staA study conducted by the Center for Population Economics at the University of Chicago studied the birth weights of 1000 babies born in New York. The mean weight was 3234 grams with a standard deviation of 871 grams. Assume that the shape of birth weight data distribution is unimodal and symmetric. Find the approximate percentage of newborns who weighted less than 4105 grams. Find the nearest answer.

Answer:

25

Step-by-step explanation:

By Geometric mean theorem:

\( {12}^{2} = y \times 16 \\ \\ 144 = y \times 16 \\ \\ y = \frac{144}{16} \\ \\ y = 9 \\ \\ x = y + 16 \\ \\ x = 9 + 16 \\ \huge \red{ \boxed{ x = 25}}\)

The missing length AC is approximately 25.61 units.

In the right-angled triangle ABC, the hypotenuse is AC, and the other two sides are AB and BC. We are given that BD = 12, DC = 16, and triangle ABC is right-angled at B.

To find the missing length AC, we can use the Pythagorean theorem:

\(AC^2 = AB^2 + BC^2\)

Since triangle ABC is right-angled at B, we can use the given lengths to find AC.

We know that BD is perpendicular to AC, so it divides triangle ABC into two smaller right-angled triangles: ABD and BDC.

In triangle ABD:

\(AB^2 = BD^2 + AD^2\\\\AB^2 = 12^2 + AD^2\\\\AB^2 = 144 + AD^2\)

In triangle BDC:

\(BC^2 = BD^2 + DC^2\\\\BC^2 = 12^2 + 16^2\\\\BC^2 = 144 + 256\)

We also know that AB = BC (since they are opposite sides in an isosceles right-angled triangle), so we can equate the two expressions:

\(144 + AD^2 = 144 + 256\)

Simplifying, we get:

\(AD^2 = 256\)

Taking the square root of both sides:

AD = 16

Now we can substitute the value of AD back into the expression for \(AB^2\):

\(AB^2 = 144 + AD^2\\\\AB^2 = 144 + 16^2\\\\AB^2 = 144 + 256\\\\AB^2 = 400\)

Again, since AB = BC (in an isosceles right-angled triangle), we can say:

\(400 = 144 + BC^2\\\\BC^2 = 400 - 144\\\\BC^2 = 256\)

Taking the square root of both sides:

BC = 16

Finally, we can substitute the values of AB and BC into the Pythagorean theorem equation to find AC:

\(AC^2 = AB^2 + BC^2\\\\AC^2 = 400 + 256\\\\AC^2 = 656\)

Taking the square root of both sides:

AC = √656

AC ≈ 25.61

Therefore, the missing length AC is approximately 25.61.

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Answer:

Can't explain but

9:12

12:16?

a) The probability that no defective component will be chosen when selecting 5 components at random is approximately 0.7738, or 77.38%.

b) The probability that exactly 2 out of the 5 components will be defective is approximately 0.0874, or 8.74%.

a) To calculate the probability that no defective component will be chosen when selecting 5 components at random, we need to determine the probability of selecting a non-defective component for each selection.

Since 5% of the components are faulty, the probability of selecting a non-defective component in each selection is 1 - 0.05 = 0.95.

The probability of selecting no defective components can be calculated using the multiplication rule for independent events. We multiply the probabilities of each selection being non-defective:

Probability of selecting no defective components = (0.95)⁵

Probability of selecting no defective components = 0.7738

b) To calculate the probability that exactly 2 out of the 5 components will be defective, we need to consider the different combinations of selecting 2 defective components out of 5.

The probability of selecting exactly 2 defective components can be calculated using the binomial probability formula:

Probability = (Number of ways to choose 2 defective components) * (Probability of selecting a defective component)^(Number of defective components) * (Probability of selecting a non-defective component)^(Number of non-defective components)

Probability = (5 choose 2) * (0.05)² * (0.95)³

Probability = (5! / (2! * (5-2)!)) * (0.05)² * (0.95)³

Probability = 0.0874

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Answer:

4.2x+3y=8

=21x+15y-40=0

x-2y+3=0

=x-2y=-3

Answer:

Step-by-step explanation:

4. 2x + 3y = 8=x= -3/2y +4

x - 2y + 3 =0​= x=2y-3

Hope this helps!

Part A: The inequality representing the temperatures for benzene to remain liquid is 42°F < T < 176°F.

Part B: The graph of the inequality includes open circles at 42°F and 176°F, indicating that these temperatures are not included in the solution set. The interval between these points should be shaded, representing the temperatures within which benzene remains liquid.

Part C: No, the chemist would not have been able to conduct her research with benzene at 20°F because it is below the lower bound of the temperature range (42°F) required for benzene to remain in its liquid form.

Part A: To represent the temperatures within which benzene must remain liquid, we can use an inequality. Since the boiling point is 176°F and the freezing point is 42°F, the temperature must stay between these two values. Therefore, the inequality is 42°F < T < 176°F, where T represents the temperature in degrees Fahrenheit.

Part B: The graph of the inequality 42°F < T < 176°F represents a bounded interval on the number line. To describe the graph, we can use open circles at 42°F and 176°F to indicate that these endpoints are not included in the solution set. The interval between these two points should be shaded, indicating that the temperatures within this range satisfy the inequality. The shading should be from left to right, covering the entire interval between 42°F and 176°F.

Part C: In February, when the building's temperature fell to 20°F, the chemist would not have been able to conduct her research with benzene. This is because 20°F is below the lower bound of the temperature range required for benzene to remain liquid. The inequality 42°F < T < 176°F indicates that the temperature needs to be above 42°F for benzene to stay in its liquid form. Therefore, with a temperature of 20°F, the benzene would have frozen, making it unsuitable for the chemist's research.

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Answer:

90

Step-by-step explanation:

15+15=30

3x30=90

Therefore, the answer to what divided by 6 equals 15 is 90. You can prove this by taking 90 and dividing it by 6, and you will see that the answer is 15. Tip: For future reference, when you are presented with a problem like "What divided by 6 equals 15?", all you have to do is multiply the two known numbers together.

Therefore, n=90 is your correct answer

The probability that the person picks a 4 as the first number (event H) and the second number exceeds 5 (event K) is 1/30.

We can determine the probabilities of events H and K as follows:

P(H) = probability of picking a 4 as the first number = 1/6 (since there are 6 equally likely numbers to choose from and only 1 of them is a 4)

P(K) = probability of picking a number greater than 5 as the second number = 1/5 (since there are 5 remaining numbers to choose from and only 1 of them is greater than 5)

Now, we need to find the probability of both events H and K occurring, which is denoted as P(H ∩ K).

Since the person must randomly pick two numbers and the second number must be different from the first, the total number of equally likely outcomes is 6 * 5 = 30 (6 choices for the first number and 5 choices for the second number).

Out of these 30 equally likely outcomes, there is only 1 outcome where the first number is 4 and the second number exceeds 5, which is (4, 6). Therefore, P(H ∩ K) = 1/30.

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The velocity of the boat when it reaches the buoy is approximately 17.32 m/s. This is found using the equation v² = u² + 2as, where u is the initial velocity, a is the acceleration, and s is the displacement.

To solve this problem, we can use the equations of motion. The initial velocity of the boat, u, is 30 m/s, the acceleration, a, is -3.0 m/s² (negative because the boat is slowing down), and the displacement, s, is 100 m. We need to find the final velocity, v, when the boat reaches the buoy.

We can use the equation: v² = u² + 2as

Substituting the given values, we have:

v² = (30 m/s)² + 2(-3.0 m/s²)(100 m)

v² = 900 m²/s² - 600 m²/s²

v² = 300 m²/s²

Taking the square root of both sides, we find:

v = √300 m/s

v ≈ 17.32 m/s

Therefore, the velocity of the boat when it reaches the buoy is approximately 17.32 m/s.

The problem provides the initial velocity, acceleration, and displacement of the boat. By applying the equation v² = u² + 2as, we can find the final velocity of the boat. This equation is derived from the kinematic equations of motion. The equation relates the initial velocity (u), final velocity (v), acceleration (a), and displacement (s) of an object moving with uniform acceleration.

In this case, the boat is decelerating with a constant acceleration of -3.0 m/s². By substituting the given values into the equation and solving for v, we find that the velocity of the boat when it reaches the buoy is approximately 17.32 m/s.

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Answer:

The answer is A

-1/2

Step-by-step explanation:

y=sin330

270≤x≥360

cosine=posite

sin=negative

tan=negative

y=sin330°= -1/2

y= -1/2

The wind chill is a measure of how cold it feels when the wind is blowing.

Wind chill takes into account the combined effect of temperature and wind speed on the human body.

As wind speed increases, it carries heat away from the body more rapidly, making it feel colder than the actual temperature.

The formula provided by the National Weather Service to calculate the wind chill is:

w = 35.74 + 0.6215t + (0.4275t - 35.75) * v^0.16

where:

- w is the wind chill value

- t is the temperature in Fahrenheit

- v is the wind speed in miles per hour

Now, let's write a Python program that takes the temperature and wind speed as inputs and calculates the wind chill value:

```python

import sys

# Get temperature and wind speed from command-line arguments

t = float(sys.argv[1])

v = float(sys.argv[2])

# Calculate wind chill using the formula

w = 35.74 + 0.6215 * t + (0.4275 * t - 35.75) * v**0.16

# Print the wind chill value

print("Wind Chill:", w)

```

Example output:

```shell

$ python wind_chill.py 32 10

Wind Chill: 23.72794265120923

```

In this example, the temperature is 32 degrees Fahrenheit and the wind speed is 10 miles per hour.

The program calculates the wind chill value to be approximately 23.73.

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1. Objective Function and Constraints:

Maximize 2x1 + 3x2 + 5x3 subject to x1 + x2 + x3 ≤ 1500, x1 + x2 ≤ 1100, x2 ≤ 800, x3 ≤ 400.

2. Using Excel Solver, find the optimal allocation of funds.

3. The maximum value of the objective function is reported by Excel Solver.

We have,

Objective Function and Constraints:

Let:

x1 = amount spent on food

x2 = amount spent on shelter

x3 = amount spent on entertainment

Objective Function:

Maximize: 2x1 + 3x2 + 5x3 (since each dollar spent on food has a satisfaction value of 2, each dollar spent on shelter has a satisfaction value of 3, and each dollar spent on entertainment has a satisfaction value of 5)

Constraints:

Subject to:

x1 + x2 + x3 ≤ $1,500 (maximum disposable income)

x1 + x2 ≤ $1,100 (amount spent on food and shelter combined must not exceed $1,100)

x2 ≤ $800 (amount spent on shelter alone must not exceed $800)

x3 ≤ $400 (entertainment cannot exceed $400)

Using Excel Solver:

In Excel, set up a spreadsheet with the following columns:

Column A: Variable names (x1, x2, x3)

Column B: Objective function coefficients (2, 3, 5)

Column C: Constraints coefficients (1, 1, 1) for the first constraint (maximum disposable income)

Column D: Constraints coefficients (1, 1, 0) for the second constraint (amount spent on food and shelter combined)

Column E: Constraints coefficients (0, 1, 0) for the third constraint (amount spent on shelter alone)

Column F: Constraints coefficients (0, 0, 1) for the fourth constraint (entertainment limit)

Column G: Right-hand side values ($1,500, $1,100, $800, $400)

Apply the Excel Solver tool with the objective function and constraints to find the optimal allocation of funds.

Once the Excel Solver completes, it will report the maximum value of the objective function, which represents the optimal satisfaction value achieved within the given budget constraints.

Thus,

Objective Function and Constraints: Maximize 2x1 + 3x2 + 5x3 subject to x1 + x2 + x3 ≤ 1500, x1 + x2 ≤ 1100, x2 ≤ 800, x3 ≤ 400.

Using Excel Solver, find the optimal allocation of funds.

The maximum value of the objective function is reported by Excel Solver.

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The parabola's equation is therefore for A: y  = 7x2 - 5$1.

As a result, the parabola's equation for B  is y = -1/2(x - 1)² + 4.

A parabola with vertex (h,k) has the basic equation y = a(x-h)² + k². In this instance, h=0 and k=-5 are equal because the vertex is at $(0,-5)$. Moreover, the parabola has a point at (-1 . 2). When we substitute these numbers in the general parabola equation, we obtain:

2 = a(-1-0)² -5

7 = a

The parabola's equation is therefore y = 7x2 - 5$1.

A cone and a plane perpendicular to its side cross to create a symmetrical open plane curve known as a parabola1. It is a U-shaped curve whose vertex and axis of symmetry characterise it. The line dividing the parabola into its two mirror counterparts is known as the axis of symmetry. The parabola's vertex is where the axis of symmetry intersects it.

A parabola's vertex is represented by the equation y = a(x - h)²+ k, where (h,k) represents the vertex. We may utilise the vertex version of the equation and substitute the values of h, k, and the x and y coordinates of the point to determine the equation of a parabola that passes through a particular point and has a certain vertex.

B.  Here, the vertex is (1, 4), and the point (5,-4). In the vertex form equation, we may therefore substitute h=1, k=4, x=5, and y=-4 to obtain:

-4 = a(5 - 1)²2 + 4 \s-8 = 16a

a = -1/2

As a result, the parabola's equation is y = -1/2(x - 1)² + 4.

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Answer: A. 1/9, 3, 27

Step-by-step explanation:

3^x for x = -2

= 3^-2 = 1/9

3^x for 1

= 3^1 = 3

3^x for 3

= 3^3 = 27

hope this helped

Therefore, the volume of the solid bounded by the coordinate planes and the plane 6x + 4y + z = 24 is 96 cubic units.

To find the volume of the solid bounded by the coordinate planes (xy-plane, xz-plane, and yz-plane) and the plane 6x + 4y + z = 24, we need to determine the region in space enclosed by these boundaries.

First, let's consider the plane equation 6x + 4y + z = 24. To find the x-intercept, we set y = 0 and z = 0:

6x + 4(0) + 0 = 24

6x = 24

x = 4

So, the plane intersects the x-axis at (4, 0, 0).

Similarly, to find the y-intercept, we set x = 0 and z = 0:

6(0) + 4y + 0 = 24

4y = 24

y = 6

So, the plane intersects the y-axis at (0, 6, 0).

To find the z-intercept, we set x = 0 and y = 0:

6(0) + 4(0) + z = 24

z = 24

So, the plane intersects the z-axis at (0, 0, 24).

We can visualize that the solid bounded by the coordinate planes and the plane 6x + 4y + z = 24 is a tetrahedron with vertices at (4, 0, 0), (0, 6, 0), (0, 0, 24), and the origin (0, 0, 0).

To find the volume of this tetrahedron, we can use the formula:

Volume = (1/3) * base area * height

The base of the tetrahedron is a right triangle with sides of length 4 and 6. The area of this triangle is (1/2) * base * height = (1/2) * 4 * 6 = 12.

The height of the tetrahedron is the z-coordinate of the vertex (0, 0, 24), which is 24.

Plugging these values into the volume formula:

Volume = (1/3) * 12 * 24

= 96 cubic units

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The condition for the first dark fringe through a single slit of width w is when the path difference between the light waves at the edges of the slit equals a half wavelength= (λ/2).

This can be expressed mathematically as:
w * sin(θ) = (m + 1/2) * λ, where m = 0 for the first dark fringe, w is the slit width, θ is the angle of the dark fringe from the central maximum, and λ is the wavelength of light.
When light passes through a single slit, it diffracts and creates an interference pattern with alternating bright and dark fringes on a screen. The dark fringes occur when light waves from the edges of the slit interfere destructively, which means their path difference must be an odd multiple of half a wavelength (λ/2).
For the first dark fringe, we set m = 0 in the equation:
w * sin(θ) = (0 + 1/2) * λ
So, the condition for the first dark fringe is:
w * sin(θ) = λ/2

Hence, The condition for the first dark fringe through a single slit of width w is when the path difference between the light waves at the edges of the slit equals a half wavelength (λ/2). This can be represented by the equation w * sin(θ) = λ/2.

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The total area between f(x) = 4x - 10 and the x-axis over the interval [-1, 4] is -14 square units. Since area cannot be negative, it suggests that the graph of f(x) lies below the x-axis within this interval.

To find the total area between the function f(x) = 4x - 10 and the x-axis over the interval [-1, 4], we need to calculate the definite integral of the absolute value of f(x) with respect to x within the given interval.

Since the function f(x) = 4x - 10 is a linear function, the area between f(x) and the x-axis can be represented as the absolute value of the function integrated over the interval.

The total area A is given by:

A = ∫[a to b] |f(x)| dx

In this case, the interval is [-1, 4], so a = -1 and b = 4. Substituting the function into the integral:

A = ∫[-1 to 4] |4x - 10| dx

To calculate the absolute value of the function within the given interval, we need to consider the points where the function crosses the x-axis (i.e., where f(x) = 0).

Setting 4x - 10 = 0, we find x = 10/4 = 2.5. So, the interval [-1, 4] is divided into two subintervals: [-1, 2.5] and [2.5, 4].

Now, we can calculate the total area by splitting the integral into these two subintervals:

A = ∫[-1 to 2.5] (4x - 10) dx + ∫[2.5 to 4] -(4x - 10) dx

Evaluating each integral:

A = [2x^2 - 10x] from -1 to 2.5 + [-2x^2 + 10x] from 2.5 to 4

A = [(2(2.5)^2 - 10(2.5)) - (2(-1)^2 - 10(-1))] + [(-2(4)^2 + 10(4)) - (-2(2.5)^2 + 10(2.5))]

Simplifying further:

A = [(10 - 25) - (-2 + 10)] + [(-32 + 40) - (-10 + 25)]

= [-15 - (-8)] + [8 - 15]

= [-15 + 8] + [8 - 15]

= -7 + (-7)

= -14

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The p-value of the test statistic will be equal to -1.5.

A statistic employed in statistical hypothesis testing is known as a test statistic. A hypothesis test is often defined in terms of a test statistic, which can be thought of as a numerical summary of a data set that distils the information into a single value that can be used to conduct the test.

The formula for test statistics:

Test statistic = z =

A sample of patients n = 160

Patients insured =x =80

P = x/n

P = 80/ 160

P = 1/2

P = 0.5

P₀ = 56/100

P₀ = 0.56

Putting the values we get:

\(Z =\dfrac{P-P_o}{\sqrt{\dfrac{P_o-(1-P_o)}{n}}}\)

\(Z =\dfrac{0.5-0.56}{\sqrt{\dfrac{0.56-(1-0.56)}{150}}}\)

Z=  -0.06/0.04

Z = - 1.5

Therefore, the p-value of the test statistic will be equal to -1.5.

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Answer:  

Here we answer the question: "What two numbers multiply to -36 and add up to -16?"

The answer to your question is:

2 and -18

Step-by-step explanation:     2 x -18 = -36

2 + (-18) = -16

Are you asking because you are trying to figure out how to factor the following quadratic equation?

x2 - 16x - 36 = 0

If so, the solution to factor the quadratic equation above is:

(X + 2 ) (X - 18)

To summarize, since 2 and -18 multiply to -36 and add up -16, you know that the following is true:

x2 - 16x - 36 = (X + 2 ) (X - 18)

The two numbers are -4 and -9.

A number is a count or measurement.

We are asked to find two numbers that multiply to 36 and add to -13,

Let the two numbers be x and y,

x+y = -13.............(i)

xy = 36

x = 36/y...............(ii)

Put eq(ii) in eq(i)

36/y + y = -13

Multiply the whole equation by y,

36 + y² = -13y

y²+13y+36 = 0

Solving for y,

y²+9y+4y+36 = 0

y(y+9)+4(y+9) = 0

(y+4)(y+9) = 0

y = -4 or y = -9

If we put y = -4 then we will get x = -9 similarly if y = -9 then x = -4

Hence the two numbers are -4 and -9.

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The form of power series solutions near x = 0 for the given differential equation is y(x) = Σ (n=0 to ∞) cnx^n.

The given differential equation is 2xy" + (x + 5)y' - (1/x)y = 0. We need to find the form of power series solutions near x = 0.

To find the power series solution, we assume that the solution can be expressed as a power series of the form:

y(x) = Σ (n=0 to ∞) cnx^n,

where cn represents the coefficients of the series.

First, we differentiate y(x) twice to find y' and y":

y'(x) = Σ (n=0 to ∞) ncnx^(n-1),

y"(x) = Σ (n=0 to ∞) n(n-1)cnx^(n-2).

Now, we substitute y, y', and y" into the given differential equation:

2x(Σ (n=0 to ∞) n(n-1)cnx^(n-2)) + (x + 5)(Σ (n=0 to ∞) ncnx^(n-1)) - (1/x)(Σ (n=0 to ∞) cnx^n) = 0.

Next, we simplify the equation by expanding the series and regrouping terms according to the powers of x. Then we equate the coefficients of each power of x to zero to obtain a recurrence relation for the coefficients cn.

By analyzing the equation term by term, we can find the recurrence relations and calculate the coefficients cn for each power of x. However, the calculation of the coefficients and recurrence relations is not requested in the question.

The form of power series solutions near x = 0 for the given differential equation is y(x) = Σ (n=0 to ∞) cnx^n, where cn are the coefficients that can be determined by solving the recurrence relations obtained from the equation.

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Answer:

Step-by-step explanation:

Answer:

solution given:

let hammers be H

now

an expression the hardware store could use to represent the number of hammers it sold in June is 2H-5

Answer:

6x^2-6x-4

Step-by-step explanation:

9,603.75 cm² is the area of the given triangle.

So, the perimeter of the triangle is 392 cm.

Then, let the rest 2 sides be x, where x = x.

Now, we know that the rest 2 equal sides are of length 98.5 cm.

Now, calculate the area as follows:

Therefore, 9,603.75 cm² is the area of the given triangle.

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Answer:

r= 1

Step-by-step explanation:

There ya go Plz give me brainiest if it helped!!!

Answer:

r = 1

Step-by-step explanation:

-3r + 10 = 15r - 8

Add 8 to both sides.

-3r + 18 = 15r

Add 3r to both sides.

18 = 18r

Divide both sides by 18.

r = 1

I was in a rush so I couldn't write fown the steps. If you need the explanations for how they were gotten, please say.

Percentage of data within 2 population standard deviations of the mean is 68%.

To calculate the percentage of data within two population standard deviations of the mean, we need to first find the mean and standard deviation of the data set.

The mean can be found by summing all the values and dividing by the total number of values:

Mean = (20*2 + 22*8 + 28*9 + 34*13 + 38*16 + 39*11 + 41*7 + 48*0)/(2+8+9+13+16+11+7) = 32.68

To calculate standard deviation, we need to calculate the variance first. Variance is the average of the squared differences from the mean.

Variance = [(20-32.68)^2*2 + (22-32.68)^2*8 + (28-32.68)^2*9 + (34-32.68)^2*13 + (38-32.68)^2*16 + (39-32.68)^2*11 + (41-32.68)^2*7]/(2+8+9+13+16+11+7-1) = 139.98

Standard Deviation = sqrt(139.98) = 11.83

Now we can calculate the range within two population standard deviations of the mean. Two population standard deviations of the mean can be found by multiplying the standard deviation by 2.

Range = 2*11.83 = 23.66

The minimum value within two population standard deviations of the mean can be found by subtracting the range from the mean and the maximum value can be found by adding the range to the mean:

Minimum Value = 32.68 - 23.66 = 9.02 Maximum Value = 32.68 + 23.66 = 56.34

Now we can count the number of data points within this range, which are 45 out of 66 data points. To find the percentage, we divide 45 by 66 and multiply by 100:

Percentage of data within 2 population standard deviations of the mean = (45/66)*100 = 68% (rounded to the nearest percent).

Therefore, approximately 68% of the data falls within two population standard deviations of the mean.

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Answer:

Your Answer Is 2/3

Step-by-step explanation:

First you have to find what number are both divisible by, in this case 2.

Next you divide: 4÷2=2 and 6÷2=3

Since 4 was your numerator you put 2 on the top and Since 6 was the denominator you put 3 on the bottom.

Answer:

836m

Step-by-step explanation:

Answer:

b:845

Step-by-step explanation:

use the calculator and divide 533 by the inside

Answer:

20

Step-by-step explanation:

180 - 90 - 70  = 20

The sum of all angles in a triangle equals 180.

Triangle Sum Theory

Step-by-step explanation:

as we have a point and the slope, we can start with the point-slope form and then transform.

the point-slope form is

y - y1 = a(x - x1)

(x1, y1) being a point on the line, a being the slope.

the slope-interceot form is

y = ax + b

a being the slope again, b being the y-intercept (the y value for x = 0).

so, we have

y - -5 = 5/2 × (x - -4)

y + 5 = 5/2 × (x + 4) = 5x/2 + 5×4/2 = 5x/2 + 10

y = 5x/2 + 5

or

y = (5/2)x + 5

and this is already the slope-intercept form. all done.

The force F = r^2k(r^) is not conservative because its curl is nonzero. The force is central because it depends only on r and acts along the radial direction. Since it is not conservative, there is no potential energy function V(r) associated with this force

To determine whether the force F = r^2k(r^) is conservative and central, let's analyze its properties.

A force is conservative if it satisfies the condition ∇ × F = 0, where ∇ is the gradient operator. In Cartesian coordinates, the force can be written as F = Fx i + Fy j + Fz k, where Fx, Fy, and Fz are the components of the force in the x, y, and z directions, respectively. The curl of F is given by:

∇ × F = (∂Fz/∂y - ∂Fy/∂z)i + (∂Fx/∂z - ∂Fz/∂x)j + (∂Fy/∂x - ∂Fx/∂y)k.

Calculating the components of F = r^2k(r^):

Fx = 0, since there is no force component in the x-direction.

Fy = 0, since there is no force component in the y-direction.

Fz = r^2kr^.

Taking the partial derivatives, we have:

∂Fz/∂x = ∂/∂x (r^2kr^) = 2rkr^2(∂r/∂x) = 2rkr^2(x/r) = 2xkr^3.

∂Fz/∂y = ∂/∂y (r^2kr^) = 2rkr^2(∂r/∂y) = 2rkr^2(y/r) = 2ykr^3.

Substituting these values into the curl equation, we get:

∇ × F = (2ykr^3 - 2xkr^3)k = 2k(r^3y - r^3x).

Since the curl of F is not zero, ∇ × F ≠ 0, we conclude that the force F = r^2k(r^) is not conservative.

Now let's determine if the force is central. A force is central if it depends only on the distance from the origin (r) and acts along the radial direction (r^).

For F = r^2k(r^), the force is indeed central because it depends solely on r (the magnitude of the position vector) and acts along the radial direction r^. Hence, it can be written as F = Fr(r^), where Fr is a function of r.

Since the force is not conservative, it does not possess a potential energy function. In conservative forces, the potential energy function V(r) can be defined, and the force can be expressed as the negative gradient of the potential energy, i.e., F = -∇V. However, since F is not conservative, there is no potential energy function associated with it.

Learn more about force from this link:

https://brainly.com/question/12785175

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