Use Stokes' Theorem to evaluate
∫
C
F � dr
where F(x, y, z) = x2yi + 1/3x3j + xyk and C is the curve of intersection of the hyperbolic paraboloid z = y2 ? x2
and the cylinder x2 + y2 = 1 oriented counterclockwise as viewed from above.
Find parametric equations for C,Let x and y be in terms of t where
0 ? t ? 2?
Answer:
[tex]\int_C F . dr = \pi[/tex]
[tex]C : x = cost , y = sin t, z = sin^2 t - cos^2 t , 0 \leq t \leq 2 \pi[/tex]
Step-by-step explanation:
Given that:
[tex]F(x,y,z) = x^2yi + \dfrac{1}{3}x^3j +xyk[/tex]
Here C is the curve of intersection of the hyperbolic parabolic [tex]z = y^2 - x^2[/tex] and the cylinder [tex]x^2 +y^2 =1[/tex]
Using Stokes' Theorem
[tex]\int_C F . dr =\int \int \limits_s \ curl \ F. \ds[/tex]
From above ;
S = the region under the surface [tex]z = y^2 -x^2[/tex] and above the circle [tex]x^2+y^2 =1[/tex]
Suppose, we consider [tex]f(x,y,z) =z-y^2+x^2[/tex]
therefore, S will be the level curve of f(x,y,z) = 0
Recall that:
[tex]\bigtriangledown f (x,y,z)[/tex] is always normal to the surface S at the point (x,y,z).
∴
This implies that the unit vector [tex]n = \dfrac{\bigtriangledown f}{|| \bigtriangledown ||}[/tex]
So [tex]\bigtriangledown f = <2x, -2y,1 >[/tex]
Also, [tex]|| \bigtriangledown f ||= \sqrt{4x^2+4y^2+1}[/tex]
Similarly ;
[tex]curl \ F = \begin {vmatrix} \begin{array} {ccc}{\dfrac{\partial }{\partial x} }&{\dfrac{\partial }{\partial y} }& {\dfrac{\partial }{\partial z} }\\ \\ x^2y& \dfrac{1}{3}x^3&xy \end {array} \end{vmatrix}[/tex]
[tex]curl \ F = \langle x ,-y,0 \rangle[/tex]
Then:
[tex]\int \int_s curl \ F .ds = \int \int_s curl \ F .nds[/tex]
[tex]\int \int_s curl \ F .ds = \iint_D curl \ F \dfrac{\bigtriangledown f}{ || \bigtriangledown f||} \sqrt{ (\dfrac{\partial z}{\partial x }^2) + \dfrac{\partial z}{\partial x }^2)+1 } \ dA[/tex]
[tex]\int \int_s curl \ F .ds = \iint_D \dfrac{\langle x,-y,0 \rangle * \langle 2x,-2y,1 \rangle }{\sqrt{4x^2 +4y^2 +1 }} \times \sqrt{4x^2 +4y^2 +1 }\ dA[/tex]
[tex]\int \int_s curl \ F .ds = \iint_D (2x^2 + 2y^2) \ dA[/tex]
[tex]\int \int_s curl \ F .ds = 2 \iint_D (x^2 + y^2) \ dA[/tex]
[tex]\int \int_s curl \ F .ds = 2 \int \limits ^{2 \pi} _{0} \int \limits ^1_0r^2.r \ dr \ d\theta[/tex]
converting the integral to polar coordinates
This implies that:
[tex]\int \int_s curl \ F .ds = 2 \int \limits ^{2 \pi} _{0} \int \limits ^1_0r^2.r \ dr \ d\theta[/tex]
⇒ [tex]\int_C F . dr = 2(\theta) ^{2 \pi} _{0} \begin {pmatrix} \dfrac{r^4}{4}^ \end {pmatrix}^1_0[/tex]
[tex]\int_C F . dr = 2(2 \pi) (\dfrac{1}{4})[/tex]
[tex]\int_C F . dr =(4 \pi) (\dfrac{1}{4})[/tex]
[tex]\int_C F . dr = \pi[/tex]
Therefore, the value of [tex]\int_C F . dr = \pi[/tex]
The parametric equations for the curve of intersection of the hyperbolic paraboloid can be expressed as the equations of the plane and cylinder in parametric form . i.e
[tex]z = y^2 - x^2 \ such \ that:\ x=x , y=y , z = y^2 - x^2[/tex]
[tex]x^2 +y^2 =1 \ such \ that \ : x = cos \ t , y= sin \ t, z = z, 0 \leq t \leq 2 \pi[/tex]
Set them equal now,
the Parametric equation of [tex]C : x = cost , y = sin t, z = sin^2 t - cos^2 t , 0 \leq t \leq 2 \pi[/tex]
A lake has a surface area of 16.0 square miles and an average depth of 51.0 feet. what is it's volume in liters
Answer:
Volume of the lake = 6.44 × 10¹¹ Litres
Step-by-step explanation:
First, let us unify the units of measurement in the problem by converting the depth from feet to miles
1 mile = 5280 feet
1 foot = 1/5280 miles
∴ 51.0 Feet = 51.0 × (1/5280) miles
= 0.009659 miles
Next, let us calculate the volume of the lake.
Volume = surface area × depth
surface area = 16.0 (mi)²
depth = 0.009659 mi
volume = 16.0 × 0.009659 = 0.15454 (mi)³
Next, since we are asked to report the volume in litres, let us convert from cubic miles to litres
1 cubic mile = 4.168 × 10¹² litres
∴ 0.15454 cubic miles = 4.168 × 10¹² × 0.15454
= 6.44 × 10¹¹ Litres
∴ Volume of the lake = 6.44 × 10¹¹ Litres
The volume of the lake is 6.46×10¹¹ liters.
To get the volume of the lake in liters, we need to calculate the volume in square miles and then convert it to liters.
Formula:
V = Ah................... Equation 1Where:
V = Volume of the lakeA = Surface area of the lakeh = height of the lake.From the question,
Given:
A = 16.0 square milesh = 51.0 feet = (51×0.00019) feet = 0.00969 milesSubstitute these values into equation 1
V = 16(0.00969)V = 0.15504 milesV = (0.15504×4.168×10¹²) litersV = 6.46×10¹¹ litersHence, the volume of the lake is 6.46×10¹¹ liters
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Please help me thank you so much. Just not sure of my answers
Answer:
An interesting experiment is given. We need to address various questions based on our knowledge of calculus.
Step 2
Part (a)
Time taken for the radius to grow to 2 cm = t1 = r/0.5 = 2/0.5 = 4 hours
Time taken for the radius to become 0 = t2 and the same can be obtained by solving:
r = 2 - √t2 = 0
Hence, t2 = 22 = 4 hours
Hence, the time duration of the entire experiment (from the introduction of the bacteria until its disappearance) = t1 + t2 = 4 + 4 = 8 hours
Step 3
Part (b)
r(t) = 0.5t for 0 ≤ t ≤ 4
and
r(t) = 2 - √(t - 4) for t > 2
Step-by-step explanation:
A ladder leans against the side of a house. The angle of elevation of the ladder is 65, and the top of the ladder is 13 from the ground. Find the length of the ladder. Round your answer to the nearest tenth.
Answer:
SOHCAHTOA.
we have to use SOH(Sin) here because the theta is 65° and the opposite is 13 while the hypotenuse is x.
which is Sin 65°=13/x.
xSin65°=13.
0.906307787x=13.
x=13/0.9063=14.34403619~14.
A triangle has vertices at F (8, 3), G (3, 5), and H (1, 7). What are the coordinates of each vertex if the triangle is rotated 180° about the origin counterclockwise?
Question 1 options:
F ¢(8, 3), G¢(-3, 5), H ¢(-1, -7)
F ¢(8, -3), G ¢(3, -5), H ¢(1, -7)
F ¢(-8, 3), G¢(-3, 5), H ¢(-1, 7)
F ¢(-8, -3), G ¢(-3, -5), H ¢(-1, -7)
Answer: F (-8, -3), G (-3, -5) and H (-1, -7)
Step-by-step explanation:
A rotation of 180° around the origin is equivalent to a reflection over the x-axis, and then another reflection over the y-axis.
Then, if we have a point (x, y) and we do a rotation of 180°, the point will transform into (-x, -y)
Then if at the start the vertices of the triangle are:
F (8, 3), G (3, 5), and H (1, 7).
After a rotation of 180°, the vertices will be:
F (-8, -3), G (-3, -5) and H (-1, -7)
The correct option is the last one.
factorize d² +13d + 22
Answer:
(d+9/2)(d-9/2)
Step-by-step explanation:
Use the formula. x=
[tex] + \sqrt({b { }^{2} } - 4ac) \div (2a) \ \\ - \sqrt({b { }^{2} } - 4ac) \div (2a) [/tex]
where x=d, a =1 , b=13 , c=22.
since,
[tex]ax {}^{2} + bx + c = 0[/tex]
the cube of the sum of 4 and 9 times x divided by the product of 5 times x and the difference of x and 1
Answer:
Step-by-step explanation:
(4 + 9x)^3 represents "the cube of the sum of 4 and 9 times x"
and if we divide by "the product of 5 times x and the difference of x and 1," we get
(4 + 9x)^3
-----------------------
5x(x - 1)
What exactly do you need to know, or to do?
What is the solution Set to 2a+6=2a+5+1
Answer:
6=6
True for all a
Step-by-step explanation:
[tex]2a+6=2a+5+1\\\mathrm{Subtract\:}2a\mathrm{\:from\:both\:sides}\\\mathrm{Simplify}\\6=5+1\\\mathrm{Simplify\:}5+1:\quad 6\\\\6 = 6[/tex]
Answer:
infinite solutions
Step-by-step explanation:
2a+6=2a+5+1
Combine like terms
2a+6 = 2a+6
Subtract 2a from each side
6 =6
Since this is always true, we have infinite solutions
A family is planning a three-week vacation for which they will drive across the country. They have a van that gets 12 miles per gallon, and they have a sedan that gets 36 miles per gallon. How much more will they pay for gasoline if they take the van? (Assume that the family will drive 2500 miles and that gas costs $2.50 a gallon. Round your answers to the nearest cent.)
Answer:
The family will pay 34722 cents more if they take the van
Step-by-step explanation:
Given
Van = 12 miles per gallon
Sedan = 36 miles per gallon
Distance = 2500 miles
Gas = $2.50 per gallon
First, we need to determine the number of gallons that'll be used by both vehicles
This is done by dividing total distance by number of miles per gallon
[tex]Van = \frac{2500}{12} \ gallon[/tex]
[tex]Sedan = \frac{2500}{36}\ gallon[/tex]
Next, is to multiply this by the cost of gas per gallon;
This gives the total spendable amount on both vehicles
[tex]Van = \frac{2500}{12} * \$2.50[/tex]
[tex]Van = \frac{\$6250}{12}[/tex]
[tex]Van = \$520.833[/tex]
[tex]Sedan = \frac{2500}{36}* \$2.50[/tex]
[tex]Sedan = \frac{\$6250}{36}[/tex]
[tex]Sedan = \$173.611[/tex]
Next is to get the difference between these amounts
[tex]Difference = \$520.833 - \$173.611[/tex]
[tex]Difference = \$347.222[/tex]
Multiply by 100 to convert to cents
[tex]Difference = 347.222 * 100 cents[/tex]
[tex]Difference = 34722.2 \ cents[/tex]
[tex]Difference = 34722\ cents[/tex] (Approximated)
Hence;
The family will pay 34722 cents more if they take the van
let f (x) =- 3x and g (x) = 2x - 1 Find the following f (x) + g (x) Pleas show steps
Answer:
See below.
Step-by-step explanation:
So we have the two functions:
[tex]f(x)=-3x\text{ and } g(x)=2x-1[/tex]
And we want to find f(x) + g(x).
So, substitute:
[tex]f(x)+g(x)\\=(-3x)+(2x-1)[/tex]
Combine like terms:
[tex]=(-3x+2x)+(-1)[/tex]
Simplify:
[tex]=-x-1[/tex]
So:
[tex]f(x)+g(x)=-x-1[/tex]
A polygon is shown:
The area of polygon MNOPQR = Area of a rectangle that is 9 square units + Area of a rectangle that is ___ square units. (Input whole numbers only, such as 8.)
Answer:
10 square units
Step-by-step explanation:
Matteo makes raspberry punch. The table shows how many parts ginger ale and raspberry juice to use for a batch. Raspberry Punch Parts Ginger Ale 2 Parts Raspberry Juice 3 Matteo decides to add one part of raspberry juice. What is the new ratio of ginger ale to raspberry juice? 2 parts ginger ale to 3 parts raspberry juice 2 parts ginger ale to 4 parts raspberry juice 3 parts ginger ale to 3 parts raspberry juice 3 parts ginger ale to 4 parts raspberry juice
Answer:
Answer: There is 1 1/2 times more juice than ginger ale; there is 2/3 as much ginger ale as there is punch.
Step-by-step explanation:
Answer:
Answer A:
Step-by-step explanation:
2 parts ginger ale to 3 parts raspberry juice.
Please help me with these questions
Answer:
Step-by-step explanation: you sjxhbtdtbd
Tell which angles are congruent to the given angle measure.
Answer:
2, 5, & 6
Step-by-step explanation:
2, because it’s the opposite angle.
5, because it’s parallel.
6, because it’s opposite to 5.
2, 5, & 6 all equal 100°.
1, 3, 4, & 7 all equal 80°.
The angles formed that are congruent to angle measure of 100 degrees when transversal t intersects parallel lines m and n are:
<2, <5, and <6.Angle measuring 100 degrees is the given angle measure formed at the point of intersection between line m and transversal t.
Thus, angles that are congruent to 100 degrees will be equal in measure to 100 degrees.
The following are angles congruent to 100 degrees.
<2 is congruent to 100 degrees (vertically opposite angles are congruent).
<5 is congruent to 100 degrees (corresponding angles are congruent).
<6 is congruent to 100 degrees (alternate exterior angles are congruent).
Therefore, the angles formed that are congruent to angle measure of 100 degrees when transversal t intersects parallel lines m and n are:
<2, <5, and <6.Learn more here:
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What is the number in standard form? 5.708 • 10^-8 Drag the answer into the box to match the number. 100 POINTS!!! HELP ME!!!
Answer:
.00000005708 is your answer. If it's asking you to solve the problem, 49.08
Answer:
[tex]\Huge \boxed{0.00000005708}[/tex]
[tex]\rule[225]{225}{2}[/tex]
Step-by-step explanation:
[tex]5.708 \cdot 10^{-8}[/tex]
Solving for exponent and evaluating:
[tex]\displaystyle 5.708 \cdot \frac{1}{10^8 }[/tex]
[tex]\displaystyle \frac{5.708}{10^8 }[/tex]
[tex]\Rightarrow \ \displaystyle \frac{5.708}{100000000}[/tex]
The decimal place moves 8 units to the left side.
[tex]\Rightarrow \ 0.00000005708[/tex]
[tex]\rule[225]{225}{2}[/tex]
The value of a 4 in the one thousands place is what FRACTION of the value of a 4 in the ten thousands place?
Answer:
1/10 = 0.1
Step-by-step explanation:
The value of a 4 in the one thousands place = 4000
the value of a 4 in the ten thousands
= 40000
In fraction,The value of a 4 in the one thousands place to the value of a 4 in the ten thousands place
= 4000/40000
= 4/40
= 1/10
= 0.1
help with math homework?
Answer:
The range of the curve is [tex][-9,9][/tex].
Step-by-step explanation:
The domain of the curve corresponds to the values on horizontal axis (x-Axis), where the range of curve corresponds to the values on vertical axis (y-Axis). In addition, the curve is continuous in [tex](-5, 8)[/tex], so that images exists within interval.
The range of the curve is [tex][-9,9][/tex].
Name the 5 ways/methods/techniques we can use to find a limit.
Answer:
you have four techniques to choose from: plugging in the x value, factoring, rationalizing the numerator, and finding the lowest common denominator.
hope that helps : )
Plz Help!!!!!!!!!
15. Men need to intake between 2200 and 2800 calories daily. Women need 600 fewer calories than this. Write and solve an inequality to discover how many calories women should be taking in per day.
A. 2200 x < 2800
B. 2200 > x < 2800
C. 600 < x < 1200
D. 1600 < x < 2800
E. 1600 < x < 2200
F. 2800 < x < 3200
Answer:
A
Step-by-step explanation:
17. x^2 + 2x + 1
O A. This polynomial could be factored by finding the GCF, then by grouping or using the perfect squares method.
O B. This polynomial could be factored by using the difference of squares method, perfect squares method, or grouping.
C. This polynomial could be factored only by using the perfect squares method.
O D. This polynomial could be factored only by using the difference of squares method.
E. This polynomial could be factored by using grouping or the perfect squares methods.
O F. This polynomial cannot be factored by any of the methods used in this lesson.
Answer:
E. This polynomial could be factored by using grouping or the perfect squares methods.
Step-by-step explanation:
x^2 + 2x + 1
There is no greatest common factor
This is a perfect square
a^2 + 2ab+ b^2 = ( x+1)^2
We can factor this by grouping
x^2 + 2x + 1
(x^2 +x) + (x+1)
x( x+1) + x+1
Factor out x+1
( x+1) ( x+1)
This is not the difference of squares since there is no subtraction
Andre says that x is 7 becuase he can move the two 1s with the x to the other side. True or false
I will mark you brainiest
Answer:
False!
Step-by-step explanation:It is not possible.
Answer:
False
Step-by-step explanation: there is no possible way
plz brainliest
Solve the equation 6x + 2x - 5= 19
Answer:
x = 3
Step-by-step explanation:
6x + 2x - 5= 19
8x -5 = 19
8x = 24
x = 3
Answer:
x = 3
Step-by-step explanation:
6x + 2x - 5 = 19
Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS.
First, combine like terms:
(6x + 2x) - 5 = 19
(8x) - 5 = 19
Next, isolate the variable, x. First, add 5 to both sides:
8x - 5 (+5) = 19 (+5)
8x = 19 + 5
8x = 24
Next, divide 8 from both sides:
(8x)/8 = (24)/8
x = 24/8
x = 3
3 is your answer for x.
~
what is the next greatest number of tens 384
Answer:
9
step by step explanation :
390
Hope this helps :)
384 the next greatest number of tens is 390.
What are Arithmetic operations?Arithmetic operations can also be specified by subtracting, dividing, and multiplying built-in functions.
The operator that performs the arithmetic operation is called the arithmetic operator.
Operators which let do a basic mathematical calculation
Note that the tens place is two moves to the right of the decimal point (if it exists).
To determine the greatest number to the nearest ten (nearest 10),
we use the place of the whole number to determine whether the tens place rounds up or stays the same.
The tens place (8) then look at the digit to the right (4):
⇒ 384
The digit to the right (4) is 4 or below. So, we replace it with a 0 (zero) to get 390 as the answer.
So, 384 the next greatest number of tens is 390.
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F/5 = -8. F=? please solve! I'm stuck!
Answer:
-40
Step-by-step explanation:
F/5 = -8 ⇒ multiply both sides by 5F = - 8*5F = -40 ⇒ answer which expression is equivalent to 6^3 x 6 1/4?
1. 6 11/4
2. 6 13/4
3. 6 3/4
4. 6
Answer:
number 3 I think I'm not sure
I need help ASAP and I need to show my work
Answer:
Hey there!
Total students: 7+9+5+3+12=36
Students that like math: 7
7/36=19.4% of students like math.
Let me know if this helps :)
Answer:
19% (Math)
Step-by-step explanation:
7 divide by total number of students (36) X100% =19%
Find the answer to zero point 73 repeating times square root of 147 end square root and select the correct answer below. Also, is the product of a nonzero rational number and an irrational number classified as rational or irrational? start fraction 80 over 99 end fraction square root of three end square root comma rational start fraction 511 over 99 end fraction square root of three end square root comma irrational 511 over 33, rational start fraction 73 over 99 end fraction plus seven square root of three end square root comma irrational
Answer:
[tex]\frac{511\,\,\sqrt{3} }{99}[/tex] which is an irrational number
Step-by-step explanation:
Recall that the repeating decimal 0.7373737373... can be written in fraction form as: [tex]\frac{73}{99}[/tex]
Now, let's write the number 147 which is inside the square root in factor form to find if it has some perfect square factors:
[tex]147=7^2\,3[/tex]
Then, 7 will be able to go outside the root when we compute the final product requested:
[tex]\frac{73}{99} \,*\,7\,\sqrt{3} =\frac{511\,\,\sqrt{3} }{99}[/tex]
This is an irrational number due to the fact that it is the product of a rational number (quotient between 511 and 99) times the irrational number [tex]\sqrt{3}[/tex]
i need help with number 4
Answer:
x = 9, AR = 25 , AM = 40
Step-by-step explanation:
Since AM = AR + RM and AM also = 7x-23
7x - 23 = (2x + 7) + 15
7x - 23 = 2x + 22
+23 +23
7x = 2x + 45
-2x -2x
5x = 45
x = 9
Now plug x = 9 into 2x + 7 to find AR
AR = 2x + 7
= 2 (9) + 7
= 18 + 7
AR = 25
Now to find AM plug x = 9 into 7x - 23
AM = 7x - 23
= 7 (9) - 23
= 63 - 23
AM = 40
To double check, we already know that RM = 15, so add AR + RM to find AM
AM = 25 + 15
AM = 40
Answer:
9 = x
AR = 25
AM = 40
Step-by-step explanation:
AR + RM = AM
2x+7 +15 = 7x - 23
Combine like terms
2x+22 = 7x -23
Subtract 2x from each side
2x+22 -2x = 7x-2x -23
22 = 5x - 23
Add 23 to each side
22+23 = 5x-23+23
45 = 5x
Divide each side by 5
45/5 = 5x/5
9 = x
AR = 2x+7 = 2*9 +7 = 18+7 = 25
AM = 7x-23 = 7*9 -23 = 63-23 = 40
a painter chargers $12 an hour while his son charges $6 an hour. if the father and son worked the same amount of time together on a job, how many hours did each of them work if their combined charge for their labor was 108
Answer: 6 hours
Step-by-step explanation:
Let x= the number of hours they've worked
12x+6x=108
Combine like terms
18x=108
Divide 18 on both sides
18/18 x=108/18
x=6
Hope this helps!! :)
Please let me know if you have any question
both the father and son worked for 6 hours each.
Let's denote the number of hours that both the father and son worked as "h".
The father charges $12 per hour, so his earnings for h hours of work would be 12h dollars.
The son charges $6 per hour, so his earnings for h hours of work would be 6h dollars.
Since their combined charge for labor was $108, we can set up the equation:
12h + 6h = 108
Combining like terms:
18h = 108
To solve for h, we divide both sides of the equation by 18:
h = 108 / 18
h = 6
Therefore, both the father and son worked for 6 hours each.
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For each hour he babysits, Anderson earns $1 more than half of Carey’s hourly rate. Anderson earns $6 per hour. Which equation can be used to solve for Carey’s hourly rate, c? One-half c plus 1 equals 6 One-half c minus 1 equals 6 One-half c plus 6 equals 1 One-half c minus 6 equals 1
Answer:
The equation used to solve Carey's hourly rate is:
1 + (c/2) = 6
Step-by-step explanation:
Answer:
The equation used to solve Carey's hourly rate is:
1 + (c/2) = 6
Step-by-step explanation:
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