The surface integral in terms of ρ and θ ∫∫S.((6y - 5)e^(x^2))
To evaluate ∬s.curl F•nds using Stokes' Theorem, we first need to find the curl of the vector field F and then compute the surface integral over the given surface S.
Given vector field F = (6yz)i + (5x)j + (yz(e^(x^2)))k, let's find its curl:
∇ × F = ∂/∂x (yz(e^(x^2))) - ∂/∂y (5x) + ∂/∂z (6yz)
Taking the partial derivatives, we get:
∇ × F = (0 - 0) i + (0 - 0) j + (6y - 5)e^(x^2)
Now, let's parametrize the surface S, which is the portion of the paraboloid z = (1/4)x^2 + y^2 for 0 ≤ z ≤ 4. We can use cylindrical coordinates for this parametrization:
r(θ, ρ) = ρcos(θ)i + ρsin(θ)j + ((1/4)(ρcos(θ))^2 + (ρsin(θ))^2)k
where 0 ≤ θ ≤ 2π and 0 ≤ ρ ≤ 2.
Next, we need to find the normal vector n to the surface S. Since S is oriented upward, the normal vector points in the positive z-direction. We can normalize this vector to have unit length:
n = (∂r/∂θ) × (∂r/∂ρ)
Calculating the partial derivatives and taking the cross product, we have:
∂r/∂θ = -ρsin(θ)i + ρcos(θ)j
∂r/∂ρ = cos(θ)i + sin(θ)j + (1/2)(ρcos(θ))k
∂r/∂θ × ∂r/∂ρ = (-ρsin(θ)i + ρcos(θ)j) × (cos(θ)i + sin(θ)j + (1/2)(ρcos(θ))k)
Expanding the cross product, we get:
∂r/∂θ × ∂r/∂ρ = (ρcos(θ)(1/2)(ρcos(θ)) - (1/2)(ρcos(θ))(-ρsin(θ)))i
+ ((1/2)(ρcos(θ))sin(θ) - ρsin(θ)(1/2)(ρcos(θ)))j
+ (-ρsin(θ)cos(θ) + ρsin(θ)cos(θ))k
Simplifying further:
∂r/∂θ × ∂r/∂ρ = ρ^2cos(θ)i + ρ^2sin(θ)j
Now, we can calculate the surface integral using Stokes' Theorem:
∬s.curl F•nds = ∮c.F•dr
= ∫∫S.((∇ × F)•n) dS
Substituting the values we obtained earlier:
∫∫S.((∇ × F)•n) dS = ∫∫S.((6y - 5)e^(x^2))•(ρ^2cos(θ)i + ρ^2sin(θ)j) dS
We can now rewrite the surface integral in terms of ρ and θ:
∫∫S.((6y - 5)e^(x^2))
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the sum of two numbers is 59. the didference between the two numbers is 11 which is the smaller of the two numbers
The larger number is 35.Let's assume the two numbers are represented by the variables "x" and "y," where "x" is the smaller number.
Equation 1: x + y = 59 (The sum of the two numbers is 59)
Equation 2: x - y = 11 (The difference between the two numbers is 11, with the smaller number being y)
According to the given information, the sum of the two numbers is 59, which can be expressed as:
x + y = 59
The difference between the two numbers is 11, with "x" being the smaller number. This can be expressed as:
y - x = 11
We can now solve these two equations simultaneously to find the values of "x" and "y."
Rearranging the first equation to solve for "y" gives us:
y = 59 - x
Substituting this value of "y" into the second equation:
(59 - x) - x = 11
Simplifying the equation:
59 - 2x = 11
Subtracting 59 from both sides:
-2x = 11 - 59
-2x = -48
Dividing both sides by -2:
x = -48 / -2
x = 24
Therefore, the smaller number is 24. To find the larger number, we substitute the value of "x" into the first equation:
24 + y = 59
y = 59 - 24
y = 35.
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Show your work please help me it’s due tomorrow!!!!
Answer: Consider me the brainiest. The answer is... Decimal form =4.083
Exact form= 49/12
The mixed number form=4 1/12
How do we solve fractions step by step?
Conversion a mixed number 2 1/3 to a improper fraction: 2 1/3 = 2 1/3 = 2 3 + 1/3 = 6 + 1/3 = 7/3
To find a new numerator
A; Multiply the whole number 2 by the denominator 3. Whole number 2 equally 2 * 3/3 = 6/3
b) Add the answer from the previous step 6 to the numerator 1. The new numerator is 6 + 1 = 7
c) Write a previous answer (new numerator 7) over the denominator 3.
Two and one-third are seven-thirds.
Conversion a mixed number 1 3/4 to a improper fraction: 1 3/4 = 1 3/4 = 1 · 4 + 3/4 = 4 + 3/4 = 7/4
To find a new numerator:
a) Multiply the whole number 1 by the denominator 4. Whole number 1 equally 1 * 4/4 = 4/4
b) Add the answer from the previous step 4 to the numerator 3. The new numerator is 4 + 3 = 7
c) Write a previous answer (new numerator 7) over the denominator 4. One and three quarters are seven quarters.
Add: 7/3 + 7/4 = 7 · 4/3 · 4 + 7 · 3/4 · 3 = 28/12 + 21/12 = 28 + 21/12 = 49/12 It
is suitable to adjust both fractions to a common (equal, identical) denominator for adding, subtracting, and comparing fractions. The common denominator you can calculate as the least common multiple of both denominators - LCM(3, 4) = 12. It is enough to find the common denominator (not necessarily the lowest) by multiplying the denominators: 3 × 4 = 12. In the following intermediate step, it cannot further simplify the fraction result by cancelling. In other words - seven thirds plus seven quarters is forty-nine twelfths.
Complete the table. Answer should be T or F.
P Q
T F P V Q P ^ Q P -> Q -P -Q -P V -Q -P -> Q -P -> -Q P <-> Q
F T P V Q P ^ Q P -> Q -P -Q -P V -Q -P -> Q -P -> -Q P <-> Q
P Q P V Q P ^ Q P -> Q -P -Q -P V -Q -P -> Q -P -> -Q P <-> Q
T T T T T F F F F F T T
F T T F F T T T T T F F
P F T F F T T T T T F F
-P T T F F T T T T T F F
-Q T T T T F T F F F T T
-P V -Q T T F F T T T T T F
-P -> Q T T F F T T T T T F
-P -> -Q T T F F T T T T T F
P <-> Q T T T T F T T T T F
Here is a more detailed explanation of how I filled out the table:
P | Q : This column is simply the truth value of P and Q. If P and Q are both true, then the entry in this column is T. If P is true and Q is false, then the entry in this column is F. If P is false and Q is true, then the entry in this column is F. And if P and Q are both false, then the entry in this column is T.
P V Q : This column is the truth value of P or Q. If P is true, then the entry in this column is T. If Q is true, then the entry in this column is T. And if P and Q are both false, then the entry in this column is F.
P ^ Q : This column is the truth value of P and Q. If P and Q are both true, then the entry in this column is T. And if P and Q are both false, then the entry in this column is F.
P -> Q : This column is the truth value of P implies Q. If P is true and Q is false, then the entry in this column is F. And if P is false or Q is true, then the entry in this column is T.
-P : This column is the negation of P. If P is true, then the entry in this column is F. And if P is false, then the entry in this column is T.
-Q : This column is the negation of Q. If Q is true, then the entry in this column is F. And if Q is false, then the entry in this column is T.
-P V -Q : This column is the truth value of not P or not Q. If P and Q are both true, then the entry in this column is F. If P and Q are both false, then the entry in this column is T. And if P or Q is true, then the entry in this column is T.
-P -> Q : This column is the truth value of not P implies Q. If P is true and Q is false, then the entry in this column is T. And if P is false or Q is true, then the entry in this column is F.
-P -> -Q : This column is the truth value of not P implies not Q. If P and Q are both true, then the entry in this column is T. If P is false or Q is false, then the entry in this column is T. And if P is true and Q is true, then the entry in this column is F.
P <-> Q : This column is the truth value of P if and only if Q. If P and Q are both true or P and Q are both false, then the entry in this column is T. And if P and Q have different truth values, then the entry in this column is F.
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Find the sum of the first 27 terms
of the arithmetic sequence.
First, fill in the equation.
a₁
= 5 and a27
Sn = 2/(a₁ + an)
Sn
=
[?]
2
+
=
83
Answer:
S₂₇ = 1188
Step-by-step explanation:
using the given formula for [tex]S_{n}[/tex] , that is
[tex]S_{n}[/tex] = [tex]\frac{n}{2}[/tex] (a₁ + [tex]a_{n}[/tex] )
with a₁ = 5 and [tex]a_{n}[/tex] = a₂₇ = 83 , then
S₂₇ = [tex]\frac{27}{2}[/tex] (5 + 83) = 13.5 × 88 = 1188
A circle has a radius of 4 ft.
What is the area of the sector formed by a central angle measuring 260°?
Use 3.14 for pi.
Enter your answer, rounded to the nearest tenth in the box.
The area of the sector is approximately 36.0 ft²
What is the area of the sector?To find the area of the sector formed by a central angle of 260° in a circle with a radius of 4 ft, you can use the formula:
Area of sector = (θ/360°) * π * r²
where θ is the central angle in degrees, π is the value of pi, and r is the radius.
Plugging in the values:
θ = 260°
r = 4 ft
π = 3.14
Area of sector = (260/360) * 3.14 * 4²
Area of sector = (0.7222) * 3.14 * 16 = 36.3 square feet
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Need help with top problem. Maybe bottom too
1) The area of a circle circumscribed about a square is 307.7 cm².
2.a.) The angle ACB is 39 degrees.°.
2b.) The value of x is 5.42.
How to determine the area of a circle?We shall find the radius to determine the area of a circle.
First, find the side length of the square:
Since the perimeter of the square = 56 cm, then, each side of the square is 56 cm / 4 = 14 cm.
Next, find the diagonal of the square, using the Pythagorean theorem:
Diagonal = the diameter of the circumscribed circle.
Diagonal² = side length² + side length²
= 14 cm² + 14 cm²
= 196 cm² + 196 cm²
= 392 cm²
Take the square root of both sides:
Diagonal = √392 cm ≈ 19.80 cm (rounded to two decimal places)
Then, the radius of the circle which is half the diagonal:
Radius = Diagonal / 2 ≈ 19.80 cm / 2 ≈ 9.90 cm (rounded to two decimal places)
Finally, compute the area of the circle using the formula:
Area = π * Radius²
Area = 3.14 * (9.90 cm)²
Area ≈ 307.7 cm² (rounded to two decimal places)
Therefore, the area of the circle that is circumscribed about a square with a perimeter of 56 cm is 307.7 cm².
2. a) We use the property of angles in a circle to solve for angle ACB: an angle inscribed in a circle is half the measure of its intercepted arc.
Given that arc AB has a measure of 78°, we can find angle ACB as follows:
Angle ACB = 1/2 * arc AB
= 1/2 * 78°
= 39°
Therefore, the angle ACB is 39 degrees.
2b.) To solve for the value of x, we use the information that the angle ADB = (3x - 12)⁴.
Given that angle ADB is (3x - 12)⁴, we can equate it to the measure of the intercepted arc AB, which is 78°:
(3x - 12)⁴ = 78
Solve the equation for x, by taking the fourth root of both sides:
∛∛((3x - 12)⁴) = ∛∛78
Simplify,
3x - 12 = ∛(78)
Isolate x by adding 12 to both sides:
3x - 12 + 12 = ∛(78) + 12
3x = ∛(78) + 12
Finally, divide both sides by 3:
x = (∛(78) + 12) / 3
x = (4.27 +12) / 3
x = 5.42
So, x is 5.42
Therefore,
1) The area of the circle is 154 cm².
2a.) Angle ACB is equal to 102°.
2b.) The value of x is 5.42
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24. What is the cost for 5 t-shirts that cost $10 with a sales tax of 6%?
$3
$56
$50
$53
The cost for 5 t-shirts, including Sales tax, is $53.
The cost of 5 t-shirts with a price of $10 each and a sales tax of 6%, we need to multiply the total price of the t-shirts by the sales tax rate and add it to the original price.
The original price of one t-shirt is $10. Therefore, the total price of 5 t-shirts is:
$10 * 5 = $50
To calculate the sales tax, we need to multiply the total price by the sales tax rate of 6%:
$50 * 6% = $50 * 0.06 = $3
Adding the sales tax to the total price:
$50 + $3 = $53
Therefore, the cost for 5 t-shirts, including sales tax, is $53.
The correct answer is: $53.
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