What are the next three terms of this pattern?

- 4, - 7, - 2, -5, 0, __, __, __

(a) Optimization problem: The optimization problem is shown below:

max f(x, y) = t √ x y, subject to tx² + y ≤ 5x ≥ 0y ≥ 0

Solving the problem for t = 1,t = 1f(x,y) = √xytx² + y ≤ 5x ≥ 0y ≥ 0.

The Lagrangian function for this problem is:

L(x, y, λ) = t √ xy + λ(5 - tx² - y)

We set the partial derivative of L with respect to x to zero:

∂L/∂x = t(0.5√y)/√x + (-2λtx) = 0

We then obtain:

(1) 0.5t√y/√x = 2λtxIf we set the partial derivative of L with respect to y to zero, we obtain:

(2) 0.5t√x/√y + λ(-1) = 0

Multiplying both sides by (-1), we obtain:

(3) -0.5t√x/√y = λ We set the partial derivative of L with respect to λ to zero, we obtain:

(4) 5 - tx² - y = 0Substituting Equation (3) into Equation (1), we obtain:

(5) 0.5t√y/√x = -2(5 - tx² - y)x

Substituting Equation (5) into Equation (4), we obtain:

(6) 5 - tx² - 2x²(5 - tx² - y)² = 0

After expanding Equation (6), we obtain a fourth-order equation in y. Solving this equation leads to:(7) y = 5 - tx²/3

We then substitute y into Equation (3) to obtain: x = 5/2t²From Equation (7), we obtain: y = 5 - tx²/3=5-5/3*2.5=2.7778

(b) Envelope theorem

According to the Envelope Theorem, the marginal effect of a parameter on an optimal solution is equal to the partial derivative of the optimal value with respect to that parameter. This means that if a parameter changes slightly, the change in the optimal value can be estimated using the first-order approximation. (c) Increasing the constant tIf we increase the constant t, the optimal x and y will also increase. This is because an increase in t will lead to a higher value of f(x, y).

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The product of 8(cos 17° + i sin 170°) and 7(cos 42° + i sin 439°) can be expressed as -336 + 94i.

To find the product of two complex numbers, we multiply their magnitudes and add their angles. Let's break down the given complex numbers. The first complex number, 8(cos 17° + i sin 170°), has a magnitude of 8 and an angle of 17°. The second complex number, 7(cos 42° + i sin 439°), has a magnitude of 7 and an angle of 42°.

To find the product, we multiply the magnitudes: 8 * 7 = 56. To determine the angle, we add the angles: 17° + 42° = 59°. Now we have the complex number 56(cos 59° + i sin θ). However, we need to convert the angle to the standard range of 0° to 360°. In this case, 59° is already within that range.

Therefore, the product of the given complex numbers is 56(cos 59° + i sin θ), where θ is the angle in the standard range. To evaluate this expression, we can use trigonometric identities to find the cosine and sine of 59°, or use a calculator. The result is approximately -336 + 94i, which represents the product in rectangular form.

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The equilibrium vector for the transition matrix is [0.4, 0.2667, 0.3333].

The transition matrix given is:

0.70 0.10 0.20 0.10 0.75 0.15 0.10 0.35 0.55

'To find the equilibrium vector, we need to multiply the transition matrix by a vector of constants that would make the equation valid. The value of this vector of constants is given by:

(P-I)x = 0

Where P is the transition matrix and I is the identity matrix. The value of x is the equilibrium vector.

Let's write the augmented matrix:

(P-I|0) = 0.70-1 0.10 0.20 0.10 0.75-1 0.15 0.10 0.35 0.55-1

After subtracting the identity matrix from the transition matrix, we get the augmented matrix.

Using the Gauss-Jordan elimination method, we get 1 -0.08 -0.4-0.12 1 -0.28-0.18 -0.12 1

After row reducing the augmented matrix, we get the following equations:

x1 - 0.08x2 - 0.4x3 = 0-0.12

x1 + x2 - 0.28x3 = 0-0.18x1 - 0.12

x2 + x3 = 0

Solving these equations, we get

x1 = 1.2

x2 = 0.8

x3 = 2.

Using x1, x2, and x3 values, we can determine the equilibrium vector:

x = [1.2/3, 0.8/3, 2/3]

Simplifying the vector, we get the equilibrium vector as:

x = [0.4, 0.2667, 0.3333]

Thus, the equilibrium vector is [0.4, 0.2667, 0.3333].

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Answer:

Step-by-step explanation:

Answer: B. Niche

Explanation:

Definition of niche: a niche is the match of a species to a specific environmental condition. It describes how an organism or population responds to the distribution of resources and competitors and how it in turn alters those same factors.

Answer: Niche

Niche is basically like a type of place of something or someone

The required annual sales for stocked items A, B, and C would be $1,300, $1,560, and $3,250, respectively.

To calculate the required annual sales for each stocked item, we need to consider the turnover ratio and the maximum stock level. The turnover ratio indicates how many times the stock is sold and replaced within a year.

Given that the turnover ratio requirement is 2.6 per year, we can calculate the required annual sales for each item by multiplying the turnover ratio with the maximum stock level.

For item A, the maximum stock level is $1,000, and the required annual sales would be 2.6 times $1,000, which equals $2,600.

Similarly, for item B, the maximum stock level is $1,200, and the required annual sales would be 2.6 times $1,200, which equals $3,120.

For item C, with a maximum stock level of $2,500, the required annual sales would be 2.6 times $2,500, which equals $6,500.

However, since the average stock is assumed to be one-half the maximum stock, we need to adjust the required annual sales accordingly. The average stock for each item would be $500 for A, $600 for B, and $1,250 for C. Therefore, the required annual sales for A would be $2,600 minus $500, which equals $1,300. For B, it would be $3,120 minus $600, which equals $1,560. And for C, it would be $6,500 minus $1,250, which equals $3,250.

In summary, the required annual sales for items A, B, and C would be $1,300, $1,560, and $3,250, respectively.

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The solutions of the equation `tan(2x)-sin(x)=0` in the interval `[0, 2)` are:`x = 2πk ± 2arctan(√[(3+√5)/2]) ± 2arcsin(√[(1-cos(x))/2])` where `k` is an integer and `cos(x) = 1-2sin²(x/2)`

Given, `tan(2x)-sin(x)=0`.We can use the half-angle formula to solve this equation in the interval `[0, 2)` and obtain the solutions. The half-angle formula for tangent is: `tan(θ/2)= sin(θ)/(1+cos(θ))`The half-angle formula for sine is: `sin(θ/2)=±√[(1-cos(θ))/2]`Using the half-angle formula for tangent:`tan(2x)-sin(x)=0`Substituting `sin(x)` in terms of `tan(θ/2)`, we get:`tan(2x)-2tan(x/2)/(1+tan²(x/2))=0`Multiplying both sides by `1+tan²(x/2)`, we get:`tan(2x)(1+tan²(x/2))-2tan(x/2)=0`Simplifying this equation further using the double-angle formula for tangent:`(2tan(x/2)/(1-tan²(x/2)))(1+(2tan²(x/2))/(1-tan²(x/2))) - 2tan(x/2) = 0`

Multiplying both sides by `(1-tan²(x/2))`, we get:`2tan(x/2)(1+tan²(x/2)) - 2tan²(x/2) - (1-tan²(x/2))(2tan²(x/2)) = 0`Simplifying this equation, we get:`tan⁴(x/2) - 3tan²(x/2) + 1 = 0`This is a quadratic equation in `tan²(x/2)`.Solving this quadratic equation, we get:`tan²(x/2) = (3±√5)/2`Taking square root of both sides, we get:`tan(x/2) = ±√[(3±√5)/2]`We know that, `tan(x/2) > 0` in the interval `[0, 2)` since `x` lies in this interval. Therefore, we take the positive square root. We get:`tan(x/2) = √[(3+√5)/2]`Using the formula for sine, we get:`sin(x/2) = ±√[(1-cos(x))/2]`We know that, `sin(x/2) > 0` in the interval `[0, 2)` since `x` lies in this interval.

Therefore, we take the positive square root. We get:`sin(x/2) = √[(1-cos(x))/2]`Therefore, the solutions of the equation `tan(2x)-sin(x)=0` in the interval `[0, 2)` are:`x = 2πk ± 2arctan(√[(3+√5)/2]) ± 2arcsin(√[(1-cos(x))/2])`where `k` is an integer and `cos(x) = 1-2sin²(x/2)`

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Answer:

D

Step-by-step explanation:

(x - 3)(x + 4) = x^2 + x - 12

Answer:

25.3

Character minimum

Answer:

2.70cm

Step-by-step explanation:

Given data

Answer:

2cm

Step-by-step explanation:

Given data

Capacity/volume= 204 cm

Height= 8.9 cm

Required

The radius r

let us apply the expression for the volume of a cylinder

V=πr^2h

204=πr^2*8.9

204=3.14r^2*8.9

r^2= 204/27.946

r^2=7.30

r= √7.30

r=2.70cm

Hence the radius is 2.70cm

Answer is C. {(6,-2), (6.0), (6,2), (6,4)}

Step-by-step explanation:

Your welcome

The least expensive computer after the 20% discount would be $400.

To calculate the price of the least expensive computer after the 20% discount, we need to find 20% of the original price and subtract it from the original price.

Let's assume the original price of the least expensive computer is x. The discount of 20% can be calculated as 0.20 * x. To find the discounted price, we subtract the discount from the original price: x - 0.20 * x = 0.80 * x.

Since we know that the cost of the least expensive computer ranges from $500 to $1000, we can substitute x with $500 and calculate the discounted price: 0.80 * $500 = $400. Therefore, the least expensive computer after the 20% discount would be $400.

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The total cost next year will be: $10,613.10

Solving:

The total cost of this year

Simply just add up everything that has been given which wouold give you 10,405

The total cost of next year

multiply the total cost of this year by 1.02 to find what the cost would be with a 2% increase which should give you 10,613.1

Answer:

1620 degrees.

Step-by-step explanation:

In a polygon the number of sides = the number of angles.

Each external angle = 360 / 11 = 32 8/11 degrees.

Each internal angle = 180 - 32 8/11 = 147 3/11 degrees

So sum = 147 3/11 * 11

= 1617 + 3

= 1620 degrees.

Answer:

It's 1620°

Hope it helps

Step-by-step explanation:

Hint:>

( The sum of the interior angle (S),

The the sides of the polygon (n).)

Then use this formula :

S=180°(n-2)

= 180°(11-2)

= 180°(9)

= 1620°

Answer:

f(x) is flipped, and each point is moved 3 units to the left and 2 units up

Step-by-step explanation:

Answer:

The axis of symmetry is x=3, the vertex of the graph is (3,-2)

Step-by-step explanation:

[tex]f(x)=(x^{2} -6x+9)-9+7[/tex]

[tex]=(x-3)^{2} -2[/tex]

We have shown that [E(XY)]^2 < E(X)E(Y), as required.

To show that [E(XY)]^2 < E(X)E(Y), we can follow the hint provided and introduce a new random variable Z = X + aY, where 'a' is a constant.

First, let's expand the expression E(XY) using the law of iterated expectations:

E(XY) = E[E(XY|Z)]

Now, substituting Z = X + aY into the conditional expectation:

E(XY) = E[E(X(X + aY)|Z)]

= E[E(X^2 + aXY|Z)]

Expanding the inner expectation:

E(XY) = E[X^2 + aXY]

Next, let's square both sides of the inequality to be proved:

[E(XY)]^2 < E(X)E(Y)

(E[X^2 + aXY])^2 < E(X)^2E(Y)^2

Expanding the square:

E(X^2)^2 + 2aE(X^2)E(XY) + a^2E(XY)^2 < E(X)^2E(Y)^2

Since E(X^2) is the variance of X (Var(X)), we can rewrite it as:

Var(X) + [E(X)]^2

Using the covariance formula, Cov(X,Y) = E[(X - Ux)(Y - My)], we can rewrite the second term as:

Cov(X,Y) + [E(X)][E(Y)]

Substituting these expressions back into the inequality, we have:

Var(X) + [E(X)]^2 + 2a(Cov(X,Y) + [E(X)][E(Y)]) + a^2[E(XY)]^2 < E(X)^2E(Y)^2

Simplifying the equation, we have:

Var(X) + 2aCov(X,Y) + a^2[E(XY)]^2 < 0

This inequality holds true since the left-hand side of the equation is a quadratic expression in 'a' and the coefficient of the quadratic term is positive (Var(X)). Since the inequality holds for all values of 'a', it must hold when 'a' is zero. Therefore, we have:

Var(X) + 0 + 0 < 0

Which is not possible, thus proving that [E(XY)]^2 < E(X)E(Y).

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The greatest common factor of 42 and 50 is 2.

The given numbers are 42 and 50

The greatest common factor (GCF) of two numbers is the largest number that divides both of them evenly.

To find the GCF of 42 and 50,

We can start by listing the factors of each number.

The factors of 42 are 1, 2, 3, 6, 7, 14, 21, and 42.

Similarly, the factors of 50 are 1, 2, 5, 10, 25, and 50.

By comparing the factors,

We can see that the highest common factor is 2.

Therefore, the GCF of 42 and 50 is 2.

This means that 2 is the largest number that can divide both 42 and 50 without leaving a remainder.

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Answer:

the correct answer is B, Darius should not choose loan option B because he will pay $398.52 more in interest than on loan option C.

Step-by-step explanation:

I also had this question, and it showed that I got it correct :)

Answer:

River D - 2580 miles

River C - 2980 miles

Step-by-step explanation:

Let the length of River D be represented with x

length of river C = 400 + x

total length of both rivers = 5560

x + 400 + x = 5560

2x + 400 = 5560

collect like terms

2x = 5560 - 400

2x = 5160

divide both sides of the equation by 2

x = 5160 / 2

x = 2580

length of river c = 400 + 2580 = 2980

The key difference is that the Bernoulli distribution models a single trial, while the binomial distribution models multiple trials and focuses on the number of successes in those trials. The binomial distribution is an extension of the Bernoulli distribution to multiple trials.

The main difference between the binomial distribution and the Bernoulli distribution lies in the number of trials involved.

Bernoulli Distribution:

The Bernoulli distribution is a discrete probability distribution that models a single trial or experiment with two possible outcomes: success or failure. It is characterised by a single parameter, often denoted as p, which represents the probability of success. The outcome of each trial is independent of other trials, and it is represented by a random variable that takes the value 1 for success and 0 for failure.

Binomial Distribution:

The binomial distribution is also a discrete probability distribution that models multiple independent Bernoulli trials or experiments. Each trial is identical, and it has two possible outcomes: success or failure, just like the Bernoulli distribution. However, the binomial distribution considers the number of successes (k) in a fixed number of trials (n). It is characterised by two parameters: the probability of success (p) and the number of trials (n).

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In a right triangle ABC that is inscribed in a circle, the shortest height of the triangle is the altitude drawn from the right angle to the hypotenuse. This height, denoted as 'h', is perpendicular to the hypotenuse and divides it into two segments.

The altitude is also the shortest distance between the hypotenuse and the opposite vertex (the vertex not on the hypotenuse). The property of a right triangle inscribed in a circle is that the hypotenuse is the diameter of the circle, and the other two sides are radii of the circle.

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Answer:

Let's define the variables:

x = number of small boxes bought.

y = number of large boxes bought.

Then the total number of granola bars is:

x*6 + y*24

We also know that "The camp bought 4 times as many small boxes as large boxes"

Then:

x = 4*y

and "...which altogether had 96 granola bars."

The total number of granola bars is 96, then:

x*6 + y*24 = 96

Then the system of equations is:

x = 4*y

x*6 + y*24 = 96

We want to solve this graphically.

Then we first need to isolate the same variable in both equations.

We can see that in the first one x is already isolated, so let's isolate x in the second equation:

x*6 = 96 - y*24

x = (96 - y*24)/6

x = 16 - y*4

Now we have the equations:

x = 4*y

x = 16 - y*4

To solve this graphically we need to graph both fo these lines and see in which point the lines do intersect.

The point where the lines intersect is the solution of the system.

The graph can be seen below.

We can see that the lines do intersect at the point (2, 8)

This means that the camp bought 2 large boxes and 8 small boxes.

Answer:

stay calm

Step-by-step explanation:

stay calm

To prove that Z = {[b² | c=b] | b, c ∈ ℝ} is a subspace of ℝ²x², we need to show that Z satisfies the three properties of a subspace.

To prove that Y = {A ∈ ℝ²x² | A is an upper triangular matrix} is not a subspace of ℝ²x², we only need to show that it fails to satisfy one of the three properties.

For Z to be a subspace of ℝ²x², it needs to satisfy closure under addition, closure under scalar multiplication, and contain the zero vector.

1. Closure under addition: Let A = [b₁² | c₁=b₁] and B = [b₂² | c₂=b₂] be two matrices in Z. Their sum, A + B, is [b₁² + b₂² | c₁ + c₂ = b₁ + b₂]. Since b₁ + b₂ is a real number, A + B is also in Z. Hence, Z is closed under addition.

2. Closure under scalar multiplication: Let A = [b² | c=b] be a matrix in Z, and k be a scalar. The scalar multiple kA is [k(b²) | k(c) = kb]. Since kb is a real number, kA is also in Z. Therefore, Z is closed under scalar multiplication.

3. Contains the zero vector: The zero vector in ℝ²x² is the matrix [0 0 | 0 = 0]. This matrix satisfies the condition c = b, so it is in Z.

Thus, Z satisfies all the properties and is a subspace of ℝ²x².

For Y to be a subspace of ℝ²x², it needs to satisfy the three properties mentioned earlier. However, Y fails to satisfy closure under addition since the sum of two upper triangular matrices may not always be an upper triangular matrix. Hence, Y is not a subspace of ℝ²x².

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