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From Hardcover Book, Marsden/Tromba, Vector Calculus, 6th ed., Section 2.1., # 40) Using polar coordinates, describe the level curves of the function defined by f (x, y) = - 2xy (22+y2) if (x, y) + (0,0) and f(0,0) = 0.
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The level curves of the function f(x, y) = -2xy / (2^2 + y^2) in polar coordinates consist of lines θ = π/2 + kπ and θ = kπ, as well as the upper half and lower half of the unit circle depending on the sign of the function. These level curves represent the points (r, θ) where the function f(r, θ) is constant.
To describe the level curves of the function f(x, y) = -2xy / (2^2 + y^2), we can first express the function in terms of polar coordinates. Let's substitute x = r cos(θ) and y = r sin(θ) into the function:
f(r, θ) = -2(r cos(θ))(r sin(θ)) / (r^2 + (r sin(θ))^2)
Simplifying this expression, we get:
f(r, θ) = -2r^2 cos(θ) sin(θ) / (r^2 + r^2 sin^2(θ))
Now, we can further simplify this expression:
f(r, θ) = -2r^2 cos(θ) sin(θ) / (r^2(1 + sin^2(θ)))
f(r, θ) = -2 cos(θ) sin(θ) / (1 + sin^2(θ))
The level curves of this function represent the points (r, θ) in polar coordinates where f(r, θ) is constant. Let's consider a few cases:
1. When f(r, θ) = 0:
This occurs when -2 cos(θ) sin(θ) / (1 + sin^2(θ)) = 0. Since the numerator is zero, we have either cos(θ) = 0 or sin(θ) = 0. These correspond to the lines θ = π/2 + kπ and θ = kπ, where k is an integer.
2. When f(r, θ) > 0:
In this case, the numerator -2 cos(θ) sin(θ) is positive. For the denominator 1 + sin^2(θ) to be positive, sin^2(θ) must be positive. Therefore, the level curves lie in the regions where sin(θ) > 0, which corresponds to the upper half of the unit circle.
3. When f(r, θ) < 0:
Similar to the previous case, the level curves lie in the regions where sin(θ) < 0, which corresponds to the lower half of the unit circle.
In summary, the level curves of the function f(x, y) = -2xy / (2^2 + y^2) in polar coordinates consist of lines θ = π/2 + kπ and θ = kπ, as well as the upper half and lower half of the unit circle depending on the sign of the function.
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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
Answers
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.
A rectangular swimming pool is 6 it deep. One side of the pool is 3.5 times longer than the other. The amount of water needed to fill the swimming pool is
1344 cubic feet. Find the dimensions of the pool.
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Answer:
8 feet by 28 feet by 6 feet
Step-by-step explanation:
So volume is length times width times height
It tells us that the volume is 1344 cubic feet (the water used to fill it)
And it also tells us that the height/depth (which are the same thing in this case) is 6ft
All we need now are length and width
We know that one of the sides is 3.5 times the other one. So we can just say length is x and width is 3.5x
So plugging that in, the equation becomes
[tex]3.5x*x*6=1344[/tex]
3.5 x times x is just 3.5x squared so
[tex]3.5x^2*6=1344[/tex]
divide both sides by 6
[tex]3.5x^2=244[/tex]
divide by 3.5
[tex]x^2 =64[/tex]
[tex]x=\sqrt{64}[/tex]
x = 8
So that means the one side is 8 feet long and the other side is 3.5 times that, which is 28 feet long.
So the dimensions of the pool are 8 feet by 28 feet by 6 feet
solve for x. round your answer to the nearest tenth
Answers
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
In the process of completing the square, 3x^2+7x-12 becomes x^2+7/4x=4. True or False
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Answer: False
Step-by-step explanation:
the radius of a circle is 8 miles. what is the area of a sector bounded by a 144° arc
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Answer:
Step-by-step explanation:
The area of a sector and the properties of circles bounded by a 144° arc in a circle with a radius of 8 miles can be calculated using the formula: Area of sector = (θ/360°) * π * r² where θ is the central angle of the sector and r is the radius of the circle.
In this case, the central angle is 144° and the radius is 8 miles. Plugging these values into the formula, we get: Area of sector = (144°/360°) * π * (8 miles)². Simplifying the equation, we have: Area of sector = (0.4) * π * (8 miles)².
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(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?
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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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Please help me with my math( if you help i will give you brainliest)
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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:
increase 50$ by 15%
Can you say how to do it and answer?
Answers
0.15 • 50 = 7.5
50 + 7.5 = 57.5
Someone please help!!! will give brainliest!!!
Round your answer to the nearest hundredths, if necessary.
Find the surface area of the figure
Answers
Answer:161.56
Step-by-step explanation:
8 x5=40
8 x 7.07=56.56
1/2 x 5 x 5 x 2= 25
8 x 5=40
Add that all together
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
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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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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
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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
For 25 pts
Pls Help this is hard as hell
Answers
Answer: For the first one Independent variable would be Cars age and the dependent would be cars price. For the second one, independent variable would be number of training miles and dependent would be Time to finish the race in minutes.
Step-by-step explanation:
Answer:
First one:
The independent variable is the car’s age
The dependent variable is the car’s price according tot he age
Second one:
The independent variable is the number of training miles
The dependent variable is the time it takes to finish
Step-by-step explanation:
Just think of the independent variable as the cause and the dependent variable as the effect.
Which of the following sets of angle measures can be used to draw an acute isosceles triangle? Select all that apply. 75°, 30°, 75° 80°, 55°, 45° 80°, 80°, 40° 60°, 60°, 60° 50°, 50°, 80° 20°, 140°, 20°
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Answer:
1. 80°, 80°, 40°
2. 60°, 60°, 60°
3. 20°, 140°, 20°
4. 50°, 50°, 80°
5. 75°, 30°, 75°
6. 80°, 55°, 45°
6, 2, and 4
Answer:
I think its 6,2, and 4
Step-by-step explanation: hope that helps! °∪°
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
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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.
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(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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PLEASE HELP! all u have to do is determine if it is positive or negative!
Answers
Answer:
I think it is positive.
Step-by-step explanation:
Iam soory if Iam wrong.
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
Answers
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
helppppppppppp meeeeeeeeeee
Answers
Answer:
330
Step-by-step explanation:
Answer:
335.5
Step-by-step explanation:
Solve the system of equations using the substitution method. Show your work and be sure to include the solution to the system.
Answers
(3.2)
Equation form:
X
x=3,y=2
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?
Answers
Answer:
0.8
Step-by-step explanation:
got it right on edg
Answer:
5/7
Step-by-step explanation:
determine the slope given the two points. (-19,4) (17,11) PLEASE SHOW WORK
Answers
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
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) =
Answers
Answer:2,6
Step-by-step explanation:Edge2021
Answer
2,6
Step-by-step explanation:
which of the following expressions is equivalent to -10?
a.-7 3
b.-3 - 7
c.3 - 7
d.7 - 3
Answers
The expression which is equivalent to -10 is the option b, -3 - 7.
Explanation:
We can use subtraction and addition of integers to get the value of the given expression. We can write the given expression as;
-3 - 7 = -10 (-3 - 7)
The addition of two negative integers will always give a negative integer. When we subtract a larger negative integer from a smaller negative integer, we will get a negative integer.
If we add -3 and -7 we will get -10. This makes the option b the correct answer.
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Can someone please help me with math.
Answers
and i think that’s a vertical angle?
Simple word problem. 40 POINTS!!!!Thank you.
Answers
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
if y = .5x + 2, what is the value of x when y=4
Answers
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!!
Find the Wronskian for the set of functions (3x^2, e^x, xe^x}, then determine if they are linearly dependent or independent.
Answers
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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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?
Answers
Answer:
$57.289
Step-by-step explanation:
18% of 48.55
48.55 × 18 ÷ 100
= 8.739
48.55 + 8.739
=$57.289
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