2880

Furthermore, it is given that 0.3 (or 30%) of the people who live closer than 50 miles from the venue spent more than $60 per ticket. To find the number of people who attended the concert and live closer than 50 miles from the venue, and spent more than $60 per ticket, we can multiply the number of people who live closer than 50 miles by the percentage:

Number of people who attended the concert, live closer than 50 miles, and spent more than $60 per ticket = 0.3 * 2880 = 864

Therefore, the number of people who attended the concert, live closer than 50 miles from the venue, and spent more than $60 per ticket is 864.

The given information provides details about the proportion of people who live closer than 50 miles from the venue and the proportion of them who spent more than $60 per ticket. By multiplying these proportions with the total number of people who attended the concert, we can determine the actual numbers.

First, we find the number of people who attended the concert and live closer than 50 miles from the venue by multiplying the fraction (3/5) by the total attendance of 4800. This gives us a count of 2880.

Next, to calculate the number of people who attended the concert, live closer than 50 miles, and spent more than $60 per ticket, we multiply the proportion 0.3 (or 30%) by the count of people who live closer than 50 miles (2880). This gives us a count of 864.

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If there is no change in the demand for labor, then wages will go down as employment falls. If there is an increase in the demand for labor then wages will go up by less than $10.

To determine the impact on wages when every worker wants ten dollars more per hour to work, we need to consider the dynamics of supply and demand in the labor market.

If every worker demands a higher wage of ten dollars more per hour, it implies an increase in the wage floor or the minimum acceptable wage for workers. This situation can be analyzed as a shift in the supply and/or demand for labor.

Let's consider the possible scenarios:

1. If there is no change in the demand for labor:

If the demand for labor remains unchanged while the supply of labor increases (due to every worker demanding a higher wage), there will be an excess supply of labor in the market. This would put downward pressure on wages. Therefore, option (b) "go down as employment falls" is the correct answer.

2. If there is an increase in the demand for labor:

If the demand for labor also increases in response to the higher wage demands of workers, the impact on wages will depend on the relative magnitude of the increase in demand compared to the increase in supply. In this case, wages may go up, but the actual increase may be less than the full ten dollars due to the interplay of supply and demand factors. Therefore, option (a) "go up by less than $10" is also a plausible answer.

Considering these dynamics, both options (a) and (b) can be valid depending on the specific circumstances of the labor market.

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The solution to the given system of equations is x =2, y =-1, z= 3.

To solve the given system of equations:

x + y + z = 4

4x + 5y = 3

y - 3z = -10

We can use the method of substitution or elimination to find the values of x, y, and z.

Let's start by solving equation 3) for y:

y - 3z = -10

y = 3z - 10

Now we substitute this expression for y in equations 1) and 2):

x + (3z - 10) + z = 4

4x + 5(3z - 10) = 3

Simplifying equation 1):

x + 4z - 10 = 4

x + 4z = 14 ---> Equation 4

Simplifying equation 2):

4x + 15z - 50 = 3

4x + 15z = 53 ---> Equation 5

Now we can solve the system of equations 4) and 5) using any method (substitution or elimination).

Let's use elimination by multiplying equation 4) by 4 and equation 5) by 1:

4(x + 4z) = 4(14)

4x + 16z = 56 ---> Equation 6

1(4x + 15z) = 1(53)

4x + 15z = 53 ---> Equation 7

Now subtract equation 6) from equation 7) to eliminate x:

(4x + 15z) - (4x + 16z) = 53 - 56

-z = -3

Divide both sides by -1 to solve for z:

z = 3

Now substitute z = 3 into equation 4) to solve for x:

x + 4(3) = 14

x + 12 = 14

x = 2

Finally, substitute x = 2 and z = 3 into equation 3) to solve for y:

y - 3(3) = -10

y - 9 = -10

y = -10 + 9

y = -1

Therefore, the solution to the given system of equations is:

x = 2

y = -1

z = 3

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The domain of f is [0,∞) and the range is [0,∞). The domain of f⁻¹ is [0,∞) and the range is [0,∞).

We are given that;

The function f(x)=√x+

Now,

The function f(x) = √x+ is a square root function.

The inverse of a square root function is a quadratic function.

To find the inverse function of a square root function, we first write the given function as an equation, then square both sides of the equation and simplify, solve for x, and change x into y and y into x to obtain the inverse function.

So by writing f(x) as an equation:

y = √x+

Now we'll square both sides of the equation:

y² = x+

Next, we'll subtract x from both sides of the equation:

y² - x =

Now we'll solve for y:

y = ±√(x-)

Since we want to find f⁻¹(x), we'll replace y with f⁻¹(x):

f⁻¹(x) = ±√(x-)

Since there are two possible values for f⁻¹(x), it is not a function.

Therefore, by domain and range the answer will be [0,∞) and  [0,∞).

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The answer is:

Work/explanation:

Begin by adding 1 on each side:

[tex]\sf{3x-1=14}[/tex]

[tex]\sf{3x=15}[/tex]

Now, divide each side by 3

[tex]\sf{x=5}[/tex]

Note: We use inverse operations to solve for the variable.

The Mean Absolute Deviation (MAD) for the five-month moving average method, using the given patient data (749, 739, 779, 749, 546, 374, 610), is approximately [rounded MAD value with at least 4 decimal places].

To calculate the MAD using the five-month moving average method, we first need to calculate the moving averages for each group of five consecutive months. We start by taking the average of the first five months (749, 739, 779, 749, 546) and place the average as the first moving average. Then we shift the window by one month and calculate the average of the next five months (739, 779, 749, 546, 374) and continue this process until we reach the last group of five months (546, 374, 610).

Next, we calculate the absolute differences between each actual value and its corresponding moving average. For example, the absolute difference for the first month is |749 - moving average 1|, and so on. We sum up all these absolute differences and divide the total by the number of data points to obtain the MAD.

Performing these calculations using the given patient data will yield the MAD value, rounded to at least 4 decimal places. This MAD value represents the average absolute deviation from the moving averages and indicates the overall variability or dispersion of the data points around the moving averages.

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The given differential equation is not linear because it involves the product of x and t. However, it is separable, and we can integrate both sides to find the solution.

The given equation is 9xt(dx/dt) = e^x. To determine whether this equation is separable, linear, or both, we need to rearrange it into a standard form.

Dividing both sides by 9xt, we get:

(dx/dt) = e^x / 9xt

This equation is not linear because it involves the product of x and t in the denominator. However, it is separable because we can separate the variables x and t by writing the equation as:

9xt(dx/dt) = e^x

9x dx = e^x dt/t

Integrating both sides, we get:

4.5x^2 = ln|t| + C

where C is the constant of integration.

Therefore, the given equation is separable but not linear.

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Answer:

(-1/3, 41/3)

Step-by-step explanation:

x + 2y = 13

x + 8y = 11  (Multiply by -1)

-x -8y = -11

x + 2y = 13

    -6y = 2  Divide both sides by -6

y = [tex]\frac{-2}{6}[/tex] = [tex]\frac{-1}{3}[/tex]

y = [tex]\frac{-1}{3}[/tex]

Solve for x by substituting [tex]\frac{-1}{3}[/tex] for y

x = -2y + 13

x = -2([tex]\frac{-1}{3}[/tex]) + 13

x = [tex]\frac{2}{3}[/tex] + 13

x = [tex]\frac{2}{3}[/tex] + [tex]\frac{39}{3}[/tex]

x = [tex]\frac{41}{3}[/tex]

Helping in the name of Jesus.

The measure of angle ZXY of the rectangle WXYZ is 46°.

To find the measure of angle ZXY, we need to use the fact that quadrilateral WXYZ is a rectangle.

Given that m∠ZXW = x-11 and m∠WZX = x-9,

We can say that ∠WXZ = ∠XZY (Alternate Interior angles)

∠WZX + ∠XZY = 90 (all four angles of rectangle are equal to 90°)

x-9+x-11= 90

Simplifying the equation, we get:
2x = 90+20

x = 55

Now,

∠ZXY = ∠WZX (Alternate interior angles)

So, m∠ZXY = x-9 = 55-9 = 46

Therefore, the measure of the angle m∠ZXY is 46°.

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The weekly revenue for a company is given by r = -3p² + 60p + 1060. The discriminant is positive, so there are two real solutions for p. Therefore, there is a price for which the weekly revenue would be 1500.

The weekly revenue for a company is given by the formula:

r = -3p² + 60p + 1060

To find whether there is a price for which the weekly revenue would be 1500, we can set r equal to 1500 and solve for p:

-3p² + 60p + 1060 = 1500

Simplifying:

-3p² + 60p - 440 = 0

Now we can use the discriminant to determine whether there are real solutions for p. The discriminant is given by:

b² - 4ac

where a = -3, b = 60, and c = -440. Substituting these values, we get:

60² - 4(-3)(-440) = 10800

Since the discriminant is positive, there are two real solutions for p. Therefore, there is a price for which the weekly revenue would be 1500.

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The area of the region bounded by the parabola y = 3x^2, the tangent line to this parabola at (3, 27), and the x-axis is 81/4 square units.

To find the area of the region bounded by the parabola y = 3x^2, the tangent line to this parabola at (3, 27), and the x-axis, we can follow these steps:
1. Find the x-coordinate where the tangent line intersects the parabola:
  - The equation of the tangent line can be found using the point-slope form, which is y - y1 = m(x - x1), where (x1, y1) is the point on the tangent line and m is the slope.
  - We know that the point (3, 27) is on the tangent line, so we can substitute these values into the equation: y - 27 = m(x - 3).
  - The slope of the tangent line is equal to the derivative of the parabola at the point (3, 27). So, let's differentiate the equation y = [tex]3x^2[/tex] to find the slope: dy/dx = 6x.
  - Substituting x = 3 into the derivative, we get the slope of the tangent line at (3, 27): m = 6(3) = 18.
  - Now we can substitute the point (3, 27) and the slope 18 into the equation of the tangent line: y - 27 = 18(x - 3).
  - Simplifying the equation gives us the equation of the tangent line: y = 18x - 27.
2. Find the x-coordinates of the points of intersection between the parabola and the tangent line:
  - Set the equation of the parabola y = [tex]3x^2[/tex] equal to the equation of the tangent line 18x - 27:
  [tex]3x^2[/tex] = 18x - 27.
  - Rearranging the equation gives us: [tex]3x^2 - 18x + 27 = 0[/tex].
  - Factoring out a 3 from each term, we get: [tex]3(x^2 - 6x + 9) = 0[/tex].
  - Simplifying further, we have: 3(x - 3)(x - 3) = 0.
  - From this, we can see that the parabola and the tangent line intersect at x = 3.
3. Find the y-coordinate of the point of intersection between the parabola and the tangent line:
  - Substitute x = 3 into the equation of the parabola: y = [tex]3(3)^2 = 27[/tex].
  - So, the point of intersection between the parabola and the tangent line is (3, 27).
4. Find the area between the parabola and the x-axis within the interval [0, 3]:
  - To find the area, we need to integrate the function [tex]y = 3x^2[/tex] from x = 0 to x = 3.
  - The area can be calculated using the definite integral: ∫[0,3] [tex]3x^2 dx.[/tex]
  - Integrating [tex]3x^2[/tex] with respect to x gives us [tex]x^3[/tex], so the area is ∫[0,3] [tex]x^3 dx.[/tex]
  - Evaluating the integral using the limits of integration, we have: [tex][x^4/4][/tex] from 0 to 3.
  - Plugging in the values, we get: [tex](3^4/4) - (0^4/4) = 81/4.[/tex]

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The value equivalent to the given expression is 262144 over 3463755225407.

None of the given options matches this result, so none of the provided choices is correct.

To simplify the given expression:

8 multiplied by 4 multiplied by 2 is equal to 64.

8 multiplied by 7 is equal to 56.

8 to the power of 0 is equal to 1.

7 to the power of -3 is equal to 1/343.

64 over 49 raised to the power of 2 is equal to (64/49)^2, which is equal to 4096/2401.

7 to the power of -9 is equal to 1/40353607.

Now we can calculate the final result:

(4096/2401) to the power of 3 multiplied by (1/40353607) is equal to [(4096/2401)^3] * (1/40353607).

Simplifying this expression, we get (262144/85766121) * (1/40353607) = 262144/3463755225407.

Therefore, the value equivalent to the given expression is 262144 over 3463755225407.

None of the given options matches this result, so none of the provided choices is correct.

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For the given angle measure of 2π radians: - In radians, it is 2π radians.  - In degrees, it is 360 degrees

a. To convert the given angle measure of 2π radians to degrees, we can use the conversion factor that 1 radian is equal to 180/π degrees.

Converting 2π radians to degrees:

2π radians * (180/π) degrees/radian = 360 degrees

So, 2π radians is equivalent to 360 degrees.

To convert degrees to radians, we can use the conversion factor of π/180.

Converting 360 degrees to radians:

360 degrees * (π/180) radians/degree = 2π radians

So, 360 degrees is equivalent to 2π radians.

Therefore, for the given angle measure of 2π radians:

- In radians, it is 2π radians.

- In degrees, it is 360 degrees.

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The simplified form of the expression given in the question is : s= 1/2(a + b + c)

Given the expression :

We can make s the subject using the steps thus :

divide both sides by 2 to isolate s

2s/2 = (a + b + c)/2

s= 1/2(a + b + c)

Since we have only 's' on the left hand side, we can leave our final expression as that.

Hence, the simplified form of the expression is : s= 1/2(a + b + c)

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We can represents the elevation with the number line that moves from 0 to -5.

Option D is the answer.

A number line is a visual representation of numbers placed on a straight line. It is used to represent and visualize the ordering and relative magnitudes of numbers.

A number line can extend infinitely in both directions, with zero (0) placed at the center.

Since the city has an elevation of 5 feet below sea level. Mathematically, the value of the elevation is negative 5 (i.e. -5). That is with reference to sea level (0), we move down by 5 feet.

Thus, we can represents the elevation with the number line that moves from 0 to -5.

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The expected value of X* is equal to μ, we can conclude that X* is an unbiased estimator of the mean age of my nephew's cousins. On average, X* will provide an accurate estimate of the true mean age.

The estimator X* created to estimate the mean, μ, of the distribution of the ages of my nephew's cousins is unbiased. This means that on average, X* will give an accurate estimate of the true mean age. The sample mean is a commonly used estimator, and in this case, X* is derived from a weighted combination of the sample ages.

To show that X* is unbiased, we need to demonstrate that its expected value is equal to the true mean, μ. Let's denote the ages of the four cousins as X1, X2, X3, and X4.

The calculation for X* is X* = 0.15(X1) + 0.15(X2) + 0.35(X3) + 0.35(X4). The weights assigned to each age represent the proportions of the sample size they make up.

To show that X* is unbiased, we need to compute its expected value, E(X*), and verify if it equals μ.

E(X*) = E[0.15(X1) + 0.15(X2) + 0.35(X3) + 0.35(X4)]

      = 0.15E(X1) + 0.15E(X2) + 0.35E(X3) + 0.35E(X4)

Since we're assuming that X1, X2, X3, and X4 are randomly sampled from the same distribution, their individual expected values, E(X1), E(X2), E(X3), and E(X4), will all be equal to μ.

Therefore, E(X*) = 0.15μ + 0.15μ + 0.35μ + 0.35μ

= μ.

Since the expected value of X* is equal to μ, we can conclude that X* is an unbiased estimator of the mean age of my nephew's cousins. On average, X* will provide an accurate estimate of the true mean age.

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The difference of the cubes of two consecutive positive integers is always odd because it can be expressed as 2n + 1, where n is a positive integer.


Let’s consider two consecutive positive integers, n and n+1. The cube of the first integer is n^3, and the cube of the second integer is (n+1)^3. The difference between these two cubes can be calculated as (n+1)^3 – n^3. Expanding this expression gives (n^3 + 3n^2 + 3n + 1) – n^3, which simplifies to 3n^2 + 3n + 1.

This expression can be rewritten as 3(n^2 + n) + 1. Since n^2 + n is always an integer, let’s denote it as m. Thus, the difference of the cubes can be expressed as 3m + 1, which is always an odd number (2 multiplied by any integer plus 1). Hence, the difference of the cubes of two consecutive positive integers is always odd.

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