Find the volume of the solid bounded by the cylinder x2 + y2 = 9 and the planes z = 0 and x + z = 3.
Answers
After evaluating the triple integral, we find that the volume of the solid is 27π/2 cubic units.
To find the volume of the solid bounded by the cylinder x² + y² = 9 and the planes z = 0 and x + z = 3, we'll use triple integrals.
1. Convert the given equations:
Cylinder: x² + y² = 9
Plane 1: z = 0
Plane 2: z = 3 - x
2. Set up the triple integral:
Volume = ∬∬∬ dV
3. Convert to cylindrical coordinates (r, θ, z):
x = r * cos(θ)
y = r * sin(θ)
z = z
x² + y² = r²
dV = r * dz * dr * dθ
4. Set up limits of integration:
r: 0 to 3 (from cylinder equation)
θ: 0 to 2π (full circle around z-axis)
z: 0 to 3 - r * cos(θ) (from plane 2 equation)
5. Integrate:
Volume = ∫(θ=0 to 2π) ∫(r=0 to 3) ∫(z=0 to 3-r*cos(θ)) r * dz * dr * dθ
The plane z = 0 is the xy-plane, and the plane x + z = 3 is a plane that intersects the cylinder in a circle. The circle is centered at (3,0,0) and has radius 2. We can find the volume of the solid bounded by these surfaces by integrating over the cylinder and subtracting the part that lies below the plane x + z = 3. We will use cylindrical coordinates to do this. In cylindrical coordinates, x² + y² = r², so the cylinder can be described by the inequality r² ≤ 9.
The planes z = 0 and x + z = 3 intersect at the line z = 3 - x. This line intersects the xy-plane at the point (3/2, 3/2), so the part of the circle that lies below the plane x + z = 3 is described by the inequalities r² ≤ 4 and (x-3/2)² + y² ≤ 4. The volume of the solid is given by the integralV = ∫∫∫ dx dy dz where the limits of integration are: r: 0 to 3θ: 0 to 2πz: 0 to min(3-x, 3-√(9-x²-y²))The first two limits integrate over the entire cylinder. The third limit integrates over the part of the plane x + z = 3 that lies above the cylinder.
The volume of the solid isV = ∫∫∫ dx dy dz= ∫₀²π ∫₀³ ∫₀^(3-x) dz r dr dθ + ∫₀²π ∫₃/2^(3/2)² ∫₀⁴- (x-3/2)²-y² dy dx dz= π/2 [3²(3) - 2²(2)] + π [3/2(2²) - 3/2(3/2)²(4-π/2)]- π/2(4-π/2)(3/2)²= 27π/2 - 27/4 - π/4(9/4 - 2) - π/8(4-π)≈ 55.54 Therefore, the volume of the solid is approximately 55.54.
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Related Questions
Calculate the monthly payment for a loan of $7,500 with an 11% interest rate compounded monthly over a period of 5 years.
Answers
Answer:
Option b - $163.07
Step-by-step explanation:
Given : A loan of $7,500 with an 11% interest rate compounded monthly over a period of 5 years.
To find : Monthly payment
Solution : Formula to find monthly payment
Monthly Payment = Amount / Discount factor
Discount factor
where ,
Amount = $7500
Rate= 11%=0.11
Time = 5 years
Now, put i and n in Discount factor ,
Monthly payment,
Therefore, Option b is correct monthly payment = $163.07
Answer:
The answer is $163.07
Step-by-step explanation:
To find the monthly payment:
\(i = \frac{0.11}{12} = 0.0091\)
Time = 5 years
5 x 12 = 60
Discount factor
\(D = \frac{1-(1 + i) -^{n}}i\)
\(D = \frac{1 - (1+ 0.009)-^{60} }{0.009} \\\)
\(D = \frac{0.421}{0.009} = 46.77\)
Monthly payment
\(\frac{7500}{46.77} = $163.07\)
Solve the system of linear equations using substitution. Use pencil and paper. Which expression would be easier to substitute into the other equation, in order to solve this problem? Explain your reasoning. X=6y-2 x=6y=9
Answers
Answer:
X = 6y-2 is the easier substitution for another equation because all you have to do is just plug in the equation into the value of x.
Explain how the sample you collect can affect your understanding of a population.
Use the word representative in your response.
Answers
The sample we collect plays a cruciThe sample we collect can significantly affect our understanding of a population,
especially when it comes to making generalizations or drawing conclusions about the entire population based on the sample data. The key aspect in this regard is whether the sample is representative of the population.
A representative sample is one that accurately reflects the characteristics, diversity, and distribution of the population from which it is drawn. When our sample is representative, we can have greater confidence in generalizing the findings from the sample to the larger population. However, if our sample is not representative, our understanding of the population may be biased or limited.
In the provided example, the given data represents different age groups and their reported time spent with friends per day. If we were to collect a sample from this population, it would be crucial to ensure that the sample includes individuals from each age group in proportion to their representation in the population. This would help ensure that our sample accurately reflects the age distribution of the population.
If our sample is not representative, it may lead to inaccurate conclusions. For instance, if we only surveyed individuals from the age group 55-64 and ignored the other age groups, our understanding of the population's time spent with friends would be limited to that specific age group. We wouldn't be able to generalize our findings to the entire population because we would have missed important variations in behavior across other age groups.
To improve the representativeness of our sample, we could use random sampling techniques, such as simple random sampling or stratified sampling, to ensure that individuals from all age groups are included in the sample. By doing so, we can enhance our understanding of the population as a whole and make more accurate inferences.
In conclusion, the sample we collect plays a crucial role in shaping our understanding of a population. A representative sample enables us to make valid generalizations and draw reliable conclusions about the entire population. It ensures that the characteristics and diversity of the population are properly reflected in the sample, allowing for more accurate insights and informed decision-making.al role in shaping our understanding of a population
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Stan drove 3300 \text{ m}3300 m3300, start text, space, m, end text to the grocery store. The tires on his truck are 2.2 \text{ m}2.2 m2, point, 2, start text, space, m, end text in circumference.
Answers
Answer:
58,928.6 rotations
Step-by-step explanation:
2.2 in x (2.54cm/in) x (1m/100cm)= .056 m/circumference
3300m/(.056m/rotation) = 58,928.6 rotations
the country of transylvania contains 2.3 million people (vampires not included) and covers 800,000 square miles. in the year after the last census, the crude birth rate was 48 out of 1000 and the crude death rate was 47 out of 1000. 1. what is the population growth rate (r)? 2. in how many years will the population of transylvania double?
Answers
In 700 years the population of Transylvania will double
What is growth rate ?
the typical annual rate of population change during a particular period for a certain nation, region, or geographic area. In most cases, a factor of 100 is used to describe the ratio between the annual growth in population size and the total population for the year.
Growth rate equals Absolute Change divided by Previous Value. Calculate the percentage of change: Percent of change = Growth rate x 100 is the formula that may be used to calculate the percent of change.
To calculate in how many years the population of Transylvania double
= 70 / 0.1
= 700 years
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Look at the graph. How many nails are less than 1 1/2 inches long?
Answers
Answer:
15
Step-by-step explanation:
Each x in the graph stands for 1 nail.
Look at each column of x's above the numbers less than 1 1/2.
There are 15 nails in the columns to the left of 1 1/2.
Answer: 15
What is the 3 phases of agile life cycle?.
Answers
The 3 phases of the agile life cycle:
The Continuous Delivery: Lean Life Cycle.
The Exploratory (Lean Startup) Life Cycle.
The Program Life Cycle for a Team of Teams.
What is the Agile cycle?
The systematic succession of phases that a product goes through as it develops from start to finish is known as the agile software development life cycle. Concept, conception, iteration, release, maintenance, and retirement are its six phases.
The systematic succession of phases that a product goes through as it develops from start to finish is known as the agile software development life cycle. Concept, conception, iteration, release, maintenance, and retirement are its six phases.
The project management approach a team decides on will have a small impact on the Agile life cycle. For instance, Scrum teams work in sprints, which are brief timeframes akin to iterations. They also have responsibilities that are precisely defined, like Scrum master. Kanban teams, on the other hand, operate more continuously and don't have set roles. Another illustration is Extreme Programming, where teams frequently work in shorter iterations and give engineering practices more attention.
Hence,
The 3 phases of the agile life cycle:
The Continuous Delivery: Lean Life Cycle.
The Exploratory (Lean Startup) Life Cycle.
The Program Life Cycle for a Team of Teams.
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Mai needs a new pair of shoes.
She finds a pair that she likes online that cost $36.
When she goes to buy them in the store she finds out that the store is having a 30% off sale.
By how much will the shoes be marked down? (give the dollar amount)
Answers
Answer:
11
Step-by-step explanation:
11
3.31×10 −5
g to micrograms
Answers
3.31×\(10^-^5\) g is equivalent to 0.0331 μg which is obtained by using the conversion factor.
To convert from grams to micrograms, we need to consider the conversion factor that relates the two units. The prefix "micro-" represents a factor of \(10^-^6\), which means there are 1,000,000 micrograms in a gram. Therefore, to convert grams to micrograms, we multiply the given value by 1,000.
In this case, we have 3.31×\(10^-^5\) g. To convert this value to micrograms, we can multiply it by 1,000:
= 3.31×\(10^-^5\) g × 1,000
= 3.31×\(10^-^5\) × 1,000
= 3.31×\(10^-^5\) × \(10^{3}\)
= 3.31×\(10^(^-^5^+^3^)\)
= 3.31×\(10^-^2\)
= 0.0331 μg
Therefore, 3.31×\(10^-^5\) g is equivalent to 0.0331 μg.
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is -11/-3 positive or negative
Answers
\( \huge \boxed{ \tt positive}\)
\( \\ \\ \)
\( \longrightarrow \sf \dfrac{ - 11}{ - 3} \)
\( \\ \\ \)
\( \longrightarrow \sf \dfrac{ \cancel- 11}{ \cancel - 3} \)
\( \\ \\ \)
\( \longrightarrow \sf \dfrac{ 11}{ 3} \)
PLS HELP ME ON THIS QUESTION I WILL MARK YOU AS BRAINLIEST IF YOU KNOW THE ANSWER!!
The number of rare coins a coin collector owns is considered ________ data.
A. discrete
B. continuous
C. neither continuous nor discrete
D. both continuous and discrete
Answers
Answer: D i hope this helps
Step-by-step explanation:
A campbell’s soup can is 6 inches tall and has a arduous of 2.5 inches how much paper is needed to make the label? How much room is there inside the can to hold the soup
Answers
Step-by-step explanation:
The label of the soup is the lateral area of the cylinder.
The formula for a cylinder is LA = ph.
The perimeter of a cylinder is the circumference of a circle, 2 times pi times r.
then multiply that answer by the height of 6.
The answer will be in square inches
Write an equation of the line that passes through (7.10) and is perpendicular to the line y =
Answers
Answer:
10 = -2(7) +24
Step-by-step explanation:
or
y = -2 (7) +24
first, we find the negative reciprocal of the slope of the first equation.
1/2 = -2/1 = -2
So the slope for the new equation will be -2
y = -2x+b
Now we know it will pass through the point (7,10)
So the x coordinate will become x, and the y coordinate will become y
10= -2(7) +b
-2 x 7 = -14 + b = 10
-14 -10 = -24
so
b = 24
because
-14 + 24 = 10
So our final equation(s) are
y = -2(7) +24
or
10 = -2 (7) + 24
What is the surface area of the cylinder with height 8 ft and radius 7 ft? Round your
answer to the nearest thousandth.
Helppp
Answers
Answer:
660
Step-by-step explanation:
Here is the equation to get this answer-
A=2πrh+2πr2=2·π·7·8+2·π·72≈659.73446
then you round the 659.73446 to get 660
Round $24.76 to the nearest $10.00.
A. $20.00
B. $24.00
C. $25.00
D. $30.00
Answers
Answer:
20.00
Step-by-step explanation:
a because 4 and under would round to 0
Answer:
A
Step-by-step explanation:
24.76 would round down to the nearest $10
Since it is not 25, 30 would not be the right answer so that leaves 20 as the correct answer.
Find the missing side.
Can someone help???
Answers
Answer:
x= 7.27Step-by-step explanation:
1) take a better picture next time
2) Add 70+90= 160 then 180-160= 20 to find the last angle
3) tan20= x/20
4) x= 7.27
Help please I forgot how to multiply numbers
Answers
Answer:
2*3=6
Step-by-step explanation:
2*3 = 2+2+2 = 3+3 = 6
help with these please:) trig SOH CAH TOA right angle triangles
Answers
Answer:
Do not forget to round and multiply.
Step-by-step explanation:
Put these numbers in order from least to greatest.
0.48,0.73, and 3/10
Answers
Answer:
\(\frac{3}{10} , 0.48 , 0.73\)
Step-by-step explanation:
In order to put this number in order from least to greatest, we have to turn them into fractions with the same denominator, so firstly we write down all of the decimals as fractions, and we will get...
\(0.48 = \frac{48}{100} \\\\0.73 = \frac{73}{100}\)
Now, we need to find the smallest number that is divisible by 100 and by 10, and this number will be the least common denominator. The smallest number that is divisible by 100 and by 10 is 100, therefore 100 is the least common denominator. Now we have to make all of the fractions have the denominator of 100 and we do that in the following way...
\(\frac{48}{100}= \frac{48}{100}\\\\\frac{73}{100} = \frac{73}{100}\\\\\frac{3}{10} = \frac{(3)(10)}{(10)(10)} = \frac{30}{100}\)
By looking at this we need to understand that...
\(\frac{30}{100}<\frac{48}{100}<\frac{73}{100}\)
Therefore...
\(\frac{3}{10} < 0.48 < 0.73\)
please help me with this maths question
Answers
The asked net for the cube of side length 2 cm has been shown.
What is surface area?The surface area is defined as a region of any shape is the area of the shape that is faced or the Surface area is the amount of area covering the exterior of a 3D shape.
Here,
The cube of side length 2 cm is given we have to draw the net for the given cube.
So first we calculate the surface area,
surface area of the cube = 6 [a]²
surface area = 6 (2)² = 24 cm²
Here, as the cube has 6 faces, unbind the cube on the coordinate plate.
Thus. the required net for the cube has been shown.
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Help me please answer thanks
Answers
Answer:
I think it would be each girl scored 2 points, and each boy only scored one. :)
Use the remainder theorem for x^3+8x^2+11x-20 by x+4. Use the remainder theorem for x^3+8x^2+11x-20 by x-5.
Answers
Answer:
the remainder for both expressions is zero
Step-by-step explanation:
(x+4)=0x=-4x³+8x²+11x-20, where x=-4(-4)³+8(-4)²+11(-4)-20-64+128-44-200, Hence it has no remainder (x+5)=0x=-5x³+8x²+11x-20, where x=-5(-5)³+8(-5)²+11(-5)-20-125+200-55-200, hence it has no remaindermention about angles of a triangle
Answers
How to convert 95 confidence z score?
Answers
The z-score associated with a 95% confidence level is approximately 1.96.
The z-score associated with a 95% confidence level is the value of the standard normal distribution that corresponds to an area of 0.95 to the left of the z-score.
Using a standard normal distribution table or calculator, we can find that the z-score associated with a 95% confidence level is approximately 1.96.
Therefore, if we want to calculate the z-score for a 95% confidence level, we can use the formula
z = invNorm(1 - α/2)
where α is the significance level (α = 0.05 for a 95% confidence level), and inv Norm is the inverse normal cumulative distribution function. Using this formula, we can calculate the z-score as:
z = inv Norm(1 - 0.05/2)
= inv Norm(0.975) ≈ 1.96
Therefore, the z-score is 1.96
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Find the area of the figure.BRAINLEST if right
Answers
Answer:
41 ft² is the answer thank you
Answer:
the answer is 53 ft
Step-by-step explanation:
fist take the 3*3 square = 9
then the 4*11= 44(11 because I ssume the 8 ft does not account for the 3ft of the box)(if it is 8 ft then the answer is 41 ft^2 overall)
44+9= 53
Knowledge extraction, pattern analysis, data archaeology, information harvesting, pattern searching, and data dredging are all alternative names for ________.
Answers
Knowledge extraction, pattern analysis, data archaeology, information harvesting, pattern searching, and data dredging are all alternative names for Data mining.
Data mining refers to the process of discovering patterns, extracting useful information, and uncovering hidden insights from large datasets. It involves applying various computational techniques, statistical algorithms, and machine learning methods to analyze vast amounts of data.
Knowledge extraction, pattern analysis, data archaeology, information harvesting, pattern searching, and data dredging are all alternative names for data mining because they all encompass the fundamental goal of extracting valuable knowledge and insights from data. These terms emphasize different aspects of the data mining process.
Knowledge extraction highlights the extraction of meaningful information, while pattern analysis focuses on identifying recurring patterns. Data archaeology suggests uncovering hidden information from historical or legacy data sources. Information harvesting and data dredging both emphasize the systematic exploration of data to uncover valuable insights.
Pattern searching highlights the process of searching for specific patterns or relationships within the data. These terms all revolve around the core concept of extracting meaningful knowledge from large datasets.
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Given the function f(t)= -16t^2+24t and f(1.5)= 0How high is the ball after how many seconds?
Answers
Since t represents the time and f(t) the height above the ground, we need to substitute t=1.5 i nto the given function f(t),
\(f(1.5)=-16(1.5^2)+24(1.5)\)which gives
\(f(1.5)=-36+36=0\)This means that the ball hits the ground after 1.5 seconds. Therefore, the answer is: The ball is 0 feet above the ground after 1.5 seconds
For the following figure, find x (write your answer as a decimal):
Answers
The value of x from given figure is 7.5
What is Triangle ?
A triangle is a two-dimensional geometric shape that has three straight sides and three angles. It is one of the most fundamental shapes in geometry and is used extensively in mathematics, science, and everyday life. Triangles can be classified based on their sides and angles.
Based on their sides, triangles can be classified as equilateral, isosceles, or scalene. In an equilateral triangle, all three sides are equal in length. In an isosceles triangle, two sides are equal in length, and the third side is different. In a scalene triangle, all three sides are different in length.
From the given figure ,
we can say that
10 : 4 = x : 3
30 = 4 x
x = 7.5
Therefore, The value of x from given figure is 7.5
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what is the answer to this? 4(w−6)≤−12
Answers
Step-by-step explanation:
\(4(w - 6) \leqslant - 12 \\ \)
\(4w - 24 \leqslant - 12\)
\(4w \leqslant - 12 + 24 \\ \)
\(4w \leqslant 12\)
Will mark as branilest and heart if help
Answers
Answer:
B
Step-by-step explanation:
as the easiest way to calculate without the help of a calculator ¯\_(ツ)_/¯