Given the solid E that lies between the cone z2 = x2 + y2 and the sphere x2 + y2 + (z +4)2 = 8. (10 pts) Set up the triple integrals that represents the volume of the solid E in the rectangular coordinate system. b) (10 pts) Set up the triple integrals that represents the volume of the solid E in the cylindrical coordinate system. c) [10 pts) Evaluate the volume of the solid E.
Answers
The volume of the solid E is \(d\theta=\int\limits0^(2\pi )\int\limits(-1/4)^{1/4)}(r-4+\sqrt(8-r^2))drd\theta.\)
We are given that;
\(x^2 + y^2 + (z +4)^2 = 8\)
Now,
In the rectangular coordinate system, we need to find the limits of integration for x, y and z. To do this, we need to solve the equations of the cone and the sphere for z and find their points of intersection. The equation of the cone is:
\(z^2 = x^2 + y^2\)
Taking the square root of both sides, we get:
\(z = \sqrt(x^2 + y^2)\)
The equation of the sphere is:
\(x^2 + y^2 + (z + 4)^2 = 8\)
Expanding and simplifying, we get:
\(z^2 + 8z + 16 = 8 - x^2 - y^2\\z^2 + 8z + x^2 + y^2 = 0\)
To find the points of intersection, we can substitute z = ±sqrt(x^2 + y^2) into this equation and get:
\(x^2 + y^2 + 8\sqrt(x^2 + y^2) + x^2 + y^2 = 0\\\)
\(V = \int\limits\int\limits\int\limits E dV \\= \int\limits(2\pi ) \int\limits(-1/4)^{1/4} \int\limits(-r)(-4+\sqrt(8-r^2)) dz dr \\ = \int\limits0^(2\pi) \int\limits(-1/4)^{1/4} (-4+\sqrt(8-r^2) - (-r)) dr\)
Therefore, by integral the answer will be \(d\theta=\int\limits0^(2\pi )\int\limits(-1/4)^{1/4)}(r-4+\sqrt(8-r^2))drd\theta.\)
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Related Questions
T/F: The median and the average of any list are always close together.
Answers
The given statement "The median and the average of any list are always close together" is False because the median and the average of a list are not always close together. In fact, the difference between the two can vary greatly depending on the shape of the distribution.
For example, in a skewed distribution with a long tail to one side, the mean can be pulled in the direction of the tail and can be quite far away from the median. On the other hand, in a symmetrical distribution such as a normal distribution, the mean and the median are equal and are located at the center of the distribution.
Therefore, it is important to consider both the mean and the median when analyzing a dataset, as they can provide different insights into the characteristics of the distribution.
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The equation shown models the height of a 32-inch candle after lighting it, where m represents the time the candle has been burning, in minutes, and h represents the candle's height. h=32−14m If the candle's height is now 25 inches, exactly how many minutes has it been burning?
Answers
Answer:
0.5 minutes
Step-by-step explanation:
Since we are provided the equation and the current height of the candle (which would be 25 inches) we can simply replace this current height with the variable h and then solve the rest of the equation to find out the value of m when the height of the candle is at 25 inches.
h = 32−14m
25 = 32−14m ... subtract both sides by 32
-7 = -14m ... divide both sides by -14
0.5 = m
Finally, we can see that after 0.5 minutes the candle would have melted down to 25 inches.
If x= (a+2) and y = (a-2), show that xy = a²-4.
Answers
\(xy=(a+2)(a-2)=a^2-2a+2a-4=a^2-4\)
PLEASE HURRY!!!!!!!!!
The data set represents the heights, in centimeters, of ten model bridges made for an engineering competition.
13 14 14 16 16 16 16 18 18 19
What is the mean?
Answers
Answer:
Mean: 16.
Step-by-step explanation:
—Mean—
13 14 14 16 16 16 16 18 18 19
Simply start by adding the numbers together.
13 + 14 + 14 + 16 + 16 + 16 + 16 + 18 + 18 + 19
= 160.
Then divide it by how many numbers there are. Which is 10.
160 divided by 10 = 16.
—MAD—
Subtract each number by the mean.
|13-16| + |14-16| + |14-16| + |16-16| + |16-16| +|16-16| + |16-16| + |18-16| + |18-16| + |19-16|
Negatives don’t matter, everything should be positive numbers.
3 + 2 + 2 + 0 + 0 + 0 + 0 + 2 + 2 + 3
= 16
16 divided by 16 (amount of numbers).
MAD = 1
–IQR—
Seperate both sides in half.
|13 14 14 16 16| |16 16 18 18 19|
Take the median of each group.
Group 1:
|13 14 (14) 16 16|
This one is 14.
Group 2:
|16 16 (18) 18 19|
This one is 18.
Now take (number from group 2) 18 and subtract by (number from group 1) 14.
18 - 14 = 4
IQR = 4.
the differential equation 3(3x ∧2+y ∧ 2)dx−2xydy=0 is homogeneous of what degree? 1 B) 2 (C) 3 (D) 4
Answers
The given differential equation, 3(3x^2 + y^2)dx - 2xydy = 0, is homogeneous of degree 2.
A homogeneous differential equation is one in which all the terms can be expressed as a function of the same degree. In this equation, the terms involving x and y are 3x^2 and y^2, respectively. Both of these terms have a degree of 2 since the exponent of x is 2 and the exponent of y is also 2.
To determine the degree of homogeneity, we examine the highest power of x and y in each term. In this case, both terms have the same highest power of 2, indicating that the equation is homogeneous of degree 2.
Homogeneous differential equations have special properties and can often be solved using substitution or transformation methods.
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Assume that a procedure yields a binomial distribution with a trial repeated n=5n=5 times. Use either the binomial probability formula (or a technology like Excel or StatDisk) to find the probability of k=1k=1 successes given the probability p=p=9/10 of success on a single trial.
(Report answer accurate to 4 decimal places.)
P(X=k)=
Answers
By using Binomial Distribution, it can be calculated that -
P(X = 1) = 0.0005
What is Binomial Distribution?
Binomial distribution is a discrete type probability distribution whose probability density function is given by
P(X = x) = \({n \choose x} p^xq^{n - x}\)
Where p is the probability of success and q is the probability of failure
Here, Binomial Distribution of probability is used
k = 1
p = 0.9
q = 1 - 0.9 = 0.1
n = 5
P(X = 1) = \({5 \choose 1} (0.9)^1(0.1)^{5 - 1}\)
= 0.00045
= 0.0005
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What is the relationship between the sine and cosine of complementary angles?
Explain your reasoning and use the relationship to find cos 50 if sin 40 ≅ 0.64 .
Answers
The relationship between the sine and cosine of complementary angles is that they are equal. If two angles are complementary, their sum is 90 degrees. The sine of an angle is equal to the cosine of its complementary angle, and vice versa.
In trigonometry, the sine (sin) of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. The cosine (cos) of an angle is defined as the ratio of the length of the adjacent side to the length of the hypotenuse in a right triangle.
For complementary angles, the sum of the two angles is 90 degrees, forming a right triangle. In this right triangle, the hypotenuse is the same for both angles. Since the opposite side of one angle is the adjacent side of its complement, and vice versa, the sine and cosine of complementary angles are equal.
Now, let's use this relationship to find cos 50 if sin 40 ≅ 0.64. Since sin 40 is given, which represents the ratio of the length of the opposite side to the hypotenuse in a right triangle, we can use the relationship between sine and cosine to find cos 50.
Since 40 degrees and 50 degrees are complementary angles, their sine and cosine are equal. Therefore, cos 50 is also approximately 0.64.
By applying the relationship between sine and cosine for complementary angles, we can determine the cosine value based on the given sine value, allowing us to find cos 50.
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find the solution of the system of equations
Answers
\(\begin{array}{cllll} -6x+5y&=&34\\ -6x-10y&=&4 \end{array} \\\\[-0.35em] ~\dotfill\)
\(\stackrel{ \textit{using elimination method} }{\begin{array}{cllll} \text{\LARGE -1}(-6x+5y&=&34)\\ -6x-10y&=&4 \end{array}}\implies \begin{array}{clclll} 6x ~~ -5y&=&-34\\ -6x-10y&=&4\\\cline{1-3} 0 ~~ -15y&=&-30 \end{array} \\\\\\ -15y=-30\implies y=\cfrac{-30}{-15}\implies y=2 \\\\[-0.35em] ~\dotfill\\\\ \stackrel{\textit{substituting on the 1st equation}}{-6x+5(2)=34\implies} -6x+10=34\implies x=\cfrac{24}{-6}\implies x=-4 \\\\[-0.35em] ~\dotfill\\\\ ~\hfill (-4~~,~~2)~\hfill\)
a baseball parkamount of soft drink dispensed is normally distributed with standard deviation 2.0 ounces. how many travel mugs should be included in a study if it is desiredthat the sample mean soft drink dispensed be within 0.5 ounces of the
Answers
If we randomly select 62 travel mugs and measure the amount of soft drink dispensed in each mug, the sample mean will be within 0.5 ounces of the population mean with 95% confidence.
To determine the number of travel mugs required for the study, we need to use the formula for sample size determination for means. The formula is:
n = (Z^2 * σ^2) / E^2
Where:
n = sample size
Z = Z-score for the desired confidence level (e.g., 1.96 for 95% confidence level)
σ = standard deviation of the population
E = desired margin of error (i.e., the maximum difference between the sample mean and the population mean)
In this case, we are given that the standard deviation of the soft drink dispensed is 2.0 ounces and we want the sample mean to be within 0.5 ounces of the population mean. Let's assume a 95% confidence level, which corresponds to a Z-score of 1.96. Plugging in the values, we get:
n = (1.96^2 * 2^2) / 0.5^2 = 61.6
We need to round up to the nearest whole number, so the final sample size should be 62 travel mugs. The larger the sample size, the smaller the margin of error will be, but it also means more time and resources required for the study.
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Complete question is:
a baseball parkamount of soft drink dispensed is normally distributed with standard deviation 2.0 ounces. how many travel mugs should be included in a study if it is desiredthat the sample mean soft drink dispensed be within 0.5 ounces of the population mean with 95% confidence.
Cho hệ các vector
U ={(1,2,0);(0,-1,0);( (2,3,1);(1,0,2)} .
a. Tìm số chiều và một cơ sở W của không gian con sinh bởi hệ vector U .
b. Tìm tham số k để u=(2,3,k^2+1) là một tổ hợp tuyến tính của W, và suy ra [ u]W.
Answers
Answer:
SACAN MÃ QR VÀ NHẬN CÂU TRẢ LỜI
VUI LÒNG ĐƯA TÔI Ở BRAINLIEST
Step-by-step explanation:
What is the solution for x?
5
16
5
Please help me
Answers
Answer:
do you have a equation to go with x?
Step-by-step explanation:
the diagonals of a quadrilateral bisect each other at right angles. What is the quadrilateral best described as?
(a) parallelogram
(b) kite
(c) rhombus
(d) trapezium
Answers
Answer:
trapezium fit the most
Please help!
Are these triangles similar? Yes or no. Why?
Answers
Answer:
yes
Step-by-step explanation:
Answer:
Yes these triangles are similar. All angles of a triangle have to equal 180 degrees, a strait line. When you have a right triangle, that already gives you 90 degrees. whatever comes next has to equal to 90. The angles of both triangles match up, DEF is just a scaled down version of ABC
Step-by-step explanation:
Which of the following lists of ordered pairs is a function? A. (2,5), (3, 6), (6, 9) B. (-1,2), (2, 3), (3, 1), (2,5) C. (1,2), (4,0), (3,5),(4,3) D. (2,5), (3, 6), (2, 1)
Answers
Answer:
A. (2,5), (3,6), (6,9)
Step-by-step explanation:
In order to be a function, all inputs (x values) must go to only one output (y value). List A is the only list that obeys this rule. Therefore, List A is a function.
What is the value of the expression 2x2 + 5x – 6 when x = 2?
Answers
Answer:
2×2^2 +5×2 -6
= 2×4 +10 -6
= 8+10-6
= 18-6
= 12
if the midpoint of ST is (5, -8),what is S if T is (1, 12)?
Answers
Answer:
(9, -28)
Step-by-step explanation:
Let S be (x,y)
\(( \frac{x + 1}{2} . \frac{y + 12}{2} ) = (5. - 8) \\ x = 5(2) - 1 = 9 \\ y = - 8(2) - 12 = - 28\)
Therefore, S is (9, -28)
which scatterplot does NOT suggest a linear relationship between x and y?
A) II only
B) III only
C) I and II only
D) II and III only
Answers
Answer:
B
Step-by-step explanation:
Tyrion, Cersei, and ten other people are sitting at a round table, with their seatingarrangement having been randomly assigned. What is the probability that Tyrion andCersei are sitting next to each other? Find this in two ways:(a) using a sample space of size 12!, where an outcome is fully detailed about the seating;(b) using a much smaller sample space, which focuses on Tyrion and Cersei
Answers
(a) In a seating arrangement with 12 people, there are 12! (factorial of 12) possible seating arrangements. The outcome is fully detailed about the seating. 2 people can be seated in 2! Ways. There are 10 people left to seat and there are 10! Ways to seat them. So, we get the following:(2! × 10!)/(12!) = 1/6. Therefore, the probability that Tyrion and Cersei are sitting next to each other is 1/6.
(b) In this smaller sample space, we will only focus on Tyrion and Cersei. There are only 2 possible ways they can sit next to each other:
1. Tyrion can sit to the left of Cersei
2. Tyrion can sit to the right of CerseiIn each case, the other 10 people can be seated in 10! Ways.
So, the probability that Tyrion and Cersei are sitting next to each other in this smaller sample space is:(2 × 10!)/(12!) = 1/6, which is the same probability we got using the larger sample space.
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Select the correct answer.
What is the equation of a line that passes through (7, 8) and has a slope of -3?
OA y=-3x+29
OB y=3x + 13
Ocy=((3)x - 299
D. 'y=(1/(3)x- 13
Answers
Answer:
A
Step-by-step explanation:
The equation of a line in slope- intercept form is
y = mx + c ( m is the slope and c the y- intercept )
Here m = - 3 , thus
y = - 3x + c ← is the partial equation
To find c substitute (7, 8) into the partial equation
8 = - 21 + c ⇒ c = 8 + 21 = 29
y = - 3x + 29 → A
Tyler and Gloria are rock climbing on separate walls across the room from each other. Each has scaled the wall 26.17 ft in the air. Their friend James is standing on the ground between the two walls. From above, the angle of depression from Tyler to James is 37°. The angle of depression from Gloria to James is 46°. The horizontal distance between Tyler and Gloria is 60 ft. Find the distance between Tyler and James.
Answers
Using the angle of depression, we found that the distance between Tyle and James is 43.47 ft.
What is meant by the angle of depression?
When the observer is higher than the thing being studied, an angle of depression results. An angle is created below the horizontal line drawn with the observer's eye level and the line connecting the object and the observer's eye when an observer looks at something that is closer to them than they are. The angle of elevation, on the other hand, is the angle created when a person stands on the ground and looks up at an item that is raised above the ground at a height greater than the person's. The main distinction between the two angles is this. The notion of trigonometry is used to determine both angles.
The figure is given below,
The height h = 26.17 ft
From the right triangle, we can write,
sin 37 = 26.17/z
where z = hypotenuse
0.602 = 26.17 / z
z = 26.17 / 0.602 = 43.47 ft
Therefore using the angle of depression, we found that the distance between Tyle and James is 43.47 ft.
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Aston left his house and ran 4 miles East and then 3 miles north. He then took the diagonal path back home. If he burned 105 calories every mile that he ran, how many total calories did burn on his run
Answers
Answer:
he do be running tho
Answer:
He burned a total of 735 calories.
Step-by-step explanation:
4 miles + 3 miles = 7 miles
105 calories/1 mile = 7 miles/ ? calories
105 calories times 7= 735 calories
Suppose a and n are relatively prime such that g.c.da, n=1, prove that \/ b 1 b) If n = 1, we cannot conclude that x=a (mod n) has solutions.
Answers
If a and n are relatively prime (gcd(a, n) = 1), it does not guarantee that the equation x ≡ a (mod n) has solutions.
If a and n are relatively prime, denoted by gcd(a, n) = 1, it means that a and n do not have any common factors other than 1. However, this does not guarantee that the equation x ≡ a (mod n) has solutions.
The equation x ≡ a (mod n) represents a congruence relation, where x is congruent to a modulo n. This equation implies that x and a have the same remainder when divided by n.
To have solutions for this congruence equation, it is necessary for a to be congruent to some number modulo n. In other words, a must lie in the residue classes modulo n. However, the fact that gcd(a, n) = 1 does not ensure that a is congruent to any residue modulo n, hence not guaranteeing the existence of solutions for the equation.
Therefore, when n = 1, we cannot conclude that the equation x ≡ a (mod n) has solutions.
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what percent of $90,000 is equal to $45,000
Answers
Answer:
50%
Step-by-step explanation:
50% = 0.5
$90000 times 0.5 = $45000
So, 50% of $90000 is equal to $45000
Answer: 50%
Hope this helps :)
Step-by-step explanation:
45,000 / 90,000 = .5 (50 out of 100)
45 * 2 = 90
I think of a number, double it and subtract 1 and then square the answer. The result is 64. What could the number be?
Answers
Let the number be x
(2x - 1)^2 = 64
Find the square root of both sides
2x - 1 = 8
2x = 8 + 1
2x = 9
x = 4.5
If you doubled an unknown number and subtract 1 and then square the answer to get 64, the unknown number is 4.5.
Let the unknown number be n.Given the following data:
Output = 64To find an unknown number:
In this exercise, you're required to solve for an unknown number while performing some arithmetic operations such as multiplication, subtraction and square.
Translating the word problem into an algebraic expression, we have;
Doubling the unknown number, we would multiply by 2:
\(2n\)
Subtracting 1 from the above and squaring, we have:
\((2n-1)^2 = 64\\\\2n-1=8\\\\2n =8+1\\\\2n=9\\\\n=\frac{9}{2}\)
n = 4.5
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Please help
Write the equation in standard form
Answers
The equation 7y = 4x - 7.
What is an equation?
An equation is a statement that asserts the equality of two mathematical expressions. It contains an equals sign (=) and typically has variables, constants, and mathematical operations on both sides.
Equations are used in many areas of mathematics, science, and engineering to represent relationships between variables and to solve problems. In algebra, equations are often used to solve for the values of variables that make the equation true.
We have;
y = 4/7x - 1
Multiplying through by 7
7y = 4x - 7
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The Pretty Pool company is building a pool in
Michael's back yard. The pool is to be 8 ft shorter than
it is wide and cover 240 square feet of the back yard.
What is the width of the pool?
Answers
Answer:
20 feetStep-by-step explanation:
The sides of rectangle are:
w and w - 8The area is the product of adjacent sides:
A = w(w - 8) = 240Solve the equation for w:
w² - 8w = 240w² - 8w + 16 = 256(w - 4)² = 16²w - 4 = 16w = 20Reflect the triangle over the y-axis and find the coordinate of the image of point K.
K’(2, 1)
K’(–2, 1)
K’(–2, –1)
K’(2, –1)
Answers
K'(-2, 1) is the coordinate of the image of point K, found by multiplying the original x-coordinate of the point K (2) by -1. To reflect the triangle over the y-axis, the x-coordinate of each point is multiplied by -1.
What is triangle?Triangle is a three-sided geometric shape which has three angles and three sides. The three sides of a triangle are typically referred to as the base, the height, and the hypotenuse. A triangle can be classified as either right, obtuse, or acute depending on the measure of its angles. Right triangles have one right angle, obtuse triangles have one obtuse angle, and acute triangles have three acute angles.
The coordinate of the image of point K is K’(-2, 1). This means that the x-coordinate of the point is -2 and the y-coordinate of the point is 1. This can be found by multiplying the original x-coordinate of the point K (2) by -1. The y-coordinate of point K remains the same, so the y-coordinate of the image of point K is 1.
Therefore, the coordinate of the image of point K is K’(-2, 1).
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if two standard, fair 6-sided dice are rolled, what is the probability that the product of the two numbers rolled is a perfect square? express your answer as a common fraction.
Answers
8/36 = 2/9 represents the likelihood that the sum of the two rolled numbers is a perfect square.
Describe fraction?A fraction is a piece of the entire. The quantity is represented as a quotient by dividing the numerator by the denominator. Both are integers in a straightforward fraction. A fraction appears in a complex fraction's numerator or denominator. The numerator of a valid fraction is smaller than the denominator. Six times six is equal to 36 possible outcomes when two dice are rolled. Only 8 of these 36 possible outcomes—the result of two diced integers producing a perfect square—are beneficial for the occurrence of the essential occurrences. The six diagonal elements (a, a) from the sample space are shown here, along with two additional elements (1, 4) and (4, 1). probabilities are 8/36, or 2/9.
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Simplify:
5(3x+2) help
Answers
Answer:
15x+10
Step-by-step explanation:
All you need to do is multiply 5 by 3x and 5 by 2
Step-by-step explanation:
\(5(3x + 2) \\ 15x + 10\)
if you want to find the value of x then
\(15x + 10 = 0 \\ 15x = - 10 \\ \\ x = - \frac{10}{15} \\ \\ x = - \frac{2}{3} \)
What is the value of 4.3X 10^5
Answers
Answer: 430000
Step-by-step explanation: you have to do pemdas
Answer: 430,000. /four hundred thirty thousand
Step-by-step explanation:
The value of 4.3 x 10^5 is 430,000.
To understand this, the notation 4.3 x 10^5 represents a number in scientific notation, where 4.3 is the coefficient or significand, and 10^5 is the exponent. The exponent of 10 tells us how many places to move the decimal point to the right. In this case, the exponent 5 means we need to move the decimal point five places to the right.
Starting with 4.3, we move the decimal point five places to the right to get 430,000. So, 4.3 x 10^5 is equivalent to 430,000.