Final answer:
To verify the divergence theorem for the given vector field and region e, we need to calculate the flux through each face of the cube and sum them up. By calculating the flux through each face and summing them, we can verify that the flux of the vector field through the region e is 0.
Explanation:
The divergence theorem states that the flux of a vector field through a closed surface is equal to the volume integral of the divergence of the vector field over the volume enclosed by the surface.
In this case, the vector field is given by f(x, y, z) = 4xi + xyj + 4xzk. The region e is a cube bounded by the planes x = 0, x = 2, y = 0, y = 2, z = 0, and z = 2.
To verify the divergence theorem, we need to calculate the flux of the vector field through each face of the cube and sum them up.
Let's go step by step to calculate the flux through each face:
Flux through the x = 0 plane: The unit normal vector of this plane is -i. The flux through this plane is given by the surface integral of f dot dS, where dS is the area element of the plane. Since f(x, y, z) = 4xi + xyj + 4xzk, the dot product is -4x. The integral becomes: integral[0 to 2] integral[0 to 2] -4x dy dz = -16.
Flux through the x = 2 plane: Similar to the previous case, the unit normal vector of this plane is i. The flux through this plane is given by the surface integral of f dot dS, where dS is the area element of the plane. Since f(x, y, z) = 4xi + xyj + 4xzk, the dot product is 4x. The integral becomes: integral[0 to 2] integral[0 to 2] 4x dy dz = 16.
Continue with steps 3-6, calculating the flux through the rest of the faces and summing them up.
Flux through the y = 0 plane: The unit normal vector of this plane is -j. The flux through this plane is given by the surface integral of f dot dS, where dS is the area element of the plane. Since f(x, y, z) = 4xi + xyj + 4xzk, the dot product is -xy. The integral becomes: integral[0 to 2] integral[0 to 2] -xy dx dz = -8.
Flux through the y = 2 plane: Similar to the previous case, the unit normal vector of this plane is j. The flux through this plane is given by the surface integral of f dot dS, where dS is the area element of the plane. Since f(x, y, z) = 4xi + xyj + 4xzk, the dot product is xy. The integral becomes: integral[0 to 2] integral[0 to 2] xy dx dz = 8.
Flux through the z = 0 plane: The unit normal vector of this plane is -k. The flux through this plane is given by the surface integral of f dot dS, where dS is the area element of the plane. Since f(x, y, z) = 4xi + xyj + 4xzk, the dot product is -4xz. The integral becomes: integral[0 to 2] integral[0 to 2] -4xz dx dy = -16.
Flux through the z = 2 plane: Similar to the previous case, the unit normal vector of this plane is k. The flux through this plane is given by the surface integral of f dot dS, where dS is the area element of the plane. Since f(x, y, z) = 4xi + xyj + 4xzk, the dot product is 4xz. The integral becomes: integral[0 to 2] integral[0 to 2] 4xz dx dy = 16.
Finally, to verify the divergence theorem, we sum up the flux through each face:
-16 + 16 + (-8) + 8 + (-16) + 16 = 0
The flux of the vector field f through the region e is 0
in a city of 88,000 people, there are 33,000 people under 25 years of age. What percent of the population is under 25 years of age?
Can anyone help ME? PLEASE HELP IF I DONT TURN THIS IN BY TOMMOROW I DONT GET TO GO TO NEW YORK AND THAT IS MY DREAM
0.04 × 0.61 =
Tell me how to work this out
In the diagram, what is the measure of angle 1?
A. 135°
B. 125°
C. 15°
D. 45°
somebody please help so I can pass, please
Rewrite the following equation in the form y = a(x - h)2 + k. Then, determine the x-coordinate of the minimum.
y=2x^2 - 32x + 56
The rewritten equation is y = ____ (x - _____ )2 + ____ .
The x-coordinate of the minimum is _____
Answer:
y = 2 (x - 8 )2 + (-72)) .
The x-coordinate of the minimum is 8.
Step-by-step explanation:
I just took this test on plato and I got it correct.
whats 70/100 as a decimal
An analogy makes a comparison between objects based on their similar qualities. Cassidy wanted to create an analogy for the motion of atoms in solids, liquids, and gases, so she compared them to marbles in a tray. Which best compares the phases of matter to marbles in a tray? A solid is like the tray being shaken and the marbles moving around it, and a liquid is like the tray being shaken slowly and all the marbles moving in their positions. A solid is like the tray being shaken slowly and all the marbles moving in their positions, a liquid is like the tray being shaken and the marbles moving around it, and a gas is like the tray being shaken hard and the marbles moving vigorously around it. A gas is like the tray being shaken slowly and all the marbles moving in their positions, and a solid is like the tray being shaken hard and the marbles moving vigorously around it. A liquid is like the tray being shaken hard and the marbles moving vigorously around it, and a gas is like the tray being shaken slowly and all the marbles moving in their positions.
Answer:
B
Step-by-step explanation:
What is the difference between 2386 and 7000?
an aquarium weighs 22.5 lb when empty the aquarium is 30 in long 14 in wide and is filled with water to a depth of 18 in. Water weighs 0.036 pounds per cubic inch. How much does the aquarium weigh when it's full of water
The weight of the aquarium when it's full of water is 294.66 pounds.
To get the weight of the aquarium we first calculate the volume of the aquarium.
[tex]Volume=(length\times width\times height)[/tex]
[tex]V=30\times 14\times18[/tex]
[tex]V=7560 \ in^3[/tex]
But weight of water is such that for every cubic inches, the weight is [tex]0.036\ lb[/tex]
thus the weight of water will be:
[tex]0.036×7560=272.16 lb[/tex]
Thus the total weight of the aquarium is:
[tex]272.16+22.5[/tex]
[tex]=294.66 \l b[/tex]
The weight of the aquarium when it's full of water is 294.66 pounds.
A six-sided die in which each side is equally likely to appear is repeatedly rolled until the total of all rolls exceed 400
Approximately 0.2266, or 22.66%, is the probability that rolling the die more than 140 times is needed to exceed a total of 400.
To approximate the probability that rolling the die more than 140 times is needed to exceed a total of 400, we can use a normal approximation to the binomial distribution since the number of rolls is large.
First, let's calculate the mean (μ) and standard deviation (σ) of the number of rolls needed to exceed 400:
[tex]\[ \text{Mean (μ)} = \frac{\text{Total target}}{\text{Expected value per roll}} = \frac{400}{\frac{7}{2}} \][/tex]
[tex]\[ \text{Standard deviation (σ)} = \sqrt{\frac{\text{Total target} \times (\text{Sides}^2 - 1)}{12}} = \sqrt{\frac{400 \times (6^2 - 1)}{12}} \][/tex]
Now, we'll use the normal approximation and the z-score formula to find the probability:
[tex]\[ z = \frac{\text{X} - \text{μ}}{\text{σ}} \][/tex]
[tex]\[ z = \frac{140 - \text{μ}}{\text{σ}} \][/tex]
Then, we look up the z-score in a standard normal distribution table or use a calculator to find the probability associated with that z-score.
Let's calculate these values.
First, let's calculate the mean (μ) and standard deviation (σ):
[tex]\[ \text{Mean (μ)} = \frac{400}{\frac{7}{2}} = \frac{800}{7} \approx 114.29 \][/tex]
[tex]\[ \text{Standard deviation (σ)} = \sqrt{\frac{400 \times (6^2 - 1)}{12}} = \sqrt{\frac{400 \times 35}{12}} \approx \sqrt{\frac{14000}{12}} \approx \sqrt{1166.67} \approx 34.16 \][/tex]
Now, let's find the z-score for rolling the die more than 140 times:
[tex]\[ z = \frac{140 - \text{μ}}{\text{σ}} = \frac{140 - 114.29}{34.16} \approx \frac{25.71}{34.16} \approx 0.75 \][/tex]
Using a standard normal distribution table or calculator, we find the probability associated with a z-score of 0.75, which represents the probability that rolling the die more than 140 times is needed to exceed a total of 400.
The Correct Question is :
A six-sided die, in which each side is equally likely to appear, is repeatedly rolled until the total of all rolls exceeds 400. Approximate the probability that this will require more than 140 rolls.
The approximate probability that it will require more than 140 rolls for the total to exceed 400 is 0.005.
To find the approximate probability that it will require more than 140 rolls for the total to exceed 400, we can relate it to the probability that the sum of the first 140 rolls is less than 400.
Let X be the random variable representing the sum of the rolls. We want to find P(X > 400), which is the probability that it will require more than 140 rolls.
We can calculate this by finding the complement of the event that the sum of the first 140 rolls is less than 400.
Let A be the event that the sum of the first 140 rolls is less than 400. Then, P(A) is the probability that we're interested in.
Now, we calculate P(A):
Since each side of the die is equally likely, the expected value of the roll is [tex]\( \frac{1+6}{2} = 3.5 \)[/tex].
The expected value of the sum of the first 140 rolls is [tex]\( 140 \times 3.5 = 490 \)[/tex].
Therefore, P(A) can be approximated using the normal distribution, since the sum of the rolls follows approximately a normal distribution due to the Central Limit Theorem.
Using the properties of the normal distribution, we can standardize the value:
[tex]\[ Z = \frac{400 - 490}{\sqrt{140 \times \left(\frac{1}{12}\right)}} \][/tex]
Here, [tex]\( \frac{1}{12} \)[/tex] is the variance of a single roll of the die.
Now, we find P(A) using the standardized value of Z:
P(A) = P(X < 400) = P(Z > z)
We can then find the probability from a standard normal distribution table or calculator.
[tex]\[ P(A) \approx P(Z > -2.589) \][/tex]
From a standard normal distribution table, we find that [tex]\( P(Z > -2.589) \approx 0.995 \)[/tex].
So, the approximate probability that it will require more than 140 rolls for the total to exceed 400 is 1 - 0.995 = 0.005.
The probable question may be:
A six-sided die, in which each side is equally likely to appear, is repeatedly rolled until the total of all rolls exceeds 400. What is the approximate probability that this will require more than 140 rolls? (Hint: Relate this to the probability that the sum of the first 140 rolls is less than 400.)
Mr. Smith earns 6% commission on every house he sells. If he earns 9,000, what was the price of the house?
Consider a large population with a mean of 150 and standard deviation of 27. a random sample of size 25 is taken from the population. calculate the standard error of the sampling distribution of this sample mean and round your answer to the hundredths place.
Answer:
Standard error = 5.4
Step-by-step explanation:
We are given that Population Mean, [tex]\mu[/tex] = 150 and Population Standard deviation, [tex]\sigma[/tex] = 27 and sample size, n = 25
Standard error formula for the sampling distribution of this sample mean is given by;
Standard error = [tex]\frac{\sigma}{\sqrt{n} }[/tex] = [tex]\frac{27}{\sqrt{25} }[/tex] = 5.4
a monument stands at level ground. The angle of elevation to the top of the monument taken at a point 405 feet away is 32°. Find the height of the monument.
The height of the monument is approximately 253.2 feet.
To find the height of the monument, we can use trigonometry, specifically the tangent function, which relates the angle of elevation to the height and distance from the observation point.
Step-by-Step Solution:
Identify the given values:
Distance from the point: 405 feetAngle of elevation: 32°Use the tangent function:
tan(θ) = opposite / adjacent
Where θ is the angle of elevation, the opposite side represents the height of the monument (h), and the adjacent side represents the distance from the point (405 feet).
Set up the equation and solve for h:
tan(32°) = h / 405 feet
Using a calculator, find tan(32°):
tan(32°) ≈ 0.6249
So the equation is:
0.6249 = h / 405
Multiply both sides by 405 to solve for h:
h ≈ 0.6249 * 405
h ≈ 253.2 feet
Thus, the height of the monument is approximately 253.2 feet.
hey can you please help me posted picture of question
Suppose that a sample of size 44 is drawn from a population with mean 36 and standard deviation 47, find the standard deviation of the distribution of sample means
Answer:
Standard deviation of the distribution of sample means = 7.0855
Step-by-step explanation:
We are given that a sample of size 44 is drawn from a population with mean 36 and standard deviation 47.
Using Central Limit Theorem, it is stated that;
Standard deviation of the distribution of sample means = [tex]\frac{Population S.D.}{\sqrt{n} }[/tex]
= [tex]\frac{\sigma}{\sqrt{n} }[/tex] = [tex]\frac{47}{\sqrt{44} }[/tex] = 7.0855
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The diameter of a certain planet is approximately 3x10^7 meters (aka 30000000 meters). The length of a certain city is approximately 5x10^4 meters (aka 50000 meters).
How many times greater is the diameter of the planet compared to the length of the city?
The polygon circumscribes a circle find the perimeter of the polygon
Polygons that encircle a circle contain tangents of similar length, as such the perimeter of the polygon that circumscribes the circle is 76 cm.
The act of circumscribing a circle inside a polygon involves the process of drawing a circle that perfectly fits in the polygon by bisecting two arcs on each side of the polygon and then drawing a circle where all the sides meet in the middle of the polygon.
Polygons that encircle a circle contain tangents of similar length that originate at the same vertex, which explains the computation of the dimensions.
Taking a look at the figure attached, we have:
Two tangents with a length of 19 cmTwo tangents with a length of 9 cmTwo tangents with a length of 4 cmTwo tangents with with a length of 6 cmThus, the perimeter of the polygon can be calculated as:
= (19 + 19 + 6 + 6 + 4 + 4 + 9 + 9) cm
= 76 cm
Learn more about the perimeter of a polygon here:
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exponential function for 2,6,18,54
Without doing any computation, decide which has a higher probability, assuming each sample is from a population that is normally distributed with mu equals100 and sigma equals 15. explain your reasoning. (a) p(90less than or equals x overbarless than or equals110) for a random sample of size nequals 10 (b) p(90less than or equals x overbarless than or equals110) for a random sample of size nequals 20
Click an item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the correct position in the answer box. Release your mouse button when the item is place. If you change your mind, drag the item to the trashcan. Click the trashcan to clear all your answers.
Find the area of an equilateral triangle (regular 3-gon) with the given measurement.
6-inch apothem
A = sq. in
Answer: 12√3 square inches
Step-by-step explanation:
By the property of equilateral triangle,
Apothem = √3/2 × Side
⇒ Side = 2/√3 × Apothem
Here, apothem = 6 inches
Thus, the side of the given equilateral triangle = [tex]\frac{2}{\sqrt{3}}\times 6[/tex]
= [tex] \frac{12}{\sqrt{3}}[/tex]
= [tex]4\sqrt{3}[/tex] unit
Since, For an equilateral triangle,
[tex]\text{ Area} = \frac{\sqrt{3}}{4}\times (\text{ side})^2[/tex]
⇒ The area of the given equilateral triangle = [tex] \frac{\sqrt{3}}{4}\times (4\sqrt{3})^2[/tex]
[tex]=\frac{\sqrt{3}}{4}\times 48[/tex]
[tex]=12\sqrt{3}[/tex] square inches
at a shopping mall,4/9 of the shoppers were children and 5/7 of the adults were woman.They were 1/2 as many boys as men.They were 90 more girls than boys. (a) what fraction of the shoppers in the shopping mall were men ? (b) How many shoppers were in the shopping mall ?
Identify all of the root(s) of g(x) = (x2 + 3x - 4)(x2 - 4x + 29)
Answer:
x=-4,1,2+5i,2-5i
Step-by-step explanation:
Given is an algebraic expression g(x) as product of two functions.
Hence solutions will be the combined solutions of two quadratic products
[tex]g(x) = (x^2 + 3x - 4)(x^2 - 4x + 29)\\[/tex]
I expression can be factorised as
[tex](x+4)(x-1)[/tex]
Hence one set of solutions are
x=-4,1
Next quadratic we cannot factorize
and hence use formulae
[tex]x=\frac{4+/-\sqrt{16-116} }{2} =2+5i, 2-5i[/tex]
Answer:
The answer above is correct:
B. 1
C. 4
E. 2+5i
F. 2-5i
Step-by-step explanation:
I got this right on Edg. 2021
3 times as much as the sum of 3/4 and 2/6
The result of 3 times as much as the sum of 3/4 and 2/6 is; 13/4
Fraction and ArithmeticsFirst, we must evaluate the sum of 3/4 and 2/6; we have;
3/4 + 2/6Using the lowest common multiple; 12
We have; (9 +4)/12 = 13/12.
Therefore, 3 times 13/12 = 39/12 = 13/4
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1. WHAT DOES IT MEAN TO SOLVE A RIGHT TRIANGLE?
2. HOW CAN YOU SOLVE RIGHT TRIANGLES USING SIMILARITY? THE PYTHAGOREAN THEOREM?
3. WHAT IS THE RELATIONSHIP BETWEEN THE SIDES AND ANGLES MEASURES OF RIGHT TRIANGLES THAT
DESCRIBE TRIGONOMETRIC FUNCTIONS?
4. WHAT ARE THE INVERSE TRIGONOMETRIC FUNCTIONS? HOW ARE THE INVERSE FUNCTIONS USED WHEN
SOLVING RIGHT TRIANGLES?
5. WHAT IS THE RELATIONSHIP BETWEEN THE ANGLE OF ELEVATION AND THE ANGLE OF DEPRESSION?
GIVE AN EXAMPLE OF AN APPLICATION USING THE ANGLE OF ELEVATION OR THE ANGLE OF DEPRESSION.
A hotel has $504 to buy new pillows. If the cost of each pillow is $6, how many pillows will the hotel be able to buy? A) 78 B) 84 C) 88 D) 96
What is the median of the data set: 50, 54,62,48,49,52
Please help me with this
On a bulletin board, the principal, Ms.Gomez, put 115 photos of the fourth grade students in her school. She put the photo in 5 esqueleto rows. How many photos did she put in each row?
23 photos per row. Got it? Good.
what is 30% of 250 =
Compute with percents