The solution to the triple integral in spherical coordinates is 22000π. This can be obtained by evaluating the integral in three steps: integrating with respect to r, then with respect to θ, and finally with respect to φ.
To change the triple integral to spherical coordinates, we need to express the integrand and the limits of integration in terms of spherical coordinates.
The given integrand is f(x, y, z) = 6x² + y² + z².
In spherical coordinates, the integrand becomes f(r, θ, φ) = 6(rsinθcosφ)² + (rsinθsinφ)² + (rcosθ)².
The limits of integration are as follows:
- The bounds for r are from 0 to 10, as the region Q is bounded by the upper hemisphere x² + y² + z² = 100.
- The bounds for θ are from 0 to π/2, as we are considering the upper hemisphere.
- The bounds for φ are from 0 to 2π, as φ covers a complete revolution around the z-axis.
The triple integral in spherical coordinates is then given by:
∭Q f(r, θ, φ) r² sinθ dr dθ dφ,
which becomes:
∫(φ=0 to 2π) ∫(θ=0 to π/2) ∫(r=0 to 10) [6(rsinθcosφ)² + (rsinθsinφ)² + (rcosθ)²] r sinθ dr dθ dφ.
To solve the given triple integral, we'll start by evaluating the innermost integral with respect to r, then the middle integral with respect to θ, and finally the outer integral with respect to φ.
The integrand is:
[6(rsinθcosφ)² + (rsinθsinφ)² + (rcosθ)²] r² sinθ
First, let's evaluate the innermost integral with respect to r, while treating θ and φ as constants:
∫(r=0 to 10) [6(rsinθcosφ)² + (rsinθsinφ)² + (rcosθ)²] r² sinθ dr
= ∫(r=0 to 10) [6(sin²θcos²φ)r⁴ + (sin²θsinφ)r⁴ + (cos²θ)r⁴] sinθ dr
= ∫(r=0 to 10) [(6sin²θcos²φ + sin²θsin²φ + cos²θ) r⁴] sinθ dr
= [(6sin²θcos²φ + sin²θsinφ + cos²θ) ∫(r=0 to 10) r⁴] sinθ dr
= [(6sin²θcos²φ + sin²θsin²φ + cos²θ) * (10^5/5)] sinθ
= [(6sin²θcos²φ + sin²θsin²φ + cosθ) * 2 × 10⁵] sinθ
Next, let's evaluate the middle integral with respect to θ, while treating φ as a constant:
∫(θ=0 to π/2) [(6sin²θcos²φ + sin²θsin²φ + cos²θ) * 2 × 10⁵] sinθ dθ
= 2 × 10⁵ ∫(θ=0 to π/2) [6sin²θcos²φ + sin²θsin²φ + cos²θ] sinθ dθ
= 2 × 10⁵ [2/3cos²φ + 1/4 + 1/3]
= 2 × 10⁵ [2/3cos²φ + 7/12]
Finally, let's evaluate the outer integral with respect to φ:
[tex][\int_{0}^{2\pi} 2\times10^5 \left( \frac{2}{3}\cos^2\phi + \frac{7}{12} \right) d\phi \\\\= 2\times10^5 \left( \frac{2}{3}\pi + \frac{7}{12}(2\pi) \right)][/tex]
= 22π × 10000
= 22000π
Therefore, the solution to the given triple integral is 22000π.
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Complete question :
Change the triple integral to spherical coordinates: SIS 6x2 + y2 + z2) av (༴ AV Where Q is bounded by the upper hemisphere: x2 + y2 +22=100 : 21 10 ("S", p's p. sino dpdooo 2T pº sino dododo 21 10 2 3 sino doopde 0 0 0 10 ["S" p2 sino apdoce
The term to term rule for a sequence is Multiply by 2 the sequence starts a 2a ___ ___ the total value of the first three terms is 63 work out the total value of the first four terms
Answer:
135
Step-by-step explanation:
The sequence are:
a, 2a, 4a, 8a, 16a.....
the total value of the first three terms is 63
That is,
a + 2a + 4a = 63
7a = 63
a = 63/7
a = 9
work out the total value of the first four terms
First four terms are: a, 2a, 4a, 8a
Where,
First term, a = 9
Second term, 2a = 2*9 = 18
Third term, 4a = 4*9 = 36
Fourth term, 8a = 8*9 = 72
The total value of the first four terms = 9 + 18 + 36 + 72
= 135
The total value of the first four terms = 135
(q17) A geologist finds out that a radioactive substance A that he found in the caves of Africa decays at a rate of 0.03 percent every year. What is the probability that an atom of this substance chosen at random will decay in the next 70 years?
None of the given options is the answer.
To calculate the probability of decay for substance A over the next 70 years, we need to consider the decay rate of 0.03 percent per year.
The decay rate of 0.03 percent per year can be converted to a decimal by dividing it by 100: 0.03 / 100 = 0.0003.
The probability of an atom decaying in a given year is equal to the decay rate, which is 0.0003.
To calculate the probability of an atom not decaying in a given year, we subtract the decay rate from 1: 1 - 0.0003 = 0.9997.
The probability of an atom not decaying over the next 70 years can be calculated by multiplying the probability of not decaying in each year together: (0.9997)^70 ≈ 0.9704.
Therefore, the probability of an atom decaying in the next 70 years is equal to 1 minus the probability of not decaying: 1 - 0.9704 ≈ 0.0296.
So, the probability that an atom of substance A chosen at random will decay in the next 70 years is approximately 0.0296 or 2.96%.
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helppppppppppp meeeeeeeeeee
Answer:
330
Step-by-step explanation:
Answer:
335.5
Step-by-step explanation:
Let S = {3,4,5,6,7,8,9) be a sample space such that the following are true. Use the information to answer the questions. E = (8,9) F = {7,8) G = {4,6,9) a) Are E and F mutually exclusive? Ο Nο Yes O Cannot be determined b) Are F and G mutually exclusive? Ο Nο Yes Cannot be determined.
(a) E and F are not mutually exclusive due to the overlapping value of 8. So the answer is No or option A.
(b) F and G are mutually exclusive since they have no common outcomes. So the answer is No or option A.
a) E and F are not mutually exclusive. To be mutually exclusive, two events cannot occur simultaneously. In this case, both E and F have an overlapping value of 8. The interval (8,9) is included in both E and F, indicating that there is a common outcome (8) between the two events.
Therefore, E and F are not mutually exclusive.
b) F and G are mutually exclusive. In order for two events to be mutually exclusive, they must have no common outcomes. Looking at the intervals (7,8) and (4,6,9), there is no overlapping value between F and G. F includes the value 7, which is not present in G, and G includes the values 4, 6, and 9, which are not present in F.
Therefore, there are no common outcomes between F and G, making them mutually exclusive.
In summary, E and F are not mutually exclusive due to the overlapping value of 8, while F and G are mutually exclusive since they have no common outcomes.
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Write a simplified polynomial expression that can be used to represent the perimeter of the rectangle. 3x-7 and x-7
Answer:
P = 8x-28
Step-by-step explanation:
Given that,
Length = (3x-7)
Width = (x-7)
We need to find the perimeter of the rectangle. The formula for the perimeter of a rectangle is given by :
[tex]P=2 (l+b)\\\\P=2(3x-7+x-7)\\\\P=2(4x-14)\\\\P=8x-28[/tex]
So, the perimeter of the rectangle is equal to 8x-28.
6.
At its first stop a bus picked up 10
people. At the next stop, 8 people got
on and 3 people got off. At the third
stop, 5 people got on and 12 people
got off. How many passengers were
then on the bus?
Answer:
8
Step-by-step explanation:
10+8-3+5-12=
PLEASE ANSWER!! Write the equation of a line that passes through the points (-2,-9) and (2,-9).
Answer:
Step-by-step explanation:
the equation in the point slope form is
and reducing the equation (slope-intercept form)
Step-by-step explanation:
first we calculate the slope of the line with the formula:
where is a point where the line passes, and is another point where the line passes.
Since we have the following points:
(8, -2)
(5,5)
we conclude that
now we substitute this values to find the slope:
to find the equation now that we know the slope we use the point-slope equation:
and we subtitute the slope and the values of and :
we reduce this equation:
the equation in the point slope form is
and reducing the equation (slope-intercept form)
Solution(s) of the differential equation *y'= 2y
y = 2x only
А. y = 0 and Y = 22
y=0 only
y = 0 and 2x
The solutions to the differential equation y' = 2y are y = 0 and y = 2x. The solution y = 0 represents a constant function. The solution y = 2x represents a family of exponential functions.
The given differential equation is y' = 2y, where y' represents the derivative of y with respect to x. To solve this equation, we can separate variables by moving all terms involving y to one side and terms involving x to the other side:
dy/y = 2dx
Next, we integrate both sides of the equation. The integral of dy/y is ln|y|, and the integral of 2dx is 2x:
ln|y| = 2x + C
Here, C is the constant of integration. To simplify the equation, we can rewrite it as:
|y| = e^(2x + C)
Since e^(2x + C) is always positive, we can remove the absolute value sign:
y = ±e^(2x + C)
Now, let's consider the two cases separately.
Case 1: y = 0
If y = 0, then the exponential term becomes e^C, which is a constant. This implies that y remains zero for all values of x. Therefore, y = 0 is a solution to the differential equation.
Case 2: y ≠ 0
If y ≠ 0, we can rewrite the solution as:
y = ±e^C * e^(2x)
Since e^C is a constant, we can replace it with another constant, let's call it K:
y = ±K * e^(2x)
Here, ±K represents a family of exponential functions that grow or decay exponentially with a rate proportional to 2. Each value of K corresponds to a different solution to the differential equation.
In summary, the solutions to the differential equation y' = 2y are y = 0 and y = ±K * e^(2x), where K is a constant. The solution y = 0 represents a constant function, while y = ±K * e^(2x) represents a family of exponential functions.
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A number,
n
n, is multiplied by
−
0.7
−0.7. The product is
−
1
2
−
2
1
. What is the value of
n
n?
Answer:
0.8
Step-by-step explanation:
got it right on edg
Answer:
5/7
Step-by-step explanation:
Two boats start their journey from the same point A and travel along directions AC and AD, as shown below:
What is the distance, CD, between the boats?
230.9 ft
284.3 ft
115.5 ft
173.2 ft
Answer:
Option (1)
Step-by-step explanation:
By applying tangent rule in ΔABD,
tan(30°) = [tex]\frac{\text{Opposite side}}{\text{Adjacent side}}[/tex]
= [tex]\frac{AB}{BD}[/tex]
BD = [tex]\frac{AD}{\text{tan}(30)}[/tex]
BD = [tex]\frac{200}{\frac{1}{\sqrt{3} } }[/tex]
BD = 200√3 ft
By applying tangent rule in ΔABC,
tan(60°) = [tex]\frac{AB}{BC}[/tex]
[tex]\sqrt{3}=\frac{200}{BC}[/tex]
BC = [tex]\frac{200}{\sqrt{3}}[/tex]
Since, CD = BD - BC
CD = 200√3 - [tex]\frac{200}{\sqrt{3}}[/tex]
= 346.41 - 115.47
= 230.94 ft
≈ 230.9 ft
Therefore, Option (1) will be the correct option.
Answer:
230.9 ft
Step-by-step explanation:
person above said it was correct answer
7.) Jessica took her parents out to dinner. The total
bill was $48.55. She left an 18% tip. What was
Jessica's total cost for dinner?
Answer:
$57.289
Step-by-step explanation:
18% of 48.55
48.55 × 18 ÷ 100
= 8.739
48.55 + 8.739
=$57.289
PLEASE HELP! all u have to do is determine if it is positive or negative!
Answer:
I think it is positive.
Step-by-step explanation:
Iam soory if Iam wrong.
The units for square centimeters are written as
Check all that apply.
O A. cm2
B. sq. cm
C. km2
D. sq.m
E cm
Worth 50 points
The table shows the inputs and corresponding outputs for the function f(x) = StartFraction 1 Over 8 EndFraction(2)x. A 2-column table with 5 rows. Column 1 is labeled x with entries 0, 2, 4, 6, 8. Column 2 is labeled f (x) with entries StartFraction 1 Over 8 EndFraction, one-half, 2, 8, 32. Find the following values of the function. f -1 (one-half) = f -1 (8) =
Answer:2,6
Step-by-step explanation:Edge2021
Answer
2,6
Step-by-step explanation:
A student suggests the following algorithm for calculating 72 - 38. 72 Two minus eight equals negative six. -38 -6 Seventy minus thirty equals forty. Forty plus negative six equals thirty-four, 34 which therefore is the result. As a teacher, what is your response? Does this procedure always work? Explain.
The student's suggested algorithm for subtracting numbers is incorrect. The algorithm produces the correct result in this specific case (72 - 38), but it does not work consistently for all subtraction problems.
The student's algorithm suggests subtracting the ones digit first and then subtracting the tens digit. While this approach may give the correct answer in some cases, it does not work for all subtraction problems. Subtraction is an operation where we need to consider the place value of the digits being subtracted.
In the case of 72 - 38, the student's algorithm produces the correct result of 34. However, if we apply the same procedure to a different subtraction problem, such as 43 - 29, we would get an incorrect result of 14 instead of the correct answer, 14. The student's algorithm fails to consider borrowing or regrouping when subtracting digits from different place values.
As a teacher, it is important to guide the student in understanding the standard algorithm for subtraction, which involves subtracting digits starting from the rightmost place value and borrowing when necessary. By teaching the correct procedure, students can consistently obtain accurate results for subtraction problems. It is crucial to explain the limitations of the student's suggested algorithm and emphasize the importance of understanding and applying the appropriate method for subtracting numbers.
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(2 x 10^4) + (7 X 10^4) =
Answer:
90,000
Step-by-step explanation:
10^4=10,000.
2*10,000=20,000
7*10,000=70,000
20,000+70,000=90,000
determine the slope given the two points. (-19,4) (17,11) PLEASE SHOW WORK
Answer:
m=7/36
Step-by-step explanation:
we need two points in order to find the ratio of the change in y and change in x, which is the slope
m=(y-y1)/(x-x1) (you can choose any point as y1 but be careful that you use the corresponding x1 value in the denominator)
m=(11-4)/(17-(-19))
m=7/36
Which equation best represents the relationship between x and y in the graph?
A. y = -2x + 1.5
B. y = -2x + 3
C. y = -1/2x + 3
D. y = -1/2x + 1.5
In the process of completing the square, 3x^2+7x-12 becomes x^2+7/4x=4. True or False
Answer: False
Step-by-step explanation:
Can someone please help me with math.
Please help me with my math( if you help i will give you brainliest)
Answer:
4. 50
5. 35
6. 45
7. 30
8. No mode
9. 42
10. 22, 25, 45, 73, 80
11. 15, 25, 30, 48, 50
12. 58
13. 35
14. 51
15. 24
16. 22.13
17. 10.22
18. Iffy's team had a lower performance than Kaiya's team. Iffy's team collected an average of 35 cans, whereas, Kaiya's team collected an average of 50 cans!! Kaiya's team also had very versatile and active players who were able to collect more, individually, unlike Iffy's team.
Step-by-step explanation:
99999 help me plz plz plz plz
Answer:
hi
Step-by-step explanation:
i think 10 in
hope it helps
Answer:
10
Step-by-step explanation:
The triangles are the same size, and if you look at the picture, you can see that MN and QR are the same, making QR 10 in.
if y = .5x + 2, what is the value of x when y=4
y=0.5x+2
y=4
4=0.5x+2
-2 -2
2=0.5x
/0.5 /0.5
4=x
---
hope it helps
Answer:
[tex]x[/tex] = 4
Step-by-step explanation:
If [tex]y[/tex] = 4 then it would look like:
[tex]y[/tex] = .5[tex]x[/tex] + 2
Since .5 is 0.5, Half of 4 equal 2, PLUS the other 2 making it 4!!
Calculate the perimeter of the composite figure. Round your answer to the nearest hundredth. Use 3.14 for $\pi$ .
th sides are 8 and 10
1. Perimeter is 3 + 3 +3 +4 +5 = 18 feet
Area = 3*3 = 9, 1/2*3*4 = 6, 9 + 6 =15 square feet
2. perimeter = 2.5 +2.5+ 2.5+2.5+0.5+0.5 = 11 meters
Area = 3*.5 = 1.5, 3*2=6, 6+1.5 = 7.5 square meters
3. perimeter = 3.14*2*3 = 18.84 +8 = 26.8 inches
Area = 6*4 = 24 + 3.14*3^2 = 28.26 = 28.26 +24 = 52.3 inches
4. surface area = 2*π*6*20+2*π*6^2= 980.2 yards
Volume = π*6^2*20 = 2261.9 cubic yards
5.surface area = 2*(9*7+2*2+2*9) = 190 cm
Volume = 2*7*9 = 126 cubic cm
6. surface area = 2*(11*11+11*11+11*11) = 726 mm
Volume = 11 *11*11 = 1331 cubic cm
Find the Wronskian for the set of functions (3x^2, e^x, xe^x}, then determine if they are linearly dependent or independent.
The Wronskian for the set of functions (3[tex]x^2[/tex], [tex]e^x[/tex], x[tex]e^x[/tex]) is W(0) = 1 and the set of functions (3[tex]x^2[/tex], [tex]e^x[/tex], x[tex]e^x[/tex]) are linearly independent.
To find the Wronskian for the set of functions (3[tex]x^2[/tex], [tex]e^x[/tex], x[tex]e^x[/tex]) and determine if they are linearly dependent or independent, we calculate the determinant of the matrix formed by taking the derivatives of these functions and evaluating them at a specific point.
The Wronskian is a determinant that helps determine if a set of functions is linearly dependent or independent.
For the given set of functions (3[tex]x^2[/tex], [tex]e^x[/tex], x[tex]e^x[/tex]), we need to calculate the Wronskian.
First, we take the derivatives of the functions:
f₁(x) = 3[tex]x^2[/tex]
f₂(x) = [tex]e^x[/tex]
f₃(x) = x[tex]e^x[/tex]
Taking the first derivatives, we get:
f₁'(x) = 6x
f₂'(x) = [tex]e^x[/tex]
f₃'(x) = [tex]e^x[/tex] + x[tex]e^x[/tex]
Next, we form a matrix with these derivatives:
| 6x [tex]e^x[/tex] [tex]e^x[/tex] + x[tex]e^x[/tex] |
To calculate the Wronskian, we evaluate this matrix at a specific point, let's say x = 0, and take the determinant:
W(0) = | 6(0) [tex]e^0[/tex] [tex]e^0[/tex] + 0[tex]e^0[/tex] |
| 0 1 1 |
| 1 1 1 |
Simplifying, we find:
W(0) = | 0 1 1 |
| 1 1 1 |
| 1 1 1 |
Calculating the determinant, we have:
W(0) = (0)(1)(1) + (1)(1)(1) + (1)(1)(1) - (1)(1)(1) - (1)(1)(0) - (1)(1)(1) = 1
Since the Wronskian is non-zero (W(0) ≠ 0), the set of functions (3[tex]x^2[/tex], [tex]e^x[/tex], x[tex]e^x[/tex]) are linearly independent.
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1. Claretta works part-time at a coffee shop. Her
weekly paychecks in March are: $87.00, $96.00,
$84.25, and $100.75. Find the median of her
paychecks.
Let f(x) = (x + 7)^2 Find a domain on which f is one-to-one and non-decreasing. Find the inverse of f restricted to this domain
The function f(x) = (x + 7)^2 is one-to-one and non-decreasing for any domain on the real numbers. The inverse of f(x) restricted to this domain is y = √x - 7.
To find a domain on which the function f(x) = (x + 7)^2 is one-to-one and non-decreasing, we need to determine where the function is strictly increasing or non-decreasing and has a one-to-one correspondence.
First, let's examine the graph of f(x) = (x + 7)^2 to understand its behavior. The function is a parabola that opens upward, centered at x = -7, and the vertex is the lowest point on the graph.
Since the vertex is the lowest point and the parabola opens upward, the function is non-decreasing for all x-values. Therefore, the function is non-decreasing over its entire domain.
To find a domain on which the function is one-to-one, we observe that the function is not symmetric about the y-axis. Hence, the domain can be any subset of the real numbers.
Now, let's find the inverse of f(x) restricted to this domain. Since f(x) is non-decreasing, the inverse will also be non-decreasing. The inverse function can be found by interchanging the roles of x and y in the original equation and solving for y.
Let's proceed with finding the inverse:
Start with the equation f(x) = (x + 7)^2.
Interchange x and y: x = (y + 7)^2.
Solve for y:
Take the square root of both sides: √x = y + 7.
Subtract 7 from both sides: y = √x - 7.
The inverse function of f(x) restricted to any domain on which it is one-to-one and non-decreasing is given by y = √x - 7.
Note that the domain can be any subset of the non-negative real numbers, since the square root function is defined only for non-negative values.
In summary, the function f(x) = (x + 7)^2 is one-to-one and non-decreasing for any domain on the real numbers. The inverse of f(x) restricted to this domain is y = √x - 7.
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solve for x. round your answer to the nearest tenth
Answer:
11.9
Step-by-step explanation:
Use sin
Sin ratio is opposite over hypotenuse
Sin [tex]57^{o}[/tex] = [tex]\frac{10.8}{x}[/tex]
x = [tex]\frac{10.8}{sin57^{o} }[/tex]
x = 11.9
Which relations represent functions? Choose all that apply.
{(-2, 6), (-5, -1), (3, 7), (-5, 0)}
help me please-
Answer:
its 5,1
Step-by-step explanation:
just took the test
Simple word problem. 40 POINTS!!!!Thank you.
Answer:
$50
Step-by-step explanation:
Hello There!
We are given that for 1 hour of work 250 dollars is charged and for 3 hours of work 350 dollars is charged
This could also be represented in two points (1,250) and (3,350)
The question wants us to find the hourly charge rate (slope)
we can easily find the slope ( hourly charge rate ) by using the slope formula
[tex]slope=\frac{y_2-y_1}{x_2-x_1}[/tex]
we have our two points so all we need to do is plug in the values (remember y values go on top and x values go on the bottom.)
[tex]slope=\frac{350-250}{3-1} \\350-250=100\\3-1=2\\slope=\frac{100}{2} or50[/tex]
So we can conclude that the hourly charge rate is $50