a group is celebrating the chinese new year eve. they make 40 dumplings and they make 3 of them to be lucky dumplings by putting coins in. assume that all dumplings look the same and they will eat the dumplings one by one. what is the expected number of dumplings to be eaten to find the first lucky dumplings?

The calculated measure of the angle θ is 1/3π rad

From the question, we have the following parameters that can be used in our computation:

The length of the arc is calculated as

Length = Radius * Angle

Substitute the known values in the above equation, so, we have the following representation

6 * θ = 2

Divide by 2

θ = 1/3

Hence, the measure of the angle is 1/3π rad

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AB is a straight line ⇒ m(∡BXA) = 180°

m(∡BXA) = m(∡BXY) + m(∡YXA) ⇔

⇔ 180° = 60° + m(∡YXA) ⇔

⇔ 60° + m(∡YXA) = 180° ⇔

⇔ m(∡YXA) = 180° - 60° ⇒ m(∡YXA) = 120°

∆XYZ is an isosceles triangle and has an angle of 90° ⇒ m(∡ZXY) = m(∡XYZ) = 45°, because a triangle has 180°

m = m(∡AXZ) ⇒ m(∡AXZ) = ?

m(∡BXA) = m(∡BXY) + m(∡YXZ) + m(∡AXZ) ⇔

⇔ 180° = 60° + 45° + m(∡AXZ) ⇔

⇔ 105° + m(∡AXZ) = 180° ⇔

⇔ m(∡AXZ) = 180° - 105° ⇒ m(∡AXZ) = 75° ⇔

⇔ m = 75°

Good luck! :)

The particular solution that satisfies the differential equation and initial condition is:

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

To find the particular solution that satisfies the differential equation and initial condition, we need to integrate the given differential equation and use the initial condition to find the constant of integration.

f'(x) = 24x^3 - 6x

Integrating both sides with respect to x:

f(x) = ∫(24x^3 - 6x) dx

f(x) = 4x^4 - 3x^2 + C

Using the initial condition f(1) = 3, we can find the value of the constant C:

3 = 4(1)^4 - 3(1)^2 + C

C = 2

Therefore, the particular solution that satisfies the differential equation and initial condition is:

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

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The calculated value of dy/dx at the point (1, 3) for the equation f(x)g(y) = 24 + x + y, where f is a differentiable function of x and g is a differentiable function of y of dy/dx is -3, None of the given options (A, B, C, D) match the calculated value. It is possible there is an error in the options provided.

To find the value of dy/dx at the point (1, 3) for the equation f(x)g(y) = 24 + x + y, where f is a differentiable function of x and g is a differentiable function of y, follow these steps:

1. Differentiate both sides of the equation with respect to x, treating y as a function of x:

  d/dx [f(x)g(y)] = d/dx [24 + x + y]

2. Use the product rule for differentiation on the left side of the equation:

  [f'(x)g(y) + f(x)g'(y)dy/dx] = [1 + dy/dx]

3. Plug in the given values of f(1), f'(1), g(3), and g'(3):

  [-2 * 7 + (-4) * 1 * dy/dx] = [1 + dy/dx]

4. Simplify and solve for dy/dx:

  [-14 - 4dy/dx] = [1 + dy/dx]

  Add 4dy/dx to both sides:

  [-14] = [1 + 5dy/dx]

  Subtract 1 from both sides:

  [-15] = [5dy/dx]

  Divide by 5:

  dy/dx = -3

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The odds for the in a red slot are 18/38, which simplifies to 9/19 or approximately 47.37%. The odds against the ball landing in a red slot are 20/38, which simplifies to 10/19 or approximately 52.63%.

1. Determine the total number of slots (38).
2. Determine the number of red slots (18).
3. Calculate the probability of landing on a red slot by dividing the number of red slots by the total number of slots: 18/38.

The odds for the ball landing in a red slot are 18/38, or approximately 0.474 (47.4%).

To find the odds against the ball landing in a red slot:

1. Determine the number of non-red slots (20), which includes 18 black and 2 green slots.
2. Calculate the probability of not landing on a red slot by dividing the number of non-red slots by the total number of slots: 20/38.

The odds against the ball landing in a red slot are 20/38, or approximately 0.526 (52.6%).

To simplify, the odds of landing in a red slot are 18/38 (47.4%), and the odds of landing in a red slot are 20/38 (52.6%).

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The height of the frog's jump when it first jumps (t = 0) is 0 inches.

We have,

To find h(0), we need to substitute 0 for t in the equation:

h(t) = 60t - 75t^2.

So,

h(0) = 60(0) - 75(0)²

Simplifying:

h(0) = 0 - 75(0)

h(0) = 0

Therefore,

The height of the frog's jump when it first jumps (t = 0) is 0 inches.

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The present value of an annuity due of $6,000 per quarter for 8 years with an annual interest rate of 6% (APR with quarterly compounding) is approximately $144,967.53.

Given,

Payment (P) = $6,000 per quarter

Time (n) = 8 years = 32 quarters

Interest rate (r) = 6% per year

As the interest rate is given in annual terms, we need to convert it into quarterly terms.

So, the quarterly interest rate (i) = 6% / 4 = 1.5%

Let PV be the present value of the annuity due.

Using the formula for the present value of an annuity due, we get

PV = P * [(1 - (1 + i)⁻ⁿ)/(i)] * (1 + i)

Substituting the given values, we get

PV = $6,000 * [(1 - (1 + 0.015)⁻³²)/(0.015)] * (1 + 0.015)

PV = $6,000 * [22.5307] * [1.015]

PV = $144,967.53 (rounded to the nearest cent)

Therefore, the present value of the annuity due is $144,967.53.

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The sample proportion is 0.68 or 68% when rounded to two decimal places.

To calculate the sample proportion, you can use the formula:
Sample proportion (p) = x/n where x is the number of successes and n is the sample size. In this case, p = 0.62 (population proportion), x = 68, and n = 100.
Now, plug in the given values:
p = 68/100
p = 0.68

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Once we find Q(t), we can substitute it back into the expression for y_p(t) to get the particular solution of the differential equation.

To find a particular solution of the differential equation y(4) − y = g(t), we can use the method of undetermined coefficients. This method assumes that the particular solution has the same form as the forcing function g(t).

Let's assume that the forcing function is of the form g(t) = P(t)e^t, where P(t) is a polynomial. Then, we can guess that the particular solution is of the form y_p(t) = Q(t)e^t, where Q(t) is also a polynomial.

Now, let's find the derivatives of y_p(t):

y_p(t) = Q(t)e^t

y'_p(t) = (Q'(t) + Q(t))e^t

y''_p(t) = (Q''(t) + 2Q'(t) + Q(t))e^t

y'''_p(t) = (Q'''(t) + 3Q''(t) + 3Q'(t) + Q(t))e^t

y''''_p(t) = (Q''''(t) + 4Q'''(t) + 6Q''(t) + 4Q'(t) + Q(t))e^t

Substituting these derivatives into the differential equation, we get:

(Q''''(t) + 4Q'''(t) + 6Q''(t) + 4Q'(t) + Q(t))e^t - Q(t)e^t = P(t)e^t

Simplifying and canceling the exponential terms, we get:

Q''''(t) + 4Q'''(t) + 6Q''(t) + 3Q'(t) = P(t)

Now, we can use integration by parts to find a formula for Q(t) in terms of integrals of P(t):

Q(t) = (1/3) ∫∫∫[P(t) - 6Q''(t) - 12Q'(t) - 8Q(t)] dt

We can repeat this process of integration by parts until we get a formula for Q(t) in terms of integrals of P(t) only. Once we find Q(t), we can substitute it back into the expression for y_p(t) to get the particular solution of the differential equation.

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We do not have sufficient evidence to conclude that the promotion significantly increased the proportion of sales over $50.

Let's go through each part:

3a) The null hypothesis (H0) is that the promotion did not significantly increase the proportion of sales over $50. In other words, H0: p_new = 0.26.

3b) The sample proportion (p_hat) is the proportion of sales over $50 during the sale, which is 127/400 = 0.3175.

3c) The standard error (SE) can be calculated using the formula: SE = √[(p * (1-p))/n], where p is the proportion from the null hypothesis (0.26) and n is the sample size (400). So, SE = √[(0.26 * (1-0.26))/400] = 0.0223.

3d) The test statistic (z) is calculated using the formula: z = (p_hat - p) / SE. In this case, z = (0.3175 - 0.26) / 0.0223 = 2.58.

3e) To find the p-value, look up the z-score (2.58) in a standard normal distribution table. The p-value is approximately 0.01.

3f) Conclusions: Since the p-value (0.01) is less than the alpha level (0.05), we reject the null hypothesis. This means that there is evidence to suggest that the promotion significantly increased the proportion of sales over $50.

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Proof:

Let [tex]x[/tex] be the length of segment AB. By the power of a point formulas,

[tex]CB \cdot CA = CD \cdot CE\\\implies (2+\sqrt{5})(2+\sqrt{5}+x)=(2\sqrt{5})(4+3\sqrt{5}).[/tex]

Expanding both sides, we get:

[tex]4+2\sqrt{5}+2x+2\sqrt{5}+5+x\sqrt{5}=8\sqrt{5}+30\\\implies 9+4\sqrt{5}+2x+x\sqrt{5} = 8\sqrt{5}+30\\\implies x(2+\sqrt{5})=21+4\sqrt{5}\\\implies x = \frac{21+4\sqrt{5}}{2+\sqrt{5}} = \frac{(21+4\sqrt{5})(2-\sqrt{5})}{(2+\sqrt{5})(2-\sqrt{5})} = \frac{42-21\sqrt{5}+8\sqrt{5}-20}{-1} = -(22-13\sqrt{5}) = 13\sqrt{5}-22.[/tex]

Hence, it is possible for the length of AB to be in the form [tex]p\sqrt{5}+q[/tex] for integers [tex]p[/tex] and [tex]q.[/tex]

It should be noted that to solve the problem of Myra picking six roses from a group of 12 roses, we employ the formula for combination.

When four individuals pose for pictures on the Sampaloc Lake view deck, the way they position themselves matters. This circumstance calls for permutation where order plays an important role in arriving at a solution. To find out how many arrangements can be made, the formula for permutation must be used.

A row of six seats with six Grade 10 students seated requires the use of the permutation formula to determine each possible individual's seating arrangement relative to one another as their positioning does matter in this situation.

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The third, fourth and fifth partial sum is -26841, 894648, and -27732491 respectively. For the given sequence the first five terms are  6, 7, 503, 35760, 2548083.

What is partial sum?

A partial sum is the total of a limited number of the series' terms. To see how the infinite sum behaves, we can examine a collection of these sums. Sn is used to represent each of these partial sums, where n stands for the index of the final term in the sum.

(a) The third partial sum is given by:

S3 = 9(−31) + 9(−31)² + 9(−31)³

S3 = 9(-31 + 961 - 29791)

S3 = -26841

(b) The fourth partial sum is given by -

[tex]S4 = 9(-31)^1 + 9(-31)^2 + 9(-31)^3 + 9(-31)^4[/tex]

S4 = 9(-31 + 961 - 29791 + 923521)

S4 = 894648

(c) The fifth partial sum is given by -

[tex]S5 = 9(-31)^1 + 9(-31)^2 + 9(-31)^3 + 9(-31)^4 + 9(-31)^5[/tex]

S5 = 9(-31 + 961 - 29791 + 923521 - 28629151)

S5 = -27732491

Therefore, the third partial sum is -26841, the fourth partial sum is 894648, and the fifth partial sum is -27732491.

To find the first five terms of the sequence defined recursively, we can use the formula given to generate each term.

Starting with a0 = 6 and a1 = 7, we have -

a2 = a0 + 71a1 = 6 + 717 = 503

a3 = a1 + 71a2 = 7 + 71503 = 35760

a4 = a2 + 71a3 = 503 + 7135760 = 2548083

a5 = a3 + 71a4 = 35760 + 712548083 = 181903833

Therefore, the first five terms of the sequence are: 6, 7, 503, 35760, 2548083.

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The Probability Density Function (PDF) of W is fW(w) = { (1/144) * w * [tex]e^{(-w/12)}[/tex] if w >= 0 ; 0 otherwise.

The Probability Density Function (PDF) of the sum of two independent exponential random variables is a convolution of their individual PDFs.

Since e[x] = e[y] = 1/λ, we have λ = 1/e[x] = 1/e[y] = 1/12.

The PDF of X and Y are:

fX(x) = λ[tex]e^{(-\lambda x)[/tex] = (1/12)[tex]e^{(-x/12)[/tex]

fY(y) = λ[tex]e^{(-\lambda y)[/tex] = (1/12)[tex]e^{(-y/12)[/tex]

The PDF of W, the sum of X and Y, is:

fW(w) = (fX * fY)(w) = ∫(0 to w) (1/12)[tex]e^{(-x/12)[/tex] * (1/12)[tex]e^{(-(w-x)/12)[/tex] dx

= (1/144) ∫(0 to w) [tex]e^{(-x/12)[/tex] * [tex]e^{(-(w-x)/12)[/tex] dx

= (1/144) ∫(0 to w) [tex]e^{(-w/12)[/tex] dx

= (1/144) * [tex]e^{(-w/12)[/tex] * ∫(0 to w) 1 dx

= (1/144) * w * [tex]e^{(-w/12)[/tex]

Therefore, the PDF of W is fW(w) = { (1/144) * w * e^(-w/12) if w >= 0 ; 0 otherwise.

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The limit is not equal to zero, the series is divergent. Therefore, the answer is DIVERGES.

The given series is the sum of terms arctan(3n) from n=0 to infinity:

Σ(arctan(3n)) from n=0 to infinity

To determine if this series is convergent or divergent, we can use the Test for Divergence. If the limit of the individual terms as n approaches infinity is not equal to zero, then the series diverges:

lim (n→∞) arctan(3n)

Since arctan(x) approaches π/2 as x approaches infinity, we have:

lim (n→∞) arctan(3n) = π/2

Since the limit is not equal to zero, the series is divergent. Therefore, the answer is DIVERGES.

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The surface area of revolution about the x-axis of y = 3sin(3x) over the interval 0 ≤ x ≤ π/3 is approximately 72.66 square units.

To find the surface area of revolution about the x-axis of y = 3sin(3x) over the interval 0 ≤ x ≤ π/3, we need to use the formula:

S = ∫[a,b] 2πy√(1+(dy/dx)²) dx

where a = 0, b = π/3, y = 3sin(3x), and dy/dx = 9cos(3x).

Substituting these values into the formula, we get:

S = ∫[0,π/3] 2π(3sin(3x))√(1+(9cos(3x))²) dx

Simplifying the integrand:

S = ∫[0,π/3] 6πsin(3x)√(1+81cos²(3x)) dx

Next, we can use the substitution u = cos(3x), du/dx = -3sin(3x), and dx = du/-3sin(3x) to simplify the integrand further:

S = -2π∫[1,0] √(1+81u²) du

Note that we changed the limits of integration because cos(3x) goes from 1 (at x = 0) to 0 (at x = π/3) as x goes from 0 to π/3, and so u goes from 1 to 0.

Now, we can use the substitution u = (1/9)tan(θ), du = (1/9)sec²(θ) dθ, and simplify the integrand even further:

S = -18π∫[π/4,0] sec³(θ) dθ

Using integration by parts with u = sec(θ) and dv = sec²(θ) dθ, we get:

S = -18π[(sec(θ)tan(θ) + ln|sec(θ) + tan(θ)||_0^π/4

Simplifying using trigonometric identities, we get:

S = 18π(√2 + ln(√2 + 1))

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The Equation that represents the blue graph is y = f(x + 2)

A horizontal shift of a function is a transformation that moves the graph of the function left or right without changing its shape.

The transformation involves adding or subtracting a constant value to the independent variable (usually denoted as x) in the function's equation.

Here we have

The graph y = f(x) represents the red line

Here we can clearly see that the blue graph is formed by shifting the red graph by 2 units to the left side

Hence, the equation corresponds to the blue graph can be found by adding 2 to x - coordinate the equation of the red graph  

Hence,

The Equation that represents the blue graph is y = f(x + 2)

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To find the work done when moving the particle along the x-axis, we can use the formula for work, which is the integral of the force function F(x) with respect to x over the given interval. In this case, the force function is F(x) = 2x, and the interval is from x = 0 to x = 2 meters.

So, the work done can be calculated as:

Work = ∫(F(x) dx) from 0 to 2 = ∫(2x dx) from 0 to 2

To find the integral of 2x, we can use the power rule:

∫(2x dx) = x^2 + C

Now, we can evaluate the integral over the interval [0, 2]:

Work = (2^2 - 0^2) = 4 - 0 = 4

Therefore, the work done when the particle is moved from x = 0 to x = 2 meters is 4 Newton-meters. As a whole number answer without units, the answer is 4.

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Answer:

[tex]\frac{31}{41}[/tex]

Step-by-step explanation:

36-5 = 31

Helping in the name of Jesus.

Answer:

Step-by-step explanation:

Cost per candle = $42 / 14 candles

Cost per candle = $3

Therefore, the cost per candle is $3

Answer:

Step-by-step explanation:

George is sitting 18 meters from the base of the tree.

Angle is a geometric shape that is formed when two lines or rays intersect each other. An angle is measured in degrees and is represented by the symbol ∠. It is a measure of the amount of rotation that occurs between two lines. Angles are used in mathematics and engineering for various calculations, such as trigonometry and the calculation of force and velocity.

Using trigonometry, we can calculate the distance between George and the base of the tree. First, we need to use the tangent of the angle of elevation to calculate the opposite side of the triangle. The tangent of the angle of elevation is equal to the opposite side divided by the adjacent side. Therefore, the opposite side is equal to the tangent of the angle of elevation multiplied by the adjacent side.

Tangent of 27° = Opposite side / Adjacent side
Opposite side = Tangent (27°) x Adjacent side

Since we know the angle of elevation and the height of the tree, we can use the adjacent side as the height of the tree, which is 36 meters.

Opposite side = Tangent (27°) x 36 meters
Opposite side = 0.5 x 36 meters
Opposite side = 18 meters

Therefore, George is sitting 18 meters from the base of the tree.

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The probability that the diameter of a selected bearing is greater than 46 millimeters is 0.8413.

To solve this problem, we need to use the standard normal distribution and standardize the diameter values to find the corresponding z-score. The formula for standardizing a value is:

z = (x - μ) / σ

Where:

x = the value we want to standardize (in this case, 46 millimeters)

μ = the mean diameter of the ball bearings (51 millimeters)

σ = the standard deviation (which is the square root of the variance, so σ = sqrt(25) = 5)

Substituting the values we have:

z = (46 - 51) / 5 = -1

Now, we need to find the probability that a selected bearing has a diameter greater than 46 millimeters, which is equivalent to finding the area to the right of z = -1 on the standard normal distribution curve. This can be done using a table or calculator.

Using a calculator or software, we can find that the area to the right of z = -1 is 0.8413 (rounded to four decimal places). Therefore, the probability that the diameter of a selected bearing is greater than 46 millimeters is 0.8413.

Answer: 0.8413 (rounded to four decimal places).

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1. The value of DC is 25cm and perimeter is 100cm

2. The value of BD is 12

A diagonal is a line that connects two vertices of a polygon or a solid, whose vertices are not on the same edge.

The property of diagonal of rhombus is that they bisect 90°.

Therefore , this means that ∆CDE is a right angled triangle.

since AC = 40cm, EC = 40/2 = 20cm

BD = 30cm , ED = 30/2 = 15cm

1. therefore DC = √20²+15²

= √400+225

= √625

= 25cm

The sides of a rhombus are equal, therefore,

Perimeter = 25×4 = 100cm

2. The diagonal of a rectangle are equal, therefore,

4x-60 = 30-x

4x+x = 30+60

5x = 90

x = 90/5

x = 18

BD = 4x-60 = 4(18)-60

= 12

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The linearization of the function f(x) = 13/x + 2 at x = −1 is -13x - 26.

The function is f(x) = 13/(x + 2).

We have to determine the linearization of the function at x = -1.

L(x) = f(-1) + f'(-1)(x - (-1))

L(x) = f(-1) + f'(-1)(x + 1) ...(1)

Now we first determine the value of f(-1) and f'(-1).

To determine f(-1) we substitute the x = -1 in the function f(x).

f(-1) = 13/(-1 + 2)

f(-1) = 13/1

f(-1) = 13

To determine the value of f'(-1), we first differentiate the function f(x).

f'(x) = d/dx[13/(x + 2)]

After differentiating

f'(x) = -13/(x + 2)²

Now substitute x = -1

f(-1) = -13/(-1 + 2)²

f(-1) = -13/(1)²

f(-1) = -13

Now substitute the value of f(-1) and f'(-1) in equation 1

L(x) = -13 + (-13)(x + 1)

Simplify

L(x) = -13 - 13x - 13

L(x) = -13x - 26

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The complete question is:

Find the linearization of the function f(x) = 13/x + 2 at x = −1.

The cost of the carpet for the figure is

The floor is made up of two trapezoids that are joined togerher

The area of the figure helps to say the quantity of carpets that will be used. The quantity multiplied by the price gives the cost

The area of the figure is solved using the formula for trapezoid

= 1/2 (sum of parallel lines ) x height

= 1/2 (29 + 21) x 11

= 0.5 * 50 * 11

= 275 square feet

Total area

= 2 * 275 square feet

= 550 square feet

Cost of carpet

= 550 x 3

= $ 1650

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A function is a relation between a set of inputs and a set of possible outputs, where each input is uniquely associated with a single output.the average rate of change of f(x) over the interval  [tex][-3, 3][/tex]  is:

[tex](6 - (-12)) / 6 = 18/6 = 3[/tex] Thus, option C is correct.

The average rate of change of a function over an interval is given by the difference in the function values at the endpoints of the interval, divided by the length of the interval.

Therefore, to find the average rate of change of  [tex]f(x) = − + 3x + 6[/tex] over the interval  [tex]−3 ≤ x ≤ 3[/tex], we need to evaluate the function at the endpoints of the interval and then divide by the length of the interval.

[tex]f(-3) = + 3(-3) + 6 = -9 - 9 + 6 = -12[/tex]

[tex]f(3) = + 3(3) + 6 = -9 + 9 + 6 = 6[/tex]

The length of the interval is 3 - (-3) = 6.

Therefore, the average rate of change of f(x) over the interval   [tex][-3, 3][/tex] is:

[tex](6 - (-12)) / 6 = 18/6 = 3[/tex]

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The cross product evaluestes to  to c × j is: -4i + 3k.

To find the cross product c × j, where c = -4i - 2j - 3k, follow these steps:

1. Write the components of vectors c and j as 3x3 determinants:
  |  i   j   k  |
  |-4  -2  -3  |
  | 0   1   0  |

2. Evaluate the determinant for each unit vector (i, j, k) component:

  i-component: ((-2)(0) - (-3)(1)) = 3
  j-component: ((-4)(0) - (0)(-3)) = 0
  k-component: ((-4)(1) - (-2)(0)) = -4

3. Combine the evaluated components: (3)i + (0)j + (-4)k = 3i - 4k.

However, there was a typo in the question, and it should be c × j, which is -4i + 3k.

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Complete question:

if c=-4i-2j-3k then find c×j

This can be read as "the cost of making x gift baskets is greater than 6 dollars" during inequality.

Both mathematical phrases, equations and inequalities, are created by connecting two expressions. The equal sign (=) indicates that two expressions in an equation are believed to be equivalent. The characters >,,, or indicate that the two expressions in an inequality are not always equal.

The equation-like form of the formula 5x 4 > 2x + 3 has an arrowhead in lieu of the equals sign. It is an illustration of inequity. This shows that the left half, 5x 4, is bigger than the right part, 2x + 3.

Let x be the number of gift baskets Mae wants to make. Then, the total amount of money she will spend can be expressed as 15.50x.

Since Mae wants to make more than 6 gift baskets, the inequality would be:

15.50x > 6

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The value of a non-zero polynomial raised to 0 is always 1.

The value of a non-zero polynomial raised to 0 is 1. This is because any non-zero polynomial can be written in the form of a sum of monomials, where each monomial is a coefficient multiplied by a variable raised to a power. When the polynomial is raised to the 0th power, each of these monomials is raised to the 0th power.

Since any variable raised to the 0th power equals 1, each monomial becomes its coefficient, and the polynomial becomes a sum of its coefficients. Therefore, the value of a non-zero polynomial raised to 0 is the sum of its coefficients, which is always equal to 1 since the polynomial is assumed to be non-zero.

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