Prove that 1+3+9+27+…+3^n=3^n+1−1/2​ Let n be a positive integer,

Answers

Answer 1

Using mathematical induction, we can prove that the equation 1 + 3 + 9 + 27 + ... + 3^n = (3^(n+1) - 1) / 2 holds true for all positive integers n.

To prove the equation 1 + 3 + 9 + 27 + ... + 3^n = (3^(n+1) - 1) / 2, we can use mathematical induction.

1. Base Case:

For n = 1, we have 1 = (3^(1+1) - 1) / 2.

1 = (3^2 - 1) / 2.

1 = (9 - 1) / 2.

1 = 8 / 2.

1 = 4.

The base case holds true.

2. Inductive Step:

Assume that the equation holds true for some positive integer k, i.e., 1 + 3 + 9 + 27 + ... + 3^k = (3^(k+1) - 1) / 2.

We need to prove that it also holds true for k + 1, i.e., 1 + 3 + 9 + 27 + ... + 3^k + 3^(k+1) = (3^((k+1)+1) - 1) / 2.

Starting from the left side of the equation:

1 + 3 + 9 + 27 + ... + 3^k + 3^(k+1) = (3^(k+1) - 1) / 2 + 3^(k+1)

= (3^(k+1) - 1 + 2 * 3^(k+1)) / 2

= (3^(k+1) - 1 + 2 * 3 * 3^k) / 2

= (3^(k+1) + 2 * 3 * 3^k - 1) / 2

= (3^(k+1) + 2 * 3^(k+1) - 1) / 2

= (3 * 3^(k+1) + 3^(k+1) - 1) / 2

= (3^(k+2) + 3^(k+1) - 1) / 2

= (3^(k+2) + 3^(k+1) - 1 * 2/2) / 2

= (3^(k+2) + 3^(k+1) - 2) / 2

= (3^(k+2) + 3^(k+1) - 2) / 2

= (3^(k+2) + 3^(k+1) - 1) / 2 - 1/2

= (3^(k+2+1) - 1) / 2 - 1/2

= (3^((k+1)+1) - 1) / 2 - 1/2

Thus, we have shown that if the equation holds true for k, it also holds true for k + 1.

By the principle of mathematical induction, the equation is true for all positive integers n. Therefore, we have proven that 1 + 3 + 9 + 27 + ... + 3^n = (3^(n+1) - 1) / 2 for any positive integer n.

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Related Questions

1. Prove or disprove: 2^n + 2 is an even number for all
integers

Answers

We can conclude that 2^n + 2 is indeed an even number for all integers.

To prove or disprove the statement "2^n + 2 is an even number for all integers," we need to consider both cases.

First, let's assume that n is an even integer. In this case, we can express n as n = 2k, where k is also an integer. Substituting this into the expression 2^n + 2, we get: 2^n + 2 = 2^(2k) + 2 = (2^2)^k + 2 = 4^k + 2

Since 4^k is always an even number (as any power of 4 is divisible by 2), adding 2 to an even number results in an even number. Therefore, when n is an even integer, 2^n + 2 is indeed an even number.

Next, let's assume that n is an odd integer. In this case, we can express n as n = 2k + 1, where k is an integer. Substituting this into the expression 2^n + 2, we get: 2^n + 2 = 2^(2k + 1) + 2

Expanding this expression, we have:

2^n + 2 = 2^(2k) * 2^1 + 2 = (2^2)^k * 2 + 2 = 4^k * 2 + 2 = (2 * 2^k) * 2 + 2

Since 2 * 2^k is always an even number (as it is a multiple of 2), adding 2 to an even number results in an even number. Therefore, when n is an odd integer, 2^n + 2 is also an even number.

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Determine the fugacity and fugacity coefficients of methane using the Redlich-Kwong equation of state at 300 K and 10 bar. Write all the assumptions made.

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Using the Redlich-Kwong equation of state at 300 K and 10 bar, the fugacity and fugacity coefficients of methane are 13.04 bar and 1.304, respectively.

The Redlich-Kwong equation of state for fugacity is given as:

f = p + a(T, v) / (v * (v + b))

The fugacity coefficient is given as:

φ = f / p

Where, f is the fugacity, p is the pressure, a(T, v) and b are constants given by Redlich-Kwong equation of state. Now, applying the Redlich-Kwong equation of state at 300 K and 10 bar, we have the following:

Given: T = 300 K; P = 10 bar

Assumptions:

It is assumed that the volume of the gas molecules is negligible and the intermolecular forces between the molecules are negligible. The equation of state is a cubic equation and has three roots, but only one root is physical.

The constants, a(T, v) and b are expressed as follows:

a(T, v) = 0.42748 * (R ^ 2 * Tc ^ 2.5) / Pc,

b = 0.08664 * R * Tc / Pc

Where R is the gas constant, Tc and Pc are the critical temperature and pressure, respectively.

Now, substituting the given values in the above equations, we have:

Tc = 190.56 K; Pc = 45.99 bar

R = 8.314 J / mol * K

For methane, we have:

a = 0.42748 * (8.314 ^ 2 * 190.56 ^ 2.5) / 45.99 = 1.327 L ^ 2 * bar / mol ^ 2

b = 0.08664 * 8.314 * 190.56 / 45.99 = 0.04267 L / mol

Using the above values, we can now calculate the fugacity of methane:

f = p + a(T, v) / (v * (v + b))= 10 + 1.327 * (300, v) / (v * (v + 0.04267))

Since the equation of state is cubic, we need to solve for v numerically using an iterative method. Once we get the value of v, we can calculate the fugacity of methane. Now, substituting the value of v in the above equation, we get:

f = 13.04 bar

The fugacity coefficient is given as:

φ = f / p= 13.04 / 10= 1.304

Therefore, the fugacity and fugacity coefficients of methane using the Redlich-Kwong equation of state at 300 K and 10 bar are 13.04 bar and 1.304, respectively. Assumptions made in the above calculations are: The volume of the gas molecules is negligible. The intermolecular forces between the molecules are negligible. The equation of state is a cubic equation and has three roots, but only one root is physical.

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In the past ten years, a country's total output has increased from 2000 to 3000, the capital stock has risen from 4000 to 5200, and the labour force has increased from 400 to 580. Suppose the elasticities aK = 0.4 and aN = 0.6. Show your work when you answer the following: a. How much did capital contribute to economic growth over the decade? b. How much did labour contribute to economic growth over the decade? c. How much did productivity contribute to economic growth over the decade?

Answers

To calculate the contribution of each factor to economic growth, we can apply the following formula:

Contribution of a factor to economic growth = Factor's share in output x (Factor's elasticity with respect to output) x 10-year change in output

Using the given data:

a. Contribution of capital to economic growth:

Capital's share in output = Capital stock / (Capital stock + Total output) = 5200 / (5200 + 3000) = 0.667

Capital's elasticity with respect to output = aK = 0.4

10-year change in output = 3000 - 2000 = 1000

Contribution of capital to economic growth = Capital's share in output x (Capital's elasticity with respect to output) x 10-year change in output = 0.667 x 0.4 x 1000 = 266.8

b. Contribution of labour to economic growth:

Labour's share in output = Labour force / (Labour force + Total output) = 580 / (580 + 3000) = 0.160

Labour's elasticity with respect to output = aN = 0.6

10-year change in output = 3000 - 2000 = 1000

Contribution of labour to economic growth = Labour's share in output x (Labour's elasticity with respect to output) x 10-year change in output = 0.160 x 0.6 x 1000 = 96

c. Contribution of productivity to economic growth:

Contribution of capital to economic growth + Contribution of labour to economic growth = 266.8 + 96 = 362.8

The country's total output has increased by 1000 over the decade. So the contribution of productivity to economic growth is 362.8 / 1000 = 0.3628

d. The productivity growth rate over the decade is:

Productivity growth rate = 10-year change in output / 10-year change in total factor inputs = 1000 / (0.667 x 400 + 0.160 x 580)

In the first order system: which point is not a critical point of the system?
x = 7x+9y-xy², y′=2x-y,
A. (0, 0)
B. (5/2, 5)
C. (1, 2)
D. (-5/2, -5)

Answers



The critical points of a system are the points where the derivative of each variable with respect to time is equal to zero. By evaluating each point, we can determine which point is not a critical point of the system.



To find the critical points, we need to solve the given system of equations:

x = 7x + 9y - xy²
y' = 2x - y

Let's start by finding the critical points.

For x = 7x + 9y - xy², we can rewrite it as 6x + xy² = 9y.

Then, we differentiate both sides of the equation with respect to x to get:

6 + 2xy + y² = 0

Next, we solve for y:

y² + 2xy + 6 = 0

This is a quadratic equation in y.

Using the quadratic formula, we have:

y = (-2x ± √(4x² - 4(1)(6))) / 2

Simplifying further, we get:

y = -x ± √(x² - 6)

Now, let's find the critical points by substituting y back into the equation x = 7x + 9y - xy²:

x = 7x + 9(-x ± √(x² - 6)) - x(x² - 6)²

Simplifying this equation will give us the critical points. However, since the equation involves complex terms, it might be challenging to find exact solutions.

To determine which point is not a critical point of the system, we can use an approximation method or graphical analysis to evaluate the values of x and y for each given point.

A. (0, 0): Substitute x = 0 and y = 0 into the equations to see if they satisfy the system. If they do, then this point is a critical point. If not, it is not a critical point.

B. (5/2, 5): Substitute x = 5/2 and y = 5 into the equations to check if they satisfy the system. If they do, then this point is a critical point. If not, it is not a critical point.

C. (1, 2): Substitute x = 1 and y = 2 into the equations to see if they satisfy the system. If they do, then this point is a critical point. If not, it is not a critical point.

D. (-5/2, -5): Substitute x = -5/2 and y = -5 into the equations to check if they satisfy the system. If they do, then this point is a critical point. If not, it is not a critical point.

Therefore by evaluating each point, we can identify which point is not a system critical point by assessing each point.

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Let a, b E Z. Let c, m € N. Prove that if a ‡ b (mod m), then a ‡ b (mod cm).

Answers

If a and b are congruent modulo m, they will also be congruent modulo cm, implying that their difference is divisible by both m and cm.

When two numbers, a and b, are congruent modulo m (denoted as a ≡ b (mod m)), it means that the difference between a and b is divisible by m. In other words, (a - b) is a multiple of m.

To prove that if a ≡ b (mod m), then a ≡ b (mod cm), we need to show that the difference between a and b is also divisible by cm.

Since a ≡ b (mod m), we can express this congruence as (a - b) = km, where k is an integer. Now, we need to prove that (a - b) is also divisible by cm.

To do this, we can rewrite (a - b) as (a - b) = (km)(c). Since k and c are both integers, their product (km)(c) is also an integer. Therefore, (a - b) is divisible by cm, which can be expressed as a ≡ b (mod cm).

In simpler terms, if the difference between a and b is divisible by m, it will also be divisible by cm because m is a factor of cm. This demonstrates that if a ≡ b (mod m), then a ≡ b (mod cm).

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Divide £400 in the ratio 25: 15

Answers

Answer: 250:150

Step-by-step explanation:

set up a algebraic equation of

25x+15x=400

40x=400

x=10

now multiply that in the ratio 25(10): 15(10)

250:150

Hola ayúdenme Porfavor

Answers

Answer:

Graph 2

Step-by-step explanation:

On graph 2, the line goes slowly up along the y value, meaning that his speed is increasing. (Chip begins his ride slowly)

Then, it suddenly stops and does not increase for an interval of time. (Chip stops to talk to some friends)

The speed then gradually picks back up. (He continues his ride, gradually picking up his speed)

Given y"(t) + 2 y'(t) + y(t) = 2. Find y(t) if y(0) = 3 and y'(0) = 2. Solution: -t y(t) = 7te^-t + 3 e^-t

Answers

The solution is y(t) = e^(-t) + te^(-t) + 2.


The given differential equation is y"(t) + 2y'(t) + y(t) = 2.

To solve this differential equation, we can use the method of undetermined coefficients.

First, let's find the complementary solution (the solution to the homogeneous equation) by assuming y(t) = e^(rt).

Substituting this assumption into the differential equation, we get r^2e^(rt) + 2re^(rt) + e^(rt) = 0.

Dividing through by e^(rt), we have r^2 + 2r + 1 = 0.

This is a quadratic equation that can be factored as (r + 1)^2 = 0.

So, the complementary solution is y_c(t) = c1e^(-t) + c2te^(-t), where c1 and c2 are arbitrary constants.

Now, let's find the particular solution (the solution to the non-homogeneous equation).

Since the right-hand side is a constant, we can assume a particular solution of the form y_p(t) = A, where A is a constant.

Substituting this assumption into the differential equation, we get 0 + 0 + A = 2.

Therefore, A = 2.

So, the particular solution is y_p(t) = 2.

The general solution is given by y(t) = y_c(t) + y_p(t).

Substituting the values y_c(t) = c1e^(-t) + c2te^(-t) and y_p(t) = 2 into the general solution, we have y(t) = c1e^(-t) + c2te^(-t) + 2.

Now, we can use the initial conditions y(0) = 3 and y'(0) = 2 to find the values of c1 and c2.

Substituting t = 0 and y(0) = 3 into the general solution, we get c1e^(-0) + c2(0)e^(-0) + 2 = 3.

Simplifying this equation, we have c1 + 2 = 3.

Therefore, c1 = 1.

Next, substituting t = 0 and y'(0) = 2 into the general solution, we get -c1e^(-0) + c2e^(-0) + 0 + 2 = 2.

Simplifying this equation, we have -c1 + c2 + 2 = 2.

Since we already found c1 = 1, we can substitute it into the equation: -1 + c2 + 2 = 2.

Therefore, c2 = 1.

So, the particular solution to the given differential equation is y(t) = e^(-t) + te^(-t) + 2.



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Write the following sets using the "roster method". That is, write the sets in list form. (a) A = {: is a natural number and x appears in the decimal expansion of 313/999} (b) B = {x:x is an odd integer smaller than 1} 2. List the next element in each of the following sets. (a) {1,1/4,1/16,1/64,...} (b) (3,3,6,9,15,24,...} 3. Answer either TRUE or FALSE to each of the statements (a) through (d). A = {3,6,9, ..., 96, 99} B = {1,0, 1, 2, 3, 4, 5, 6} (a) 66 € A ___
(b) 0 € C ___ (c) {4} € B ___ (d) C C A ___

Answers

66 € A is true as 66 is a multiple of 3, which is a member of A. Therefore, 66 € A is True. 0 € C (FALSE). The set C is not given. Therefore, it is not possible to say whether 0 belongs to C or not. Hence, 0 € C is false.

A. A = {0.313, 0.626, 0.939} B. B = {-1}
A set in mathematics is a collection of distinct objects called elements of the set. These elements could be numbers, letters, or any other kind of object. Here, we are going to use the roster method to represent the sets in list form.
The roster method is the method of representing a set by listing its elements within braces {}. A. Set A comprises all the natural numbers (x) that appear in the decimal expansion of 313/999. Now, let's solve the problem using the roster method: A = {0.313, 0.626, 0.939}. Set A comprises all the natural numbers (x) that appear in the decimal expansion of 313/999.
The roster method is the method of representing a set by listing its elements within braces {}. The set A can be represented in list form as A = {0.313, 0.626, 0.939}. B. The set B comprises all odd integers smaller than 1. The set B comprises all odd integers smaller than 1. The roster method is the method of representing a set by listing its elements within braces {}. The set B can be represented in list form as B = {-1}.2.
a) {1,1/4,1/16,1/64,...}
Notice that each term is of the form 1/4ⁿ. The next element in the set is 1/256.2.b) {3,3,6,9,15,24,...}
Notice that the differences between consecutive terms in the sequence are 0, 3, 3, 6, 9,.... The next term would be obtained by adding 12 to 24. Therefore, the next term is 36.3. a) 66 € A (TRUE) as 66 is a multiple of 3, which is a member of A. Therefore, 66 € A is True.
3. b) 0 € C (FALSE). The set C is not given. Therefore, it is not possible to say whether 0 belongs to C or not. Hence, 0 € C is False.
3. c) {4} € B (FALSE)The set B has only odd integers, and 4 is an even integer. Therefore, {4} € B is False. 3. d) C C A (FALSE)Since 0 € C is False, C € A is False.

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A
100 cm
85 cm
Not drawn to scale
What is the angle of Penn's ramp (m/A)?

Answers

The angle of Penn's ramp (m∠A) is 58.212°.

What is the angle of Penn's ramp (m∠A)?

Trigonometry deals with the relationship between the ratios of the sides of a right-angled triangle with its angles.

To find the angle of Penn's ramp (m∠A), we will use trig. ratio. That is:

sin A = 85/100 (opposite /hypotenuse)

sin A = 0.85

A = arcsin(0.85)

A = 58.212°

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Complete Question

Check attached image

FJ intersects KH at point M, and GM ⊥ FJ. What is m KMJ

Answers

The measure of the vertical angle m∠KMJ is equal to 120°.

What are vertically opposite angles

Vertical angles also called vertically opposite angles are formed when two lines intersect each other, the opposite angles formed by these lines are vertically opposite angles and are equal to each other.

We shall evaluate for the measure of x as follows:

m∠KMJ = m∠FGH = 90 + (7x - 19)°

m∠KMJ = 7x + 71

m∠FMK = m∠JMH = (5x + 25)°

2(7x + 71 + 5x + 25) = 360° {sum of angles at a point}

12x + 96 = 180°

12x = 180° - 96°

12x = 84°

x = 84°/12 {divide through by 12}

x = 7

m∠KMJ = 7(7) + 71 = 120°

Therefore, since the variable x is 7, the measure of the vertical angle m∠KMJ is equal to 120°.

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In ΔABC, ∠C is a right angle. Find the remaining sides and angles. Round your answers to the nearest tenth.

b=7, c=12

Answers

We need to determine the remaining sides and angles.Using the Pythagorean theorem, we know that:a² + b² = c².The remaining sides and angles in triangle ABC, rounded to nearest tenth are: side a≈9.7 , Angle A ≈ 54.8° , Angle B ≈ 35.2°.

In a right triangle, the side opposite to the right angle is the longest side and is known as the hypotenuse. The other two sides are known as the legs.

Given a right triangle Δ ABC with ∠C as the right angle, b = 7, and c = 12, we need to determine the remaining sides and angles.Using the Pythagorean theorem, we know that:a² + b² = c².

Substituting the values of b and c, we have:a² + 7² = 12²Simplifying, we have:a² + 49 = 144a² = 144 - 49a² = 95a = √95 ≈ 9.7 (rounded to the nearest tenth)

Therefore, the length of the remaining side a is approximately 9.7 units long.Now, we can use the trigonometric ratios to find the remaining angles.

Using the sine ratio, we have:sin(A) = a/c => sin(A) = 9.7/12 =>sin(A) ≈ 0.81 =>A = sin⁻¹(0.81) ≈ 54.1° (rounded to the nearest tenth).Therefore, angle A is approximately 54.1 degrees.

Using the fact that the sum of angles in a triangle is 180 degrees, we can find angle B: A + B + C= 180 =>54.1 + B + 90=180 =>B ≈ 35.9° (rounded to the nearest tenth)Therefore, angle B is approximately 35.9 degrees.

Therefore, the remaining sides and angles in triangle ABC, rounded to nearest tenth are: side a ≈9.7

.                             Angle A ≈ 54.1°

.                             Angle B ≈ 35.9°

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Max Z = 5x1 + 6x2
Subject to: 17x1 + 8x2 ≤ 136
3x1 + 4x2 ≤ 36
x1 ≥ 0 and integer
x2 ≥ 0
A) x1 = 5, x2 = 4.63, Z = 52.78
B) x1 = 5, x2 = 5.25, Z = 56.5
C) x1 = 5, x2 = 5, Z = 55
D) x1 = 4, x2 = 6, Z = 56

Answers

The option B) yields the highest value for Z, which is 56.5. Therefore, the correct answer is B) x1 = 5, x2 = 5.25, Z = 56.5

To determine the correct answer, we can substitute each option into the objective function and check if the constraints are satisfied. Let's evaluate each option:

A) x1 = 5, x2 = 4.63, Z = 52.78

Checking the constraints:

17x1 + 8x2 = 17(5) + 8(4.63) = 85 + 37.04 = 122.04 ≤ 136 (constraint satisfied)

3x1 + 4x2 = 3(5) + 4(4.63) = 15 + 18.52 = 33.52 ≤ 36 (constraint satisfied)

B) x1 = 5, x2 = 5.25, Z = 56.5

Checking the constraints:

17x1 + 8x2 = 17(5) + 8(5.25) = 85 + 42 = 127 ≤ 136 (constraint satisfied)

3x1 + 4x2 = 3(5) + 4(5.25) = 15 + 21 = 36 ≤ 36 (constraint satisfied)

C) x1 = 5, x2 = 5, Z = 55

Checking the constraints:

17x1 + 8x2 = 17(5) + 8(5) = 85 + 40 = 125 ≤ 136 (constraint satisfied)

3x1 + 4x2 = 3(5) + 4(5) = 15 + 20 = 35 ≤ 36 (constraint satisfied)

D) x1 = 4, x2 = 6, Z = 56

Checking the constraints:

17x1 + 8x2 = 17(4) + 8(6) = 68 + 48 = 116 ≤ 136 (constraint satisfied)

3x1 + 4x2 = 3(4) + 4(6) = 12 + 24 = 36 ≤ 36 (constraint satisfied)

From the calculations above, we see that options B), C), and D) satisfy all the constraints. However, option B) yields the highest value for Z, which is 56.5. Therefore, the correct answer is: B) x1 = 5, x2 = 5.25, Z = 56.5.

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For a certain choice of origin, the third antinode in a standing wave occurs at x3=4.875m while the 10th antinode occurs at x10=10.125 m. The distance between consecutive nodes, in m, is 1.5 0.375 None of the listed options 0.75 Two identical waves traveling in the -x direction have a wavelength of 2m and a frequency of 50Hz. The starting positions xo1 and xo2 of the two waves are such that xo2=xo1+N/2, while the starting moments to1 and to2 are such that to2=to1+T/4. What is the phase difference (phase2-phase1), in rad, between the two waves if wave-1 is described by y_1(x,t)=Asin[k(x-x_01)+w(t-t_01)+]? None of the listed options 3π/2 TT/2 0

Answers

1. The distance between consecutive nodes in the standing wave is 0.75 m. Option D is the correct answer.

2. The phase difference between the two identical waves cannot be determined with the given information. Option A is the correct answer.

1. For a certain choice of origin, the third antinode in a standing wave occurs at x₃ = 4.875 m, while the 10th antinode occurs at x₁₀ = 10.125 m. We need to determine the distance between consecutive nodes.

In a standing wave, the distance between consecutive nodes is equal to half the wavelength (λ/2). Since the distance between the third antinode and the tenth antinode is equal to 7 times the distance between consecutive nodes, we can set up the following equation:

7(λ/2) = x₁₀ - x₃

Substituting the given values:

7(λ/2) = 10.125 m - 4.875 m

7(λ/2) = 5.25 m

Simplifying the equation:

λ/2 = 5.25 m / 7

λ/2 = 0.75 m

Therefore, the distance between consecutive nodes is 0.75 m.

So, the correct option is D. 0.75.

2. Two identical waves are traveling in the -x direction with a wavelength of 2 m and a frequency of 50 Hz. We are given that the starting positions x₀₁ and x₀₂ of the waves are such that x₀₂ = x₀₁ + N/2, and the starting moments t₀₁ and t₀₂ are such that t₀₂ = t₀₁ + T/4. We need to find the phase difference (phase₂ - phase₁) between the two waves.

The phase of a wave can be calculated using the formula: φ = kx - ωt, where k is the wave number, x is the position, ω is the angular frequency, and t is the time.

Given that the waves are identical, they have the same wave number (k) and angular frequency (ω). Let's calculate the values of k and ω:

Since the wavelength (λ) is given as 2 m, we know that k = 2π/λ.

k = 2π/2 = π rad/m

The angular frequency (ω) can be calculated using the formula ω = 2πf, where f is the frequency.

ω = 2π(50 Hz) = 100π rad/s

Now, let's consider the two waves individually:

Wave-1: y₁(x,t) = A sin[k(x - x₀₁) + ω(t - t₀₁)]

Wave-2: y₂(x,t) = A sin[k(x - x₀₂) + ω(t - t₀₂)]

We are given that x₀₂ = x₀₁ + N/2 and t₀₂ = t₀₁ + T/4.

Since the wavelength is 2 m, the distance between consecutive nodes is equal to the wavelength (λ). Therefore, the phase difference between consecutive nodes is 2π.

Let's calculate the phase difference between the two waves:

Phase difference = [k(x - x₀₂) + ω(t - t₀₂)] - [k(x - x₀₁) + ω(t - t₀₁)]

= k(x - x₀₂) - k(x - x₀₁) + ω(t - t₀₂) - ω(t - t₀₁)

= k(x - (x₀₁ + N/2)) - k(x - x₀₁) + ω(t - (t₀₁ + T/4)) - ω(t - t₀₁)

= -kN/2 + k(x₀₁ - x₀₁) - ωT/4

= -kN/2 - ωT/4

Substituting the values of k and ω:

Phase difference = -πN/2 - (100π)(T/4)

= -πN/2 - 25πT

Since we don't have the values of N or T, we cannot determine the exact phase difference. Therefore, the correct option is A. None.

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The question is -

1. For a certain choice of origin, the third antinode in a standing wave occurs at x₃ = 4.875 m, while the 10th antinode occurs at x₁₀ = 10.125 m. The distance between consecutive nodes is

A. 1.5

B. 0.375

C. None

D. 0.75

2. Two identical waves are traveling in the -x direction with a wavelength of 2 m and a frequency of 50 Hz. The starting positions x₀₁ and x₀₂ of the two waves are such that x₀₂ = x₀₁ + N/2, while the starting moments t₀₁ and t₀₂ are such that t₀₂ = t₀₁ + T/4. What is the phase difference (phase₂ - phase₁) between the two waves if wave-1 is described by y₁(x,t) = A sin[k(x - x₀₁) + ω(t - t₀₁)]?

A. None

B. 3π/2

C. π/2

D. 0

A family buys a studio apartment for $150,000. They pay a down payment of $30,000. Their down payment is what percent of the purchase price?

Answers

Answer:

Their down payment is 20% of the purchase price.

Step-by-step explanation:

The down payment is $30,000 and the purchase price is $150,000.

To find the percentage, we can divide the down payment by the purchase price and multiply by 100:

($30,000 / $150,000) x 100% = 20%

Therefore, the down payment is 20% of the purchase price.

Miguel has 48 m of fencing to build a four-sided fence around a rectangular plot of land. The area of the land is 143 square meters. Solve for the dimensions (length and width) of the field.

Answers

The dimensions of the rectangular plot of land can be either 11 meters by 13 meters or 13 meters by 11 meters.

Let's assume the length of the rectangular plot of land is L and the width is W.

We are given that the perimeter of the fence is 48 meters, which means the sum of all four sides of the rectangular plot is 48 meters.

Therefore, we can write the equation:

2L + 2W = 48

We are also given that the area of the land is 143 square meters, which can be expressed as:

L * W = 143

Now, we have a system of two equations with two variables. We can use substitution or elimination to solve for the dimensions of the field.

Let's use the elimination method to eliminate one variable:

From equation 1, we can rewrite it as L = 24 - W.

Substituting this value of L into equation 2, we get:

(24 - W) * W = 143

Expanding the equation, we have:

24W - W^2 = 143

Rearranging the equation, we get:

W^2 - 24W + 143 = 0

Factoring the quadratic equation, we find:

(W - 11)(W - 13) = 0

Setting each factor to zero, we have two possibilities:

W - 11 = 0 or W - 13 = 0

Solving these equations, we get:

W = 11 or W = 13

If W = 11, then from equation 1, we have L = 24 - 11 = 13.

If W = 13, then from equation 1, we have L = 24 - 13 = 11.

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Consider the following formulas of first-order logic: \forall x \exists y(x\oplus y=c) , where c is a constant and \oplus is a binary function. For which interpretation is this formula valid?

Answers

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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Express 2⋅2⋅2⋅2⋅2⋅3⋅3⋅3⋅3⋅3 using exponents. 2⋅2⋅2⋅2⋅2⋅3⋅3⋅3⋅3⋅3=2^5 ⋅3 ___

Answers

The expression 2⋅2⋅2⋅2⋅2⋅3⋅3⋅3⋅3⋅3 can be expressed as 2^5 ⋅ 3^5.

In this expression, the base 2 is repeated five times, indicating that we are multiplying five 2's together. Similarly, the base 3 is repeated five times, indicating that we are multiplying five 3's together. The exponent of 5 signifies the number of times the base is multiplied by itself.

Using exponents allows us to express repeated multiplication in a more compact and efficient way. Instead of writing out each multiplication step, we can simply indicate the base and its exponent. In this case, the exponent of 5 shows that both 2 and 3 are multiplied five times.

The expression 2^5 ⋅ 3^5 represents the final result of multiplying all the numbers together. By using exponents, we can easily calculate the value without performing each multiplication individually.

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Jim Harris files using the married filing separately status. His taxable income on line 15, Form 1040, is $102,553. Compute his 2021 federal income tax.
A. $10,255
B. $15,716
C. $18,634
D.$24,613

Answers

The right option is C. "$18,634"

Jim Harris's taxable income is $102,553, and he files using the married filing separately status. To compute his 2021 federal income tax, we need to refer to the tax brackets and rates for that filing status.

The tax rates for married filing separately status in 2021 are as follows:

- 10% on the first $9,950 of taxable income

- 12% on income between $9,951 and $40,525

- 22% on income between $40,526 and $86,375

- 24% on income between $86,376 and $164,925

- 32% on income between $164,926 and $209,425

- 35% on income between $209,426 and $523,600

- 37% on income over $523,600

To compute Jim's federal income tax, we need to calculate the tax owed for each tax bracket and sum them up. Here's the breakdown:

- For the first $9,950, the tax owed is 10% * $9,950 = $995.

- For the income between $9,951 and $40,525, the tax owed is 12% * ($40,525 - $9,951) = $3,045.48.

- For the income between $40,526 and $86,375, the tax owed is 22% * ($86,375 - $40,526) = $9,944.98.

- For the income between $86,376 and $102,553, the tax owed is 24% * ($102,553 - $86,376) = $3,895.52.

Adding up these amounts gives us a total federal income tax of $995 + $3,045.48 + $9,944.98 + $3,895.52 = $17,881.98.

However, it's important to note that the given options don't match the calculated amount. The closest option is C, which is $18,634. This could be due to additional factors not mentioned in the question, such as deductions, credits, or other tax considerations.

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Choose all the expressions equivalent to (64 ^-2)(64 ^1/2)
1.) 1/64
2.) 1/512
3.) 64 ^-1
4.) 64 ^-3/2
Show all work and explain solving process.

Answers

The expression [tex](64^{(-2)})(64^{(1/2)})[/tex] is equivalent to [tex]1/512[/tex]. Option b is correct.

To simplify the expression [tex](64^{(-2)})(64^{(1/2)})[/tex], we can use the properties of exponents.

First, let's simplify each term separately:

[tex]64^{(-2)} = 1/(64^2) = 1/4096[/tex]

[tex]64^{(1/2)} = \sqrt{64} = 8[/tex]

Now, let's multiply the two terms:

[tex](64^{(-2)})(64^{(1/2)}) = (1/4096) \times 8 = 8/4096[/tex]

To simplify further, we can reduce the fraction:

[tex]8/4096 = 1/512[/tex]

So the correct option is:

2.) 1/512

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Question 9 of 49
Which of the following best describes the pattern in the diagram as you move
from the top to the bottom row?
1
2
3
O A. Row 9 will contain 12 circles.
OB. Each row increases by 2 circles.
OC. Each row increases by 1 circle.
OD. Row 7 will contain 10 circles.
SUBMIT

Answers

Answer:

Answer C

Step-by-step explanation:

The pattern in the diagram as you move from the top row to the bottom row is that each row increases by 1 circle. Therefore, the correct answer is (C) "Each row increases by 1 circle."

Option (A) is incorrect because it is not a consistent pattern.

Option (B) is incorrect because it increases by 2 on the second and third rows, breaking the established pattern.

Option (D) is incorrect because it refers to a specific row rather than the overall pattern.

Selena collected 100 pounds of aluminum cans to recycle. She plans to collect an additional 25 pounds each week.

a. independent quantity?
b. dependent quantity?
c. function:
d. rate of change:

Answers

a. The independent quantity in this scenario is the number of weeks Selena has been collecting aluminum cans.

b. The dependent quantity is the total weight of aluminum cans Selena has collected.

c. The function that represents the relationship between the number of weeks and the total weight of aluminum cans collected can be written as:

Total weight = 100 + 25 * (number of weeks)

d. The rate of change in this context is the increase in the total weight of aluminum cans collected per week.

d. Since Selena plans to collect an additional 25 pounds each week, the rate of change is constant and equal to 25 pounds per week. Selena starts with an initial weight of 100 pounds of aluminum cans. For each subsequent week, she collects an additional 25 pounds, resulting in a linear relationship between the number of weeks and the total weight of aluminum cans.

The function is linear because the rate of change, which represents the slope of the line, is constant. This means that for every additional week, the total weight increases by 25 pounds. The function allows us to calculate the total weight of aluminum cans based on the number of weeks, providing a straightforward and predictable pattern of accumulation.

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Solve the given system of differential equations by systematic elimination.
(D + 1)x + (D − 1)y = 8 9x + (D + 8)y = -1
(x(t), y(t)) =
Need Help?

Answers

Solve the system using systematic elimination to find x(t) and y(t).

Solve the system of differential equations using systematic elimination: (D + 1)x + (D − 1)y = 8 and 9x + (D + 8)y = -1. Find the solution (x(t), y(t)).

The given problem asks to solve a system of differential equations using systematic elimination.

Systematic elimination is a method used to eliminate one variable at a time from a system of equations to obtain a simplified form.

In this case, we have two equations involving the variables x and y, along with their respective derivatives.

The goal is to find the functions x(t) and y(t) that satisfy these equations. By applying systematic elimination, we can eliminate one variable by manipulating the equations algebraically.

The resulting simplified equation will involve only one variable and its derivative.

Solving this simplified equation will yield the solution for that variable.

Repeat the process for the remaining variable to obtain the complete solution for the system of differential equations.

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In the diagram below, points E, F, and G are collinear. If FH bisects ZEFI and m/IFG=38°, then which
of the following is the measure of ZHFG?

Answers

Where the above conditions are given, note that ∠AFB  and ∠EFD are not vertical angles neither are they linear pair angles.

How is this so?

Vertical angles are a pair of non-adjacent angles formed by the intersection of two lines.

They are equal in measure and are formed opposite to each other. An example of vertical angles is when two intersecting roads create an "X" shape, and the angles formed at the intersection points are vertical angles.

Linear pair angles are a pair of adjacent angles formed by intersecting lines. They share a common vertex and a common side.

An example of linear pair angles is when two adjacent walls meet at a corner, and the angles formed by the walls are linear pair angles.

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perfect square number less than 10​

Answers

Answer:

2

Step-by-step explanation:

if that is not it please let me know i like feedback

Find the line of intersection between the lines: <3,−1,2>+t<1,1,−1> and <−8,2,0>+t<−3,2,−7>. (3) (10.2) Show that the lines x+1=3t,y=1,z+5=2t for t∈R and x+2=s,y−3=−5s, z+4=−2s for t∈R intersect, and find the point of intersection. (10.3) Find the point of intersection between the planes: −5x+y−2z=3 and 2x−3y+5z=−7. (3)

Answers

Solving given equations, we get line of intersection as  t = -11/4, t = -1, and t = 1/4, respectively. The point of intersection between the given lines is (-8, 2, 0). The point of intersection between the two planes is (2, 2, 86/65).

(10.2) To find the line of intersection between the lines, let's set up the equations for the two lines:

Line 1: r1 = <3, -1, 2> + t<1, 1, -1>

Line 2: r2 = <-8, 2, 0> + t<-3, 2, -7>

Now, we equate the two lines to find the point of intersection:

<3, -1, 2> + t<1, 1, -1> = <-8, 2, 0> + t<-3, 2, -7>

By comparing the corresponding components, we get:

3 + t = -8 - 3t   [x-component]

-1 + t = 2 + 2t   [y-component]

2 - t = 0 - 7t    [z-component]

Simplifying these equations, we find:

4t = -11   [from the x-component equation]

-3t = 3     [from the y-component equation]

8t = 2      [from the z-component equation]

Solving these equations, we get t = -11/4, t = -1, and t = 1/4, respectively.

To find the point of intersection, substitute the values of t back into any of the original equations. Taking the y-component equation as an example, we have:

-1 + t = 2 + 2t

Substituting t = -1, we find y = 2.

Therefore, the point of intersection between the given lines is (-8, 2, 0).

(10.3) Let's solve for the point of intersection between the two given planes:

Plane 1: -5x + y - 2z = 3

Plane 2: 2x - 3y + 5z = -7

To find the point of intersection, we need to solve this system of equations simultaneously. We can use the method of substitution or elimination to find the solution.

Let's use the method of elimination:

Multiply the first equation by 2 and the second equation by -5 to eliminate the x term:

-10x + 2y - 4z = 6

-10x + 15y - 25z = 35

Now, subtract the second equation from the first equation:

0x - 13y + 21z = -29

To simplify the equation, divide through by -13:

y - (21/13)z = 29/13

Now, let's solve for y in terms of z:

y = (21/13)z + 29/13

We still need another equation to find the values of z and y. Let's use the y-component equation from the second plane:

y - 3 = -5s

Substituting y = (21/13)z + 29/13, we have:

(21/13)z + 29/13 - 3 = -5s

Simplifying, we get:

(21/13)z - (34/13) = -5s

Now, we can equate the z-components of the two equations:

(21/13)z - (34/13) = 2z + 4

Simplifying further, we have:

(21/13)z - 2z = (34/13) + 4

(5/13)z = (34/13) + 4

(5/13)z = (34 + 52)/13

(5/13)z =

86/13

Solving for z, we find z = 86/65.

Substituting this value back into the y-component equation, we can find the value of y:

y = (21/13)(86/65) + 29/13

Simplifying, we have: y = 2

Therefore, the point of intersection between the two planes is (2, 2, 86/65).

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15 176 points ebook Hint Print References Required information A car with mass of 1160 kg accelerates from 0 m/s to 40.0 m/s in 10.0 s. Ignore air resistance. The engine has a 22.0% efficiency, which means that 22.0% of the energy released by the burning gasoline is converted into mechanical energy. What is the average mechanical power output of the engine? kW

Answers

The average mechanical power output of the car's engine is 24.65 kW.

To calculate the average mechanical power output of the car's engine, we need to determine the work done and the time taken. First, we find the work done by the engine, which is equal to the change in kinetic energy of the car. The initial kinetic energy is zero, and the final kinetic energy can be calculated using the formula KE = 0.5 * mass * velocity^2. Plugging in the values (mass = 1160 kg, velocity = 40.0 m/s), we find that the final kinetic energy is 928,000 J.

Next, we calculate the time taken for the car to accelerate from 0 m/s to 40.0 m/s, which is given as 10.0 s. The work done by the engine is equal to the change in kinetic energy divided by the time taken. Therefore, the work done is 928,000 J / 10.0 s = 92,800 W.

Since the engine's efficiency is 22.0%, only 22.0% of the energy released by the burning gasoline is converted into mechanical energy. Thus, the average mechanical power output of the engine is 0.22 * 92,800 W = 20,416 W, or 20.42 kW (rounded to two decimal places). Therefore, the average mechanical power output of the car's engine is 24.65 kW.

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Use the 18 rules of inference to derive the conclusion of the following symbolized argument:
1) G ⊃ A
2) G ⊃ L / G ⊃ (A · L)

Answers

Using the 18 rules of inference to derive the conclusion of the following symbolized argument is G ⊃ (A · L)

How to explain the information

In order to derive the conclusion using the 18 rules of inference, we can follow these steps:

Start with the premises:

G ⊃ A

G ⊃ L

Apply the rule of hypothetical syllogism (HS) to premises 1 and 2:

3. G ⊃ (A · L)

Therefore, the conclusion of the given argument is G ⊃ (A · L).

In conclusion, using the 18 rules of inference to derive the conclusion of the following symbolized argument is G ⊃ (A · L).

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Using the 18 rules of inference, we can derive the conclusion of the symbolized argument: 1) G ⊃ A, 2) G ⊃ L / G ⊃ (A · L).

To derive the conclusion G ⊃ (A · L) from the premises G ⊃ A and G ⊃ L, we can utilize the rules of inference.

Assume G (Assumption),

Apply Modus Ponens to premise 1 and assumption G: A.

Apply Modus Ponens to premise 2 and assumption G: L.

Apply Conjunction Introduction to A and L: (A · L).

Apply Conditional Introduction to the assumption G and the derived (A · L): G ⊃ (A · L).

By utilizing the rules of inference, we have successfully derived the conclusion G ⊃ (A · L) from the given premises G ⊃ A and G ⊃ L. This demonstrates the logical validity of the argument, showing that the conclusion follows from the premises using valid reasoning.

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If C. P = Rs480, S. P. = Rs 528, find profit and profit percent​

Answers

Answer:

Step-by-step explanation:

To find the profit and profit percentage, we need to know the cost price (C.P.) and the selling price (S.P.) of an item. In this case, the cost price is given as Rs480, and the selling price is given as Rs528.

The profit (P) can be calculated by subtracting the cost price from the selling price:

P = S.P. - C.P.

P = 528 - 480

P = 48

The profit percentage can be calculated using the following formula:

Profit Percentage = (Profit / Cost Price) * 100

Substituting the values, we get:

Profit Percentage = (48 / 480) * 100

Profit Percentage = 0.1 * 100

Profit Percentage = 10%

Therefore, the profit is Rs48 and the profit percentage is 10%.

Given three sets A, B, C. Determine whether each of the following propositions is always true.
(a) (AUB) NC = A U(BNC)
(b) If A UB = AUC, then B = C.
(c) If B is a subset of C, then A U B is a subset of AU C.
(d) (A \ B)\C = (A\ C)\B.

Answers

(a) The proposition (AUB) NC = A U(BNC) is always true.

(b) The proposition "If A UB = AUC, then B = C" is not always true.

(c) The proposition "If B is a subset of C, then A U B is a subset of AU C" is always true.

(d) The proposition "(A \ B)\C = (A\ C)\B" is not always true.

(a) The proposition (AUB) NC = A U(BNC) is always true. In set theory, the complement of a set (denoted by NC) consists of all elements that do not belong to that set. The union operation (denoted by U) combines all the elements of two sets. Therefore, (AUB) NC represents the elements that belong to either set A or set B, but not both. On the other hand, A U(BNC) represents the elements that belong to set A or to the complement of set B within set C. Since the union operation is commutative and the complement operation is distributive over the union, these two expressions are equivalent.

(b) The proposition "If A UB = AUC, then B = C" is not always true. It is possible for two sets A, B, and C to exist such that the union of A and B is equal to the union of A and C, but B is not equal to C. This can occur when A contains elements that are present in both B and C, but B and C also have distinct elements.

(c) The proposition "If B is a subset of C, then A U B is a subset of AU C" is always true. If every element of set B is also an element of set C (i.e., B is a subset of C), then it follows that any element in A U B will either belong to set A or to set B, and hence it will also belong to the union of set A and set C (i.e., A U C). Therefore, A U B is always a subset of A U C.

(d) The proposition "(A \ B)\C = (A\ C)\B" is not always true. In this proposition, the backslash (\) represents the set difference operation, which consists of all elements that belong to the first set but not to the second set. It is possible to find sets A, B, and C where the difference between A and B, followed by the difference between the resulting set and C, is not equal to the difference between A and C, followed by the difference between the resulting set and B. This occurs when A and B have common elements not present in C.

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Which of the following information should the nurse include QUESTION 9 The olfactory cortex where sensations of smell are picked up from olfactory neurons is located in this lobe of the cerebrum temporal lobe frontal lobe parietal lobe occipital lobe QUESTION 18 In the rhodopsin molecule contained within the photoreceptor cells called rods, the so called retinal portion changes from retinal as to light activates the molecule, causing the associated sodium and calcium channels to cis-retinal/trans-retinal / open trans-retinal / cis-retinal / close cis-retinal/trans-retinal / close trans-retinal / cis-retinal / open QUESTION 19 In the light configuration photoreceptor cells stop the release of the neurotransmitter that causes of cells glutamate/hyperpolarization / bipolar. glutamate / depolarization / bipolar Oglycine/hyperpolarization / ganglionic glutamate/hyperpolarization / ganglionic When white light reflects off of a green surface, which of the following occurs?1. All wavelengths of light are absorbed.2. Only the green wavelengths of light are absorbed.3. Only the green wavelengths of light are reflected.4. All wavelengths of light are reflected. What is the cash value of a lease requiring payments of $1,380.00 at the beginning of every three months for 13 years, if interest is 12% compounded semi-annually?The cash value of the lease is $(Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.) An emf of 15.0 mV is induced in a 513-turn coil when the current is changing at the rate of 10.0 A/s. What is the magneticflux through each turn of the coil at an instant when the current is 3.80 A? (Enter the magnitude.) 254 kg/h of sliced fresh potato (82.19% moisture, the balance is solids) is fed to a forced convection dryer. The air used for drying enters at 86C, 1 atm, and 10.4% relative humidity. The potatoes exit at only 2 43% moisture content. If the exiting air leaves at 93.0% humidity at the same inlet temperature and pressure, what is the mass ratio of air fed to potatoes fed?Type your answer in 3 decimal places. Exercise 1.20. What would be the final temperature of one litre of water produced by adding 500 mL of hot water at 92.0 C to 500 mL of cold water at 18.0 C in a calorimeter? Exercise 1.21. What would be the final temperature if 52.2 grams of silver, heated to 102.0 C, were added to a calorimeter containing 24.0 grams of water at 16.6 C? Exercise 1.22. When 33.6 grams of an unknown metal was heated to 98.8 C and placed in a calorimeter containing 75.0 grams of water at 14.8 C the temperature increased to 18.9 C and then underwent no further changes. (a) What is the calculated value for the specific heat of the unknown metal? (b) What is the likely identity of the metal? 9 (10 points) A planet orbits a star. The period of the rotation of 400 (earth) days. The mass of the star is 6.00 * 1030 kg. The mass of the planet is 8.00*1022 kg What is the orbital radius? On Friday, September 13, 1992, the lira was worth DM 0.015. Over the weekerd the lira devalued anainst the DM to DM 012. By wisat percent has the DM changed in value relative to the Lita? 20% 25% 25% 20% 1) Create a vector of from F(x,y,z) such that the x,y,&z components contain at least two variables (x,y,&z). The solve for the gradient, divergence, and curl of the vector, by hand. Show all of your work. 2) Create a problem of common ODE Form #1 or #2 with boundary values you define (see the notes for a refresher). Solve the equation using the boundary values you provide, by hand. Show all of your work. 3) Create a problem of common ODE Form #3 with boundary values you define (see the notes for a refresher). Solve the equation using the boundary values you provide, by hand. Show all of your work. 4) Create a problem of common ODE Form #5 with boundary values you define (see the notes for a refresher). Solve the equation using the boundary values you provide, by hand. Show all of your work. A pole-vaulter approaches the takeoff point at a speed of 9.15m/s. Assuming that only this speed determines the height to which they can rise, find the maximum height which the vaulter can clear the bar