PLEASE HELP ME!!!! I WILL MARK!!!

The solution to the initial value problem is:

x(t) = [tex](1/4) * e^{(3t)} - (1/4) * e^{(4t)}[/tex] and y(t) = [tex]e^{(3t)} + (1/6) * e^{(4t)}[/tex]

To find the solution to the initial value problem Y' = AY, where A is the matrix with eigenvalues and eigenvectors given, we can use the eigenvalue-eigenvector method.

Let's denote the eigenvectors as v₁ and v₂:

v₁ = [2]

[8]

v₂ = [-3]

[2]

The general solution to the system of linear differential equations can be written as:

Y(t) = [tex]c_1 * e^{(X_1t)} * v_1 + c_2 * e^{(X_2t)} * v_2[/tex]

where c₁ and c₂ are constants.

Substituting the given eigenvalues and eigenvectors:

Y(t) = c₁ * [tex]e^{(3t)[/tex] * [2] + c₂ * [tex]e^(4t)[/tex] * [-3]

[8] [2]

Simplifying further:

Y(t) = [2c₁ * [tex]e^{(3t)[/tex] - 3c₂ * [tex]e^{(4t)[/tex]]

[8c₁ * [tex]e^{(3t)[/tex] + 2c₂ * [tex]e^{(4t)[/tex]]

To find the specific solution that satisfies the initial condition Y(0) = [0] and [8], we can substitute t = 0 into the general solution:

Y(0) = [2c₁ - 3c₂]

[8c₁ + 2c₂]

Equating this to the initial condition:

[2c₁ - 3c₂] = [0]

[8c₁ + 2c₂] [2]

Solving this system of equations will give us the values of c₁ and c₂:

2c₁ - 3c₂ = 0 ----(1)

8c₁ + 2c₂ = 2 ----(2)

Multiplying equation (1) by 4 and adding it to equation (2), we get:

16c₁ = 2

Solving for c₁, we find:

c₁ = 1/8

Substituting c₁ back into equation (1), we can solve for c₂:

2(1/8) - 3c₂ = 0

1/4 - 3c₂ = 0

3c₂ = 1/4

c₂ = 1/12

Therefore, the specific solution to the initial value problem is:

x(t) = [tex](1/8) * e^{(3t)} * 2 + (1/12) * e^{(4t) }* (-3)[/tex]

y(t) = [tex](1/8) * e^{(3t)} * 8 + (1/12) * e^{(4t)} * 2[/tex]

Simplifying further:

x(t) = [tex](1/4) * e^{(3t)} - (1/4) * e^{(4t)}[/tex]

y(t) = [tex]e^{(3t)} + (1/6) * e^{(4t)}[/tex]

Therefore, the solution to the initial value problem is x(t) = [tex](1/4) * e^{(3t)} - (1/4) * e^{(4t)}[/tex] and y(t) = [tex]e^{(3t)} + (1/6) * e^{(4t)}[/tex].

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(a) The area of triangle PQR is 9.165 units².

(b) The formula for a plane given three points is: 10x - 7y - 2z = 0

(c) The volume of the parallelepiped with edges PO, PR, and PS is 148 units³.

(d)  A point on the line through P and Q which is two units away from P is (4/5, 6/5, 2/5).

(a) The area of triangle PQR is 9.165 units².

The formula for the area of a triangle given three points is:

Area = 1/2 | ((x2 − x1) × (y3 − y1)) − ((y2 − y1) × (x3 − x1)) |

The coordinates for P, Q, and R are: P (0, 0, -2)Q (2, 3, 4)R (4, 6, 5)

Substituting into the formula gives us:

Area = 1/2 | ((2 - 0) × (5 + 2)) − ((3 - 0) × (4 - 0)) |

Area = 9.165 units² (rounded to three decimal places)

(b) An equation of the form ax + by + cz = d for the plane containing points P, Q, and R is:

10x - 7y - 2z = 0

The formula for a plane given three points is: ax + by + cz = d

To find a, b, c, and d, we first need to find two vectors on the plane.

We can use PQ and PR.

PQ = Q - P = (2 - 0)i + (3 - 0)j + (4 + 2)k = 2i + 3j + 6k

PR = R - P = (4 - 0)i + (6 - 0)j + (5 + 2)k = 4i + 6j + 7k

Now we can find the normal vector by taking the cross product of PQ and PR:

PQ x PR = <3i - 26j - 12k>

So the equation of the plane is:3x - 26y - 12z = 0

We can simplify this by multiplying all terms by -2, which gives:10x - 7y - 2z = 0

(c) The volume of the parallelepiped with edges PO, PR, and PS is 148 units³.

The volume of a parallelepiped is given by the scalar triple product of three vectors.

We can use OP, PR, and PS.

OP = P - O = -i - j - 2k = < -1, -1, -2 >

PR = R - P = 4i + 6j + 7k = < 4, 6, 7 >

PS = S - P = 6i + 11j + 12k = < 6, 11, 12 >

The scalar triple product is:

OP ⋅ (PR x PS)OP ⋅ (PR x PS) = < -1, -1, -2 > ⋅ (< 54, -20, -10 >)OP ⋅ (PR x PS) = -148

The volume of the parallelepiped is 148 units³.

(d) A point on the line through P and Q which is two units away from P is (4/5, 6/5, 2/5).

The equation of the line through P and Q is:x = 2t, y = 3t, z = 4 + 6t

A point on the line that is two units away from P is given by:

t = 2/5

Substituting into the equations for x, y, and z gives:(4/5, 6/5, 2/5)

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Answer:

D

Step-by-step explanation:

Answer:

4 is choice C. THis is because the y intercept is 1 and when you go up 2 units, and go right once, you go to another point (rise over run)

Step-by-step explanation:

If f(x) is reducible over Zp, then f(x) is reducible over Q.

If a polynomial f(x) with integer coefficients is reducible over Zp (the integers modulo p), where p is a prime number, then it is also reducible over Q (the rational numbers).

This result follows from the fact that if a polynomial is reducible over a smaller field (Zp), it must also be reducible over a larger field (Q).

Since Zp is a subset of Q, any factorization of the polynomial in Zp can also be used in Q. Therefore, if f(x) is reducible over Zp, it is also reducible over Q.

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Answer:

Formula for circumference: 2 x pie(3.14 or 22/7) x radius(half of diameter or already given radius). Area of a circle: pie(3.14 or 22/7) x radius squared

Radius: 3, 6 is diameter and half of that is 3.

3 x 3.14 x 2 = 18.84 <------ circumference

3.14 x 3^2

3.14 x 9 = 28.26 <------ area of the circle

Answer: n=-14

Step-by-step explanation:

CONCEPT:

SOLVE:

The expression in the questions is [tex](x^{3} )(x^{-17})[/tex] which is MULTIPLYING, which means we should add the exponents together

[tex](x^{3} )(x^{-17})[/tex]=[tex]x^n[/tex] ⇔ Given

[tex]x^{3+(-17)}[/tex]=[tex]x^n[/tex] ⇔ Adding Exponents Together

[tex]x^{3-17}[/tex]=[tex]x^n[/tex] ⇔ Simplify

[tex]x^{-14}[/tex]=[tex]x^n[/tex] ⇔ Simplify

[tex]n[/tex]=[tex]-14[/tex] ⇔ Correspondingly

Hope this helps!! :)

Please let me know if you have any questions

Answer:

[tex] \frac{9}{10} = 0.9[/tex]

Answer:

13.9% Approximately

The problem involves calculating the probability that a patient who tests positive for HIV is not actually infected with HIV. Given that 8% of the patients tested are infected and that the test has a 92% true positive rate for infected patients and a 9% false positive rate for non-infected patients, we need to determine the probability of a false positive result.

Let's denote the events as follows:

A: Patient is infected with HIV

B: Patient tests positive for HIV

We are interested in finding P(A'|B), which represents the probability that a patient is not infected (A') given that they test positive (B).

According to Bayes' theorem, we can express this probability as:

P(A'|B) = (P(B|A') * P(A')) / P(B)

First, let's calculate P(B|A'), which represents the probability of testing positive given that the patient is not infected. Since the false positive rate is given as 9%, we have P(B|A') = 0.09.

Next, we need to calculate P(A'), which is the probability of not being infected. Since 8% of the patients are infected, the complement event (not being infected) has a probability of 1 - 0.08 = 0.92.

To calculate P(B), the probability of testing positive, we need to consider the total probability of testing positive, which includes both infected and non-infected patients. Therefore, we have:

P(B) = P(B|A) * P(A) + P(B|A') * P(A') = 0.92 * 0.08 + 0.09 * 0.92.

Finally, substituting these values into Bayes' theorem, we can calculate P(A'|B), the probability that a patient testing positive is not infected with HIV.

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Answer:

The answer is d

Step-by-step explanation:

The markup would be 133.33%

30is 200% of 15

5 is 33.33333333% of 15

33.33333333%+100+=133.33333333%

The markup is 133.33% or 133.33333333%

The sets X and Y are defined based on specific conditions on positive integers. In functional form, X n Y can be represented as X n Y = {x € Z + : x ≤ 20, x ≤ 24, and x^2 € Z +, sqrt(x) € Z +}.

a) To find X U Y (the union of X and Y), we need to identify all the positive integers that satisfy either the condition for X or the condition for Y. In enumeration form, X U Y = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 24}. In functional form, X U Y can be represented as X U Y = {x € Z + : (x ≤ 20 and x^2 € Z +) or (x ≤ 24 and sqrt(x) € Z +)}.

To find X n Y (the intersection of X and Y), we need to identify the positive integers that satisfy both the condition for X and the condition for Y. In enumeration form, X n Y = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}. In functional form, X n Y can be represented as X n Y = {x € Z + : x ≤ 20, x ≤ 24, and x^2 € Z +, sqrt(x) € Z +}.

b) To determine the sets X n Z + and Y n Z +, we need to identify the positive integers that satisfy the conditions for X and Y, respectively, and also belong to the universal set of positive integers, Z +. Since X and Y are subsets of Z +, X n Z + = X and Y n Z + = Y.

To find X U Z +, we need to identify all the positive integers that satisfy either the condition for X or belong to Z +. In this case, X U Z + = Z + since all positive integers are included in X. Similarly, Y U Z + = Z + since all positive integers are included in Y.

In summary, X U Y = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 24} and X n Y = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16}. The sets X n Z + and Y n Z + are equal to X and Y, respectively, while X U Z + and Y U Z + are both equal to Z +.

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The operation that results in the simplified expression x⁵ + x⁴ - 5x³ - 3x² + x - 8 is; P + Q

We are given the Polynomials as;

P(x) = x⁴ + 3x³ + 2x² - x + 2

Q(x) = (x³ + 2x² + 3)(x² - 2)

We want to find the combination of P and Q that would yield;

x⁵ + x⁴ - 5x³ - 3x² + x - 8

Let us expand Q(x) to get;

Q(x) = x⁵ + 2x⁴ + 3x² - 2x³ - 4x² - 6

Q(x) = x⁵ + 2x⁴ - 2x³ - x² - 6

Now, the combined polynomial shows us that coefficient of x⁴ is 1 and coefficient of x³ is - 5.

By inspection, we can say that the combination that would produce the required result is;

Q(x) - P(x) = x⁵ + 2x⁴ - 2x³ - x² - 6 -  x⁴ - 3x³ - 2x² + x - 2

Q(x) - P(x) = x⁵ + x⁴ - 5x³ - 3x² + x - 8

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Answer:

Option C (None Of The Above)

Step-by-step explanation:

Given expression = [tex] - (\frac{ - e}{ - f} )[/tex]

Lets multiply the numerator and denominator by -1.

[tex] = > - ( \frac{ - e \times - 1}{ - f \times - 1} )[/tex]

[tex] = > - ( \frac{e}{f} )[/tex]

Now , lets open the brackets.

[tex] = > \frac{ - e}{f} [/tex]

But as this expression is not there in the given options , the correct answer will be Option C.

Answer:

C. None Of The Above

Step-by-step explanation:

Hope this helps

Answer:

0

Step-by-step explanation:

hope this helps :)

SJxz dV where E is the region in the first octant inside the ball of radius 2.

∭E xz dV = ∫₀² ∫₀√(4-x²-z²) ∫₀√(4-x²-y²) xz dy dz dx.

Integrating with respect to y first, then z, and finally x, we can calculate the value of the integral.

To evaluate the integral ∭E xz dV, where E is the region in the first octant inside the ball of radius 2, we need to set up the limits of integration.

Since we are integrating over the region inside the ball of radius 2 in the first octant, we can set up the limits as follows:

0 ≤ x ≤ 2,

0 ≤ y ≤ √(4 - x^2 - z^2),

0 ≤ z ≤ √(4 - x^2 - y^2).

Note that we are using the equation of the sphere x^2 + y^2 + z^2 = 4 to determine the limits of integration for y and z.

Now we can evaluate the integral as follows:

∭E xz dV = ∫₀² ∫₀√(4-x²-z²) ∫₀√(4-x²-y²) xz dy dz dx.

Integrating with respect to y first, then z, and finally x, we can calculate the value of the integral.

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Therefore, the total payment made today by the payee is $2,149.01

To total payment made today would place the payee in the same financial position as the scheduled payments if money can earn 6.25% we will use the  formula below.

PV = FV /(1 + Rr)^n

r =6.25%

FV = $1,260

PV = $ 1,260 / (1+ 0.0625) ^(7/12)

PV = $ 1,179.01

The value of the second payment is  $ 1,179.01.

Lets find the total payment. We can represent the  total payment by X.

X - $ 970 = $ 1,179.01.

To isolate X, we will add $ 970  to both sides.

X = $ 970 + $ 1,179.01.

X = $2,149.01

Therefore, the total payment made today by the payee is $2,149.01

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Answer:

Supplementary Angles are 180 degrees. A triangle has 180 degrees. One angle is already 90 degrees, so 2 more angles with 180 degrees is impossible.

Step-by-step explanation:

Answer:

Step-by-step explanation:

25(w+10) + 13.5(w +10)

Answer:

25(w+10)+13.5(w+10)

Step-by-step explanation:

hope it helps

Answer:

The pythagorean theorem is a theorem which states that the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.

Pythagorean theorem

in the right angle triangles there is a relationship that governs the length of the three sides. if we knows the length of any two sides, the third side may be calculated by the use of the Pythagorean theorem.

This theorem states that the square on the hypotenuse of a right angled triangle is equal to the sum of the squares on the remaining two sides.

we only use Pythagorean theorem to find the length of the missing side of a right angled triangle only.

Answer:

           The slope-intercept form of the equation:  y = 0.5x + 5

           The slope:    m = 0.5

           The y-intercept:     b = 5

Step-by-step explanation:

Slope-intercept form is  y = mx + b, where m is the slope and b is y-intecept

4y = 2x + 20           {divide both sides by 4}

y = 0.5x + 5   ⇒   m = 0.5  and  b = 5

Answer:

5

Step-by-step explanation:

Distance Equation Solution:

[tex]d=\sqrt{(2-(-1)^2+(6-2)^2}\\d=\sqrt{(3)^2+(4)^2} \\d= \sqrt{9+6}\\d= \sqrt{25}\\d=5[/tex]

Answer:

2k

Step-by-step explanation:

-k+3k

2k

The blue region indicated by the Venn diagram can be notated using set notation as A ∩ B or using symbolic representation as C = A ∩ B.

When representing the blue region of a Venn diagram, there are two common ways to notate it: using set notation and using symbolic representation.

1. Set Notation: In set notation, each circle in the Venn diagram represents a set. Let's assume the sets represented by the circles are A and B. The blue region corresponds to the intersection of sets A and B, meaning the elements that are common to both sets. To notate this, we use the symbol ∩, which represents the intersection. Therefore, the blue region can be notated as A ∩ B.

2. Symbolic Representation: Another approach is to use a symbolic representation to notate the blue region. In this case, we can assign a variable, such as C, to represent the blue region. To indicate that C represents the intersection of sets A and B, we write C = A ∩ B. This notation clarifies that C represents the elements that belong to both sets A and B.

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Answer:

26

Step-by-step explanation:

Given

4x - a ← substitute a = - 2, x = 6 into the expression

= 4(6) - (- 2) [ note - (- ) is equivalent to + ]

= 24 + 2

= 26

Answer:

8 students

Step-by-step explanation:

There are 20 boxes and there are a total of 40 students interviewed. So, each box is worth 2 students. Since blue has 4 boxes, 4*2 = 8 students chose blue as their favorite shoe color.

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The mass flow rate is 0.56 kg/s

The formula for calculating the mass flow rate is expressed as;

m = W / (hf - hg)

Such that the parameters are expressed as;

Substitute the values, we get;

m = 90 MW / (2418 - 2257)

Subtract the values, we have;

m = 90/161

Divide the values, we get;

m=  0.56 kg/s

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Answer:

118degrees

Step-by-step explanation:

Assuming we are given the following and m<k and ,<N lies on the same straight line, hence;

m<K = 62 degrees

n<N = ?

Since are on the same straight line, hence;

m<N + m<K = 180

m<N + 62 = 180

m<N = 180 - 62

m<N  = 118

Hence the measure of m<N is 118degrees

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