Consider the steady state temperature u(r, z) in a solid cylinder of radius r = c with bottom z = 0 and top z= L. Suppose that u= u(r, z) satisfies Laplace's equation. du lou d'u + = 0. + dr² r dr dz² [6 Marks] We can study the problem such that the cylinder is semi-infinte, i.e. L= +0o. If we consider heat transfer on this cylinder we have the boundary conditions u(r,0) = o. hu(c,z)+ Ur(C,z)=0, and further we require that u(r, 2) is bounded as z-+00. Find an expression for the steady state temperature u = u(r, z). End of assignment

The auxiliary equation for the homogeneous problem y" + 2y + 5y = 0 is ar² + br + c = 0.The roots of the auxiliary equation are complex conjugates with no real roots.A fundamental set of solutions for the homogeneous problem is ye = C₁e^(αx)cos(βx) + C₂e^(αx)sin(βx), where α and β are constants.

To solve the homogeneous problem y" + 2y + 5y = 0, we first find the auxiliary equation by substituting y = e^(rx) into the differential equation.

This gives us ar² + br + c = 0.

In this case, the coefficients a, b, and c are 1, 2, and 5, respectively.

Solving the auxiliary equation, we find that the roots are complex conjugates with no real roots.

Let's denote the roots as α ± βi, where α and β are real numbers.

Then, a fundamental set of solutions for the homogeneous problem is given by ye = C₁e^(αx)cos(βx) + C₂e^(αx)sin(βx), where C₁ and C₂ are arbitrary constants.

Next, to find a particular solution to the non-homogeneous problem y" + 2y + 5y = 20cos(x), we use the method of undetermined coefficients. We assume a particular solution of the form yp = Acos(x) + Bsin(x), where A and B are coefficients to be determined.

By substituting yp into the differential equation, we solve for the coefficients A and B.

After finding the particular solution yp, the general solution to the non-homogeneous problem is given by y = ye + yp.
Finally, to solve the initial value problem (IVP) with the given initial conditions y(0) = 5 and y'(0) = 5, we substitute these values into the general solution and solve for the arbitrary constants.

This will give us the unique solution to the IVP.

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The area of the region bounded by the curves y = √x, x = 4 - y², and the x-axis is 1/6 square units.

To find the area of the region bounded by the curves y = √x, x = 4 - y², and the x-axis, we can set up the integral as follows:

A = ∫[a,b] (f(x) - g(x)) dx

where f(x) is the upper curve and g(x) is the lower curve.

In this case, the upper curve is y = √x and the lower curve is x = 4 - y².

To find the limits of integration, we set the two curves equal to each other:

√x = 4 - y²

Solving for y, we get:

y = ±√(4 - x)

To find the limits of integration, we need to determine the x-values at which the curves intersect.

Setting √x = 4 - y², we have:

x = (4 - y²)²

Substituting y = ±√(4 - x), we get:

x = (4 - (√(4 - x))²)²

Expanding and simplifying, we have:

x = (4 - (4 - x))²

x = x²

This gives us x = 0 and x = 1 as the x-values of intersection.

So, the limits of integration are a = 0 and b = 1.

Now, we can calculate the area using the integral:

A = ∫[0,1] (√x - (4 - y²)) dx

To simplify the integral, we need to express (4 - y²) in terms of x.

From the equation y = ±√(4 - x), we can solve for y²:

y² = 4 - x

Substituting this into the integral, we have:

A = ∫[0,1] (√x - (4 - 4 + x)) dx

A = ∫[0,1] (√x - x) dx

Integrating, we get:

A = [(2/3)x^(3/2) - (1/2)x²] evaluated from 0 to 1

A = (2/3 - 1/2) - (0 - 0)

A = 1/6

Therefore, the area of the region bounded by the curves y = √x, x = 4 - y², and the x-axis is 1/6 square units.

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The graph of sinusoidal functions f (x) and g (x) are shown in graph.

And, the transformation of each function is shown below.

We have,

Two sinusoidal functions,

a. f(x) = - 3 cos(45(x - 2°)) + 5

b. g(x) = 2.5 sin(- 3(x+90° )) - 1

Now, Let's break down the transformations for each function:

a. For the function f(x) = -3⋅cos(45(x-2°)) + 5:

The coefficient in front of the cosine function, -3, represents the amplitude.

It determines the vertical stretching or compression of the graph. In this case, the amplitude is 3, but since it is negative, the graph will be reflected across the x-axis.

And, The period of the cosine function is normally 2π, but in this case, we have an additional factor of 45 in front of the x.

This means the period is shortened by a factor of 45, resulting in a period of 2π/45.

And, The phase shift is determined by the constant inside the parentheses, which is -2° in this case.

A positive value would shift the graph to the right, and a negative value shifts it to the left.

So, the graph is shifted 2° to the right.

Since, The constant term at the end, +5, represents the vertical shift of the graph. In this case, the graph is shifted 5 units up.

b. For the function g(x) = 2.5⋅sin(-3(x+90°)) - 1:

Here, The coefficient in front of the sine function, 2.5, represents the amplitude. It determines the vertical stretching or compression of the graph. In this case, the amplitude is 2.5, and since it is positive, there is no reflection across the x-axis.

Period: The period of the sine function is normally 2π, but in this case, we have an additional factor of -3 in front of the x.

This means the period is shortened by a factor of 3, resulting in a period of 2π/3.

Phase shift: The phase shift is determined by the constant inside the parentheses, which is +90° in this case.

A positive value would shift the graph to the left, and a negative value shifts it to the right.

So, the graph is shifted 90° to the left.

Vertical shift: The constant term at the end, -1, represents the vertical shift of the graph.

In this case, the graph is shifted 1 unit down.

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In a linear regression model, the linearity assumption refers to the relationship between the independent variables and the dependent variable.

It assumes that the dependent variable is a linear combination of the independent variables, with the coefficients representing the effect of each independent variable on the dependent variable.

In the given model, y^2 = ax_1 + bx_2 + u, the dependent variable y is squared, which introduces a non-linearity to the model. The presence of y^2 in the equation makes the model non-linear, as it cannot be expressed as a linear combination of the independent variables.

If you want to include quadratic or cubic terms for the dependent variable y, you would need to transform the model accordingly. For example, you could use a quadratic or cubic transformation of y, such as y^2, y^3, or even log(y), and include those transformed variables in the linear regression model along with the independent variables. This would allow you to capture non-linear relationships between the dependent variable and the independent variables in the model.

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The value of [tex](313 mod 14)^2[/tex] mod 21 is 4.

To calculate the given expression, let's break it down step by step:

Calculate (313 mod 14):

The modulus operator (%) returns the remainder when dividing the number 313 by 14.

So, 313 mod 14 = 5.

Calculate[tex](5^2 mod 21):[/tex]

Here, "^" denotes exponentiation. We need to calculate 5 raised to the power of 2, and then find the remainder when dividing the result by 21.

5^2 = 25.

25 mod 21 = 4.

Therefore, the value of[tex](313 mod 14)^2[/tex]mod 21 is 4.

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Answer:

  A, D

Step-by-step explanation:

You want to identify the pair of congruent triangles among those shown in the figure.

We observe all of the triangles are right triangles. For the purpose here, it is convenient to identify the triangles by the lengths of their legs:

Triangles A and D have the same leg lengths, so are congruent.

__

Additional comment

The LL or SAS congruence theorems apply.

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The positive integer isthat if gcd(a, b) = 4 and a2 + b2c2, then gcd(a, c) = 4.

a, b, and c are positive integers and we have to prove that if gcd(a, b) = 4 and a2+b2c2, then god(a, c)=4.So, assume that a, b, and c are positive integers where gcd(a, b) = 4 and a2+b2c2.

If we factor out 4 from a and b, we will get a = 4a' and b = 4b'.

Then a2 + b2c2 becomes (4a')2 + (4b')2c2 which simplifies to 16a'2 + 16b'2c2.

We can further simplify 16a'2 + 16b'2c2 by factoring out 16 and getting 16(a'2 + b'2c2).

Now, we know that gcd(a, b) = 4, so we can say that a and b are both divisible by 4.

Since a = 4a', we can say that 4|a and similarly since b = 4b', we can say that 4|b.

Now, let us assume that gcd(a, c) = k where k > 4.

We can say that a = ka' and c = kc' where k > 4.

Now, since a = 4a', we can say that 4|ka' or in other words, 4|a.

Also, we know that a2 + b2c2, so we can say that 4|a2.

Next, we can say that c = kc', so 4|kc'.Now, since a2 + b2c2, we know that 4 divides b2c2, so we can say that 4|b2 and 4|c2.

Now, we have 4|a2 and 4|b2c2, so we can say that 4|a2 + b2c2.

Now, we have already simplified a2 + b2c2 to 16(a'2 + b'2c2), so we can say that 4|16(a'2 + b'2c2).But, 4|16, so we can say that 4|a'2 + b'2c2, which means that gcd(a, b) >= 4

which contradicts our original assumption that gcd(a, b) = 4.

So, we can conclude that if gcd(a, b) = 4 and a2 + b2c2, then gcd(a, c) = 4.

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It is proven that both c and z as multiples of 2. This means gcd(a, c) = 2, and that gcd(a, c) = 4.

Let's prove statement (2) step by step:

Given information:

gcd(a, b) = 4

a² + b² = c²

To prove:

gcd(a, c) = 4

Proof by contradiction:

Assume that gcd(a, c) ≠ 4.

Since gcd(a, b) = 4, we can express a and b as:

a = 4x

b = 4y

Substituting these values in the given equation a² + b² = c², we have:

(4x)² + (4y)² = c²

16x² + 16y² = c²

4(4x² + 4y²) = c²

4(4(x² + y²)) = c²

We can see that c² is divisible by 4. Since a perfect square is divisible by 4 if and only if each of its prime factors appears with an even exponent, it means that c must also be divisible by 2.

Now, consider the prime factorization of c. Since c is divisible by 2, we can express it as c = 2z, where z is an integer.

Substituting this in the equation c^2 = 4(4(x² + y²)), we have:

(2z)² = 4(4(x² + y²))

4z² = 4(4(x² + y²))

z² = 4(x² + y²)

From this equation, we can see that z^2 is divisible by 4. This implies that z must also be divisible by 2.

Therefore, we have expressed both c and z as multiples of 2. This means gcd(a, c) = 2, contradicting our assumption that gcd(a, c) ≠ 4.

Hence, our assumption was incorrect, and we can conclude that gcd(a, c) = 4.

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Valid logical arguments are those where the conclusion logically follows from the premises, avoiding fallacies and being supported by evidence or reasoning. Option A and Option B are valid arguments, while Option C is invalid due to the fallacy of equivocation.

To determine which of the following options are valid logical arguments, we need to understand what makes an argument valid. A valid argument is one where the conclusion logically follows from the premises.

1. An argument is valid if it has a clear and valid logical structure, meaning that the conclusion logically follows from the premises. The argument must be structured in a way that ensures that if the premises are true, then the conclusion must also be true.

2. An argument is valid if it avoids logical fallacies, such as circular reasoning, false cause, straw man, or ad hominem attacks. Logical fallacies can weaken an argument and make it invalid.

3. An argument is valid if it is supported by evidence or reasoning. The premises of the argument should be true or highly probable, and the reasoning used to reach the conclusion should be sound.

Based on these criteria, let's evaluate the options:

- Option A: "All cats are mammals. Fluffy is a mammal. Therefore, Fluffy is a cat." This is a valid logical argument because the conclusion follows logically from the premises.

- Option B: "If it rains, the ground gets wet. The ground is wet. Therefore, it rained." This is also a valid logical argument because the conclusion logically follows from the premises.

- Option C: "Apples are fruits. Oranges are fruits. Therefore, apples are oranges." This is not a valid logical argument because the conclusion does not logically follow from the premises. It commits the fallacy of equivocation by equating two different things (apples and oranges).

In conclusion, the valid logical arguments are Option A: "All cats are mammals. Fluffy is a mammal. Therefore, Fluffy is a cat." and Option B: "If it rains, the ground gets wet. The ground is wet. Therefore, it rained." Option C: "Apples are fruits. Oranges are fruits. Therefore, apples are oranges." is not a valid logical argument.

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The maximum height reached by the projectile is 12 feet, and it hits the ground approximately 1.228 seconds and 3.772 seconds after being launched.

To find the maximum height reached by the projectile and the time it takes to hit the ground, we can analyze the given quadratic function H(t) = -16t^2 + 72t + 12.

The function H(t) represents the height of the projectile at time t seconds after its launch. The coefficient of t^2, which is -16, indicates that the path of the projectile is a downward-facing parabola due to the negative sign.

To determine the maximum height, we look for the vertex of the parabola. The x-coordinate of the vertex can be found using the formula x = -b / (2a), where a and b are the coefficients of t^2 and t, respectively. In this case, a = -16 and b = 72. Substituting these values, we get x = -72 / (2 * -16) = 9/2.

To find the corresponding y-coordinate (the maximum height), we substitute the x-coordinate into the function: H(9/2) = -16(9/2)^2 + 72(9/2) + 12. Simplifying this expression gives H(9/2) = -324 + 324 + 12 = 12 feet.

Hence, the maximum height reached by the projectile is 12 feet.

Next, to determine the time it takes for the projectile to hit the ground, we set H(t) equal to zero and solve for t. The equation -16t^2 + 72t + 12 = 0 can be simplified by dividing all terms by -4, resulting in 4t^2 - 18t - 3 = 0.

This quadratic equation can be solved using the quadratic formula: t = (-b ± √(b^2 - 4ac)) / (2a), where a = 4, b = -18, and c = -3. Substituting these values, we get t = (18 ± √(18^2 - 4 * 4 * -3)) / (2 * 4).

Simplifying further, we have t = (18 ± √(324 + 48)) / 8 = (18 ± √372) / 8.

Using a calculator, we find that the solutions are t ≈ 1.228 seconds and t ≈ 3.772 seconds.

Therefore, the projectile hits the ground approximately 1.228 seconds and 3.772 seconds after its launch.

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The minimum amount of money Harrington must save each month to have enough money for his first year of tuition at a private university is $875.

To calculate this, we subtract the amount his parents will give him ($8,000) from the total tuition cost ($21,500). This gives us the remaining amount Harrington needs to save, which is $13,500. Since he plans to save money for the next 4 years, we divide the remaining amount by 48 (4 years x 12 months) to find the monthly savings goal. Therefore, Harrington needs to save at least $875 per month to cover his first-year tuition expenses.

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The logical equivalence of the statement ¬p∧(p→q) is option b. ¬p, which is the negation of p.

To determine the logical equivalence of the statement ¬p∧(p→q), we can simplify it using logical equivalences and truth tables.

Using the definition of the implication (p→q ≡ ¬p∨q), we can rewrite the statement as ¬p∧(¬p∨q).

Applying the distributive law (¬p∧(¬p∨q) ≡ (¬p∧¬p)∨(¬p∧q)), we get (¬p∧¬p)∨(¬p∧q).

Using the idempotent law (¬p∧¬p ≡ ¬p) and the distributive law again ((¬p∧¬p)∨(¬p∧q) ≡ ¬p∨(¬p∧q)), we simplify it to ¬p∨(¬p∧q).

From the truth table, we can see that the expression ¬p∨(¬p∧q) evaluates to T (true) only when p is false (F) regardless of the value of q. Otherwise, it evaluates to F (false).

Therefore, Option b, which is the negation of p, is the logical equivalent of the statement "p" (pq).

Now, let's analyze the truth table for the expression ¬p∨(¬p∧q):

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The solution of v = (17 - b) / (a + 4)

1. Start with the given equation: av + 17 = -4v - b.

2. Move all terms containing v to one side of the equation: av + 4v = -17 - b.

3. Combine like terms: (a + 4)v = -17 - b.

4. Divide both sides of the equation by (a + 4) to solve for v: v = (-17 - b) / (a + 4).

5. Simplify the expression: v = (17 + (-b)) / (a + 4).

6. Rearrange the terms: v = (17 - b) / (a + 4).

Therefore, the solution for v is (17 - b) / (a + 4).

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Using the distributive propert the sum of the areas of these rectangles would give us the result, 756

To illustrate the distributive property using the situation of six cars traveling together, we can consider the total number of people in the cars. If each car has two people in front and three people in the back, we can calculate the total number of people by multiplying the number of cars by the sum of people in front and people in the back.

Using the distributive property, we can express this calculation as follows:

Total number of people = (2 + 3) × 6

This simplifies to:

Total number of people = 5 × 6

Total number of people = 30

Therefore, using the distributive property, we can calculate that there are 30 people in total among the six cars.

Regarding the 10% off sale at your favorite store, the discount will be the same regardless of the order in which the items are purchased. The distributive property of multiplication over addition states that multiplying a sum by a number is the same as multiplying each term in the sum by the number and then adding the results together. In this case, the discount applies to each item individually, so it does not matter if you apply the discount to each item separately or calculate the total cost and then apply the discount. The result will be the same.

Therefore, you will get the same discount regardless of the method you use, and this is related to the distributive property of arithmetic.

For the multiplication problem 27×28, using the partial-products method, we can break down the calculation as follows:

27 × 20 = 540

27 × 8 = 216

Then, we add the partial products together:

540 + 216 = 756

On graph paper or a tangle, we can draw an array with 27 rows and 28 items in each row. Subdividing the array to correspond to the steps in the partial-products method, we would have one large rectangle representing 27 × 20 and one smaller rectangle representing 27 × 8. The sum of the areas of these rectangles would give us the result, 756.

Using expanded forms and the distributive property, we can also express the calculation as follows:

27 × 28 = (20 + 7) × 28

= (20 × 28) + (7 × 28)

= 560 + 196

= 756

This equation relates to the steps in the partial-products method, where we multiply each term separately and then add the partial products together to obtain the final result of 756.

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The probability that a randomly chosen test-taker will score between 135 and 159 is 0.8185.

The probability that a randomly chosen test-taker will score between 135 and 159 can be found by standardizing the values of X to the corresponding Z-scores and then finding the probabilities from the normal distribution table. Let X be the LSAT test score of a randomly chosen test-taker.

We have;

Z₁ = (X₁ - μ) / σ = (135 - 151) / 8 = -2

Z₂ = (X₂ - μ) / σ = (159 - 151) / 8 = 1

The probability that a randomly chosen test-taker will score between 135 and 159 is the area under the standard normal curve between the corresponding Z-scores.

Z₁ = -2 and Z₂ = 1.

Using the standard normal distribution table, the probability is;

P(-2 ≤ Z ≤ 1) = P(Z ≤ 1) - P(Z ≤ -2)

P(Z ≤ 1) = 0.8413

P(Z ≤ -2) = 0.0228

P(-2 ≤ Z ≤ 1) = 0.8413 - 0.0228 = 0.8185

Therefore, the probability that a randomly chosen test-taker will score between 135 and 159 is 0.8185 (rounded to four decimal places).

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The volume of the rubberized material that makes up the holder is 111.78 cubic centimeters.

To calculate the volume of the rubberized material, we need to subtract the volume of the can from the volume of the holder. The volume of the can can be calculated using the formula for the volume of a cylinder, which is given by V_can = π * r_can^2 * h_can, where r_can is the radius of the can and h_can is the height of the can. In this case, the can has a height of 12 centimeters and we can assume it has the same radius as the holder.

The volume of the holder can be calculated by subtracting the volume of the can from the volume of the entire holder. The volume of the entire holder is equal to the volume of a cylinder, which is given by V_holder = π * r_holder^2 * h_holder, where r_holder is the radius of the holder and h_holder is the height of the holder. In this case, the height of the holder is 11.5 centimeters, including 1 centimeter for the thickness of the base.

To find the radius of the holder, we subtract the thickness of the rim from the radius of the can. The thickness of the rim is 1 centimeter, so the radius of the holder is 11.5 - 1 = 10.5 centimeters.

Now we can calculate the volume of the can using the given values: V_can = π * (10.5)^2 * 12 = 1385.44 cubic centimeters.

Finally, we can calculate the volume of the rubberized material by subtracting the volume of the can from the volume of the holder: V_rubberized_material = V_holder - V_can = π * (10.5)^2 * 11.5 - 1385.44 = 111.78 cubic centimeters.

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Answer:

A) x=-2

Step-by-step explanation:

We can solve this equation for x:

-12x-2(x+9)=5(x+4)

distribute

-12x-2x-18=5x+20

combine like terms

-14x-18=5x+20

add 18 to both sides

-14x=5x+38

subtract 5x from both sides

-19x=38

divide both sides by -19

x=-2

So, the correct option is A.

Hope this helps! :)

To find the value of x in the equation 243^x = 3^2, we can rewrite both sides of the equation using the same base.

Since 243 = 3^5, we can rewrite the equation as: (3^5)^x = 3^2

Now, we can simplify the equation by applying the exponent rule: 3^(5x) = 3^2

Since the bases are the same, the exponents must be equal: 5x = 2

To solve for x, we divide both sides of the equation by 5: x = 2/5

Therefore, the value of x is 2/5.

The value of x in the equation 243^x = 3^2 is 2/5.

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Answer:

Step-by-step explanation:

El certificado de depósito ganó un interés de aproximadamente $1.72. Cabe mencionar que este cálculo se basa en la suposición de que el certificado de depósito no tiene ninguna penalización o retención de impuestos.

Para calcular el interés ganado en el certificado de depósito, necesitamos utilizar la fórmula del interés simple: Interés = (Principal × Tasa de interés × Tiempo).

En este caso, el principal es de $450 y la tasa de interés es del 4.6% anual. Sin embargo, debemos convertir la tasa de interés a una tasa mensual, ya que el certificado de depósito es mensual.

Para convertir la tasa anual a una tasa mensual, dividimos la tasa anual entre 12: 4.6% / 12 = 0.3833% (aproximadamente). Ahora tenemos la tasa mensual: 0.3833%.

El tiempo es de un mes, por lo que sustituimos los valores en la fórmula del interés simple: Interés = ($450 × 0.3833% × 1 mes).

Calculando el interés: Interés = ($450 × 0.003833 × 1) = $1.72 (aproximadamente).

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If The circumference of a circle is 37. 68 inches. The circle's radius is approximately 6 inches.

The circumference of a circle is given by the formula:

C = 2πr

Where C is the circumference, π (pi) is a mathematical constant approximately equal to 3.14, and r is the radius of the circle.

Given that the circumference of the circle is 37.68 inches, we can set up the equation as:

37.68 = 2 * 3.14 * r

To solve for r, we can divide both sides of the equation by 2π:

37.68 / (2 * 3.14) = r

r ≈ 37.68 / 6.28

r ≈ 6 inches

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The full solution to the non-homogeneous differential equation y" - 2y' + 2y = 2x with initial conditions y(0) = 4 and y'(0) = 8 is:

y(x) = 3e^x cos(x) + 7e^x sin(x) + x + 1

The given differential equation is y" - 2y' + 2y = 2x, which is a second-order linear non-homogeneous differential equation. The complementary solution (yc) is obtained by finding the roots of the characteristic equation associated with the homogeneous equation, which is obtained by setting the right-hand side of the differential equation to zero.

The characteristic equation is r^2 - 2r + 2 = 0, and its roots are complex conjugates: r₁ = 1 + i and r₂ = 1 - i. Using Euler's formula, we can rewrite the roots as e^(1+ix) and e^(1-ix), respectively.

The complementary solution is yc = C₁e^x cos(x) + C₂e^x sin(x), where C₁ and C₂ are arbitrary constants determined by the initial conditions.

To find the particular solution (yp), we assume it has the form yp = ax + b, where a and b are constants to be determined. Substituting yp into the original differential equation, we get:

2a - 2a + 2(ax + b) = 2x

2ax + 2b = 2x

By comparing coefficients, we find a = 1 and b = 1. Therefore, the particular solution is yp = x + 1.

The full solution is obtained by adding the complementary and particular solutions:

y(x) = C₁e^x cos(x) + C₂e^x sin(x) + x + 1

Using the initial conditions y(0) = 4 and y'(0) = 8, we can determine the values of C₁ and C₂. Substituting x = 0 into the full solution, we get:

4 = C₁e^0 cos(0) + C₂e^0 sin(0) + 0 + 1

4 = C₁ + 1

From this, we find C₁ = 3. Differentiating the full solution and substituting x = 0, we have:

8 = -C₁e^0 sin(0) + C₂e^0 cos(0) + 1

8 = C₂ + 1

From this, we find C₂ = 7.

Therefore, the full solution with the given initial conditions is:

y(x) = 3e^x cos(x) + 7e^x sin(x) + x + 1

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Answer:

Step-by-step explanation:

The missing values of the given triangle DEF would be listed below as follows:

<D = 40°

<E = 90°

line EF = 50.6

To determine the missing part of the triangle, the Pythagorean formula should be used and it's giving below as follows:

C² = a²+b²

where;

c = 80

a = 62

b = EF = ?

That is;

80² = 62²+b²

b² = 80²-62²

= 6400-3844

= 2556

b = √2556

= 50.6

Since <E= 90°

<D = 180-90+50

= 180-140

= 40°

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The statement ¹[]-[8] S a holds true for n ≥ 1 by mathematical induction.

The given statement, "¹[]-[8] S a," can be explained using mathematical induction.

For the base case, when n = 1, we can see that ¹[]-[8] S 1 holds true since 1 is equal to 8 - 7. Next, assuming that the statement holds true for an arbitrary value k, we can derive the inequality ¹[] S k + 7.

To prove the statement for k + 1, we show that k + 7 is less than or equal to k + 1. By considering the properties of the numbers involved, we can conclude that ¹[]-[8] S k+1 is true.

Therefore, based on the principles of mathematical induction, we have established that for n ≥ 1, the given statement ¹[]-[8] S a holds true.

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Steady-state temperature distribution: u(x) = 25 - (5/3)x.

The steady-state temperature distribution in the heat conduction problem is given by u(x) = 25 - (5/3)x.

To find the steady-state temperature distribution, we need to solve the heat conduction problem with the given boundary conditions. The equation Uxx = ut represents the heat conduction equation, where U is the temperature distribution, x is the spatial variable, and t is the time variable.

The boundary conditions are u(0,t) = 20, u(30,t) = 50, and u(x, 0) = 60 - 2x. The first two boundary conditions specify the temperatures at the ends of the domain, while the third boundary condition specifies the initial temperature distribution.

To find the steady-state temperature distribution, we assume that the temperature does not change with time, which means the derivative with respect to time, ut, is zero. Therefore, the heat conduction equation simplifies to Uxx = 0. This is a second-order linear differential equation.

By solving this differential equation subject to the given boundary conditions, we find that the steady-state temperature distribution is u(x) = 25 - (5/3)x. This equation represents a linear temperature profile that decreases linearly from 25 at x = 0 to 10 at x = 30.

The heat conduction problem and steady-state temperature distribution in mathematical physics and engineering applications.

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A. The solution to the given system of differential equations is x = 2t + 1 and y = -10t^2 + 20t + C, where C is an arbitrary constant.

B. To solve the system of differential equations, we'll use a combination of separation of variables and integration.

Let's start with the first equation, dx/dt - y = -10t. Rearranging the equation, we have dx/dt = y - 10t.

Next, we integrate both sides with respect to t:

∫ dx = ∫ (y - 10t) dt

Integrating, we get x = ∫ y dt - 10∫ t dt.

Using the second equation, 16x - dy/dt = 10, we substitute the value of x from the previous step:

16(2t + 1) - dy/dt = 10.

Simplifying, we have 32t + 16 - dy/dt = 10.

Rearranging, we get dy = 32t + 6 dt.

Integrating both sides, we have:

∫ dy = ∫ (32t + 6) dt.

Integrating, we get y = 16t^2 + 6t + C.

Therefore, the general solution to the system of differential equations is x = 2t + 1 and y = -10t^2 + 20t + C, where C is an arbitrary constant.

Note: It's worth mentioning that the arbitrary constant C is introduced due to the integration process.

To obtain specific solutions, initial conditions or additional constraints need to be provided.

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The given solution is incorrect. Therefore, the correct factors are (x - 3)(x - 1)². The errors and solution are tabulated below:ErrorsSolution(x + 1) is not a factor of g(x)g(x) = (x - 3)(x - 1)²

Without dividing, to determine the remainder when x³ + 2x² − 6x + 1 is divided by x + 2:According to the remainder theorem, when a polynomial f(x) is divided by (x - a), the remainder is equal to f(a).

Therefore, we need to substitute -2 in place of x in the polynomial to get the remainder when x³ + 2x² − 6x + 1 is divided by x + 2.

Hence, (-2)³ + 2(-2)² - 6(-2) + 1 = -8 + 8 + 12 + 1 = 13.

Therefore, the remainder is 13. Hence, the main answer is "13".b) The possible factors of g(x) are 1, 3, 9. On trying g(1) = 1³ – 9(1)² – 1 +9 = 0, we observe that the given polynomial g(x) is not divisible by (x - 1).

Thus, we have errors as follows:According to the factor theorem, if x = -1 is a root of the polynomial g(x), then (x + 1) is a factor of the polynomial.

The value of g(-1) can be computed as follows: g(-1) = (-1)³ - 9(-1)² - (-1) + 9 = 1 - 9 + 1 + 9 = 2Thus, (x + 1) is not a factor of g(x).Therefore, the fully factored expression of g(x) is g(x) = (x - 3)(x - 1)².

Thus, the given solution is incorrect. Therefore, the correct factors are (x - 3)(x - 1)². The errors and solution are tabulated below:ErrorsSolution(x + 1) is not a factor of g(x)g(x) = (x - 3)(x - 1)²

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The second order equation that transforms into single equation , has initial condition equation ---  3 cos(√(8) t) - (5/(√(8)))sin(√(8) t).

The given system is: x₁ = 9x₁ + 4x² - x²

= 4x₁ + 9x²

Let's convert it into a second-order equation:

x₁ = 9x₁ + 4x² - x²

⇒ 9x₁ + 4x² - x² - x₁ = 0

⇒ 9x₁ - x₁ + 4x² - x² = 0

⇒ (9 - 1)x₁ + 4(x² - x₁) = 0

⇒ 8x₁ + 4x² - 4x₁ = 0

⇒ 4x₁ + 4x² = 0

⇒ x₁ + x² = 0

Now, we have two equations:

x₁ + x² = 0

9x₁ + 4x² - x²

= 4x₁ + 9x²

To solve it, let's substitute x² in terms of x₁ :

x₁ + x² = 0

⇒ x² = -x₁

Substituting it in the second equation:

9x₁ + 4x² - x² = 4x₁ + 9x²

⇒ 9x₁ + 4(-x₁) - (-x₁) = 4x₁ + 9(-x₁)

⇒ 9x₁ - 4x₁ + x₁ = -9x₁ - 4x₁

⇒ 6x₁ = -13x₁

= -13/6

Since, x² = -x₁

⇒ x² = 13/6

Now, let's find x₁(t) and x²(t):

x₁(t) = x₁(0) cos(√(8) t) + (13/(6√(8)))sin(√(8) t)x²(t)

= x²(0) cos(√(8) t) - (x₁(0)/(6√(8)))sin(√(8) t)

Putting x₁(0) = 10 and x²(0) = 3x₁

(t) = 10 cos(√(8) t) + (13/(6√(8)))sin(√(8) t)x²

(t) = 3 cos(√(8) t) - (5/(√(8)))sin(√(8) t)

Therefore, the solution of the system is  

 x₁(t) = 10 cos(√(8) t) + (13/(6√(8)))sin(√(8) t)x²(t)

= 3 cos(√(8) t) - (5/(√(8)))sin(√(8) t).

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The number of people in this town who are under the age of 18 is 3224. option C is the correct answer.

Given that in 2008, a small town has 8500 people. At the 2018 census, the population had grown by 28%.

At this point, 45% of the population is under the age of 18.

To calculate the number of people in this town who are under the age of 18, we will use the following formula:

Population in the year 2018 = Population in the year 2008 + 28% of the population in 2008

Number of people under the age of 18 = 45% of the population in 2018

= 0.45 × (8500 + 0.28 × 8500)≈ 3224

Option C is the correct answer.

15. Let the current ages of two relatives be 7x and x respectively, since the ratio of their ages is given as 7:1.

Let's find the ratio of their ages after 6 years. Their ages after 6 years will be 7x+6 and x+6, so the ratio of their ages will be (7x+6):(x+6).

We are given that the ratio of their ages after 6 years is 5:2, so we can write the following equation:

(7x+6):(x+6) = 5:2

Using cross-multiplication, we get:

2(7x+6) = 5(x+6)

Simplifying the equation, we get:

14x+12 = 5x+30

Collecting like terms, we get:

9x = 18

Dividing both sides by 9, we get:

x=2

Therefore, the current ages of two relatives are 7x and x which is equal to 7(2) = 14 and 2 respectively.

Hence, option B is the correct answer.

16. The formula for H is given as:

H = 3 - ³

Given that z = -4.

Substituting z = -4 in the formula for H, we get:

H = 3 - ³

   = 3 - (-64)

   = 3 + 64

   = 67

Therefore, option D is the correct answer.

17.  We are to identify the equation that does not pass through the point (3,-4).

Let's check the options one by one, taking the first option into consideration:

2x - 3y = 18

Putting x = 3 and y = -4,

we get:

2(3) - 3(-4) = 6+12

                 = 18

Since the left-hand side is equal to the right-hand side, this equation passes through the point (3,-4).

Now, taking the second option:

y = 5x - 19

Putting x = 3 and y = -4, we get:-

4 = 5(3) - 19

Since the left-hand side is not equal to the right-hand side, this equation does not pass through the point (3,-4).

Therefore, option B is the correct answer.

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