Please help! I need the answers with an explanation to these questions ASAP. Thanks!

Question 1. What value could complete this given equation that would make it true for no values of x? Show or explain your thinking.

Question 2. What value could complete this equation that would make it true for every value of x? Show or explain.

The differential equation for the amount of pollutant y(t) in the lake is dy/dt = 550/yr - (y(t)/12200)(67.1 km^3/yr), where y(t) is measured in tons and t is measured in years. The solution to the differential equation is y(t) = (550/67.1)(1 - exp((-67.1/12200)t)) + 200000. As t goes to infinity, y(t) approaches 20818.5 tons.

The change in pollutant in the lake over a small time interval is given by the difference between the amount that enters the lake and the amount that leaves, plus the amount that is added directly:

dy/dt = (rate in) - (rate out) + (rate added)

The rate in and rate out are both equal to 67.1 km^3/yr, so we can substitute these values:

dy/dt = 550/yr - (y(t)/12200)(67.1 km^3/yr)

To solve the differential equation, we can use separation of variables:

dy/dt + (67.1/12200)y = 550/12200

Multiplying both sides by the integrating factor exp((67.1/12200)t), we get:

exp((67.1/12200)t)dy/dt + (67.1/12200)y exp((67.1/12200)t) = (550/12200)exp((67.1/12200)t)

This can be written as:

d/dt (exp((67.1/12200)t)y) = (550/12200)exp((67.1/12200)t)

Integrating both sides with respect to t,

exp((67.1/12200)t)y = (550/67.1)exp((67.1/12200)t) + C

where C is the constant of integration. We can find the value of C using the initial condition y(0) = 200000:

exp(0) * 200000 = (550/67.1)exp(0) + C

C = 200000 - (550/67.1)

Substituting this value back into the equation, we get:

exp((67.1/12200)t)y = (550/67.1)exp((67.1/12200)t) + 200000 - (550/67.1)

y(t) = (550/67.1)(1 - exp((-67.1/12200)t)) + 200000

As t goes to infinity, the exponential term exp((-67.1/12200)t) goes to zero, so y(t) approaches the steady state solution given by:

y(t) → (550/67.1) + 200000 ≈ 20818.5

In practical terms, this means that over time the amount of pollutants in the lake will approach a constant value of approximately 20818.5 tons. The rate at which the pollutants enter and leave the lake is balanced by the rate at which they are added directly, resulting in a steady state concentration of pollutants in the lake.

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The convolution operation will yield the main answer. Assuming the input signal x(t) is causal (i.e., x(t) = 0 for t < 0), the output signal y(t) can be computed by convolving the input signal with the impulse response:y(t) = x(t) * h(t)

To compute the output of the given input signal using the impulse response, we need to convolve the input signal with the impulse response.

The convolution integral is given by:

y(t) = ∫[x(τ) * h(t-τ)] dτ

Substituting the given values:

x(t) = 6 + 10cos(t) + 13cos(2t)

h(t) = \(5e^(-t)u(t) - 16e^(-2t)u(t) + 13e^(-3t)u(t)\)

Performing the convolution operation, we multiply the input signal with the impulse response for different values of τ and integrate over the range of τ.

y(t) = ∫[x(τ) * h(t-τ)] dτ

y(t) = ∫[(6 + 10cos(τ) + 13cos(2τ)) * (5e^(-(t-τ))u(t-τ) - 16e^(-2(t-τ))u(t-τ) + 13e^(-3(t-τ))u(t-τ))] dτ

Evaluating this integral will give us the output signal y(t).

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Answer:

Helena is correct.

Step-by-step explanation:

The slope intersept form of the line is

y = m x + c

where, m is the slope and c is the intersept.

The point slope form of the equation is

\(y - y'= \frac{y''-y'}{x''-x'}\times (x-x')\)

So, if there are two points are given, the equation of line is found by the point slope form.

So, Helena is correct.

Answer:

Step-by-step explanation:

The system of equation given is a simultaneous equation

7x + 2y = 24 ...... 1

2x + 1 = y ......... 2

Rearrange

7x + 2y = 24  .... *1

| 2x - y = -1 ..... * 2

Use elimination method

7x + 2y = 24

4x - 2y = -2

add

7x+4x = 24-2

11x = 22

x = 22/11

x = 2

Substitute x = 2 into 2

2x+1 = y

y 2(2)+1

y = 4+1

y = 5

Hence x = 2, y = 5

The system of equation is a simultaneous equation

The measured signal could have a dynamic error of approximately 21.6% of the true value due to the limitations of the 1st order system with a time constant of 0.7 seconds.

The dynamic error that could result from measuring a 2 Hz periodic waveform using a 1st order system with a time constant of 0.7 seconds can be estimated using the following formula:

Dynamic error = 100% x (1 - exp(-T/τ)),

where T is the period of the input signal, τ is the time constant of the system, and exp(-T/τ) is the system's gain at the input frequency.

Substituting T = 0.5 s (since the period of a 2 Hz waveform is 0.5 seconds) and τ = 0.7 s, we get:

Dynamic error = 100% x (1 - exp(-0.5/0.7)) ≈ 21.6%

This means that the measured signal could have a dynamic error of approximately 21.6% of the true value due to the limitations of the 1st order system with a time constant of 0.7 seconds. This error is due to the fact that the system cannot respond fast enough to the rapid changes in the input signal, leading to an output that lags behind the input. The larger the time constant of the system relative to the period of the input signal, the larger the dynamic error will be.

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Answer:

1) \(49 {a}^{2} - 16 {b}^{2} \)

2) \(251502\)

Step-by-step explanation:

a) Question :-\((7a - 4b)(7a + 4b)\)

Identity used here :-\((a + b)(a - b) = {a}^{2} - {b}^{2} \)

So,

\((7a + 4b)(7a - 4b)\)

\(=> {(7a)}^{2} - {(4b)}^{2} \)

\(=> 49 {a}^{2} - 16 {b}^{2} \)

b) Question :- \(501 \times 502\)

Identity used here :-\((x + a)(x + b) = {x}^{2} + (a + b)x + ab\)

So,

\(501 \times 502 = (500 + 1)(500 + 2)\)

\( = > {(500)}^{2} + (1 + 2)500 + 1 \times 2\)

\( = > 250000 + 1500 + 2\)

\( = > 251502\)

The first statement is false because for any invertible matrix null space contains only zero vector.

Part (b) is true as the set of solutions of a homogeneous system of equations is the null space of the matrix. here the order of the matrix is 12 by 15. we will have a linear transformation from R^15 to R^12.null space is always a subspace of domain space.

Part (d) is false. because a subspace is always a subset of vector space.

For part (c). True

You can use the rank nullity theorem for any m by n matrix null space of A and cool space of A have the zero vector in common.

A mixture consists of one or more pure substances with different compositions. There are two types of mixtures: heterogeneous mixtures and homogeneous mixtures. Heterogeneous mixtures have visually distinct components, while homogeneous mixtures appear homogeneous throughout.

Homogeneous means (1) having the same or similar properties and (2) being uniform in structure or composition. The corresponding noun is homogeneity. Homogeneity, whose corresponding noun is homogeneity, is a little-used biological term whose ancient meaning has largely been lost. is a mixture of gases, liquids, or solids equal It has a uniform composition throughout. Inhomogeneous mixtures only one phase of matter is observed.

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Answer:

  42, 38, 34, 30

Step-by-step explanation:

You want the first 4 terms of the sequence described by ...

  an = 42 -4(n -1), n ∈ ℕ

You can write the first 4 terms of the sequence by evaluating the 'an' expression for n = 1, 2, 3, 4.

Or, you can recognize the expression describes a sequence with a first term of 42 and a common difference of -4. That is, each term is 4 less than the one before.

The terms you want are ...

  42, 38, 34, 30

__

Additional comment

The equation for the n-th term of an arithmetic sequence is ...

  an = a1 +d(n -1)

where a1 is the first term, and d is the common difference. Comparing this to the given equation, we see a1 = 42, d = -4.

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Answer:

Option 2

Step-by-step explanation:

ΔPQR - ΔSTR

Because

∠S ≅ ∠P

∠T ≅ ∠Q

∠R ≅ ∠R

I hope this helps you

Answer:

c = -22, 22

Step-by-step explanation:

\( {(x - 11)}^{2} = {x}^{2} - 22x + 121\)

\( {(x + 11)}^{2} = {x}^{2} + 22x + 121\)

The ordered pairs that could be points on a parallel line are:

(-8, 8) and (2, 2)

(-2, 1) and (3, -2)

Parallel lines have the same slope. Thus, we have to find ordered pairs with a slope of -3/5.

We have:

slope of the line is -3/5.

Thus, m = -3/5

Formula for slope between two coordinates is;

m = (y₂ - y₁)/(x₂ - x₁)

A) At (–8, 8) and (2, 2);

m = (2 - 8)/(2 - (-8))

m = -6/10

m = -3/5

B) At (–5, –1) and (0, 2);

m = (2 - (-1))/(0 - (-5))

m = 3/5

C) At (–3, 6) and (6, –9);

m = (-9 - 6)/(6 - (-3))

m = -15/9

m = -5/3

D) At (–2, 1) and (3, –2);

m = (-2 - 1)/(3 - (-2))

m = -3/5

E) At (0, 2) and (5, 5);

m = (5 - 2)/(5 - 0)

m = 3/5

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When salt is added to water, the boiling point of water is increased. The boiling point elevation depends on the concentration of salt added. The addition of salt increases the boiling point of water.

The extent of the increase in boiling point is directly proportional to the amount of salt added. Boiling point is defined as the temperature at which the vapor pressure of a liquid is equal to the pressure exerted on it by the surrounding atmosphere.

In layman's terms, this means that a liquid boils when its vapor pressure equals atmospheric pressure. Boiling point of a liquid can be increased or decreased by the addition of solutes such as salt.When a solute such as salt is added to water, the boiling point of water increases due to the phenomenon of boiling point elevation.

This is because the solute particles in the water interfere with the formation of water vapor molecules at the surface of the liquid. As a result, more energy is required to boil the water, which means that the boiling point is elevated.

The amount of elevation is directly proportional to the concentration of salt added. In short, the addition of salt to water increases the boiling point of water.

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The solution to the given system of linear equations using the five methods is x₁ = -2, x₂ = -1, and x₃ = -4.

Here is the step-by-step explanation using each method:

1. Gaussian Elimination:

  - Rewrite the system in matrix form: [2 -6 -1; -8 1 -2] * [x₁; x₂; x₃] = [-38; -20].

  - Perform row operations to eliminate variables: Row₂ = Row₂ + 4 * Row₁

2. Gauss-Jordan Elimination:

  - Augment the coefficient matrix with the column vector of constants.

  - The rightmost column will contain the solutions x₁, x₂, and x₃.

3. LU Factorization:

  - Decompose the coefficient matrix A into LU form, where L is a lower triangular matrix and U is an upper triangular matrix.

  - Solve Lc = b, where c is a vector containing intermediate solutions.

  - Solve Ux = c to obtain the final solution.

4. Inverse Matrix:

  - Rewrite the system in matrix form: AX = B, where A is the coefficient matrix, X is the vector of variables, and B is the vector of constants.

  - Find the inverse matrix A⁻¹.

  - Multiply both sides of the equation by A⁻¹ to obtain X = A⁻¹B.

5. Cramer's Rule:

  - Divide the determinant of the modified A by the determinant of A to find x₁.

  - Repeat the process for x₂ and x₃, replacing the respective columns of A with B.

Note: Due to the limited word count, the step-by-step explanations provided here may not be comprehensive. It's recommended to consult a textbook or online resources for detailed instructions on each method.

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Answer:

p = 78.8 m

mass of chain = 134 kg

Step-by-step explanation:

a)  sin28 = 37/p             sin = opposite/hypotenuse

p = 37/sin28 = 78.8 m

b)  78.8 m  x 1.7 kg/m = 134 kg

Answer:

0.41

Step-by-step explanation:

Unit = 5/12

Unit = 0.41 runs per inn innings

Answer:

(x+5)-6=7

(x+5)=7+6

(x+5)=13

x+5=13

x=13-5

x=8

Answer:

Step-by-step explanation:

Answer:

v=9

Step-by-step explanation:

Answer:

v

Step-by-step explanation:

Answer:

C.) 33.60

Step-by-step explanation:

OMP. = $56

D. = 25%

0.25 * $56 = $14

SP. = $56-$14=$42

CD. = 20% of Selling price

= 0.20 * $42 = $8.40

FSP. = $42-$8.40      = $33.60

Answer:

c

Step-by-step explanation:

Answer:

y= -2

x= -1

Step-by-step explanation:

Answer:

It's (8/3) 6  I took the test and that was the answer. Your Welcome!!!!!!!!!!!!!

Answer:

Its 83 the power of 6. Also Please choose my anwer as brainliest

Answer:

150 pounds

the weight is directly proportional to the volume. hence:

138/23 = 6

now, 25*6 = 150

Answer:

B. 150 pounds

Step-by-step explanation:

The given integral ∫(-2 to 1) x^2 dx converges and its value is 3.

To determine if the integral converges or diverges, we evaluate the definite integral ∫(-2 to 1) x^2 dx.

Integrating x^2 with respect to x, we get (1/3) x^3. Evaluating the definite integral over the given bounds, we have:

(1/3) [x^3] from -2 to 1.

Substituting the bounds into the antiderivative expression, we get:

(1/3) [1^3 - (-2)^3] = (1/3) [1 - (-8)] = (1/3) [9] = 3.

Since the value of the integral is a finite number (3), we conclude that the given integral converges. The definite integral has a finite value when evaluated over the interval from -2 to 1.

Therefore, the given integral ∫(-2 to 1) x^2 dx converges and its value is 3.

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Answer:

Step-by-step explanation:

Key March Highlights: Travel spending totaled $93 billion in February—5% above 2019 levels and 9% above 2022 levels. Leisure travel demand does not appear to be abating with America’s excitement to travel at record highs and more than half (55%  Data Question 2 The following table shows the number of visitors to a park from January to April: Month January

The least common denominator of the given rational expressions is 2x^3(x + 4). This is the smallest expression that both denominators can be multiplied by to obtain equivalent fractions with a common denominator.

To find the least common denominator (LCD) of the rational expressions 1/(2x^2 + 8x)and 10/x^2, we need to determine the common factors in the denominators and then multiply them.

The first denominator, 2x^2 + 8x, can be factored as 2x(x + 4), where x + 4 is a linear factor.

The second denominator,\(x^2\), is already in its factored form.

To find the LCD, we need to consider all the factors and their highest powers. In this case, we have:

2x(x + 4) and \(x^2\)

Since the highest power of x in 2x(x + 4) is 1 and in \(x^2\) is 2, the LCD must include both factors with their highest powers. Therefore, the LCD is:

LCD = 2x(x + 4)(x^2)

Simplifying further, we can write the LCD as:

LCD = 2x^3(x + 4)

So, the least common denominator of the given rational expressions is 2x^3(x + 4). This is the smallest expression that both denominators can be multiplied by to obtain equivalent fractions with a common denominator.

Using the LCD, we can then add or subtract the rational expressions by multiplying each term by the necessary factors to achieve the common denominator.

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\(8.92 \times 10^6\)

\(8920000\)

Answer: 945.52

Step-by-step explanation:

Answer:

The answer would be -2.

Step-by-step explanation:You have to look at where the line is hitting.It has to be perfectly on the corners of the square. The bottom of the y-intercept is always negative and the top is always positive but for the x-intercept the right side is positive and the left side is negative. I hope that helped you!!

Answer:

1st=2. (-7,9) (-18,9)

2nd=4. y= 5/9x+10