x² + y²-15x+8y +50= 5x-6; area​

To find the partial derivatives of the function f(x, y) = 2x² + y² + 2xy + 4x + 2y, we need to differentiate the function with respect to each variable while treating the other variable as a constant. fₓ = 4x + 2y + 4 fᵧ = 2y + 2x + 2 fₓᵧ = 2

Let's start by finding the partial derivative with respect to x, denoted as fₓ or ∂f/∂x: fₓ = ∂f/∂x = 4x + 2y + 4 To find the partial derivative with respect to y, denoted as fᵧ or ∂f/∂y: fᵧ = ∂f/∂y = 2y + 2x + 2

Finally, let's find the mixed derivative with respect to x and y, denoted as fₓᵧ or ∂²f/∂x∂y: fₓᵧ = ∂²f/∂x∂y = 2

The partial derivatives give us information about the rate of change of the function with respect to each variable. The first-order partial derivatives (fₓ and fᵧ) indicate how the function changes as we vary only one variable while keeping the other constant.

The mixed partial derivative (fₓᵧ) indicates how the rate of change of the function with respect to one variable is affected by the other variable. To summarize: fₓ = 4x + 2y + 4 fᵧ = 2y + 2x + 2 fₓᵧ = 2

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The partial derivatives of the function f(x, y) = 2x² + y² + 2xy + 4x + 2yfₓ = 4x + 2y + 4 fᵧ = 2y + 2x + 2 fₓᵧ = 2.

Here, we have,

To find the partial derivatives of the function

f(x, y) = 2x² + y² + 2xy + 4x + 2y,

we need to differentiate the function with respect to each variable while treating the other variable as a constant.

fₓ = 4x + 2y + 4 fᵧ = 2y + 2x + 2 fₓᵧ = 2

Let's start by finding the partial derivative with respect to x, denoted as fₓ or ∂f/∂x: fₓ = ∂f/∂x = 4x + 2y + 4

To find the partial derivative with respect to y, denoted as fᵧ or ∂f/∂y:

fᵧ = ∂f/∂y = 2y + 2x + 2

Finally, let's find the mixed derivative with respect to x and y, denoted as fₓᵧ or ∂²f/∂x∂y: fₓᵧ = ∂²f/∂x∂y = 2

The partial derivatives give us information about the rate of change of the function with respect to each variable. The first-order partial derivatives (fₓ and fᵧ) indicate how the function changes as we vary only one variable while keeping the other constant.

The mixed partial derivative (fₓᵧ) indicates how the rate of change of the function with respect to one variable is affected by the other variable. To summarize: fₓ = 4x + 2y + 4 fᵧ = 2y + 2x + 2 fₓᵧ = 2

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(a) The statement "vf(a, b) is always a unit vector" is False.

(b) The statement "vf(a, b) is orthogonal to the level curve that passes through (a, b)" is True.

(c) The statement "Düf is a maximum at (a, b) when ū = vf(a, b)" is False.

(a) The vector vf(a, b) represents the gradient vector of the function f(x, y) at the point (a, b). The gradient vector provides information about the direction of the steepest ascent of the function at that point. It is not always a unit vector unless the function f(x, y) has a constant magnitude gradient at all points.

(b) The gradient vector vf(a, b) is orthogonal (perpendicular) to the level curve that passes through the point (a, b). This is a property of the gradient vector and holds true for any differentiable function.

(c) The statement suggests that the directional derivative Duf is a maximum at (a, b) when the direction ū is equal to vf(a, b). This is not generally true. The directional derivative represents the rate of change of the function f(x, y) in the direction ū. The maximum value of the directional derivative may occur at a different direction than vf(a, b), depending on the shape and behavior of the function at (a, b).

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The area of the trapezoid image attached is solved to be

Area of a trapezoid is solved using the formula given belos

= 1/2 (sum of parallel lines) * height

In the figure the parallel lines are

= 3 + 6 + 3 = 12 and 6, and the height is 8 in

Plugging in the values

= 1/2 (12 + 6) * 8

= 9 * 8

= 72 square in

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The area of the composite figure in this problem is given as follows:

A = 72 in².

The area of a composite figure is obtained as the sum of the areas of all the parts that compose the figure.

The figure in this problem is composed as follows:

Hence the area of the composite figure in this problem is given as follows:

A = 6 x 8 + 2 x 1/2 x 3 x 8

A = 72 in².

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To minimize the total weekly cost, the company should produce 23 units in Miami and 135 units in Baltimore.

To minimize the total weekly cost function C(x, y) = x^2 + (1/5)y^2 + 46x + 54y + 800, we need to find the values of x and y that minimize this function.

We can solve this problem using calculus. First, we calculate the partial derivatives of C(x, y) with respect to x and y:

∂C/∂x = 2x + 46

∂C/∂y = (2/5)y + 54

Next, we set these partial derivatives equal to zero and solve for x and y:

2x + 46 = 0 (equation 1)

(2/5)y + 54 = 0 (equation 2)

Solving equation 1 for x:

2x = -46

x = -23

Solving equation 2 for y:

(2/5)y = -54

y = -135

So, according to the partial derivatives, the critical point occurs at (x, y) = (-23, -135).

To determine if this critical point corresponds to a minimum, we need to calculate the second partial derivatives of C(x, y):

∂^2C/∂x^2 = 2

∂^2C/∂y^2 = 2/5

The determinant of the Hessian matrix is:

D = (∂^2C/∂x^2)(∂^2C/∂y^2) - (∂^2C/∂x∂y)^2 = (2)(2/5) - 0 = 4/5 > 0

Since the determinant is positive, we can conclude that the critical point (x, y) = (-23, -135) corresponds to a minimum.

Therefore, 23 units in Miami and 135 units in Baltimore should be produced to minimize the total weekly cost.

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By applying the Mean Value Theorem, we can evaluate the given limit as -3.The limit is equal to f(c), which is equal to cos(2c) + 1.

Let f(x) = cos(2x) + 1. The Mean Value Theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one c in (a, b) such that the derivative of the function evaluated at c is equal to the average rate of change of the function over [a, b].

In this case, we need to find the value of c that satisfies f'(c) = (f(2) - f(-7))/(2 - (-7)), which simplifies to f'(c) = (f(2) - f(-7))/9.

Taking the derivative of f(x), we get f'(x) = -2sin(2x). Now we can substitute c back into the derivative: -2sin(2c) = (f(2) - f(-7))/9.

Evaluating f(2) and f(-7), we have f(2) = cos(4) + 1 and f(-7) = cos(-14) + 1. Simplifying further, we obtain -2sin(2c) = (cos(4) + 1 - cos(-14) - 1)/9.

By using trigonometric identities, we can rewrite the equation as -2sin(2c) = (2cos(9)sin(5))/9.

Dividing both sides by -2, we get sin(2c) = -cos(9)sin(5)/9.

Solving for c, we find that sin(2c) = -cos(9)sin(5)/9.

Since sin(2c) = -cos(9)sin(5)/9 is satisfied for multiple values of c, we cannot determine the exact value of c. However, we can conclude that the limit lim(x→-3) cos(2x) + 1 evaluates to the same value as f(c), which is f(c) = cos(2c) + 1. Since c is not known, we cannot determine the exact numerical value of the limit.

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The solution to the given system of differential equations and with the given initial condition, is (a) x(t) = [[-2[tex]e^{t}[/tex]], [2[tex]e^{2t}[/tex]], [-[tex]e^{t}[/tex]]], and (b) x(t) = [[0], [[tex]e^{2}[/tex]], [[tex]e^{t}[/tex]]].

To find the solution to the given system of differential equations, we can use the matrix exponential method.

For (a) x(0) = [[-2], [2], [-1]]:

First, we need to find the eigenvalues and eigenvectors of the coefficient matrix [[1 0 -1], [0 2 0], [1 0 1]]. The eigenvalues are λ = 1 and λ = 2, with corresponding eigenvectors v1 = [[-1], [0], [1]] and v2 = [[0], [1], [0]], respectively.

Using the eigenvalues and eigenvectors, we can write the solution as:

x(t) = c1e^(λ1t)v1 + c2e^(λ2t)v2,

Substituting the given initial condition x(0) = [[-2], [2], [-1]], we can solve for c1 and c2:

[[-2], [2], [-1]] = c1v1 + c2v2,

Solving this system of equations, we find c1 = -2 and c2 = 0.

Therefore, the solution for (a) is x(t) = [[-2[tex]e^{t}[/tex]], [2[tex]e^{2t}[/tex]], [-[tex]e^{t}[/tex]]].

For (b) x(-π) = [[0], [1], [1]]:

Using the same procedure as above, we find c1 = 0 and c2 = 1.

Hence, the solution for (b) is x(t) = [[0], [[tex]e^{2}[/tex]], [[tex]e^{t}[/tex]]].

Thus, the solutions to the given system with the respective initial conditions are x(t) = [[-2[tex]e^{t}[/tex]], [2[tex]e^{2t}[/tex]], [-[tex]e^{t}[/tex]]], and (b) x(t) = [[0], [[tex]e^{2}[/tex]], [[tex]e^{t}[/tex]]].

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The correct question is:

Find the solution to the given system that satisfies the given initial condition.

[tex]x'(t)=\left[\begin{array}{ccc}1&0&-1\\0&2&0\\1&0&1\end{array}\right]\\\\x(0)=\left[\begin{array}{ccc}-2\\2\\-1\end{array}\right] x(-\pi )=\left[\begin{array}{ccc}0\\1\\1\end{array}\right][/tex]  

The given differential equation, [tex]\frac{dV}{dt}[/tex] [tex]- t + (1 + 2t) = 3[/tex], is a linear differential equation.

A linear differential equation is a differential equation where the unknown function and its derivatives appear linearly, i.e., raised to the first power and not multiplied together.

In the given equation, we have the term dV/dt, which represents the first derivative of the unknown function V(t).

The other terms, -t, 1, and 2t, are constants or functions of t. The right-hand side of the equation, 3, is also a constant.

To classify the given equation, we check if the equation can be written in the form:

dy/dx + P(x)y = Q(x),

where P(x) and Q(x) are functions of x. In this case, the equation can be rearranged as:

dV/dt - t = 2t + 4.

Since the equation satisfies the form of a linear differential equation, with the unknown function V(t) appearing linearly in the equation, we conclude that the given equation is a linear differential equation.

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The partial derivatives ∂z/∂u and ∂z/∂v, evaluated at u = ln(3) and v = 2, are given by :

∂z/∂u = 5/(1 + (3 + sin(2))^2) * 3 and ∂z/∂v = 5/(1 + (3 + sin(2))^2) * cos(2), respectively.

To find the partial derivatives ∂z/∂u and ∂z/∂v, we'll use the chain rule.

z = 5tan⁻¹(x), where x = eu + sin(v)

u = ln(3)

v = 2

First, let's find the partial derivative ∂z/∂u:

∂z/∂u = ∂z/∂x * ∂x/∂u

To find ∂z/∂x, we differentiate z with respect to x:

∂z/∂x = 5 * d(tan⁻¹(x))/dx

The derivative of tan⁻¹(x) is 1/(1 + x²), so:

∂z/∂x = 5 * 1/(1 + x²)

Next, let's find ∂x/∂u:

x = eu + sin(v)

Differentiating with respect to u:

∂x/∂u = e^u

Now, we can evaluate ∂z/∂u at u = ln(3):

∂z/∂u = ∂z/∂x * ∂x/∂u

= 5 * 1/(1 + x²) * e^u

= 5 * 1/(1 + (e^u + sin(v))^2) * e^u

Substituting u = ln(3) and v = 2:

∂z/∂u = 5 * 1/(1 + (e^(ln(3)) + sin(2))^2) * e^(ln(3))

= 5 * 1/(1 + (3 + sin(2))^2) * 3

Simplifying further if desired.

Next, let's find the partial derivative ∂z/∂v:

∂z/∂v = ∂z/∂x * ∂x/∂v

To find ∂x/∂v, we differentiate x with respect to v:

∂x/∂v = cos(v)

Now, we can evaluate ∂z/∂v at v = 2:

∂z/∂v = ∂z/∂x * ∂x/∂v

= 5 * 1/(1 + x²) * cos(v)

Substituting u = ln(3) and v = 2:

∂z/∂v = 5 * 1/(1 + (e^u + sin(v))^2) * cos(v)

Again, simplifying further if desired.

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To find the measures of the angles of the triangle ABC with vertices A=(-2,0), B=(2,2), and C=(2,-2), we can use the distance formula and the dot product.

First, let's find the lengths of the sides of the triangle:

AB = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(2 - (-2))² + (2 - 0)²]

= √[4² + 2²]

= √(16 + 4)

= √20

= 2√5

BC = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(2 - 2)² + (-2 - 2)²]

= √[0² + (-4)²]

= √(0 + 16)

= √16

= 4

AC = √[(x₂ - x₁)² + (y₂ - y₁)²]

= √[(2 - (-2))² + (-2 - 0)²]

= √[4² + (-2)²]

= √(16 + 4)

= √20

= 2√5

Now, let's use the dot product to find the measure of angle ZABC (angle at vertex B):

cos(ZABC) = (AB·BC) / (|AB| |BC|)

= (ABx * BCx + ABy * BCy) / (|AB| |BC|)

where ABx, ABy are the components of vector AB, and BCx, BCy are the components of vector BC.

AB·BC = ABx * BCx + ABy * BCy

= (2 - (-2)) * (2 - 2) + (2 - 0) * (-2 - 2)

= 4 * 0 + 2 * (-4)

= -8

|AB| |BC| = (2√5) * 4

= 8√5

cos(ZABC) = (-8) / (8√5)

= -1 / √5

= -√5 / 5

Using the inverse cosine function, we can find the measure of angle ZABC:

ZABC = arccos(-√5 / 5)

≈ 128.189° (rounded to the nearest thousandth)

Therefore, the measure of angle ZABC is approximately 128.189 degrees.

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The work done to move the box is approximately 182.12 Joules.

To find the work done in joules to move the box, use the formula:

Work = Force × Distance × cos(θ)

Where:

- Force is the magnitude of the constant force applied (41.5 N),

- Distance is the distance traveled by the box (5 m), and

- θ is the angle between the force and the direction of motion (28 degrees).

Let's calculate the work done:

Work = 41.5 N × 5 m × cos(28 degrees)

Using a calculator, we can evaluate cos(28 degrees) which is approximately 0.88295.

Work = 41.5 N × 5 m × 0.88295

Work ≈ 182.12 Joules

Therefore, the work done to move the box is approximately 182.12 Joules.

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To find the scalars a, b, c, and k that satisfy the equation of the plane, we can use the equation of a plane in normal form: ax + by + cz = k, where (a, b, c) is the normal vector of the plane.

Given that the normal vector n = (1, 6, 4) and a point P(1, 3, -3) lies on the plane, we can substitute these values into the equation of the plane:

1a + 6b + 4c = k.

Since P(1, 3, -3) satisfies the equation, we have:

1a + 6b + 4c = k.

By comparing coefficients, we can determine the values of a, b, c, and k. From the equation above, we can see that a = 1, b = 6, c = 4, and k can be any constant value.

Therefore, the scalars a, b, c, and k that satisfy the equation of the plane containing P(1, 3, -3) with normal n = (1, 6, 4) are a = 1, b = 6, c = 4, and k can be any constant value.

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To show that the vectors a = (3, -2, 1), b = (1, -3, 5), and c = (2, 1, -4) form a right-angled triangle, we need to verify if the dot product of any two vectors is equal to zero.

If the dot product is zero, it indicates that the vectors are perpendicular to each other, and hence they form a right-angled triangle.

First, let's calculate the dot products between pairs of vectors:

a · b = (3)(1) + (-2)(-3) + (1)(5) = 3 + 6 + 5 = 14

b · c = (1)(2) + (-3)(1) + (5)(-4) = 2 - 3 - 20 = -21

c · a = (2)(3) + (1)(-2) + (-4)(1) = 6 - 2 - 4 = 0

From the dot products, we observe that a · b ≠ 0 and b · c ≠ 0. However, c · a = 0, indicating that vector c is perpendicular to vector a. Therefore, the vectors a, b, and c form a right-angled triangle, with c being the hypotenuse.

In summary, we can determine if three vectors form a right-angled triangle by calculating the dot product between pairs of vectors. If any dot product is zero, it indicates that the vectors are perpendicular to each other and form a right-angled triangle. In this case, the dot product of vectors a and c is zero, confirming that the vectors a, b, and c form a right-angled triangle.

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There are 720 possible ways for the six workers to pick up the six tasks.

If there are six different types of tasks in a department and six workers to pick up these tasks, we can calculate the number of possible ways using the concept of permutations.

Since each worker can pick up one task, we need to calculate the number of permutations of 6 tasks taken by 6 workers.

The formula for permutations is:

P(n, r) = n! / (n - r)!

where n is the total number of items and r is the number of items taken at a time.

In this case, n = 6 (number of tasks) and r = 6 (number of workers). Substituting the values into the formula, we get:

P(6, 6) = 6! / (6 - 6)!

= 6! / 0!

= 6! / 1

= 6 x 5 x 4 x 3 x 2 x 1

= 720

Therefore, there are 720 possible ways for the six workers to pick up the six tasks.

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To find the Taylor series for the function (1+7x²) centered at 0, we can use the formula for the Taylor series expansion:

[tex]f(x)=f(a)+f'(a)\frac{x-a}{1!} +f''(a)\frac{(x-a)^{2} }{2!}+ f'''(a)\frac{(x-a)^{3}}{3!}+.........[/tex]

In this case, the function is (1+7x²) and we want to center it at 0 (a = 0). Let's find the derivatives of the function:

f(x) = (1+7x²)

f'(x) = 14x

f''(x) = 14

f'''(x) = 0 (since the third derivative of any constant is always 0)

...

Now, we can plug in the values into the Taylor series formula:

[tex]f(x) = f(0) + f'(0)\frac{(x-0)}{1!}+ f''(0)\frac{(x-0)^{2} }{2!} +f'''(0)\frac{(x-0)^{3} }{3!}+....[/tex]

f(0) = (1+7(0)²) = 1

f'(0) = 14(0) = 0

f''(0) = 14

f'''(0) = 0

...

Plugging these values into the formula, we get:

[tex]f(x) = 1 +\frac{ 0(x-0)}{1!} + \frac{14(x-0)^2}{2!} +\frac{0(x-0)^3}{3!} + ......[/tex]

Simplifying, we have:

f(x) = 1 + 0 + 7x² + 0 + ...

So, the first four nonzero terms of the Taylor series for (1+7x²) centered at 0 are: 1 + 7x²

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A) The probability that all 100 lights will last 2 years is 0.9048.

B) The probability that at least one bulb will burn out in 2 years is 0.0952.

A) To find the probability that all 100 lights will last 2 years, we assume that the success or failure of each bulb is independent.

The probability of a single bulb lasting 2 years is 0.995, so the probability of all 100 bulbs lasting 2 years is:

P(all 100 bulbs last 2 years) is (0.995)¹⁰⁰ ≈ 0.9048

B) The probability that at least one bulb will burn out in 2 years is determined using the complement rule.

P(at least one bulb burns out) = 1 - P(no bulbs burn out)

Since the probability of a single bulb lasting 2 years is 0.995, the probability of a single bulb burning out in 2 years is 1 - 0.995 = 0.005.

The probability of at least one bulb burning out in 2 years is:

P(at least one bulb burns out) = 1 - P(no bulbs burn out)

P(at least one bulb burns out) = 1 - 0.9048

P(at least one bulb burns out) ≈ 0.0952

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The quotient of the expression (15a⁴b³) / (12a²b) is (5a²b²) / 4.

Given is an expression 15a⁴b³/12a²b, we need to find the quotient, assuming the denominator no equal to zero.

To find the quotient of the expression (15a⁴b³) / (12a²b), we can simplify it by canceling out common factors in the numerator and denominator:

First, let's simplify the coefficients:

15 and 12 can both be divided by 3:

(15a⁴b³) / (12a²b) = (5a⁴b³) / (4a²b).

Next, let's simplify the variables:

a⁴ divided by a² is a² (subtract the exponents), and b³ divided by b is b² (subtract the exponents):

(5a⁴b³) / (4a²b) = (5a²b²) / 4.

Therefore, the quotient of the expression (15a⁴b³) / (12a²b) is (5a²b²) / 4.

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The maximum and minimum values of the function f(x) = (x - 8x) / (x + 2) on the interval [0,4]  is 0, and the minimum value is -8/3, occurring at x = 0 and x = 4, respectively.

To find the maximum and minimum values of the function f(x) on the interval [0,4], we need to evaluate the function at critical points and endpoints within this interval.

First, we check the endpoints:

f(0) = (0 - 8(0)) / (0 + 2) = 0

f(4) = (4 - 8(4)) / (4 + 2) = -16/6 = -8/3

Next, we find the critical points by setting the derivative of f(x) equal to zero and solving for x:

f'(x) = [(1 - 8) * (x + 2) - (x - 8x)(1)] / (x + 2)^2 = 0

Simplifying, we get:

-7(x + 2) - x + 8x = 0

-7x - 14 - x + 8x = 0

0 = 0

Since 0 = 0 is an identity, there are no critical points within the interval [0,4].

Comparing the function values at the endpoints and noting that f(x) is a continuous function, we find:

The maximum value of f(x) on [0,4] is 0, which occurs at x = 0.

The minimum value of f(x) on [0,4] is -8/3, which occurs at x = 4.

In conclusion, the maximum value of the function f(x) = (x - 8x) / (x + 2) on the interval [0,4] is 0, and the minimum value is -8/3, occurring at x = 0 and x = 4, respectively.

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Jaime Tutankhamun would need 12,500 square inches of cardboard material for 100 square pyramid packages.

To determine the amount of cardboard material needed for 100 square pyramid packages, we first calculate the surface area of a single package. Each square pyramid has a base area of 49 square inches. The four triangular faces of the pyramid are congruent isosceles triangles, and the slant height is given as 5 inches.

Using the formula for the lateral surface area of a pyramid, we find that each triangular face has an area of (1/2) * base * slant height = (1/2) * 7 * 5 = 17.5 square inches. Since there are four triangular faces, the total lateral surface area of one package is 4 * 17.5 = 70 square inches. Adding the base area, the total surface area of one package is 49 + 70 = 119 square inches. Therefore, for 100 packages, Jaime would need 100 * 119 = 11,900 square inches of cardboard material.

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The product of the given complex numbers is √3(cos149 + i sin116) and the quotient is 5√5(cos37 + i sin37).

Given, z1 = √3(cos59 + i sin59) and z2 = 1 + i sin57.

To find the product and the quotient of the above complex numbers in polar form.

Product of complex numbers is calculated by multiplying their moduli and adding their arguments (in radians).

The formula to find the quotient of two complex numbers in polar form is given as,

When two complex numbers in polar form z1 = r1(cosθ1 + isinθ1) and z2 = r2(cosθ2 + isinθ2) are divided, then the quotient is given byz1/z2 = r1/r2(cos(θ1-θ2) + isin(θ1-θ2)).

Now, let's solve the problem:

Product of z1 and z2 is given by:

zzzz = z1z2

= √3(cos59 + i sin59)(1 + i sin57)

= √3(cos59 + i sin59)(cos90 + i sin57)

= √3(cos(59 + 90) + i sin(59 + 57))

= √3(cos149 + i sin116)

Therefore, the product of zzzz is √3(cos149 + i sin116).

Quotient of z1 and z2 is given by:

z1/z2 = √3(cos59 + i sin59)/(1 + i sin57)= √3(cos59 + i sin59)(1 - i sin57)/(1 - i sin57)(1 + i sin57)= √3(cos59 + sin59 + i(cos59 - sin59))/(1 + [tex]sin^257[/tex])= √3(2cos59)/(1 + [tex]sin^257[/tex]) + i√3(2cos59 sin57)/(1 + [tex]sin^257[/tex])

Now, let's put the values and simplify,

z1/z2 = 5√5(cos37 + i sin37)

Therefore, the quotient of z1 and z2 is 5√5(cos37 + i sin37).

Hence, the product of the given complex numbers is √3(cos149 + i sin116) and the quotient is 5√5(cos37 + i sin37).

We were required to find the product and the quotient of complex numbers z1 = √3(cos59 + i sin59) and z2 = 1 + i sin57 expressed in polar form. For multiplication of two complex numbers in polar form, we multiply their moduli and add their arguments in radians. Similarly, the quotient of two complex numbers in polar form can be found by dividing their moduli and subtracting their arguments in radians. Applying the same formula, we found that the product of z1 and z2 is √3(cos149 + i sin116). On the other hand, the quotient of z1 and z2 is 5√5(cos37 + i sin37). Thus, the polar form of the required complex numbers is obtained.

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The complete question is :

Find the product z1z2 and the quotient 21. Express your answers in polar form. v3(cos( 59 ) + i sin(SA)). 1 + i sin( 57 )). 22 = 5V5(cos( 37) + i sin( % )) 37 Z1 = COS Z122 = 21 NN Il Need Help? Read it

The value of the integral ∫√₁ (x² + 2x - (x² + 2x - 8)) dx is 0.

The integral to be evaluated is ∫√₁ (x² + 2x - (x² + 2x - 8)) dx. To solve this integral, we need to simplify the expression inside the square root, evaluate the integral, and find the antiderivative of the simplified expression.

The expression inside the square root, x² + 2x - (x² + 2x - 8), simplifies to just -8. Thus, the integral becomes ∫√₁ (-8) dx.

Since the integrand is a constant, we can pull the constant outside of the integral and evaluate the integral of 1. The square root of -8 is equal to 2i√2 (where i represents the imaginary unit). Therefore, the integral becomes -8 ∫√₁ 1 dx.

Integrating 1 with respect to x gives x as the antiderivative. Evaluating this antiderivative between the limits of integration, 1 and √1, we have √1 - 1.

Thus, the evaluated integral is -8(√1 - 1). Simplifying further, we get -8(1 - 1) = -8(0) = 0.

Therefore, the value of the integral ∫√₁ (x² + 2x - (x² + 2x - 8)) dx is 0.

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When the height of the water is 7m, the rate at which the height is changing is 2/(49π) m/min.

To find how fast the height of the water is changing, we need to use the volume formula for a conical tank and differentiate it with respect to time.

The volume formula for a conical tank is V = (1/3)πr^2h, where V is the volume, r is the radius of the base, and h is the height of the water.

Given that water is being filled into the tank at a rate of 2 m/min, we have dV/dt = 2. We want to find dh/dt, the rate at which the height is changing.

Differentiating the volume formula with respect to time, we get:

dV/dt = (1/3)π(2rh)(dh/dt) + (1/3)πr^2(dh/dt)

Since the base radius and height of the tank are equal, we can substitute r = h into the equation:

2 = (1/3)π(2h^2)(dh/dt) + (1/3)πh^2(dh/dt)

Simplifying the equation:

2 = (2/3)πh^2(dh/dt) + (1/3)πh^2(dh/dt)

2 = πh^2(dh/dt)(2/3 + 1/3)

2 = πh^2(dh/dt)(1)

2 = πh^2(dh/dt)

Now, we can solve for dh/dt:

dh/dt = 2/(πh^2)

To find the value of dh/dt when the height of the water is 7m, we substitute h = 7 into the equation:

dh/dt = 2/(π(7^2))

dh/dt = 2/(49π)

Therefore, when the height of the water is 7m, the rate at which the height is changing is 2/(49π) m/min.

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The function is not a solution of the differential equation y(4) - 7y = 0. y = 7 cos(x) .

To determine if y(x) = 7cos(x) is a solution of the differential equation y(4) - 7y = 0, we need to substitute y(x) and its derivatives into the differential equation:

y(x) = 7cos(x)

y'(x) = -7sin(x)

y''(x) = -7cos(x)

y'''(x) = 7sin(x)

y''''(x) = 7cos(x)

Substituting these into the differential equation, we get:

y(4)(x) - 7y(x) = y'''(x) - 7y(x) = 7sin(x) - 7(7cos(x)) = -42cos(x) ≠ 0

Since the differential equation is not satisfied by y(x) = 7cos(x), y(x) is not a solution of the differential equation y(4) - 7y = 0.

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Only vectors v and w are perpendicular to each other.

To determine if vectors are parallel or perpendicular, we can analyze their dot products.

a) Comparing vectors u = 5i - j + k and v = i + 5k:

To check for parallelism, we'll calculate the dot product u · v:

u · v = (5i)(i) + (-j)(0) + (k)(5k)

= 5i^2 + 0 + 5k^2

= 5 + 5

= 10

Since the dot product is non-zero (10), the vectors u and v are not perpendicular.

b) Comparing vectors u = 5i - j + k and w = -15i + 3j - 3k:

To check for parallelism, we'll calculate the dot product u · w:

u · w = (5i)(-15i) + (-j)(3j) + (k)(-3k)

= -75i^2 - 3j^2 - 3k^2

= -75 - 3 - 3

= -81

Since the dot product is non-zero (-81), the vectors u and w are not perpendicular.

c) Comparing vectors v = i + 5k and w = -15i + 3j - 3k:

To check for parallelism, we'll calculate the dot product v · w:

v · w = (i)(-15i) + (5k)(3j) + (-15k)(-3k)

= -15i^2 + 15k^2

= -15 + 15

= 0

Since the dot product is zero, the vectors v and w are perpendicular.

In summary:

Vectors u and v are neither parallel nor perpendicular.

Vectors u and w are neither parallel nor perpendicular.

Vectors v and w are perpendicular.

Therefore, among the given vectors, v and w are perpendicular to each other.

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The matches would be:f(x, y) = x + 2y: D., f(x, y) = ln(x + 2y): A.,[tex]f(x, y) = e^zy: C[/tex].,[tex]f(x, y) = x^4y^3[/tex]: B.

To match the functions with the graphs of their domains, let's analyze each function and its corresponding graph:

f(x, y) = x + 2y:

This function is a linear function with variables x and y. The graph of this function is a plane in three-dimensional space. It has no restrictions on the domain, so the graph extends infinitely in all directions. The graph would be a flat plane with a slope of 1 in the x-direction and 2 in the y-direction.

f(x, y) = ln(x + 2y):

This function is the natural logarithm of the expression x + 2y. The domain of this function is restricted to x + 2y > 0 since the natural logarithm is only defined for positive values. The graph of this function would be a surface in three-dimensional space that is defined for x + 2y > 0. It would not exist in the region where x + 2y ≤ 0.

[tex]f(x, y) = e^zy[/tex]:

This function involves exponential growth with the base e raised to the power of z multiplied by y. The graph of this function would also be a surface in three-dimensional space. It does not have any specific restrictions on the domain, so the graph extends infinitely in all directions.

[tex]f(x, y) = x^4y^3[/tex]:

This function is a power function with x raised to the power of 4 and y raised to the power of 3. The graph of this function would be a surface in three-dimensional space. It does not have any specific restrictions on the domain, so the graph extends infinitely in all directions.

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The probability of exactly 4 residents supporting political party A can be calculated using the binomial probability formula, while the probability of less than 4 residents supporting party A can be obtained by summing the probabilities of 0, 1, 2, and 3 residents supporting party A.

i. To calculate the probability of exactly 4 residents supporting political party A, we use the binomial probability formula. The formula is P(X = k) = (nCk) * p^k * (1-p)^(n-k), where n is the sample size, k is the number of successes, p is the probability of success, and nCk represents the number of combinations. In this case, n = 6, k = 4, and p = 0.7. Plugging these values into the formula, we can calculate the probability.

ii. To calculate the probability of less than 4 residents supporting party A, we need to sum the probabilities of 0, 1, 2, and 3 residents supporting party A. This can be done by calculating the individual probabilities using the binomial probability formula for each value of k (0, 1, 2, 3) and then summing them up.

By performing these calculations, we can find the probabilities for both scenarios.

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Given the equation of the ellipse and thefunction f(x) values as follows. x²/4 + y²/36 = 1; f(x,y) = 5x + yNow, f(x,y) = 5x + yAlso, x²/4 + y²/36 = 1We have to find the maximum and minimum values of f(x,y) under the given conditions.

To find the maximum and minimum values of f(x,y) we need to find the values of x and y by the method of Lagrange's multiplier.Method of Lagrange's Multiplier:Lagrange's multiplier method is a method that helps to find the maximum and minimum values of a function f(x,y) subjected to the constraints g(x,y).Let, f(x,y) = 5x + y and g(x,y) = x²/4 + y²/36 - 1Hence, to maximize or minimize f(x,y), we can writeL(x, y, λ) = f(x,y) + λg(x,y)L(x, y, λ) = 5x + y + λ(x²/4 + y²/36 - 1)Now, we have to find the partial derivatives of L(x,y,λ) with respect to x, y and λ.Lx(x, y, λ) = 5 + λ(x/2) = 0Ly(x, y, λ) = 1 + λ(y/18) = 0Lλ(x, y, λ) = x²/4 + y²/36 - 1 = 0From (1) 5 + λ(x/2) = 0 ⇒ λ = -10/x ⇒ (2)From (2), 1 + λ(y/18) = 0 ⇒ -10/x(y/18) = -1 ⇒ xy = 180 ⇒ (3)From (3), we can substitute the value of y in terms of x in equation (4) to obtain the maximum and minimum values of f(x,y).x²/4 + (180/x)²/36 - 1 = 0⇒ x⁴ + 16x² - 81 × 100 = 0On solving the above equation we get,x = √360(√17 - 1) or x = - √360(√17 + 1)Now, we can use these values of x to obtain the values of y and then substitute the values of x and y in f(x,y) to get the maximum and minimum values of f(x,y).x = √360(√17 - 1) ⇒ y = 6√17 - 36Now, f(x,y) = 5x + y = 5(√360(√17 - 1)) + 6√17 - 36 = 30√17 - 6x = - √360(√17 + 1) ⇒ y = -6√17 - 36Now, f(x,y) = 5x + y = 5(-√360(√17 + 1)) - 6√17 - 36 = -30√17 - 6Hence, the maximum value of f(x,y) is 30√17 - 6 and the minimum value of f(x,y) is -30√17 - 6.

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A principal amount of P 200,000 was deposited for a period of 4 years and 6 months, and it earned an interest of P 85,649.25. To find the rate of interest compounded monthly, we can use the formula for compound interest and solve for the interest rate.

The formula for compound interest is given by A = P(1 + r/n)^(nt), where A is the final amount, P is the principal amount, r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years.

In this case, we are given the principal amount P as P 200,000, the final amount A as P 285,649.25 (P 200,000 + P 85,649.25), the time t as 4 years and 6 months (or 4.5 years), and we need to find the interest rate r compounded monthly (n = 12).

Using the given values in the compound interest formula and solving for r, we can find the rate of interest. By rearranging the formula and substituting the known values, we can isolate the interest rate r and calculate its value.

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The slope of the tangent line to the curve without eliminating the parameter `t` is `-3`.

Given the parametric equations x = t - 2t and y = 3t+1. We are to find the slope of the tangent line to the curve dy/dx without eliminating the parameter, t.

Formula for dy/dx using parametric equationsThe formula for dy/dx using parametric equations is:

dy/dx = dy/dt ÷ dx/dt

Firstly, we'll find the derivatives dy/dt and dx/dt. Then, we'll substitute the resulting values into the formula `dy/dx = dy/dt ÷ dx/dt`.

Let's find the derivatives first.`x = t - 2t`

So, `dx/dt = 1 - 2 = -1``y = 3t+1

`So, `dy/dt = 3`Substituting `dy/dt` and `dx/dt` into the formula, we have;`dy/dx = dy/dt ÷ dx/dt``dy/dx = 3/-1`

Simplifying,`dy/dx = -3`

Therefore, the slope of the tangent line to the curve without eliminating the parameter `t` is `-3`.

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The value of the given expression, I = Sc(sin x + 3y) dx + (5x + y) dy, evaluated along the nonclosed path ABCD, is equal to 100.

The given expression, I = Sc(sin x + 3y) dx + (5x + y) dy, represents a line integral over the path ABCD. To evaluate this integral, we need to substitute the coordinates of each point on the path into the expression and calculate the integral over each segment.

Starting at point A (0,0), we move along the line segment AB to point B (5,5). Along this segment, the expression becomes I = Sc(sin x + 3y) dx + (5x + y) dy. Integrating this expression with respect to x from 0 to 5 and with respect to y from 0 to 5, we obtain the value of the integral for this segment.

Next, we continue along the line segment BC to point C (5,10). The expression remains the same, and we integrate over this segment from x = 5 to y = 10. Finally, we move along the line segment CD to point D (0,15). Again, the expression remains the same, and we integrate over this segment from x = 5 to y = 15.

After evaluating the integral over each segment, we sum up the results to find the total value of the expression along the path ABCD. In this case, the value of the integral is equal to 100.

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The volume of triangular prism A, the preimage, is 68 km³.When a triangular prism is dilated, the volume of the resulting prism is equal to the scale factor cubed times the volume of the original prism.

In this case, if triangular prism B is the image of triangular prism A after dilation by a scale factor of 4 and the volume of prism B is 4352 km³, we can find the volume of prism A by reversing the dilation.

Let V₁ be the volume of prism A. Since prism B is a dilation of prism A with a scale factor of 4, we can write:

V₂ = (scale factor)³ * V₁

Substituting the given values, we have:

4352 = 4³ * V₁

Simplifying:

4352 = 64 * V₁

Dividing both sides by 64:

V₁ = 4352 / 64

V₁ = 68 km³.

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