Use the elimination method to find a general solution for the given linear system, where differentiation is with respect to t. x' = 2x-y
y' = 2y-16x Eliminate x and solve the remaining differential equation for y. Choose the correct answer below a. y(t) = C1 e^-2t + C2 e^6t
b. y(t) = C1 e^- -2t + C2 e^-6t
c. y(t) = C1 e^6t + C2 te^6t
d. y(t) = C1 e^-2t + C2 te^-2t
e. The system is degenerate.

Answer:

\(\boxed {x = 0}\)

Step-by-step explanation:

Solve for the value of \(x\):

\(x - 10 = 5x - 10\)

-Take \(5x\) and subtract it from \(x\):

\(x - 10 - 5x = 5x - 5x - 10\)

\(-4x - 10 = -10\)

-Add \(10\) to both sides:

\(-4x - 10 + 10 = -10 + 10\)

\(4x = 0\)

-Since \(4x\) does not equal to \(0\), then \(x\) must equal to \(0\):

\(\boxed {x = 0}\)

(a)The amount of water in the tank is increasing.

(b)Evaluate  \(\int\limits^{18}_0(W(t) - R(t)) dt\) to get the number of gallons of water in the tank at t = 18.

(c)Solve part (b) to get the absolute minimum from the critical points.

(d)The equation can be set up as   \(\int\limits^k_{18}-R(t) dt = 1200\) and solve this equation to find the value of k.

What is the absolute value of a number?

The absolute value of a number is its distance from zero on the number line. It represents the magnitude or size of a real number without considering its sign.

To solve the given problems, we need to integrate the given rates of water flow to determine the amount of water in the tank at various times. Let's go through each part step by step:

a)To determine if the amount of water in the tank is increasing at time t = 15, we need to compare the rate of water being pumped in with the rate of water being removed.

At t = 15, the rate of water being pumped in is given by \(W(t) = 95Vt sin^2(\frac{t}{6})\) gallons per hour. The rate of water being removed is \(R(t) = 275 sin^2(\frac{1}{3})\) gallons per hour.

Evaluate both rates at t = 15 and compare them. If the rate of water being pumped in is greater than the rate of water being removed, then the amount of water in the tank is increasing. Otherwise, it is decreasing.

b) To find the number of gallons of water in the tank at time t = 18, we need to integrate the net rate of water flow from t = 0 to t = 18. The net rate of water flow is given by the difference between the rate of water being pumped in and the rate of water being removed. So the integral to find the total amount of water in the tank at t = 18 is:

\(\int\limits^{18}_0(W(t) - R(t)) dt\)

Evaluate this integral to get the number of gallons of water in the tank at t = 18.

c)To find the time t when the amount of water in the tank is at an absolute minimum, we need to find the minimum of the function that represents the total amount of water in the tank. The total amount of water in the tank is obtained by integrating the net rate of water flow over the interval [0, 18] as mentioned in part b. Find the critical points and determine the absolute minimum from those points.

d. For t > 18, no water is pumped into the tank, but water continues to be removed at the rate R(t) until the tank becomes empty. To find the value of k, we need to set up an equation involving an integral expression that represents the remaining water in the tank after time t = 18. This equation will represent the condition for the tank to become empty.

The equation can be set up as:

\(\int\limits^k_{18}-R(t) dt = 1200\)

Here, k represents the time at which the tank becomes empty, and the integral represents the cumulative removal of water from t = 18 to t = k. Solve this equation to find the value of k.

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Answer:

the full solution is in the picture

d= -3

n= -2

x= -32

Answer:  They read for a equal amount of time.

Step-by-step explanation:

10/12 simplifies to 5/6, making these equal.

Proof:

10 divided by 2 equals 5, and 12 divided by 2 equals 6. This means that they are equal.

Answer:

The area of the circle

Step-by-step explanation:

The area of a circle is given by:

A = pi•r^2

☝️That is "Area equals pi times r squared"

Fun fact: you can use r (radius) or d (diameter) to find the circumference (distance all the way around a circle)

C = pi • d

or C = 2pi • r

But for area (all the space inside)

A = pi • r^2

You get:

Work/Explanation:

When multiplying pi by the square of a circle's radius, we get the circle's area.

This is how it looks in a formula:

\(\bf{\pi\times r^2}\)

Which can be simplified to

\(\bf{\pi r^2}\)

Which is the same as

\(\bf{A=\pi r^2}\)

Answer:

522.5

Step-by-step explanation:

you need to show me the picture, or point out which is the height, which are the bases.

I am supposed 38 is the height

8.5 & 19 are the bases.

the formula for calculating the are of the trapezoid is:

(8.5+19)*38 /2 =522.5

Using the divergence theorem we can compute that the outward flux of the vector field is 16π .

The outward flux of F over the solid cylinder and z = 0 is

∫∫F·ds  = ∫∫∫ DivF dv

F = 2xy² i   +  2x²y j + 2xy k

Div F  =  D/dx (2xy²)  +   D/dy (2x²y )

Div F  =  2y²  +  2x²

In cylindrical coordinates   dV = rdrdθdz and as z = 0 the region is a surface ds = r·dr·dθ

Using the parametric form of the surface equation

x = rcosθ        y =  r sinθ      and z = z

Div F  = 2r² sin²θ  + 2r²cos²θ

∫∫∫ DivF dv  =  ∫∫ [2r²sin²θ  + 2r²cos²θ] × rdrdθ

∫∫ 2r² [sin²θ  + cos²θ] × rdrdθdz   ⇒  ∫∫ 2r³ drdθ

Integration limits

0 < r < 2          0 < θ < 2π

2∫₀² r³ ∫dθ

(2/4) × (2)⁴ × 2π

The divergence theorem, commonly known as Gauss' theorem or Ostrogradsky's theorem, is a theorem that connects the flow of a vector field across a closed surface to the field's divergence in the volume enclosed.

In more detail, the divergence theorem states that the surface integral of a vector field across a closed surface, sometimes referred to as the "flux" through the surface, is equal to the volume integral of the divergence over the region inside the surface.

Therefore the flux is 16π .

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Answer:

-16r^2 - 10r

Step-by-step explanation:

all you got to do is distribute

We are given a path c, and a vector field f. The path c is defined by r(t) = (t³, t², -2t) for 0 ≤ t ≤ 1. The vector field f = (-4sin x, 5cos y, 10xz). We are required to evaluate the line integral ∫cf ⋅ dr using the given information.To evaluate the line integral, we use the following formula:∫cf ⋅ dr = ∫abf(r(t)) ⋅ r'(t) dt.

Here, we can see that r(t) is already in vector form, so we don't need to convert it. We just need to find r'(t).Differentiating r(t) with respect to t, we get:r'(t) = (3t², 2t, -2)Substituting the given values of f and r'(t), we get:∫cf ⋅ dr = ∫₀¹ (-4sin t³, 5cos t², -20t³) ⋅ (3t², 2t, -2) dt= ∫₀¹ (-12t⁴ sin t³ + 10t cos t² - 20t⁴) dt= (-3t⁴ cos t³ + 5t³ sin t² - 5t⁵) from 0 to 1= -3 cos 1 + 5 sin 1 - 5The final answer is -3 cos 1 + 5 sin 1 - 5.

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Answer:

Step-by-step explanation:

I would help but I struggle in the same subject

Answer:

Hello is this math i can help you with your answer

Answer:

x=5

Step-by-step explanation:

6(5)= 30

a. the direction of the acceleration is approximately 78.4° counterclockwise from the +x-axis. b. the mass of the object is approximately 12.1 kg. c. the initial speed of the object is 0 m/s. d. the velocity components after 21.0 s based solely on the given information.

To solve this problem, we will use Newton's second law, which states that the net force acting on an object is equal to the mass of the object multiplied by its acceleration:

ΣF = m * a

Given:

F₁ = (-1.85i + 6.75) N

F₂ = (-5.50i - 2.35) N

F₃ = (-43i) N

Acceleration magnitude (a) = 3 m/s²

(a) Direction of the acceleration:

To determine the direction of the acceleration, we need to find the net force acting on the object and then calculate the angle between the net force vector and the positive x-axis.

Net force (ΣF) = F₁ + F₂ + F₃

Net force = (-1.85i + 6.75) + (-5.50i - 2.35) + (-43i)

         = (-1.85i + 6.75 - 5.50i - 2.35 - 43i)

         = (-7.35i - 35.6)

The magnitude of the net force can be calculated using Pythagorean theorem:

|ΣF| = sqrt(Re² + Im²) = sqrt((-7.35)² + (-35.6)²) ≈ 36.3 N

The angle (θ) between the net force vector and the positive x-axis can be found using trigonometry:

θ = atan(Im/Re) = atan((-35.6)/(-7.35)) ≈ 78.4°

Therefore, the direction of the acceleration is approximately 78.4° counterclockwise from the +x-axis.

(b) Mass of the object:

From Newton's second law, we know that ΣF = m * a. Since we have the acceleration magnitude and the net force, we can solve for the mass (m):

|ΣF| = m * a

36.3 = m * 3

m = 36.3 / 3

m ≈ 12.1 kg

Therefore, the mass of the object is approximately 12.1 kg.

(c) Initial speed:

Given that the object is initially at rest (v₀ = 0 m/s), we can use the formula for final velocity (v) in terms of initial velocity (v₀), acceleration (a), and time (t):

v = v₀ + a * t

Since v₀ = 0 m/s and a = 3 m/s², we have:

v = 0 + 3 * 0

v = 0 m/s

Therefore, the initial speed of the object is 0 m/s.

(d) Velocity components after 21.0 s:

The problem does not provide information about the time evolution of the forces acting on the object, so we cannot determine the velocity components after 21.0 s based solely on the given information.

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Answer:

A

Step-by-step explanation: When she got to 14-4+3*4 she added 4+3 first instead of multiplying 3*4.

Answer:

m

Step-by-step explanation:

m x m x m x m x m = \(m^{5}\)

So the base would be "m", the exponent is "5".

Answer: A. m>5

Step-by-step explanation:

An interaction occurs when (D) the values of one independent variable influence how another independent variable influences a result.

A type of activity known as interaction takes place when two or more items have an impact on one another.

In contrast to a one-way causal effect, the idea of a two-way effect is crucial to the concept of interaction.

Interactivity and interconnection are closely related concepts; the latter deals with the interactions of interactions inside systems; combinations of numerous straightforward interactions might result in unexpected emergent phenomena.

In many sciences, interaction has numerous specialized meanings.

When the values of one independent variable affect how a result is affected by another independent variable, this is known as an interaction.

Therefore, an interaction occurs when (D) the values of one independent variable influence how another independent variable influences a result.

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Complete question:

An interaction occurs when:

a. a single independent variable changes the dependent variable, disregarding all other variables in the study

b. two independent variables both influence the dependent variable

c.the dependent variable does not depend on any of your independent variables

d. the effects of one of your independent variables depend on the level of the other independent variable

The following a probability model? what do we call the outcome No, the provided information is not sufficient to determine if it is a probability model. The outcome "red" is typically referred to as an event.

A probability model is a mathematical representation of a random experiment, where the sample space is defined, and probabilities are assigned to all possible outcomes. To determine if the given information is a probability model, we would need to know the complete list of possible outcomes, their corresponding probabilities, and ensure that the probabilities meet the necessary conditions (sum up to 1 and are non-negative).

Based on the limited information provided, we cannot determine if it is a probability model. The outcome "red" is called an event in the context of probability.

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Spherical harmonics are an integral part of quantum mechanics. They describe the shape of the orbitals, which electrons occupy in atoms. Moreover, the spherical harmonics provide the angular distribution of a wave in spherical coordinates. In 3D, the spherical harmonics can be written as:

Ylm(θ, φ) = √(2l + 1)/(4π) * √[(l - m)!/(l + m)!] * Plm(cosθ) * e^(imφ)

Here, l and m are known as the angular quantum numbers. They define the shape and orientation of the orbital. Plm(cosθ) represents the associated Legendre polynomial, and e^(imφ) is the exponential function. The spherical harmonics have various forms, including:

Y1,1 = -Y1,-1 = 1/2 √(3/2π) sinθe^(iφ)

Y1,0 = 1/2 √(3/π)cosθ

Y2,2 = 1/4 √(15/2π)sin²θe^(2iφ)

Y2,1 = -Y2,-1 = 1/2 √(15/2π)sinθcosφ

Y2,0 = 1/4 √(5/π)(3cos²θ-1)

Y0,0 = 1/√(4π)

The spherical harmonics have various applications in physics, including quantum mechanics, electrodynamics, and acoustics. They play a crucial role in understanding the symmetry of various systems. Hence, the spherical harmonics are an essential mathematical tool in modern physics. Thus, this is how one can show another form for spherical harmonics.

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The elementary skills have used so far in the course is explained below.

What is whole numbers?

An integer is the range zero, a high quality herbal range or a bad integer with a minus sign. The bad numbers are the additive inverses of the corresponding high quality numbers. In the language of mathematics, the set of integers is frequently denoted through the boldface Z or blackboard bold.

The cap potential to define entire numbers and differentiate them from natural numbers is a critical component of math.

complete numbers like 0, 1, and 2 are the constructing blocks to expertise greater complex wide variety identifiers like actual numbers, rational numbers, and irrational numbers.

The subset of numbers within integers is whole numbers. those are numbers that we're maximum used to working with, which include zero. We see entire numbers on nutrients labels, or signs on the toll road telling us how many miles are to the subsequent exit/city.

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Answer:

B

Step-by-step explanation:

Radius is half of diameter so r is 36

\(4/3\pi r^3\) is volume of sphere

using 3.14 as pi you get 195,333.12 cubic cm

The mathematical representation of 20P4 is \(^{20}P_4 = 20 \times 19 \times 18 \times 17\). so option C is the correct.

We can use combinations for this case,

The Total number of distinguishable things is m.

Here, Out of those m things, k things are to be chosen such that their order doesn't matter.

This can be done in total of

\(^mC_k = \dfrac{m!}{k! \times (m-k)!} ways.\)

If the order matters, then each of those choice of k distinct items would be permuted k! times,

So, total number of choices in that case would be:

\(^mP_k = k! \times ^mC_k = k! \times \dfrac{m!}{k! \times (m-k)!} = \dfrac{m!}{ (m-k)!}\\\\^mP_k = \dfrac{m!}{ (m-k)!}\)

This is called permutation of k items chosen out of m items (all distinct).

We have to find the mathematical representation of 20P4.

\(^{20}P_4\)

here, m = 20

k = 4

Now substitute;

\(^mP_k = \dfrac{m!}{ (m-k)!}\\\\^{20}P_4 = \dfrac{20!}{ (20-4)!}\\\\^{20}P_4 = \dfrac{20!}{ (16)!}\\\\\\^{20}P_4 = \dfrac{20 \times 19 \times 18 \times 17 \times 16!}{ (16)!}\\\\\\^{20}P_4 = 20 \times 19 \times 18 \times 17\)

Hence, the mathematical representation of 20P4 is \(^{20}P_4 = 20 \times 19 \times 18 \times 17\). so option C is the correct.

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Answer:

the slope is -7

Step-by-step explanation:

\(\frac{-2-5}{3-2} =\frac{-7}{1}\)

Answer:

59.11

Step-by-step explanation:

To determine the necessary sample size to obtain an interval width of 50 ppm for a confidence level of 95%, we need to use the formula for sample size calculation for estimating a population mean.

The formula for sample size calculation is:

n = (Z * σ / E)^2

In this case, the desired margin of error is 50 ppm, which means the interval width is 2 * E = 50 ppm. Therefore, E = 25 ppm.

The Z-score corresponding to a 95% confidence level is approximately 1.96.

Given that the investigators made a rough guess of 175 for the value of σ (standard deviation) before collecting data.

We can substitute these values into the sample size formula:

n = (1.96 * 175 / 25)^2

Simplifying the calculation:

n = (7 * 175)^2

n = 1225^2

n ≈ 1,500,625

Therefore, a sample size of approximately 1,500,625 would be necessary to obtain an interval width of 50 ppm for a confidence level of 95%.

To obtain an interval width of 50 ppm with a confidence level of 95%, a sample size of approximately 1,500,625 is required. This is calculated using the formula for sample size estimation, considering a desired margin of error of 25 ppm and a standard deviation estimate of 175. The Z-score corresponding to a 95% confidence level is used to determine the sample size.

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Answer:

x=-6\sqrt(3)

Step-by-step explanation:

The concept of ratio is used here to determine the number of boys in the school. Here the in a school of 425 pupils, the total number of boys in the school is 200.

Here there are 425 pupils, If the ratio of boys to girls is 8 to 9, then 8 out of 17 are boys and 9 out of every 17 are girls. Set up a proportion for the boys, where b = the total number of boys.

8 / 17 = b / 425

17 b = 8 × 425

b = 8 × 425 / 17

b = 200

So the number of boys are 200 and the number of girls are 425 - 200 = 225.

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Answer:

m<A = 90° - m<B

(AB)^2 = (AC)^2 + (BC)^2

sin A = cos B

if <A = <B, then m<A = 45°

Answer:

Step-by-step explanation:

∠A+∠B=90 complementary angles ( the sum equal 90°)

∠A=90 degrees-∠B

tan A=sinA/cos A

AB²= AC² +BC² (Pythagorean theorem)

if angle A=angle B then the angles are 45 degrees

cos A=sin(90-A)  

sin (a - b) = sin a.cos b - sin b.cos a

sin(90-A)=sin90.cosA-sinbAcos90 cos 90=0 and sin 90=1

sin(90-A)=1*cosA

sin(90-A)=cosA

the ones are  in bold are right