Prove Euler's Rotation Theorem: When a sphere is moved about its center it is always possible to find a diameter of the sphere whose direction in the displaced position is the same as in the initial position.
Prove using something related to orthogonal properties

In a randomized block design, total variation in data is decomposed into three components: variation between blocks (SSB), variation between treatments within each block (SSE), and residual variation within each treatment and block combination (SST).

We can express the total sum of squares (Total SS) in a randomized block design as:

Total SS = SSB + SSE + SST

∑Yij² = SST + SSB + SSE and ∑Yij = bYi• + b∑y•j - bY

SSE = ∑(Yij - y•j - Yi• + Y)²

       = ∑Yij² - 2∑Yijy•j - 2∑YijYi• + 2∑YijY + b∑y•j² + b∑Yi•² - 2bY∑y•j - 2bY∑Yi• + bY²

       = SST + SSB + SSE - b∑y•j² - b∑Yi•² + 2bY∑y•j + 2bY∑Yi• - bY²

Rearranging and simplifying terms:

SSE = b∑y•j² + b∑Yi•² - 2bY∑y•j - 2bY∑Yi• + bY²

Multiplying both sides by k, the number of treatments:

kSSE = bk∑y•j² + bk∑Yi•² - 2bkY∑y•j - 2bkY∑Yi• + bkY²

Also, SST = ∑(Yij - Yi•)² and SSB = ∑(Yi• - Y)²

Therefore,

Total SS = SST + SSB + bk∑y•j² + bk∑Yi•² - 2bkY∑y•j - 2bkY∑Yi• + bkY²

Expanding the terms ∑Yi•² and ∑y•j² using the fact that ∑Yij² = SST + SSB + SSE:

Total SS = SST + SSB + bk(SST + SSB + SSE) - 2bkY∑y•j - 2bkY∑Yi• + bkY²

Simplifying the terms:

Total SS = bkSST + bkSSB + bkSSE - 2bkY∑y•j - 2bkY∑Yi• + bkY

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If A and B are independent events with P(A) = 0.4 and P(B) = 0.25, then P(A È B) =The answer is d. 0.55.

Since A and B are independent events, we can use the formula for the probability of the union of two independent events: P(A ∪ B) = P(A) + P(B) - P(A)P(B). Given P(A) = 0.4 and P(B) = 0.25, we can calculate P(A ∪ B) as follows:

P(A ∪ B) = 0.4 + 0.25 - (0.4)(0.25) = 0.4 + 0.25 - 0.10 = 0.55

Or we can calculate as follows:

To find P(A È B), we use the formula: P(A È B) = P(A) + P(B) - P(A and B)

Since A and B are independent, P(A and B) = P(A) x P(B) = 0.4 x 0.25 = 0.1

Therefore, P(A È B) = 0.4 + 0.25 - 0.1 = 0.55.

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The  sum of the vectors u and v is (-1,1).

To find the sum of the vectors u and v, we need to add their corresponding components.

The vector u has initial point (2,0) and terminal point (3,2), which means its components are (3-2, 2-0) = (1,2).

The vector v has initial point (2,2) and terminal point (0,1), which means its components are (0-2, 1-2) = (-2,-1).

To find the sum of these vectors, we simply add their corresponding components:

u + v = (1,2) + (-2,-1) = (1-2, 2-1) = (-1,1).

Therefore, the sum of the vectors u and v is (-1,1).

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The total weight of all the cans that are heavier than 9 ounces is 29 ounces.

We have,

Looking at the line plot, we see that there are 4 cans weighing exactly 9 ounces, and 3 cans weighing more than 9 ounces (2 at 9.5 ounces and 1 at 10 ounces).

To find the total weight of all the cans weighing more than 9 ounces, we can calculate:

Total weight = (number of cans weighing 9.5 ounces) x (9.5 ounces per can) + (number of cans weighing 10 ounces) x (10 ounces per can)

Total weight = (2 cans x 9.5 ounces per can) + (1 can x 10 ounces per can)

Total weight = 19 + 10

Total weight = 29 ounces

Therefore,

The total weight of all the cans that are heavier than 9 ounces is 29 ounces.

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Yes, that is correct. The Gaussian elimination rules are essentially the same as the three basic row operations, which allow you to algebraically manipulate a matrix's rows.

You can interchange two rows, multiply a row by a constant, or add a row to another row. These rules are essential in solving systems of linear equations and finding the reduced row echelon form of a matrix. By applying these rules, you can transform a matrix into an equivalent matrix that is easier to work with and reveals important information about the system of equations or the matrix itself. The Gaussian elimination rules, also known as the three basic row operations, allow you to algebraically manipulate a matrix in order to solve systems of linear equations. These operations include:
1. Interchanging two rows (R2 ↔ R3)
2. Multiplying a row by a nonzero number (R1 → kR1, where k is a constant)
3. Adding a row to another row (R2 → R2 + 3R1)
These rules help simplify the matrix and ultimately obtain the unique solution for the system of equations.

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The volume of the cone is 180π cubic feet (or approximately 565.49 cubic feet if you evaluate π as 3.14159).

To calculate the volume of a cone, you can use the formula V = (1/3)πr^2h, where r is the radius of the base of the cone and h is its height.

Since the diameter of the cone is 12 feet, the radius is half of that, which is 6 feet. And the height is given as 15 feet.

Plugging these values into the formula of volume, we get:

V = (1/3)π[tex](6)^2[/tex](15)

V = (1/3)π(36)(15)

V = (1/3)(540π)

V = 180π

Thus, the answer is 180π.

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The volume of the snowball is decreasing at a rate of approximately 5.4 cm³/min when the radius is 12 cm.

The formula for the volume of a sphere is V = (4/3)πr³, where V is the volume and r is the radius. To find the rate of change of the volume with respect to time, we need to take the derivative of this formula with respect to time. Using the chain rule, we get:

dV/dt = (4/3)π(3r²)(dr/dt)

where dV/dt is the rate of change of the volume with respect to time and dr/dt is the rate of change of the radius with respect to time.

Substituting the given values, we get:

dV/dt = (4/3)π(3(12)²)(-0.3)

= -241.9π cm³/min

Since the rate of change of volume cannot be negative, we take the absolute value of the result to get:

|dV/dt| = 241.9π cm³/min ≈ 759.8 cm³/min

Therefore, the volume of the snowball is decreasing at a rate of approximately 5.4 cm³/min when the radius is 12 cm.

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Answer:

There are three ways to roll a sum of 4: (1,3), (2,2), and (3,1). There are 36 possible outcomes when rolling two dice because each die has 6 possible outcomes, so we multiply the number of outcomes of each die: 6 x 6 = 36.

Therefore, the probability of rolling a sum of 4 with two standard dice is:

P(D1 + D2 = 4) = number of ways to roll a sum of 4 / total number of outcomes

P(D1 + D2 = 4) = 3 / 36

Simplifying the fraction, we get:

P(D1 + D2 = 4) = 1 / 12

Therefore, the probability of rolling a sum of 4 with two standard dice is 1/12 or approximately 0.083.

Step-by-step explanation:

in answer

The equation to determine the velocities of boxes is given by, v² = 2*a*d

To determine the velocities of the boxes when they move a distance d down the slope after being released from rest, we can use the following terms: velocity, box, and distance (d). Here's a step-by-step explanation:

1. Since the boxes are released from rest, their initial velocity (v0) is 0.
2. Let's assume the slope has an angle (θ) and the acceleration due to gravity (g) is 9.81 m/s².
3. Calculate the acceleration (a) of the boxes down the slope using the formula: a = g * sin(θ).
4. To find the final velocity (v) of the boxes after traveling a distance (d) down the slope, we can use the equation: v² = v0² + 2*a*d.

Since the boxes are released from rest, v0 is 0. Therefore, the equation simplifies to:

v² = 2*a*d

Now, substitute the acceleration (a) and distance (d) into the equation and solve for the final velocity (v) of the boxes.

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Once it reaches the end of the journey, the vessel is approximately 13.68 nautical miles away from the port.

To solve this problem, we can use the law of cosines to find the distance from the vessel to the port. Let's label the angles and sides as follows:

- Angle A is the initial heading of 32 degrees
- Angle B is the counterclockwise change in heading of 32 degrees
- Angle C is the angle between sides a and b (the distance from the vessel to the port)
- Side a is the distance traveled in the first leg, which is 5 nautical miles
- Side b is the distance traveled in the second leg, which is 10 nautical miles
- Side c is the distance from the vessel to the port, which we want to find

Using the law of cosines, we have:

c^2 = a^2 + b^2 - 2ab cos(C)

Plugging in the values we know, we get:

c^2 = 5^2 + 10^2 - 2(5)(10) cos(180-32)

Note that we use 180-32 for the angle C because it is the supplement of angle B.

Simplifying, we get:

c^2 = 125 - 100cos(148)

Using a calculator, we find that cos(148) is approximately -0.6235. Plugging this in, we get:

c^2 = 187.35

Taking the square root, we get:

c = 13.68 nautical miles

Therefore, the vessel is approximately 13.68 nautical miles away from the port once it reaches the end of the journey.

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No, The relation is not a function.

Domain = {9, 4, 1, 0}

Range = {-5, - 3, - 1, 0, 1, 3, 5}

We have to given that;

The relation is,

{(9,−5),(4,−3),(1,−1),(0,0),(1,1),(4,3),(9,5)}

We know that;

A relation between a set of inputs having one output each is called a function.

Here, Relation have;

(9, - 5) and (9, 5)

Thus, It does not satisfy the definition of function.

And, The value of domain of relation is,

Domain = {9, 4, 1, 0}

And, The value of range of relation is,

Range = {-5, - 3, - 1, 0, 1, 3, 5}

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When a French historian spends a semester touring museums and historic battlefields in the United States, it affects U.S. exports, imports, and net exports as follows:

- U.S. Exports: The French historian's spending on tourism services (such as accommodations, guided tours, and local transportation) is considered an export of services. As the historian spends money in the U.S., it will lead to an increase in U.S. exports.
- U.S. Imports: There is no direct impact on U.S. imports, as the historian's activities do not involve the U.S. purchasing goods or services from France or any other country.
- Net Exports: Since the French historian's spending increases U.S. exports without affecting imports, this will result in an increase in U.S. net exports (which is the difference between exports and imports).

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All the possible values of x are given as follows:

26 < x < 28.

In a triangle, the sum of the lengths of the two smaller sides has to be greater than the length of the greater side.

If 27 is the greater side, we have that:

x + 1 > 27

x > 26.

If x is the greater side, we have that:

x < 27 + 1

x < 28.

Hence the interval of possible values is given as follows:

26 < x < 28.

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Solving the Inequality will give us x > -1.25

We have the inequality:-

0. 2(x + 20) – 3 > –7 – 6. 2x

Here we are given a set of instructions and need to use them to get the final answer.

First, we need to use the distributive property. Using that, we will eliminate the brackets to get

0.2x + (0.2 X 20) - 3 > - 7 - 6.2x

or, 0.2x + 4 - 3 > - 7 - 6.2x

Now, we need to combine the like terms. Here it would be 4 and -3. Hence we get

0.2x + 1 > - 7 - 6.2x

Next, we need to use the addition property of inequality, which states that for terms a, b, c

if a > b

then, a + c > b + c

Hence here we will add 6.2x to get

0.2x + 1 + 6.2x > - 7 - 6.2x + 6.2x

or, 6.4x + 1 > - 7

Similarly, applying the subtraction property of inequality states

if a > b

then, a - c > b - c

Here we need to subtract 1 to get

6.4x + 1 - 1 > - 7 - 1

or, 6.4x > - 8

Using the division property will give us

x > - 8/6.4

or, x > 1.25

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The car's speed is 60 km/hour.

Let's denote the distance from the starting point to the destination by D, and let's denote the car's speed by S.

Using the formula speed = distance / time.

S = d  / t = (D - 200) / 1  ---- (1)

S = d / t = (D - 80) / 3  ----- (2)

We can simplify equation (2) by multiplying both sides by 3:

Expanding the right-hand side:

3S = D - 80

From equation 1 and 2:

3 (D - 200) = D - 80

3D - 600 = D - 80

3D - D = 600 - 80

2D = 520

D = 260

Therefore, the distance from the starting point to the destination is 260 km.

Using equation (1), we can find the car's speed:

S = 260 - 200 / 1

S = 60 m/s

Therefore, the car's speed is 60 km/hour.

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The complete question is:

A car is 200km from its destination after 1 hour and 80km from its destination after 3 hours. At what rate is the car traveling per hour?

The scale factor of dilation from the small figure to the large figure in the question are;

3. 1 : 4

4. 5 : 6

A scale factor is the ratio of the length of a side of an image (obtained from a preimage) to the length of the corresponding side of the preimage

The scale factor of the polygons obtained from diagrams are;

3. 4.5 yd to 18 yd = 1 to 4

The scale factor is 1 to 4

4. The ratio of the corresponding sides pairs of sides on the image and the preimage are;

Ratio on the large triangle; 42 : 18 = 7 : 3

Ratio on the small triangle; 35 : 15 = 7 : 3

The ratio  of the pair of corresponding sides are equivalent, therefore, the triangles are similar.

The scale factor of the small triangle to the large triangle, obtained from the ratio of the corresponding sides is therefore;

35 : 42 = 5 : 6

15 : 18 = 5 : 6

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The circle has a diameter of 10 and a radius of 5

the radius times itself ( 5 x 5 ) = 25

25 times pi (3.14) = 78.5

so the circle is 78.5, lets say square cm.

and the circumference is 31.41593 or 31.42  

we need to find the center point and the length of the radius. The center point is at 5. 1.  so h = 5  and  k = 1

Now let’s count from the center to a point on the circumference to find the length of the radius. 5

The radius is 5 so  r = 5

Now let’s plug everything into the standard form of a circle.

(x - h)²+ (y - k)² = r²

We have found the measures of CD, CE, HD, GD, HG, and HF in triangle BCD, given that H is the circumcenter of the triangle.

In triangle BCD, the circumcenter H is the point where the perpendicular bisectors of the sides of the triangle intersect. This point is equidistant from the three vertices of the triangle.

Using the properties of the circumcenter, we can find the measures of various sides and angles of the triangle:

CD = 2FD, where FD is the foot of the perpendicular from H to CD.

CE = BE = 26, since H is equidistant from B and C.

HD = HC = 33, since H is equidistant from D and C.

GD = 1/2BD = 1/2(58) = 29, since H is equidistant from B and D.

HG = √HD² - GD² = √33² - 29² = 2√62 ≈ 15.75, using the Pythagorean theorem.

HF = √HD² - FD² = √33² - 32² = √65 ≈ 8.06, using the Pythagorean theorem.

Therefore, we have found the measures of CD, CE, HD, GD, HG, and HF in triangle BCD, given that H is the circumcenter of the triangle.

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The area of the base of the pyramid is 81 square feet.

The formulation for the volume of a square pyramid is [tex]V = (1/3) * b^2 * h[/tex], in which" b" is the length of 1 aspect of the base and" h" is the height of the pyramid.

We are suitable to use this methodology to break for the length of one facet of the base, with a purpose to give us the area of the base.

[tex]114.75 = (1/3) * b^2 * 4.25[/tex]

Multiplying each sides by 3 gives

[tex]344.25 = b^2 * 4.25[/tex]

Dividing each angles by way of 4.25 gives

[tex]b^2 = 81[/tex]

Taking the square root of both angles gives

b = 9

Thus, the area of the base of the pyramid is

[tex]A = b^2 = 9^2 = 81[/tex]  square feet.

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The average rate at which the object falls during the first 3 seconds is given as follows:

-48.

The average rate of change of a function is given by the change in the output of the function divided by the change in the input of the function.

The function for this problem is defined as follows:

h(x) = 400 - 16x².

The numeric values are given as follows:

Thus the average rate of change is obtained as follows:

r = (256 - 400)/(3 - 0)

r = -48.

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For a regular quadrilateral, each exterior angle measures 90 degrees, so the sum of the exterior angles is 4 times 90 degrees, or 360 degrees.

In a regular quadrilateral, all angles are equal. To find the sum of the measures of the exterior angles, we can follow these steps:

1. Determine the sum of the interior angles of a quadrilateral, which is always 360 degrees.
The sum of the measures of the exterior angles of any polygon, including a regular quadrilateral, is always 360 degrees. This is because each exterior angle of a polygon is formed by extending one of the sides of the polygon, and the sum of the exterior angles is equal to the sum of the measures of the angles formed by all the sides of the polygon.

2. Since it's a regular quadrilateral, divide the sum by the number of sides (4) to find the measure of each interior angle. 360 / 4 = 90 degrees.

3. To find the measure of each exterior angle, subtract the measure of the interior angle from 180 degrees (since they are supplementary). 180 - 90 = 90 degrees.

4. Multiply the measure of one exterior angle by the number of sides (4) to find the sum of the measures of the exterior angles. 90 * 4 = 360 degrees.

The sum of the measures of the exterior angles of a regular quadrilateral is 360 degrees.

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Answer: I think its 2x + c

Hope it helped :D

Let me know if I helped

Sorry if wrong

The general solution of the O.D.E y²q2 + y2 + 22 +1 = y' involves solving for y in terms of x and a constant, represented by "c". To do this, we can use the technique of separation of variables.

First, we rearrange the equation to isolate the derivative term on one side:

y²q2 + y² + 22 + 1 = y'
y²q2 + y² + 1 = y' - 2
(y²q2 + y² + 1)dy = dx

Next, we integrate both sides with respect to their respective variables:

∫(y²q2 + y² + 1)dy = ∫dx
y³/3 + y + y = x + c
y³ + 3y = 3x + c

At this point, we can use the trigonometric substitution u = tan(x/3 + c) to simplify the expression. Then, we can solve for y in terms of u:

u = tan(x/3 + c)
y = √(u² - 1/3)

Finally, we substitute back in the expression for u and simplify to obtain the general solution:

y = √(tan²(x/3 + c) - 1/3)
y = tan(x/3 + c)

Therefore, the answer is (a) y = tan(x/3 + c).

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We have shown that f(x) = f(0) for all x ∈ (-1,1).

Since f is continuous at 0, we have:

lim x → 0 f(x) = f(0)

Since f(x) = f(x³) for all x ∈ (-1,1), we can substitute x = x³ and get:

f(x) = f(x³) = f(x⁹) = f(x²⁷) = ...

Since |x| < 1, we have x² < |x| < 1, and thus:

lim x² → 0 f(x²) = f(0)

Therefore, we can apply the limit of the sequence of nested intervals to obtain:

f(x) = f(x³) = f(x⁹) = f(x²⁷) = ... = lim n → ∞ f(x^(3ⁿ)) = lim y → 0 f(y) = f(0)

where we have made the substitution y = x^(3ⁿ), which implies that x = y^(1/(3ⁿ)) → 0 as n → ∞.

Thus, we have shown that f(x) = f(0) for all x ∈ (-1,1).

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The use of a stratified random sample over a simple random sample provides a statistical advantage in ensuring that the sample accurately represents the population. Stratified random sampling involves dividing the population into subgroups or strata based on certain characteristics, such as age or income.

This allows for a more representative sample as it ensures that each stratum is represented proportionally in the sample. In contrast, a simple random sample does not take into account any characteristics or strata of the population, which may result in an unrepresentative sample. Therefore, the use of a stratified random sample provides a statistical advantage by reducing the potential for sampling bias and increasing the accuracy of the study's results.
In the context of your study, a statistical advantage of using a stratified random sample over a simple random sample is that it ensures greater representation and accuracy in the results.

In a stratified random sample, the population is first divided into distinct subgroups or strata based on specific characteristics, such as age, gender, or income. Then, a simple random sample is taken from each stratum. This method helps to better represent each subgroup in the sample, which in turn improves the overall accuracy of the results.

On the other hand, a simple random sample involves selecting individuals from the entire population without considering any specific characteristics. This approach may not adequately represent certain subgroups, leading to potential biases and less accurate results. In summary, stratified random sampling provides a statistical advantage over simple random sampling by ensuring a better representation of subgroups, leading to more accurate and reliable results.

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Answer:

$1,593

Step-by-step explanation:

I=PRT

I=$2,276×14%×5

I=1,593.2 to nearest tenth

I=$1,593

Answer:

The answer to your problem is, 2.04 inches

Step-by-step explanation:

We can assume that the width of the new phone is ‘ x ‘ inches

We know that [tex]\frac{x}{0.84} = \frac{3.4}{1.4}[/tex]

x = [tex]\frac{3.4}{1.4}[/tex] × 0.84

x = 2.04

Thus the answer to your problem is, 2.04 inches

40% of responses are considered ideal, which means that the majority of responses fall outside of the ideal range of 0 to 5 minutes.

To calculate the percentage of responses that are considered ideal, we need to determine the proportion of responses that fall between 0 and 5 minutes. We can use the Normal distribution to solve this problem by calculating the z-score for 5 minutes and for 0 minutes, and then finding the area under the curve between those two z-scores.

The formula for calculating the z-score is (x - μ) / σ, where x is the observed value, μ is the mean, and σ is the standard deviation. For 5 minutes, the z-score is (5 - 6) / 1.2 = -0.83, and for 0 minutes, the z-score is (0 - 6) / 1.2 = -5.

We can use a standard Normal distribution table or a calculator to find the area under the curve between -5 and -0.83, which is approximately 0.3997. Multiplying this by 100 gives us 39.97%, which we round to 40%.

This suggests that ski patrol rescue responders may need to re-evaluate their response times and consider ways to improve their efficiency in order to increase the percentage of ideal responses.

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Answer:

B

Step-by-step explanation:

Sara read for a constant rate for 6 days and got a total of 7 1/2 hours.

Therefore, the situation can be represented by the equation 1 1/4×6=7 1/2

The probability that the standard normal random variable z is less than -0.87 is approximately 0.1922.

To find the probability P(Z < -0.87) for a standard normal random variable or z-score, follow these steps:

1. Identify the given value: In this case, z = -0.87.

2. Look up the corresponding probability in a standard normal (Z) table or use a calculator or software with a built-in function for finding probabilities of standard normal random variables.

Using a standard normal table or software, you'll find that P(Z < -0.87) ≈ 0.1922. So, the probability that the standard normal random variable Z is less than -0.87 is approximately 0.1922.

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