Solve each proportion.

3/4 = 5/x

(e) The overall solution is given by the equation x(t) =  C1t^3 + C2/t^3,, where C1 and C2 are arbitrary constants.

(a) The Wronskian of x(1) and x(2) is given by:

W = | x1(t) x2(t) |

| x1'(t) x2'(t) |

Let's evaluate the Wronskian of x(1) and x(2) using the given formula:

W = | t 2t^2 | - | 4t t^2 |

| 1 2t | | 2 2t |

Simplifying the determinant:

W = (t)(2t^2) - (4t)(1)

= 2t^3 - 4t

= 2t(t^2 - 2)

(b) For x(1) and x(2) to be linearly independent, the Wronskian W should be non-zero. Since W = 2t(t^2 - 2), the Wronskian is zero when t = 0, t = -√2, and t = √2. For all other values of t, the Wronskian is non-zero. Therefore, x(1) and x(2) are linearly independent in the intervals (-∞, -√2), (-√2, 0), (0, √2), and (√2, +∞).

(c) Since x(1) and x(2) are linearly dependent for the values t = 0, t = -√2, and t = √2, it implies that the coefficients in the system of homogeneous differential equations satisfied by x(1) and x(2) are not all zero. At least one of the coefficients must be non-zero.

(d) The system of equations x': = 9t^2x is already given.

(e) The general solution of the differential equation x' = 9t^2x can be found by solving the characteristic equation. The characteristic equation is r^2 = 9t^2, which has roots r = ±3t. Therefore, the general solution is:

x(t) = C1t^3 + C2/t^3,

where C1 and C2 are arbitrary constants.

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The inverse Laplace transform of -15/(s² + 9) is -15sin(3t),

and the inverse Laplace transform of 15/(s² - 9) is 15sinh(3t).

To solve the given initial value problem using the Laplace transform, we'll apply the Laplace transform to the differential equation and use the initial conditions to find the solution.

Taking the Laplace transform of the differential equation y⁴ - 81y = 0, we have:

s⁴Y(s) - s³y(0) - s²y'(0) - sy''(0) - y'''(0) - 81Y(s) = 0,

where Y(s) is the Laplace transform of y(t).

Substituting the initial conditions y(0) = 14, y'(0) = 27, y''(0) = 72, and y'''(0) = 135, we get:

s⁴Y(s) - 14s³ - 27s² - 72s - 135 - 81Y(s) = 0.

Rearranging the equation, we have:

Y(s) = (14s³ + 27s² + 72s + 135) / (s⁴ + 81).

Now, we need to find the inverse Laplace transform of Y(s) to obtain the solution y(t). This can be done by using partial fraction decomposition and consulting Laplace transform tables or using symbolic algebra software.

Please note that due to the complexity of the inverse Laplace transform, the solution for y(t) cannot be calculated without knowing the specific values of the partial fraction decomposition or using specialized software.

To find the inverse Laplace transform of Y(s), we can perform partial fraction decomposition.

The denominator s⁴ + 81 can be factored as (s² + 9)(s² - 9), which gives us:

Y(s) = (14s³ + 27s² + 72s + 135) / [(s² + 9)(s² - 9)].

We can write the right side of the equation as the sum of two fractions:

Y(s) = A/(s² + 9) + B/(s² - 9),

where A and B are constants that we need to determine.

To find A, we multiply both sides by (s² + 9) and then evaluate the equation at s = 0:

14s³ + 27s² + 72s + 135 = A(s² - 9) + B(s² + 9).

Plugging in s = 0, we get:

135 = -9A + 9B.

Similarly, to find B, we multiply both sides by (s² - 9) and evaluate the equation at s = 0:

14s³ + 27s² + 72s + 135 = A(s² - 9) + B(s² + 9).

Plugging in s = 0, we get:

135 = -9A + 9B.

We now have a system of two equations:

-9A + 9B = 135,

-9A + 9B = 135.

Solving this system of equations, we find A = -15 and B = 15.

Now, we can rewrite Y(s) as:

Y(s) = -15/(s² + 9) + 15/(s² - 9).

Using Laplace transform tables or software, we can find the inverse Laplace transform of each term.

Therefore, the solution y(t) is:

y(t) = -15sin(3t) + 15sinh(3t).

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In the formulation of a mathematical model for an evaporator, the following are five key assumptions:

1. Constant volume and density of the system.

2. Evaporation takes place only from the surface of the liquid.

3. The transfer of heat takes place only through conduction.

4. The heat transfer coefficient does not change with time.

5. The properties of the liquid are constant throughout the system.

Derivation of the total mass balance equation:

The total mass balance equation relates the rate of mass flow of material entering a system to the rate of mass flow leaving the system.

It is given by:

Rate of Mass Flow In - Rate of Mass Flow Out = Rate of Accumulation

Assuming that the evaporator operates under steady-state conditions, the rate of accumulation of mass is zero.

Hence, the mass balance equation reduces to:

Rate of Mass Flow In = Rate of Mass Flow Out

Let's assume that the mass flow rate of the feed stream is represented by m1 and the mass flow rate of the product stream is represented by m₂.

Therefore, the mass balance equation for the evaporator becomes:

m₁ = m₂ + me

Where me is the mass of water that has been evaporated. This equation is useful in determining the amount of water evaporated from the system.

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The second equation after eliminating the variable x is 0.5y + 3.5z = 11.5.

To solve the system of linear equations using Gaussian elimination, we'll perform row operations to eliminate variables one by one. Let's start with the given system of equations:

Eliminate x from equations 2 and 3:

To eliminate x, we'll multiply equation 1 by -1.5 and add it to equation 2. We'll also multiply equation 1 by -2 and add it to equation 3.

New equation 3: (4x - 2y + 3z) - 2 * (2x + y - z) = -9 - 2 * 1

Simplifying the equation 3: 4x - 2y + 3z - 4x - 2y + 2z = -9 - 2

Simplifying further: -0.5y - 3.5z = -11.5

So, the second equation after eliminating the variable x is 0.5y + 3.5z = 11.5.

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The vector product (cross product) of M and N is -10j + 155k - 362j - 6k + 24i.

The vector product (cross product) of two vectors M and N is calculated using the determinant method. The cross product of M and N is denoted as M x N. To calculate M x N, we can use the following formula,

M x N = (2 * (-1) - (-4) * (-6))i + ((-4) * 61 - 31 * (-1))j + (31 * (-6) - 2 * 61)k

Simplifying the equation, we get,

M x N = -10j + 155k - 362j - 6k + 24i

Therefore, the vector product M x N is -10j + 155k - 362j - 6k + 24i.

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The eigenvalues of the matrix A = [ 3 0 1 2 2 2 -2 1 2 ] are:

λ₁ = (5 - √17)/2 and λ₂ = (5 + √17)/2

To find the eigenvalues (A) of the matrix A = [ 3 0 1 2 2 2 -2 1 2 ], we use the following formula:

Eigenvalues (A) = |A - λI

|where λ represents the eigenvalue, I represents the identity matrix and |.| represents the determinant.

So, we have to find the determinant of the matrix A - λI.

Thus, we will substitute A = [ 3 0 1 2 2 2 -2 1 2 ] and I = [1 0 0 0 1 0 0 0 1] to get:

| A - λI | = | 3 - λ 0 1 2 2 - λ 2 -2 1 2 - λ |

To find the determinant of the matrix, we use the cofactor expansion along the first row:

| 3 - λ 0 1 2 2 - λ 2 -2 1 2 - λ | = (3 - λ) | 2 - λ 2 1 2 - λ | + 0 | 2 - λ 2 1 2 - λ | - 1 | 2 2 1 2 |

Therefore,| A - λI | = (3 - λ) [(2 - λ)(2 - λ) - 2(1)] - [(2 - λ)(2 - λ) - 2(1)] = (3 - λ) [(λ - 2)² - 2] - [(λ - 2)² - 2] = (λ - 2) [(3 - λ)(λ - 2) + λ - 4]

Now, we find the roots of the equation, which will give the eigenvalues:

λ - 2 = 0 ⇒ λ = 2λ² - 5λ + 2 = 0

The two roots of the equation λ² - 5λ + 2 = 0 are:

λ₁ = (5 - √17)/2 and λ₂ = (5 + √17)/2

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g is both injective and surjective, i.e., g is bijective.

Given the function f: R → R, we define g(x) = 1/3(f(x) + 4).

Injectivity:

If f is injective, then for every x, y in R, f(x) = f(y) implies x = y.

If g(x) = g(y), then f(x) + 4 = 3g(x) = 3g(y) = f(y) + 4.

Hence, f(x) = f(y), which implies x = y.

So, g(x) = g(y) implies x = y. Therefore, g is injective.

Surjectivity:

If f is surjective, then for every y in R, there is an x in R such that f(x) = y.

For any z ∈ R, g(x) = z can be written as 1/3(f(x) + 4) = z ⇒ f(x) = 3z - 4.

Since f is surjective, there exists an x in R such that f(x) = 3z - 4.

Therefore, g(x) = z. Hence, g is surjective.

Therefore, g is bijective since it is both injective and surjective.

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The formula \forall x \exists y(x\oplus y=c) in first-order logic states that for any value of x, there exists a value of y such that the binary function \oplus of x and y is equal to a constant c.

To determine the interpretations for which this formula is valid, we need to consider the possible interpretations of the binary function \oplus and the constant c.

Since the formula does not provide specific information about the binary function \oplus or the constant c, we cannot determine a single interpretation for which the formula is valid. The validity of the formula depends on the specific interpretation of \oplus and the constant c.

To evaluate the validity of the formula, we need additional information about the properties and constraints of the binary function \oplus and the constant c. Without this information, we cannot determine the interpretation(s) for which the formula is valid.

In summary, the validity of the formula \forall x \exists y(x\oplus y=c) depends on the specific interpretation of the binary function \oplus and the constant c, and without further information, we cannot determine a specific interpretation for which the formula is valid.

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The given system of differential equations is:

Jx' = Ax + By

y' = Cx + Dy

To find the solution to the given system of differential equations, let's first rewrite the system in matrix form:

Jx' = A*x + B*y

y' = C*x + D*y

where

J = [-67 -210]

A = [21 66]

B = [0]

C = [0]

D = [1]

Now, let's solve the system using the initial conditions. We'll differentiate both sides of the second equation with respect to t:

y' = C*x + D*y

y'' = C*x' + D*y'

Substituting the values of C, x', and y' from the first equation, we have:

y'' = 0*x + 1*y' = y'

Now, we have a second-order ordinary differential equation for y(t):

y'' - y' = 0

This is a homogeneous linear differential equation with constant coefficients. The characteristic equation is:

r^2 - r = 0

Factoring the equation, we have:

r(r - 1) = 0

So, the solutions for r are r = 0 and r = 1.

Therefore, the general solution for y(t) is:

y(t) = c1*e^0 + c2*e^t

y(t) = c1 + c2*e^t

Now, let's solve for c1 and c2 using the initial conditions:

At t = 0, y(0) = -5:

-5 = c1 + c2*e^0

-5 = c1 + c2

At t = 0, y'(0) = 17:

17 = c2*e^0

17 = c2

From the second equation, we find that c2 = 17. Substituting this into the first equation, we get:

-5 = c1 + 17

c1 = -22

So, the particular solution for y(t) is:

y(t) = -22 + 17*e^t

Now, let's solve for x(t) using the first equation:

Jx' = A*x + B*y

Substituting the values of J, A, B, and y(t), we have:

[-67 -210] * x' = [21 66] * x + [0] * (-22 + 17*e^t)

[-67 -210] * x' = [21 66] * x - [0]

[-67 -210] * x' = [21 66] * x

Now, let's solve this system of linear equations for x(t). However, we can see that the second equation is a multiple of the first equation, so it doesn't provide any new information. Therefore, we can ignore the second equation.

Simplifying the first equation, we have:

-67 * x' - 210 * x' = 21 * x

Combining like terms:

-277 * x' = 21 * x

Dividing both sides by -277:

x' = -21/277 * x

Integrating both sides with respect to t:

∫(1/x) dx = ∫(-21/277) dt

ln|x| = (-21/277) * t + C

Taking the exponential of both sides:

|x| = e^((-21/277) * t + C)

Since x can be positive or negative, we have two cases:

Case 1: x > 0

x = e^((-21/277) * t + C)

Case 2: x < 0

x = -e^((-21/277) * t + C)

Therefore, the solution to the

given system of differential equations is:

x(t) = C1 * e^((-21/277) * t) for x > 0

x(t) = -C2 * e^((-21/277) * t) for x < 0

y(t) = -22 + 17 * e^t

where C1 and C2 are constants determined by additional initial conditions or boundary conditions.

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Answer:

-6x² + 6 - 2x = x

-6x² - 3x + 6 = 0

2x² + x - 2 = 0

x = (-1 + √(1² - 4(2)(-2)))/(2×2)

= (-1 + √17)/4

The maximum height the pole-vaulter can clear is approximately 4.06 meters.

To find the maximum height the pole-vaulter can clear, we can use the principle of conservation of mechanical energy. At the takeoff point, the vaulter possesses only kinetic energy, which can be converted into potential energy at the maximum height.

The formula for gravitational potential energy is:

Potential energy =[tex]mass \times gravitational acceleration \times height[/tex]

Since the vaulter's mass is not given, we can assume it cancels out when comparing different heights. Thus, we only need to consider the change in height.

Using the conservation of mechanical energy:

Kinetic energy at takeoff = Potential energy at maximum height

[tex](1/2) \times mass \times velocity^2 = mass \times gravitational acceleration \times height[/tex]

We can cancel out the mass and rearrange the equation to solve for height:

height = [tex](velocity^2) / (2 \times gravitational acceleration)[/tex]

Substituting the given values:

height = [tex](9.15^2) / (2 \times 9.8[/tex]) ≈ 4.06 meters

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The vectors related to given points are AB <6, 4> and BC <4, 6>, respectively.

In this problem we must determine the equations of two vectors represented by a figure, each vector is between two consecutive points set on Cartesian plane. The definition of a vector is introduced below:

AB <x, y> = B(x, y) - A(x, y)

Where:

Now we proceed to determine each vector:

AB <x, y> = (6, 4) - (0, 0)

AB <x, y> = (6, 4)

AB <6, 4>

BC <x, y> = (10, 10) - (6, 4)

BC <x, y> = (4, 6)

BC <4, 6>

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The probability that all 4 oranges picked are not rotten is 0.857.

To calculate the probability, we need to consider the number of favorable outcomes (picking 4 non-rotten oranges) and the total number of possible outcomes (picking any 4 oranges).

The number of favorable outcomes can be calculated using the concept of combinations. Since the customer is picking at random, the order in which the oranges are picked does not matter. We can use the combination formula, nCr, to calculate the number of ways to choose 4 non-rotten oranges from the total number of non-rotten oranges in the crate. In this case, n is the number of non-rotten oranges and r is 4.

The total number of possible outcomes is the number of ways to choose 4 oranges from the total number of oranges in the crate. This can also be calculated using the combination formula, where n is the total number of oranges in the crate (including both rotten and non-rotten oranges) and r is 4.

By dividing the number of favorable outcomes by the total number of possible outcomes, we can find the probability of picking 4 non-rotten oranges.

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To complete each ordered pair using the function y = 200 tan(x) on the interval 0° ≤ x ≤ 141°, we need to substitute different values of x within that interval and calculate the corresponding values of y. Let's calculate the ordered pairs by rounding the answers to the nearest whole number:

1. For x = 0°:

  y = 200 tan(0°) = 0

  The ordered pair is (0, 0).

2. For x = 45°:

  y = 200 tan(45°) = 200

  The ordered pair is (45, 200).

3. For x = 90°:

  y = 200 tan (90°) = ∞ (undefined since the tangent of 90° is infinite)

  The ordered pair is (90, undefined).

4. For x = 135°:

  y = 200 tan (135°) = -200

  The ordered pair is (135, -200).

5. For x = 141°:

  y = 200 tan (141°) = -13

  The ordered pair is (141, -13).

So, the completed ordered pairs (rounded to the nearest whole number) are:

(0, 0), (45, 200), (90, undefined), (135, -200), (141, -13).

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Answer:

SSS, because a segment is congruent to itself.

The future value of the ordinary annuity is $122,304.74 and n is the number of compounding periods.

To find the future value of an ordinary annuity with a given payment and interest rate, we can use the formula:

where FV is the future value, PMT is the payment amount, r is the interest rate per compounding period.

Given:

Substituting these values into the formula, we get:

Calculating this expression will give us the future value of the ordinary annuity.

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The output for t = 1.25 for the given differential equation 2y"(t) + 214y(t) = et + et with conditions is equal to y(1.25) = 0.

To solve the given differential equation 2y"(t) + 214y(t) = et + et, with initial conditions y(0) = 0 and y'(0) = 0,

find the particular solution and then apply the initial conditions to determine the specific solution.

The right-hand side of the equation consists of two terms, et and et.

Since they have the same form, assume a particular solution of the form yp(t) = At[tex]e^t[/tex], where A is a constant to be determined.

Now, let's find the first and second derivatives of yp(t),

yp'(t) = A([tex]e^t[/tex] + t[tex]e^t[/tex])

yp''(t) = A(2[tex]e^t[/tex] + 2t[tex]e^t[/tex])

Substituting these derivatives into the differential equation,

2(A(2[tex]e^t[/tex] + 2t[tex]e^t[/tex])) + 214(At[tex]e^t[/tex]) = et + et

Simplifying the equation,

4A[tex]e^t[/tex] + 4At[tex]e^t[/tex] + 214At[tex]e^t[/tex]= 2et

Now, equating the coefficients of et on both sides,

4A + 4At + 214At = 2t

Matching the coefficients of t on both sides,

4A + 4A + 214A = 0

Solving this equation, we find A = 0.

The particular solution is yp(t) = 0.

Now, the general solution is given by the sum of the particular solution and the complementary solution:

y(t) = yp(t) + y c(t)

Since yp(t) = 0, the general solution simplifies to,

y(t) = y c(t)

To find y c(t),

solve the homogeneous differential equation obtained by setting the right-hand side of the original equation to zero,

2y"(t) + 214y(t) = 0

The characteristic equation is obtained by assuming a solution of the form yc(t) = [tex]e^{(rt)[/tex]

2r² + 214 = 0

Solving this quadratic equation,

find two distinct complex roots: r₁ = i√107 and r₂ = -i√107.

The general solution of the homogeneous equation is then,

yc(t) = C₁[tex]e^{(i\sqrt{107t} )[/tex] + C₂e^(-i√107t)

Applying the initial conditions y(0) = 0 and y'(0) = 0:

y(0) = C₁ + C₂ = 0

y'(0) = C₁(i√107) - C₂(i√107) = 0

From the first equation, C₂ = -C₁.

Substituting this into the second equation, we get,

C₁(i√107) + C₁(i√107) = 0

2C₁(i√107) = 0

This implies C₁ = 0.

Therefore, the specific solution satisfying the initial conditions is y(t) = 0.

Now, to obtain the output for t = 1.25, we substitute t = 1.25 into the specific solution:

y(1.25) = 0

Hence, the output for t = 1.25 for the differential equation is y(1.25) = 0.

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This is the equation of the tangent line at t = -1 for the given parametric equation. It uses an independent variable known as a parameter and dependent variables that are defined as continuous functions of the parameter and independent of other variables.

To find the equation of a parabola with a given directrix and focus, we can use the standard form of the equation for a parabola:

1. The directrix is a vertical line, so the equation of the directrix can be written as x = -10.5.
The focus is given as (-9.5, 7).

The vertex of the parabola will lie halfway between the directrix and the focus, so the x-coordinate of the vertex is the average of -10.5 and -9.5, which is -10.
Since the parabola is symmetric with respect to its vertex, the y-coordinate of the vertex will be the same as the y-coordinate of the focus, which is 7.

Using the standard form of the equation for a parabola, we can write the equation as follows:

(x - h)^2 = 4p(y - k)

where (h, k) is the vertex and p is the distance between the vertex and the focus.

In this case, the vertex is (-10, 7) and the focus is (-9.5, 7), so p = 0.5.

Plugging in the values, we get:

(x - (-10))^2 = 4(0.5)(y - 7)

Simplifying, we have:

(x + 10)^2 = 2(y - 7)

This is the equation of the parabola.

2. To find the slope of the tangent line, we need to find the derivative of y with respect to x, dy/dx.

Using the chain rule, we have:

dy/dx = (dy/dt) / (dx/dt)

Differentiating the given parametric equations, we get:

dx/dt = 6
dy/dt = 4t^3

Plugging these values into the chain rule formula, we have:

dy/dx = (4t^3) / 6

Simplifying, we get:

dy/dx = (2/3)t^3

To find the slope of the tangent line at t = -1, we substitute t = -1 into the equation:

dy/dx = (2/3)(-1)^3
      = (2/3)(-1)
      = -2/3

So, the slope of the tangent line at t = -1 is -2/3.

To find the equation of the tangent line, we can use the point-slope form of the equation:

y - y1 = m(x - x1)

where (x1, y1) is a point on the line and m is the slope.

Since we are looking for the equation of the tangent line at t = -1, we can substitute t = -1 into the parametric equations to find the corresponding point on the curve:

x = 6t
x = 6(-1)
x = -6

y = t^4
y = (-1)^4
y = 1

Using the point (-6, 1) and the slope -2/3, we can write the equation of the tangent line as:

y - 1 = (-2/3)(x - (-6))

Simplifying, we have:

y - 1 = (-2/3)(x + 6)

This is the equation of the tangent line at t = -1 for the given parametric equation.

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a). Since both second partial derivatives are positive, we conclude that the critical points are minimum points.

In both b) and c), we have omitted the constant of integration, denoted by + C, which represents the family of antiderivatives.

a) To find the critical points of the function f(x, y) = 2x^2 + 2xyy + 2y^2 - 6x, we need to find the partial derivatives with respect to x and y and set them equal to zero.

Partial derivative with respect to x (df/dx):

df/dx = 4x + 2yy - 6

Partial derivative with respect to y (df/dy):

df/dy = 4y + 2xy

Setting df/dx = 0 and df/dy = 0, we have:

4x + 2yy - 6 = 0 ----(1)

4y + 2xy = 0 ----(2)

From equation (2), we can factor out 2y:

2y(2 + x) = 0

This gives us two possibilities:

y = 0

2 + x = 0, which means x = -2

Now we substitute these values of x and y into equation (1):

For y = 0:

4x - 6 = 0

4x = 6

x = 6/4

x = 3/2

For x = -2:

4(-2) + 2yy - 6 = 0

-8 + 2yy - 6 = 0

2yy = 14

yy = 7

y = ±√7

Therefore, the critical points are (3/2, 0) and (-2, ±√7).

To determine whether these points are minimum or maximum, we need to find the second partial derivatives and evaluate them at the critical points.

Second partial derivative with respect to x (d^2f/dx^2):

d^2f/dx^2 = 4

Second partial derivative with respect to y (d^2f/dy^2):

d^2f/dy^2 = 4

Since both second partial derivatives are positive, we conclude that the critical points are minimum points.

b) To find the critical points of the function f(x, y) = -2x^2 + 8x - 3y^2 + 24y + 7, we follow a similar process.

Partial derivative with respect to x (df/dx):

df/dx = -4x + 8

Partial derivative with respect to y (df/dy):

df/dy = -6y + 24

Setting df/dx = 0 and df/dy = 0, we have:

-4x + 8 = 0 ----(1)

-6y + 24 = 0 ----(2)

From equation (1), we can solve for x:

-4x = -8

x = 2

From equation (2), we can solve for y:

-6y = -24

y = 4

Therefore, the critical point is (2, 4).

To determine whether this point is a minimum or maximum, we again find the second partial derivatives:

Second partial derivative with respect to x (d^2f/dx^2):

d^2f/dx^2 = -4

Second partial derivative with respect to y (d^2f/dy^2):

d^2f/dy^2 = -6

Since both second partial derivatives are negative, we conclude that the critical point (2, 4) is a maximum point.

Integrals:

a) ∫(5x + 2) dx

To integrate this expression, we use the power rule of integration:

∫(5x + 2) dx = (5/2)x^2 + 2x + C

b) ∫x dx

Using the power rule of integration:

∫x dx = (1/2)x^2 + C

c) ∫2x dx

Using the power rule of integration:

∫2x dx = x^2 + C

The integration constant (+ C), which stands for the family of antiderivatives, has been left out of both b) and c).

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Jin's total assets are $8,794. Her liabilities are $6,292. Her net worth is $2,502.

To calculate Jin's net worth, we subtract her liabilities from her total assets.

Total Assets - Liabilities = Net Worth

Given:

Total Assets = $8,794

Liabilities = $6,292

Substituting the values, we have:

Net Worth = $8,794 - $6,292

Net Worth = $2,502

Therefore, Jin's net worth is $2,502.

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A. The numeral 481 in base five is written as 2011.

B. To convert the base-ten numeral 481 to base five, we need to divide it by powers of five and determine the corresponding digits in the base-five system.

Step 1: Divide 481 by 5 and note the quotient and remainder.

481 ÷ 5 = 96 with a remainder of 1. Write down the remainder, which is the least significant digit.

Step 2: Divide the quotient (96) obtained in the previous step by 5.

96 ÷ 5 = 19 with a remainder of 1. Write down this remainder.

Step 3: Divide the new quotient (19) by 5.

19 ÷ 5 = 3 with a remainder of 4. Write down this remainder.

Step 4: Divide the new quotient (3) by 5.

3 ÷ 5 = 0 with a remainder of 3. Write down this remainder.

Now, we have obtained the remainder in reverse order: 3141.

Hence, the numeral 481 in base five is represented as 113.

Note: The explanation assumes that the numeral in the indicated bases is meant to be the answer for part (a) only.

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The area and the perimeter of the compound figure are 95 square units and 43 units, respectively.

In this question we must compute the area of a compound figure formed by four squares of different size. The area formula of a square are listed below:

A = l²

Where l is the side length of the square.

Now we proceed to determine the area of the compound figure by addition of areas:

A = 1² + 2² + 3² + 9²

A = 1 + 4 + 9 + 81

A = 14 + 81

A = 95

And the perimeter of the figure is equal to:

p = 3 · 3 + 4 · 1 + 6 + 3 · 9

p = 9 + 4 + 6 + 27

p = 16 + 27

p = 43

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Correct option is y = -x-1 + e^x.

The given linear ODE:

y' - y = x; y(0) = 0 can be solved by the following method:

We first need to find the integrating factor of the given differential equation. We will find it using the following formula:

IF = e^integral of P(x) dx

Where P(x) is the coefficient of y (the function multiplying y).

In the given differential equation, P(x) = -1, hence we have,IF = e^-x We multiply this IF to both sides of the equation. This will reduce the left side to a product of the derivative of y and IF as shown below:

e^-x y' - e^-x y = xe^-x We can simplify the left side by applying the product rule of differentiation as shown below:

d/dx (e^-x y) = xe^-x We can integrate both sides to obtain the solution of the differential equation. The solution to the given linear ODE:y' - y = x; y(0) = 0 is:y = -x-1 + e^x + C where C is the constant of integration. Substituting y(0) = 0, we get,0 = -1 + 1 + C

Therefore, C = 0

Hence, the solution to the given differential equation: y = -x-1 + e^x

So, the correct option is y = -x-1 + e^x.

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There exist A, B ∈ Z that satisfy the equation A² + B² = 2(a² + b²).

To prove the statement using complex numbers, let's start by representing the integers a and b as complex numbers:

a = a + 0i

b = b + 0i

Now, we can rewrite the equation a² + b² = 2(a² + b²) in terms of complex numbers:

(a + 0i)² + (b + 0i)² = 2((a + 0i)² + (b + 0i)²)

Expanding the complex squares, we get:

(a² + 2ai + (0i)²) + (b² + 2bi + (0i)²) = 2((a² + 2ai + (0i)²) + (b² + 2bi + (0i)²))

Simplifying, we have:

a² + 2ai - b² - 2bi = 2a² + 4ai - 2b² - 4bi

Grouping the real and imaginary terms separately, we get:

(a² - b²) + (2ai - 2bi) = 2(a² - b²) + 4(ai - bi)

Now, let's choose A and B such that their real and imaginary parts match the corresponding sides of the equation:

A = a² - b²

B = 2(a - b)

Substituting these values back into the equation, we have:

A + Bi = 2A + 4Bi

Equating the real and imaginary parts, we get:

A = 2A

B = 4B

Since A and B are integers, we can see that A = 0 and B = 0 satisfy the equations. Therefore, there exist A, B ∈ Z that satisfy the equation A² + B² = 2(a² + b²).

This completes the proof.

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The solutions to the ratio problems are as follows:

1. Ratio of nonfiction to fiction 1:2

2. Number of hours rested is 175

3. Ratio of pants to shirts is 3:5

4. The ratio of medium to large shirts is 7:3

We can determine the ratio by expressing the figures as numerator and denominator and dividing them with a common factor until no more division is possible.

In the first instance, we are told to find the ratio between nonfiction and fiction will be 2500/5000. When these are divided by 5, the remaining figure would be 1/2. So, the ratio is 1:2.

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The equation of the least-squares line that best fits the given data points is y = 2 + (2/3)x.

To find the equation of the least-squares line that best fits the given data points, we can use the method of least squares to minimize the sum of the squared differences between the actual y-values and the predicted y-values on the line.

Calculate the mean of the x-values and the mean of the y-values.

[tex]\bar x[/tex] = (0 + 1 + 2 + 3) / 4 = 1.5

[tex]\bar y[/tex]= (2 + 2 + 5 + 5) / 4 = 3.5

Calculate the deviations from the means for both x and y.

x₁ = 0 - 1.5 = -1.5

x₂ = 1 - 1.5 = -0.5

x₃ = 2 - 1.5 = 0.5

x₄ = 3 - 1.5 = 1.5

y₁ = 2 - 3.5 = -1.5

y₂ = 2 - 3.5 = -1.5

y₃ = 5 - 3.5 = 1.5

y₄ = 5 - 3.5 = 1.5

Calculate the sum of the products of the deviations from the means.

Σ(xᵢ * yᵢ) = (-1.5 * -1.5) + (-0.5 * -1.5) + (0.5 * 1.5) + (1.5 * 1.5) = 4

Calculate the sum of the squared deviations of x.

Σ(xᵢ²) = (-1.5)² + (-0.5)² + (0.5)² + (1.5)² = 6

Calculate the least-squares slope (B₁) using the formula:

B₁ = Σ(xᵢ * yᵢ) / Σ(xᵢ²) = 4 / 6 = 2/3

Calculate the y-intercept (Bo) using the formula:

Bo = [tex]\bar y[/tex] - B₁ * [tex]\bar x[/tex] = 3.5 - (2/3) * 1.5 = 2

Therefore, the equation of the least-squares line that best fits the given data points is y = 2 + (2/3)x.

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The present value of the annuity is $4,813.52.

To calculate the present value of the annuity, we can use the formula for the present value of an increasing annuity:

PV = C * (1 - (1 + r)^(-n)) / (r - g)

Where:

PV = Present Value

C = Payment amount at time t=1

r = Interest rate

n = Number of payments

g = Growth rate of payments

In this case:

C = $300

r = 8% or 0.08

n = Number of payments = Last payment amount - First payment amount / Growth rate + 1 = ($1000 - $300) / $50 + 1 = 14

g = Growth rate of payments = $50

Plugging in these values into the formula, we get:

PV = $300 * (1 - (1 + 0.08)^(-14)) / (0.08 - 0.05) = $4,813.52

Therefore, the present value of this annuity is $4,813.52. This means that if we were to invest $4,813.52 today at an interest rate of 8%, it would grow to match the future cash flows of the annuity.

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The function that has a period of 4π and an amplitude of 8 is y = 8sin(2θ), which is option (H).

The general form of the equation of a sine function is given as f(θ) = a sin(bθ + c) + d

where, a is the amplitude of the function, the distance between the maximum or minimum value of the function from the midline, b is the coefficient of θ, which determines the period of the function and is calculated as:

Period = 2π / b.c

which is the phase shift of the function, which is calculated as:

Phase shift = -c / bd

which is the vertical shift or displacement from the midline. The period of the function is 4π, and the amplitude is 8. Therefore, the function that meets these conditions is given as:

f(θ) = a sin(bθ + c) + df(θ) = 8 sin(bθ + c) + d

We know that the period is given by:

T = 2π / b

where T = 4π4π = 2π / bb = 1 / 2

The equation now becomes:

f(θ) = 8sin(1/2θ + c) + d

The amplitude of the function is 8. Hence

= 8 or -8

The function becomes:

f(θ) = 8sin(1/2θ + c) + df(θ) = -8sin(1/2θ + c) + d

We can take the positive value of a since it is the one given in the answer options. Also, d is not important since it does not affect the period and amplitude of the function.

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This shows that cx is also a vector in U since it satisfies the equation Ax = Bx.

To show that U = {x in R^n | Ax = Bx} is a subspace of R^n, we need to demonstrate that it satisfies three conditions:

U is non-empty: Since A and B are matrices, there will always be at least one vector x that satisfies Ax = Bx, namely the zero vector.

U is closed under vector addition: Let x1 and x2 be any two vectors in U. We want to show that their sum, x1 + x2, is also in U.

From the definition of U, we have Ax1 = Bx1 and Ax2 = Bx2. Now, consider the sum of these two equations:

Ax1 + Ax2 = Bx1 + Bx2

Factoring out x1 and x2 on the left side gives:

A(x1 + x2) = B(x1 + x2)

This shows that x1 + x2 is also a vector in U since it satisfies the equation Ax = Bx.

U is closed under scalar multiplication: Let x be any vector in U, and let c be any scalar. We want to show that the scalar multiple cx is also in U.

From the definition of U, we have Ax = Bx. Now, consider the equation:

A(cx) = B(cx)

Using the properties of matrix multiplication and scalar multiplication, we can rewrite this as:

(cA)x = (cB)x

Since U satisfies all three conditions, it is a subspace of R^n.

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