The region bounded by \(y=r^2-x^2\), y = 0, and x = 0 is revolved about the y-axis to form a paraboloid. A hole, centered along the axis of revolution, is drilled through this solid. The hole has a radius k, 0 < k < r.
(a) Find the volume of the resulting ring by integrating with respect to x.
(b) Find the volume of the resulting ring by integrating with respect to y.
Answers
The Volume of the resulting ring by integrating with respect to y is π(r² - k²)².
a) Find the volume of the resulting ring by integrating with respect to x.
The given region can be depicted in the following diagram: Paraboloid The area of the hole is the difference of two circles; one with radius k and the other with radius r.
Therefore, the area of the hole is π(r² - k²).
To find the volume of the solid obtained by revolving this region about the y-axis, the formula for the volume of the paraboloid obtained by rotating the curve y = f(x) about the y-axis is used:
V = π ∫[f(x)]² dx Using the given function y = r² - x²,
the volume of the solid can be found by evaluating the integral:V = π ∫[r² - x²]² dx= π ∫(r⁴ - 2r²x² + x⁴) dx= π [(r⁴)x - (2r²/3)x³ + (x⁵/5)]|₀ᵣ= π [(r⁵/5) - (2r²/3) r³ + (r⁵/5)/5]Using k as the radius of the hole, the volume of the resulting ring can be found by subtracting the volume of the hole from the volume of the paraboloid. V_ring = V_paraboloid - V_hole= π [(r⁵/5) - (2r²/3) r³ + (r⁵/5)/5] - π(r² - k²) (r > k)
Therefore, the volume of the resulting ring by integrating with respect to x is π [(r⁵/5) - (2r²/3) r³ + (r⁵/5)/5] - π(r² - k²) (r > k).b) Find the volume of the resulting ring by integrating with respect to y.
The given region can also be expressed as x = ±√(r² - y), y = 0, and y = r².Using the shell method, the volume of the resulting ring can be found by adding the volumes of the cylindrical shells that have height dy and radius x = √(r² - y).
The volume of each shell is 2πxhdy, where h is the thickness of the shell, which is the distance between the circles of radius k and r along the y-axis.Therefore, h = r² - k².
The volume of the resulting ring can be found by integrating the expression for the volume of each shell over the range y = 0 to y = r².V_ring = ∫ 2πxh dy= ∫ 2π√(r² - y)(r² - k²) dy= π(r² - k²)²
Using k as the radius of the hole, the volume of the resulting ring is π(r² - k²)².
Therefore, the volume of the resulting ring by integrating with respect to y is π(r² - k²)².
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Related Questions
Item 5
Write the word sentence as an inequality.
One plus a number y is no more than −13.
Answers
Answer:
1+xy-13 that is you answer
8 less than a number
Answers
Answer:
13 is the answer to 8 less than a number
Can someone solve fast it’s 381 to the power of 5 divided by 381
Answers
Answer:
1.6056648e+12
Step-by-step explanation:
Answer:
21071715921
Step-by-step explanation:
This is what I got when i entered it into a calculator. Hopefully i'm correct and this helps :) Have a good day/night
1.1 1.2 1.3 1.4 1.5 1.6 prudence wants to paint the front of the house. She has two identical windows as well as a circular vent near the roof. Gable 1,2 m 0.6 m 1,8 m 0.5m 500 cm Vent Window 2,6 m Use Formulae, to answer the questions that follow. Area = L x B Area = 2 x r²(π = 3,142) Area = ²× b × h Convert 500 cm to m Calculate the area of one window. Calculate the area of the vent. Calculate the area of the rectangular part of the wall. Calculate the area of the gable. Prudence states that the area of the Wall that must pe painted is 14,42 m². Is this statement valid? Set your work out clearly. (PLEASE NOTE THAT THE VENT AND THE WINDOWS WILL NOT BE PAINTED)
Answers
The Area of the gable is 0.9 m²
Prudence's statement is not valid.
1. Convert 500 cm to meters:
Since 1 meter is equal to 100 centimeters,
So, 500 cm ÷ 100 = 5 meters
2. The formula for the area of a rectangle is Area = Length x Width.
Given the dimensions:
Length = 1.2 m
Width = 1.4 m
Area of one window = 1.2 m x 1.4 m = 1.68 m²
3. Calculate the area of the vent:
The formula for the area of a circle is Area = π x r².
Radius = (2.6 m) / 2 = 1.3 m
Area of the vent = 3.142 x (1.3 m)² ≈ 5.309 m²
4. Length = 1.1 m
Width = 1.6 m
Area of the rectangular part of the wall = 1.1 m x 1.6 m
= 1.76 m²
5. Calculate the area of the gable:
Given the dimensions:
Base = 0.6 m
Height = 1.8 m
Area of the gable = 0.5 m x 1.8 m = 0.9 m²
Prudence states that the area of the wall that must be painted is 14.42 m².
So, Area of the rectangular part of the wall + Area of the gable
= 1.76 m² + 0.9 m² = 2.66 m²
Therefore, Prudence's statement is not valid.
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thirty randomly selected students took the calculus final. if the sample mean and standard deviation were 92 and 8.8 respectively, construct a 99% confidence interval for the mean score of all students.
Answers
The 99% confidence interval for the mean score of all students is approximately 87.867 to 96.133.
To construct a 99% confidence interval for the mean score of all students, we can use the formula:
Confidence Interval = sample mean ± (critical value) * (standard deviation / √n)
where:
- Sample mean = 92
- Standard deviation = 8.8
- n = sample size = 30
First, we need to find the critical value corresponding to a 99% confidence level. Since we have a large sample size (n > 30), we can use the Z-distribution. The critical value for a 99% confidence level is approximately 2.576.
Now, we can calculate the confidence interval:
Confidence Interval = 92 ± (2.576) * (8.8 / √30)
Calculating the values:
Confidence Interval = 92 ± (2.576) * (8.8 / √30)
= 92 ± (2.576) * (8.8 / 5.477)
Simplifying:
Confidence Interval = 92 ± (2.576) * (1.603)
= 92 ± 4.133
The lower bound of the confidence interval is given by 92 - 4.133 = 87.867, and the upper bound is given by 92 + 4.133 = 96.133.
Therefore, the 99% confidence interval for the mean score of all students is approximately 87.867 to 96.133.
This means that we can be 99% confident that the true population mean score falls within this interval.
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For a dinner party, Myra is creating individual servings of appetizers. She has 15 carrot sticks
and 18 celery sticks. If she wants each serving to be identical, with no food left over, what is
the greatest number of servings Myra can create?
Answers
Answer:
We’ll, 18 and 15 have only 1 number in common and that’s 3. So if you go back a little (divide by 3) you get to 6 and 5. So she only gets 3 servings
Step-by-step explanation:
Given that a+b=6
And (a)square+(b)square =72
Find the value of 3b-3a
Answers
Answer:
216
Step-by-step explanation:
72= 36 +36
36(3)+36(3)=216
I'm honestly not sure ,tell me if it's wrong :/
A box contains ten cards labeled Q, R,S ,T ,U ,V ,W , X, Y, and Z . One card will be randomly chosen.
What is the probability of choosing a letter from Q toT ?
Write your answer as a fraction in simplest form.
Please help my day is going so bad ive litterally cried 5 times already and im so streesed with online school and having so much homework its just so difficult please help IM SO STRESSED :(
Answers
Answer:
10 cards. q to t is 4 of them- so 4 out of 10. 4/10. simplified 2/5
Can someone help me pleaseeee
Answers
Answer:
B
Step-by-step explanation:
Lucy rolls 2 fair dice and adds the results from each.
Work out the probability of getting a total of 10.
Answers
Answer:
Step-by-step explanation:
Discussion
The numbers on each die are 1 to 6.
They are all equally likely to turn up. There are 36 possibilities in all 6*6
So getting a 10 can be done in 3 ways.
6 + 4 = 10
4 + 6 = 10
5 + 5 = 10
You can't get 10 any other way.
So of the 36 ways of throwing 2 dice, only 3 will work.
Answer
P(10)= 3/36 = 1/12 = 0.0833333...
The work that Ryan did to find the greatest common factor of 48 and 72 is shown below.
Prime factorization of 48: 2 x 2 x 2 x 2 x 3
Prime factorization of 72: 2 x 2 x 2 x 3 x 3
The greatest common factor is 2 ´ 2 ´ 2 ´ 3 x 3
What is Ryan’s error?
Answers
Answer:
there will only be one 3
Step-by-step explanation:
see cause the first 3 of both numbers are 2 but only the last one of both numbers are 3 .
Can someone help on this? Thank youu;)
Answers
Answer:
2^(3/7)
Step-by-step explanation:
For these types of questions, what is inside the root is the numerator, and what is on top is the denominator.
3 is in the root, so it is the numerator
7 is outside the root, so it is the denominator
Therefore, The answer is 2^(3/7)
A house on the market was valued at $235,000. After several years, the value decreased by 9%. By how much did the house's value decrease in dollars? What is the current value of the house?
Decrease in value:
Current house value:
Answers
Decrease in value = $21150
Current house value = $213850
What is Depreciation?
Depreciation is accounting method used to allocate the cost of a tangible or physical asset over its useful life.
To calculate the decrease in value,
i.e 9% of $235000 = $21150
So, to calculate the current house value
i.e $235000 - $21500
= $213850
Therefore, the current house price is $213850 and it decreases in $21500
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6у = 8 - 9 + 6y
solve for y
Answers
Answer:
No solution exists
Step-by-step explanation:
Use a triple integral to find the volume of the given solid The tetrahedronenclosed by the coordinate planes and the plane 10x y z 4
Answers
The volume of the tetrahedron is 1.0667 cubic units.
The tetrahedron is enclosed by the coordinate planes and the plane 10x + y + z = 4.
To find the volume of the tetrahedron using a triple integral, we can use the following bounds:
0 ≤ z ≤ 4 - 10x - y
0 ≤ y ≤ 4 - 10x
0 ≤ x ≤ 0.4
This is because the tetrahedron is bound by the coordinate planes, so x, y, and z all must be positive, and the plane 10x + y + z = 4 sets the upper bound for z and y, and the upper bound for x is simply where the plane intersects the x-axis (which can be found by setting y and z equal to 0 and solving for x).
The triple integral to find the volume of the tetrahedron is then:
V = ∫∫∫ dV = ∫0^0.4 ∫0^(4-10x) ∫0^(4-10x-y) dz dy dx
Evaluating this integral, we get:
V = ∫0^0.4 ∫0^(4-10x) (4-10x-y) dy dx
= ∫0^0.4 [4y - 10xy - (1/2)y^2]0^(4-10x) dx
= ∫0^0.4 [4(4-10x) - 10x(4-10x) - (1/2)(4-10x)^2] dx
= ∫0^0.4 (-25x^2 + 20x + 8) dx
= [-25/3 x^3 + 10 x^2 + 8x]0^0.4
= 1.0667
Therefore, the volume of the tetrahedron is approximately 1.0667 cubic units.
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HeLp meee NeOW If DONT U WILL SUFfEr SEVERE consequences...
The lower a county's economic development, the
A) higher the literacy rate.
B) higher the cost of living.
C) higher costs of education.
D) lower the infant mortality rate.
Answers
Answer:
B
Step-by-step explanation:B
Answer:
the answer to this question is B
Q 42 - The proportion of salary of An and B is 5:3 and that of their use is 9:5. On the off chance that they spare Rs. 2600 and Rs. 1800, then their livelihoods are: A-9000, 5400 B-10000, 6000 C-6000, 3600
Answers
Answer: Let's solve this problem step by step.
First, let's assume the salaries of An and B to be 5x and 3x, respectively, where x is a common multiplier.
According to the given information, they save Rs. 2600 and Rs. 1800, respectively. Since savings come from the remaining portion of their incomes after spending, we can calculate their expenditures as follows:
For An:
Income of An = Salary of An + Savings of An
Income of An = 5x + 2600
For B:
Income of B = Salary of B + Savings of B
Income of B = 3x + 1800
Now, let's consider the proportion of their expenditures. It is given that the proportion of their expenditures is 9:5. So, we can write the following equation:
(Expenditure of An)/(Expenditure of B) = 9/5
Since expenditure is the complement of savings, we have:
[(Income of An - Savings of An)] / [(Income of B - Savings of B)] = 9/5
Substituting the previously derived expressions for income, we get:
[(5x + 2600 - 2600)] / [(3x + 1800 - 1800)] = 9/5
Simplifying the equation, we have:
5x / 3x = 9/5
Cross-multiplying, we get:
5 * 3x = 9 * 3x
15x = 27x
Subtracting 27x from both sides, we have:
0 = 12x
This implies that x = 0, which is not a valid solution. Therefore, there seems to be an error or inconsistency in the given information or equations. Please recheck the problem statement or provide additional information to help resolve the issue.
Austin is planning a canoe trip with his dad and brother. They are taking a cooler and a few supplies that weigh a total of 89 pounds with them. The canoe has a maximum weight limit of 765 pounds. Which inequality can be used to find , the average allowable weight in pounds for each person in the canoe?
A. 3p - 89 > 765
B. 3p - 89 765
C. 3p + 89 > 765
D. 3p + 89 < 765
Answers
Answer:
3p + 89 < 765
Correct answer: D.
Step-by-step explanation:
Austin, his dad, and brother make a group of three people of varying weights. Assume their average weight is p and their total weight 3p.
The cooler and other supplies weigh 89 pounds in total.
The weight of the supplies and the three people in the canoe should not be equal or exceed the maximum allowable weight of the canoe of 765 pounds.
This leads to inequality:
3p + 89 < 765
Correct answer: D.
the proportion of americans that are women is assumed to be known to be 51% (or 0.51). a marketing survey telephones 100 people at random. would a sample proportion of 70% (or 0.70) be considered unusual? make sure you select the correct justification in the answer choices. the sample proportion is not unusual because the statistic falls more than three standard errors from the mean. the sample proportion is unusual because the statistic falls within three standard errors from the mean. the sample proportion is not unusual because the statistic falls within three standard errors from the mean. the sample proportion is unusual because the statistic falls more than three standard errors from the mean.
Answers
The sample proportion of 70% (or 0.70) would be considered unusual because the statistic falls more than three standard errors from the mean.
To determine whether a sample proportion is considered unusual, we can compare it to the expected population proportion. In this case, the assumed proportion of women in the population is 51% (or 0.51).
To assess the unusualness of the sample proportion of 70% (or 0.70), we can use the concept of standard errors. The standard error measures the variability or uncertainty of a sample proportion. In general, when a sample proportion is within three standard errors of the population proportion, it is considered not unusual.
However, if the sample proportion falls more than three standard errors away from the population proportion, it suggests a significant difference between the sample and population. This could indicate that the observed proportion is unlikely to occur by random chance alone.
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Balena has a garden in the shape of a circle of radius 10 m.
He is going to cover the garden with grass seed to make a lawn.
Grass seed is sold in boxes.
Each box of grass seed will cover 46 m' of garden.
Balena wants to cover all the garden with grass seed.
Work out an estimate for the number of boxes of grass seed Balena needs.
Show your working out.
Answers
Answer:
7
Step-by-step explanation:
→ Work out area of circle
π × 10² = 100π = 314.159 (3 decimal places)
→ Divide area of circle by box area
314.159 ÷ 46 = 6.8295
→ Round answer as you can't get a decimal of a box
7
The area of a circle is given as πr².
The number of boxes of grass seeds required to cover the garden is 9 boxes.
What is a circle?A circle is a round-shaped figure that has no corners or edges.
The area of a circle is given as:
Area = πr²
We have,
The radius of the circle = 10 m
Area of the garden:
= πr²
We can take,
π = 22/7 = 3.14
= 3.14 x 10 x 10
= 314 m²
The area of a box of grass seeds:
= 46 m²
The number of boxes of grass seeds to cover the garden area:
= 314 / 46
= 6.8 boxes
Thus,
The number of boxes of grass seeds required to cover the garden is 9 boxes.
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this variable in an experiment is the one being deliberately changed by the scientist____
Answers
The variable which the scientist willfully modifies is referred to as the independent variable. Every variable observed throughout the experiment seems to be the dependent variable.
What does a math variable mean?A symbol (often a letter) used in mathematics to represent an unknowable quantitative score in such an equation or algebraic statement. A variable can be defined as a quantity that is changeable and not fixed.
Which examples of variables?Examples of variables include age, gender, company revenue and costs, the country of birth, expenditures, class grades, eye color, and vehicle kind. Because the amount may differ between data units within a population and may change over time, it is referred to as a variable.
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PLEASE HELP ME!!
find the Trinomial!
Answers
Answer:
\(a^{4}-2a^{2} -24\)
Step-by-step explanation:
We do FOIL to solve this
The average of a sample of high daily temperature in a desert is 114 degrees F, a sample standard deviation or 5 degrees F, and 26 days were sampled. What is the 90% confidence interval for the average temperature? Please state your answer in a complete sentence, using language relevant to this question.
Answers
In a sample of 26 days from the desert, the average of the high daily temperature is 114 degrees F, and the sample standard deviation is 5 degrees F.
To determine the 90% confidence interval for the average temperature, we can use the t-distribution as follows:To find the critical value of t, we can use the t-table. Since our sample has n = 26, the degrees of freedom (df) are n - 1 = 25. At a confidence level of 90%, with 25 degrees of freedom, the critical t-value is 1.708.The standard error of the mean is calculated as s / sqrt(n), where s is the sample standard deviation and n is the sample size. Therefore, the 90% confidence interval for the average temperature is (114 - 1.67, 114 + 1.67), or (112.33, 115.67) degrees F.
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Please help
Solve for x
A 110
B 105
C 25
D 78
Answers
.. still taking it all back :]
Which equation is true for the triangle?
Answers
Answer:
D
Step-by-step explanation:
the three main trig functions are
sin = opposite / hypotenuse
cos = adjacent / hypotenuse
tan = opposite / adjacent
( opposite = side opposite of angle , hypotenuse = longest side and adjacent = other side )
Sin(55) = a/c as side a is opposite of the angle and c is the hypotenuse so D might be the answer
cos(55) would be b/c as the adjacent of that angle is b and the hypotenuse is c. B and C show the wrong ratios so B and C can't be the answer
tan(55) would be a/b as side a is opposite of the angle and b is the adjacent, A says c/a which is wrong
so we can conclude the answer is D
The PTO is selling raffle tickets to raise money for classroom supplies. A raffle ticket costs $4. There is 1 winning ticket out of the 110 tickets sold. The winner gets a prize worth $82. Round your answers to the nearest cent.
What is the expected value (to you) of one raffle ticket? $
Calculate the expected value (to you) if you purchase 10 raffle tickets. $
What is the expected value (to the PTO) of one raffle ticket? $
If the PTO sells all 110 raffle tickets, how much money can they expect to raise for the classroom supplies? $
Answers
The z-score for P(? ≤ z ≤ ?) = 0.60 is approximately 0.25.
The z-score for P(z ≥ ?) = 0.30 is approximately -0.52.
How to find the Z score
P(Z ≤ z) = 0.60
We can use a standard normal distribution table or a calculator to find that the z-score corresponding to a cumulative probability of 0.60 is approximately 0.25.
Therefore, the z-score for P(? ≤ z ≤ ?) = 0.60 is approximately 0.25.
For the second question:
We want to find the z-score such that the area under the standard normal distribution curve to the right of z is 0.30. In other words:
P(Z ≥ z) = 0.30
Using a standard normal distribution table or calculator, we can find that the z-score corresponding to a cumulative probability of 0.30 is approximately -0.52 (since we want the area to the right of z, we take the negative of the z-score).
Therefore, the z-score for P(z ≥ ?) = 0.30 is approximately -0.52.
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Kadeem and Quinn both drive 25 miles. Kadeem drives at a constant speed of 50 miles and hour. Quinn drives at a constant of 75 miles an hour. Who takes longer to drive 25 miles? How much longer?
Answers
Kadeem takes 0.5 hours to travel 25 miles, and Quinn 0.33 hours, so Kadeem takes longer.
Who takes longer to drive 25 miles?
Remember the relation:
Speed*time = distance.
Notice that if the distance is fixed, then for a longer speed the time needed will be smaller.
Here we know that:
Kadeem drives at a constant speed of 50mi/h
Quinn drives at a constant speed of 75 mi/h.
Replacing that in the equation:
Kadeem's case:
(50 mi/h)*t = 25mi
t = (25 mi)/(50 mi/h) = 0.5 hours.
Quinn's case:
t = (25mi)/(75 mi/h) = 0.33 hours.
So Kadeem takes longer.
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In the table below, 1:2, 2:4, 3:6, 4:8, and so on are all equivalent ratios.
A multiplication table. In the row labeled 1, the numbers 1, 2, 3, 4, 5, 6, 7, 8 and 9 are highlighted. In the row labeled 2, the numbers 2, 4, 6, 8, 10, 12, 14, 16, and 18 are highlighted.
By what factor is the first term in each ratio multiplied to get the second term?
1
2
3
4
Answers
2
Step by step:
1•2=2
2•2=4
3•3=6
And so on...
Hope this helps!!
Answer:
2!!!!!!
Step-by-step explanation:
How many 2/3s in 3? Show your working out.
Answers
Using the division operation, the number of 2/3's in 3 is 4
Using Division3 ÷ 2/3
change the sign to multiplication and take inverse of 2/3
3 × 3/2 = 9/2 = 4.5
We need only the whole number value .
Therefore, the number of 2/3's in 3 is 4.
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When do we use the t-distribution instead of the normal distribution to construct confidence intervals for a population mean, u? a. when Population Standard Deviation 0 is known b. when Population Standard Deviation 0 is unknown c. when Sample Standard Deviation is known d. when Sample Standard Deviation is unknown
Answers
We use the t-distribution instead of the normal distribution to construct confidence intervals for a population means, u, when Population Standard Deviation 0 is known.
When we use the T distribution instead of the normal distribution to construct a confidence interval for the population mean, if you have the population standard deviation sigma, you are using the standard normal distribution.
The t-distribution, also known as the Student's t-distribution, is a probability distribution similar to the normal bell-shaped distribution, but with heavier tails. It is used to estimate population parameters with small samples or unknown variance. The T distribution is more prone to outliers than the normal distribution and therefore has fatter tails. The t-distribution is the basis for calculating t-tests in statistics.
The normal distribution is used when the population distribution is assumed to be normal. The t-distribution is similar to the normal distribution, except with thicker tails. Both assume that the population is normally distributed. Therefore, the T distribution has higher kurtosis than the normal distribution. The t-distribution is more likely to give values far from the mean than the normal distribution.
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