Evaluate triple integral_W dx dy dz/(x^2 + y^2 + z^2)^3/2, where W is the solid bounded by the two spheres x^2 + y^2 + z^2 = a^2 and x^2 + y^2 + z^2 = b^2, where 0 < b < a.
Answers
The Triple Intergral of the given expression is \(4 \pi \times ln (\frac{a}{b})\)
\(\int\limits \int\limits \int\limits_w \frac{dx dy dz}{(x^2 + y^2 + z^2)^{\frac{3}{2}}}\)
\(\text{Where } x^2 + y^2 + z^2 = a^2 \text{ and } x^2 + y^2 + z^2 = b^2, 0 < b < a\)
Changing to Spherical Coordinates
\(b \leq \rho \leq a, 0 \leq \theta \leq 2 \pi, 0 \leq \phi \leq \pi\)
\(\therefore x^2 + y^2 + z^2 = \rho^2; dx dy dz = \rho^2 sin \phi .d\phi .d\theta. d\rho\)
\(\implies \int\limits^a_b \int\limits^{2\pi}_{0} \int\limits^{\pi}_{0} \frac{1}{\rho^3} \rho^2 sin \phi .d\phi .d\theta. d\rho\)
\(=\int\limits^a_b {\frac{1}{\rho}} \, d\rho \int\limits^0_{2\pi} d\theta \int\limits^{\pi}_0 \sin{\phi} \, d\phi\)
\(= [ln \rho]^{a}_{b} [\theta]^{2\pi}_{0} [-\cos \phi]^{\pi}_{0}\)
\(= [\ln a - \ln b] \times [2\pi] \times [- \cos \pi - (-\cos 0)]\)
\(= 2\pi \times 2 \times (\ln a - \ln b)\)
\(= 4 \pi \, \ln{\frac{a}{b}}\)
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Related Questions
I need help!! Its so hard for me i tryed so many times and still no answers
Answers
7:1 2:14 3:21 4:28 5:35 6:42 7:49
Hope you do your best
The length of a rectangle is 8 inches longer than 3 times its width. The area of the rectangle is 156 square inches.
What is the width of the rectangle?
Answers
The Answer:
Width = 6
Step-by-step explanation:
Let width be w
(3w + 8) x w = 156
3 \(w^{2}\) x 8n - 156 = 0
(w + 8 x 6) (w-6) = 0
w = 6
Refer to the given terms . A pattern defines the relationship between the first two terms . Determine the relationship and identify the missing term of the second pair , such that both the pairs are analogous . AZP : ZAR :: TXK :
Answers
The missing term in the second pair is YUL.
To determine the relationship between the first two terms, let's analyze the given pattern: AZP : ZAR.
If we look closely, we can observe that each letter in the first term is shifted one position forward in the alphabet to form the corresponding letter in the second term.
So, applying the same pattern to the second pair (TXK), we need to shift each letter one position forward in the alphabet:
T + 1 = U
X + 1 = Y
K + 1 = L
Therefore, the missing term in the second pair is YUL.
The analogy between the pairs is that each letter in the first term is shifted one position forward in the alphabet to form the corresponding letter in the second term.
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Mrs. Cook is a teacher whose salary is $23,125 for a 185-day school year. In Mrs. Cook's school district, substitute teachers are paid at a rate of $90 per day. If a substitute is paid to teach Mrs. Cook's class in her absence one day, how much less does the school district pay in salary by paying a substitute teacher instead of paying Mrs. Cook for that day? *
Answers
Answer:
35
Step-by-step explanation:
23,125/185= 125
125-90 = 35
$35 less amount school district pay in salary by paying a substitute teacher instead of paying Mrs. Cook for that day
What is unitary Method?The unitary technique involves first determining the value of a single unit, followed by the value of the necessary number of units.
For example, Let's say Ram spends 36 Rs. for a dozen (12) bananas.
12 bananas will set you back 36 Rs. 1 banana costs 36 x 12 = 3 Rupees.
As a result, one banana costs three rupees. Let's say we need to calculate the price of 15 bananas.
This may be done as follows: 15 bananas cost 3 rupees each; 15 units cost 45 rupees.
Given:
Mrs. Cook is a teacher whose salary is $23,125 for a 185-day school year.
Mrs. Cook per day salary= 23125/ 185 = $125
Now, substitute teachers are paid at a rate of $90 per day.
So, if substitute teachers take class instead of Mrs. Cook, then school will pay
= 125-90
=$35 less amount.
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The population of a city was 10,000 in 2010. The population increase at an annual rate of 2.5% per year. Is the growth model function that represents the population of the city linear?
Answers
Answer:
The growth model that represents the population of this city is not linear--it is exponential:
\(f(t) = 10000( {1.025}^{t} )\)
\(t = 0 \: represents \: 2010\)
you want to create a 99 confidence interval with a margin of error of 25 assuming that the population standard deviation is equal to 1.5 what's the minimum size of the Rams Tampa
Answers
Answer:
The minimum sample size is 239.
Step-by-step explanation:
We have that to find our \(\alpha\) level, that is the subtraction of 1 by the confidence interval divided by 2. So:
\(\alpha = \frac{1 - 0.99}{2} = 0.005\)
Now, we have to find z in the Z-table as such z has a p-value of \(1 - \alpha\).
That is z with a pvalue of \(1 - 0.005 = 0.995\), so Z = 2.575.
Now, find the margin of error M as such
\(M = z\frac{\sigma}{\sqrt{n}}\)
In which \(\sigma\) is the standard deviation of the population and n is the size of the sample.
Population standard deviation is equal to 1.5
This means that \(\sigma = 1.5\)
Margin of error of 0.25
This means that \(M = 0.25\)
What's the minimum size of the sample?
\(M = z\frac{\sigma}{\sqrt{n}}\)
\(0.25 = 2.575\frac{1.5}{\sqrt{n}}\)
\(0.25\sqrt{n} = 2.575*1.5\)
\(\sqrt{n} = \frac{2.575*1.5}{0.25}\)
\((\sqrt{n})^2 = (\frac{2.575*1.5}{0.25})^2\)
\(n = 238.7\)
Rounding up:
The minimum sample size is 239.
Find a and b using the factor theorem.
\(f(x)=x^3+ax^2+bx-12\) has factor \((x-1), (x+1)\)
Answers
The values of a and b using the factor theorem for the polynomial f(x), we set f(1) and f(-1) equal to zero. Solving the resulting system of equations, we find that a = 12 and b = -1.
To find the values of a and b using the factor theorem, we need to use the given factors (x - 1) and (x + 1) and the fact that they are roots of the polynomial f(x).
The factor theorem states that if (x - c) is a factor of a polynomial, then f(c) = 0. Therefore, we can set x = 1 and x = -1 in the polynomial f(x) to get two equations.
First, let's substitute x = 1 into f(x):
f(1) = (1)^3 + a(1)^2 + b(1) - 12
f(1) = 1 + a + b - 12
Next, let's substitute x = -1 into f(x):
f(-1) = (-1)^3 + a(-1)^2 + b(-1) - 12
f(-1) = -1 + a - b - 12
Since (x - 1) and (x + 1) are factors, f(1) and f(-1) must equal zero. Therefore, we can set the two equations equal to zero and solve for a and b:
1 + a + b - 12 = 0
-1 + a - b - 12 = 0
Rearraning the equations, we have:
a + b = 11
a - b = 13
Now, we can solve this system of equations. Adding the two equations, we get:
2a = 24
a = 12
Substituting the value of a into one of the equations, we find:
12 - b = 13
b = -1
Therefore, the values of a and b are 12 and -1 respectively.
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Add −12 + 7.
options:
19
−5
−19
5
Answers
Answer:
\( - 12 + 7 = - 5\)
I hope I helped you^_^
HELPPPP simplify: 7! = ?
Answers
The solution of number 7! is,
⇒ 5,040
We have to given that,
To simplify 7! and find its value.
Now, by definition of factorial or a number, we get;
⇒ 7!
⇒ 7 × 6 × 5 × 4 × 3 × 2 × 1
Multiply it,
⇒ 5,040
Therefore, The solution of number 7! is,
⇒ 5,040
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Milo bought nine bottles of bug spray
for $44.91. Sheri bought four bottles for
$19.96. What is the constant of
proportionality?
Answers
Answer:209 by 67 ansered
(a) n= (Round up to the nearest integer.)A researcher wishes to estimate the proportion of adults who have high-speed Internet access. What size sample should be obtained if she wishes the estimate to be within 0.03 with 99% confidence if (a) she uses a previous estimate of 0.52? (b) she does not use any prior estimates? (A)n=
Answers
The required sample sizes are given as follows:
a) Previous estimate of 0.52: n = 1839.
b) No previous estimate: n = 1842.
What is a confidence interval of proportions?A confidence interval of proportions has the bounds given by the rule presented as follows:
\(\pi \pm z\sqrt{\frac{\pi(1-\pi)}{n}}\)
In which the variables used to calculated these bounds are listed as follows:
\(\pi\) is the sample proportion, which is also the estimate of the parameter.z is the critical value.n is the sample size.The confidence level is of 99%, hence the critical value z is the value of Z that has a p-value of 0.995, so the critical value is z = 2.575.
The margin of error is given as follows:
\(M = z\sqrt{\frac{\pi(1-\pi)}{n}}\)
For an estimate of 0.52, the parameters are given as follows:
\(M = 0.03, \pi = 0.52\)
Hence the sample size is obtained as follows:
\(0.03 = 2.575\sqrt{\frac{0.52(0.48)}{n}}\)
\(0.03\sqrt{n} = 2.575\sqrt{0.52 \times 0.48}\)
\(\sqrt{n} = \frac{2.575\sqrt{0.52 \times 0.48}}{0.03}\)
\((\sqrt{n})^2 = \left(\frac{2.575\sqrt{0.52 \times 0.48}}{0.03}\right)^2\)
n = 1839.
Without a previous estimate, the sample size is given as follows:
\(0.03 = 2.575\sqrt{\frac{0.5(0.4)}{n}}\)
\(0.03\sqrt{n} = 2.575\sqrt{0.5 \times 0.5}\)
\(\sqrt{n} = \frac{2.575\sqrt{0.5 \times 0.5}}{0.03}\)
\((\sqrt{n})^2 = \left(\frac{2.575\sqrt{0.52 \times 0.5}}{0.03}\right)^2\)
n = 1842.
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Which of the following is Playfair's axiom?
A. A straight line segment can be drawn between any two points.
B. A circle can be drawn with any center and radius.
C. Through a given point not on a given line, there is exactly one line
parallel to the given line.
D. All right angles are equal to one another.
Answers
Answer:
C. Through a given point not on a given line, there is exactly one line parallel to the given line.
Step-by-step explanation:
hope i helped
Answer:
Look down
Step-by-step explanation:
Thanks for reaching out for assistance! Based on the context you provided, it seems like the correct answer is C. Playfair's axiom states that there is exactly one line parallel to a given line that can be drawn through a point not on it. I hope this helps! Let me know if there's anything else I can do for you.
1. Yejin plans to retire at age 60. She wants to create an annuity fund, which will pay her a monthly allowance of $4000 during her retirement. She wants to save enough money so that the payments lasts for 30 years. A financial advisor has told her the she can expect to earn 5% interest on her funds, compounded annually.
2. Yejin has just turned 28 years old. She currently has no retirement savings. She wants to save part of her salary each month into her annuity fund. Calculate the amount Yejin needs to save each month, to meet her retirement goal.
Answers
9514 1404 393
Answer:
$3891.10
Step-by-step explanation:
This question is a bit unusual in that the interest is compounded annually, but payments and withdrawals are made monthly. The effective monthly rate is ...
1.05^(1/12) -1 = 0.407412378% = i
We assume that payments to the annuity are made at the end of the month, and that withdrawals are made at the beginning of the month. (The last payment and the first withdrawal are made on the same day.)
The amount of money required in the fund is ...
A = $4000(1 -(1.00407^-360))/(1 -1.00407^-1) = $757,712.88
The amount of money needed each month to be put into the fund is P, where ...
$757,712.88 = P(1 -1.00407)^(-12(60-28))/(1 -1.00407^-1) = 194.7297P
P = $757,712.88/194.7297 = $3891.10
Yejin needs to save $3891.10 each month to meet her retirement goal.
_____
Sanity check
Yejin wants payments for 30 years from the fund to which she has contributed for 32 years. The similarity of the time periods means that Yejin's monthly contribution will need to be very similar to the amount she plans to withdraw.
The only ways to reduce the required contribution are to earn a higher interest on deposits, or to adjust the relative time periods (retire later).
Solve the inequation x - 12 ≤ 3 - 2x and graph its solution.
Let's do these one-by-one.
The solution to x - 12 ≤ 3 - 2x is:
Answers
Answer:
Step-by-step explanation:
x - 12 ≤ 3 - 2x
3x ≤ 15
x ≤ 5
To graph this, just draw a VERTICAL line passing through x = 5, then shade the area to the LEFT
simplify(4-x) (7+x)
Answers
(4-x) ( 7+ ×)
28 +4x -7x -x²
28 -3x - x²
An archer hits a bullseye 64% of the time. What is the probability the archer hitsthe bullseye exactly 4 times during 10 total attempts?a. 242b. .365C..077d. 168
Answers
If the probability of hitting the bullseye is 64% (P(H) = 0.64), then the probability of not hitting the bullseye (P(H_bar)) is:
\(P(\bar{H})=1-P(H)=1-0.64=0.36\)Now, if we have 10 attempts, and the archer hits 4 times, then he misses 6 times.
So we have 4 cases of hitting (P(H)) and 6 cases of not hitting(P(H_bar)), and the probability is the product of the probabilities of each case:
\(P=P(H)^4\cdot P(\bar{H})^6=(0.64)^4\cdot(0.36)^6=0.16777\cdot0,0021767=0\text{.}0003652\)We also need to multiply this probability by a combination of 10 choose 4, because the 4 hits among the 10 attempts can be any of the 10, in any order:
\(C(10,4)=\frac{10!}{4!(10-4)!}=\frac{10!}{4!6!}=\frac{10\cdot9\cdot8\cdot7}{4\cdot3\cdot2}=210\)So the final probability is:
\(P^{\prime}=P\cdot C(10,4)=0.0003652\cdot210=0.077\)So the answer is C.
Brianna made 9 1/4 bags of popcorn for a movie night with some friends. Together they ate 4 bags of it. How much popcorn was left?
Answers
There were 5 1/4 bags of popcorn left after eating 4 bags.
To find out how much popcorn was left after eating 4 bags, we need to subtract the amount eaten from the total amount Brianna made.
Brianna made 9 1/4 bags of popcorn, which can be represented as a mixed number. To perform calculations, let's convert it to an improper fraction:
9 1/4 = (4 * 9 + 1) / 4 = 37/4
Now, let's subtract the 4 bags eaten from the total:
37/4 - 4
To subtract fractions, we need a common denominator. The common denominator of 4 and 1 is 4. Therefore, we can rewrite the expression as:
37/4 - 4/1
Now, let's find a common denominator and subtract the fractions:
37/4 - 16/4 = (37 - 16) / 4 = 21/4
The result is 21/4, which is an improper fraction. Let's convert it back to a mixed number:
21/4 = 5 1/4
Therefore, there were 5 1/4 bags of popcorn left after eating 4 bags.
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-2 = k+7/3 solve for x and show work this is all the info i have dont get this deleted
Answers
Answer:
\( - 2 = k + \frac{7}{3} \\ - 2 \times 3 = k + \frac{7}{3} \times 3 \\ - 6 = k + 7 \\ - 6 - 7 = k + 7 - 7 \\ - 13 = k \\ or \\ k = - 13\)
Answer:
-13/3
Step-by-step explanation:
-2=k+7/3
k=-2-7/3
k=-6/3-7/3
k=-13/3
Please mark me as Brainliest if you're satisfied with the answer.
After working satisfactorily for 6 months, Collin will be eligible for a 7% raise. How much will Collin's
gross salary be, after her raise, for each 2-week pay period? His current gross salary is $1,083.34.
Answers
Collin's gross salary, after the 7% raise, for each 2-week pay period, will be approximately $1,159.17.
To calculate Collin's gross salary after the 7% raise for each 2-week pay period, we need to find the raise amount and add it to his current gross salary.
Collin's current gross salary is $1,083.34.
To find the raise amount, we multiply his current gross salary by 7% (or 0.07):
Raise amount = $1,083.34 \(\times\) 0.07
= $75.8338 (rounded to two decimal places)
Now, we add the raise amount to his current gross salary to find the new gross salary:
New gross salary = $1,083.34 + $75.8338 = $1,159.1738 (rounded to two decimal places)
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What is greater 1.0275 or 1.029
Answers
Answer:
1.029 is greater than 1.0275
Step-by-step explanation:
Answer:
1.029
Step-by-step explanation:
What is greater 1.0275 or 1.029
.027 < .029
So, 1.029 is greater.
calculate the length of the hypotenuse
Answers
these are all problems designed to practice the law of sine :
a/sin(A) = b/sin(B) = c/sin(C)
with a, b, c being the sides, and A, B, C being the corresponding opposite angles.
I
and remember :
sin(90) = 1
a)
4/sin(30) = c/sin(90) = c
4/0.5 = c = 8 cm
b)
17/sin(53) = c/sin(90) = c
c = 21.28630619... cm
c)
6/sin(41) = c/sin(90) = c
c = 9.14551852... cm
Given the function f(x) = -5|x + 1| + 3, for what values of x is f(x) = -12? O X= -2. x = 1 O x= -2 x = 4 O x = 2. x = 4 O x = 2, x = 4
Answers
The values are x= -2 and x =4 of the given function f (x) =-5|x+1| +3
What is function?
Function is an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
According to the question,
The given function is
\(f(x) = -5|x + 1| + 3\)
Here f(x) = -12
Substitute f(x) and solve for x
\(- 12 = -5|x + 1| + 3\)
By subtracting 3 on both sides
\(- 12 - 3 = -5|x + 1| + 3 - 3\)
\(- 15 = -5|x + 1|\)
By dividing -5 on both sides
\(3 = |x + 1|\)
We get absolute values for the function.
\(x + 1 = 3 , x + 1 = -3\)
By subtracting 1 on both sides
\(x + 1 - 1 = 3 - 1 = 2\\\\x + 1 - 1 = - 3 - 1 = -4\\\\x = 2 , x = - 4\)
Therefore, the values of x are 2 and - 4 of the given function f (x) =-5|x+1| +3
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If f(x)=(-x)^3-x^2-x+15What is f(0) =
Answers
Please help answer 2 and 3 the best you possibly can
Answers
Answer:
2. x = 3
3. x = 5
Answer:
Below.
Step-by-step explanation:
(2). 6x + 51 + 111 = 180
6x = 180 - 162 = 18
x = 3.
(3). 6x = 30
x = 5.
Find the distance between the two points rounding to the nearest tenth (if necessary). (3,3) and (6,6)
Answers
Answer:
d ≈ 4.2
Step-by-step explanation:
calculate the distance d using the distance formula
d = \(\sqrt{(x_{2}-x_{1})^2+(y_{2}-y_{1})^2 }\)
with (x₁, y₁ ) = (3, 3 ) and (x₂, y₂ ) = (6, 6 )
d = \(\sqrt{(6-3)^2+(6-3)^2}\)
= \(\sqrt{3^2+3^2}\)
= \(\sqrt{9+9}\)
= \(\sqrt{18}\)
≈ 4.2 ( to the nearest tenth )
1/4(2x + 1.2) = 3(0.7x + 3) - 5 1/2
what is the solution?
show your work please!!
Answers
Answer:
Step-by-step explanation:
Simplify parentheses : 0.5x+0.3=2.1x+9-5.5
Combine like terms : 0.5x+0.3=2.1x+3.5
Isolate the variable : -1.6x=3.2
Divide by -1.6 on both sides : x=-2
Answer: the answer is in the picture
have a nice one
Step-by-step explanation:
Please Give Me The Answer
Answers
The area covered by the signal is 181.37 m²
What is the area covered by the signal?Here we can see that the area covered by the signal is the fourth of a circle of radius R = 15.2m
The area of a circle of radius R is given by:
A = 3.14*R²
Then the area of a fourth of a circle is:
A = (3.14/4)*R²
Replacing the value of the radius in the formula we will get:
A = (3.14/4)*(15.2m)² = 181.37 m²
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Which function has the same range as \(f(x)= - 2\sqrt{x} =-3 + 8\\\)
Answers
Both g(x) = 5 - x² and h(x) = \(5 - e^(6-^x^)\)have the same range as f(x) = -2√(x) - 3 + 8, which is (-∞, 5].
To determine which function has the same range as f(x) = -2√(x) - 3 + 8, we need to first find the range of f(x).
The square root function √(x) takes non-negative values as input and gives non-negative outputs, so the expression -2√(x) will always be non-positive. Therefore, the range of f(x) will be all real numbers less than or equal to -3 + 8, which is 5.
In other words, the range of f(x) is (-∞, 5].
So, we need to find a function whose range is also (-∞, 5]. One possible function is g(x) = 5 - x². We can see that when x is zero, g(x) is at its maximum value of 5, and as we increase or decrease x, g(x) will decrease, eventually approaching negative infinity.
Another possible function is h(x) = 5 - e^(-x). When x is negative infinity, e^(-x) is approaching positive infinity, so h(x) is approaching 0. As we increase x, e^(-x) is approaching zero, so h(x) is approaching 5.
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1. A home remodeling company charges a $50 consultation fee plus $150 per hour for labor.
[CG 3.0 F.IF-DOK3]
(a) Write an equation to represent the total cost of a job that takes "x" hours. (3points)
(b) How much would it cost if it takes 40 hours to complete a job? (3points)
(©) If you have $5000 to pay for the work, would it be enough for a job that takes 40
hours? Explain or show your reasoning. (2points)
(d) Would you describe this situation as linear or exponential? Explain why you think so.
(2 points)
Can someone please help me
](https://i5t5.c14.e2-1.dev/h-images-qa/contents/attachments/gbJ9acXj7rFRA3Qj1pFrtvRehQ0cPDpx.png)
Answers
An equation to represent the total cost of a job that takes "x" hours is "total cost = $50 + ($150 × x)". The situation can be described as linear.
What is an equation?An equation is formed when two equal expressions are equated together with the help of an equal sign '='.
Given the cost of consulting is $50, while the cost of labour is $150 per hour.
A.) An equation to represent the total cost of a job that takes "x" hours.
Total cost = $50 + ($150 × x)
B.) Total cost is the labour works for 40 hours,
Total cost = $50 + ($150 × 40) = $6,050
C.) As found above the cost of working 40 hours is $6,050, therefore, the amount of $5,000 will not be enough.
D.) The situation can be described as linear.
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Theprofit(indollars)earnedbysellingwappletreesisrepresentedby w2 − 4w + 14. The profit (in dollars) earned by selling w pear trees is
represented by 2w + 10. Write a polynomial that represents how much more profit is earned by selling w apple trees than w pear trees.
Answers
\(w^2 - 6w + 4\) is a polynomial that represents how much more profit is earned by selling w apple trees than w pear trees.
The profit earned by selling w pear trees is 2w + 10
To find the difference in profit, we need to subtract the profit earned by selling w pear trees from the profit earned by selling w apple trees:
\((w^2 - 4w + 14) - (2w + 10)\)
Simplifying the expression:
\(w^2 - 6w + 4\)
Therefore, the polynomial that represents how much more profit is earned by selling w apple trees than w pear trees is \(w^2 - 6w + 4\).
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