a health food store makes trail mix

If K = -1, the dilation would be a reduction. Dilation is a geometric transformation that either enlarges or reduces the size of an object. Which can be positive or negative.

When the scale factor, K, is positive, the dilation is an enlargement. This means that the image of the object is larger than the original. The positive scale factor indicates that the object is being stretched or magnified.

However, when the scale factor, K, is negative, the dilation is a reduction. In this case, the image of the object is smaller than the original. The negative scale factor indicates that the object is being compressed or diminished.

Therefore, if K = -1, it signifies that the dilation is a reduction. The object will be transformed into a smaller version of itself. It is important to note that the absolute value of the scale factor determines the magnitude of the reduction, with a larger absolute value resulting in a greater reduction in size.

In summary, if K = -1, the dilation is a reduction of the object.

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Answer:

n^3*a_n ≈ (1/27) * n^3 → non-zero limit

Step-by-step explanation:

We have the sequence given by a_n = (5+3n)^(-3), and we want to find a value of k such that n^k*a_n has a finite non-zero limit as n approaches infinity.

Let's simplify the expression n^k*a_n:

n^k*a_n = n^k*(5+3n)^(-3)

We can rewrite this as:

n^k*a_n = [n/(5+3n)]^3 * [1/(n^(-k))]

Using the fact that 1/(n^(-k)) = n^k, we can further simplify this to:

n^k*a_n = [n/(5+3n)]^3 * n^k

We want this expression to have a finite non-zero limit as n approaches infinity. For this to be true, we need the first factor, [n/(5+3n)]^3, to approach a finite non-zero constant as n approaches infinity.

To see why this is the case, note that as n gets large, the 3n term dominates the denominator and we have:

[n/(5+3n)]^3 ≈ [n/(3n)]^3 = (1/27) * n^(-3)

So we need k = 3 for n^k*a_n to have a finite non-zero limit. Specifically, as n approaches infinity, we have:

n^3*a_n ≈ (1/27) * n^3 → non-zero constant.

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The assembly time for a product is uniformly distributed between 6 to 10 minutes the standard deviation of assembly time in minutes is approximately 1.155.

To find the standard deviation of assembly time for a product that is uniformly distributed between 6 to 10 minutes, we can use the following formula for a uniform distribution:

Standard Deviation (σ) = √((b - a)² / 12)

Here, 'a' is the lower limit (6 minutes) and 'b' is the upper limit (10 minutes).

Step 1: Calculate (b - a)²
(10 - 6)² = 4² = 16

Step 2: Divide by 12
16 / 12 = 1.3333

Step 3: Find the square root
√1.3333 ≈ 1.155

So, the standard deviation of assembly time for a product in minutes is approximately 1.155.

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According to question  the value of ∫41(3f(x) 2x)dx is 73.

We know that the average value of the function f on the interval [1,4] is 8. This means that:

(1/3) * ∫1^4 f(x) dx = 8

Multiplying both sides by 3, we get:

∫1^4 f(x) dx = 24

Now, we need to find the value of ∫4^1 (3f(x) 2x) dx. We can simplify this expression as follows:

∫1^4 (3f(x) 2x) dx = 3 * ∫1^4 f(x) dx + 2 * ∫1^4 x dx

Using the average value of f, we can substitute the first integral with 24:

∫1^4 (3f(x) 2x) dx = 3 * 24 + 2 * ∫1^4 x dx

Evaluating the second integral, we get:

∫1^4 x dx = [x^2/2]1^4 = 8.5

Substituting this value back into the equation, we get:

∫1^4 (3f(x) 2x) dx = 3 * 24 + 2 * 8.5 = 73

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The sketched region enclosed by the given curves, y = 2/x, y = 8x, and y = x/8 is given below.

To sketch the region enclosed by the given curves, we'll first plot each curve separately and then identify the region between them. The curves are:

y = 2/x

y = 8x

y = x/8

Let's start by plotting these curves one by one:

y = 2/x:

Since x > 0, the curve y = 2/x is a hyperbola with the y-axis as an asymptote and passes through the point (1, 2).

y = 8x:

This is a straight line passing through the origin (0, 0) with a slope of 8. The line goes through the first quadrant.

y = x/8:

This is another straight line with a slope of 1/8. It passes through the origin (0, 0) and also goes through the first quadrant.

Therefore, the final graph is given below.

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The complete question:

Sketch the region enclosed by the given curves.

y = 2/x, y = 8x, y = x/8, x>0

The volume of Sanjay’s closet would be  82.875 ft³

It is known that a rectangular prism is a three-dimensional shape that has two at the top and bottom and four are lateral faces.

The volume of a rectangular prism=Length X Width X Height

Given parameters are;

4 1/4 ft long, 3 1/4 ft wide, and 6 ft tall.

V = Length X Width X Height

V = 3 1/4 x 4 1/4 x 6

V = 82. 7/8 ft³ or 82.875 ft³

The complete question is

Sanjay’s closet is shaped like a rectangular prism. It measures 4 1/4 ft long, 3 1/4 ft wide, and 6 ft tall. What is the volume of Sanjay’s closet?

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The event (e ∩ f ∩ g)c is equal to the set {3, 8, 13, 18}.

To find the complement of the intersection of sets e, f, and g, denoted as (e ∩ f ∩ g)c, we first need to determine the intersection of sets e, f, and g.

The intersection of sets e, f, and g is the set of elements that are present in all three sets. In this case:

e ∩ f ∩ g = {23, 28}

To find the complement of this intersection, we need to consider all the elements that are not in the set {23, 28}.

Given that the original set s = {3, 8, 13, 18, 23, 28}, the complement of the intersection can be found by subtracting {23, 28} from set s:

(e ∩ f ∩ g)c = s - {23, 28}

Calculating this, we have:

(e ∩ f ∩ g)c = {3, 8, 13, 18}

Therefore, the event (e ∩ f ∩ g)c is equal to the set {3, 8, 13, 18}.

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Given that a person walks 18.2 m straight towards the west and then 27.8 m straight towards the north, to find the vector angle which describes the person's direction from the forward direction (east).

We know that vector angle is the angle which the vector makes with the positive direction of the x-axis (East).

Therefore, the vector angle which describes the person's direction from the forward direction (east) can be calculated as follows:

Step 1: Calculate the resultant [tex]vectorR = √(18.2² + 27.8²)R = √(331.24)R = 18.185 m ([/tex]rounded to 3 decimal places)

Step 2: Calculate the angleθ = tan⁻¹ (opposite/adjacent)where,opposite side is 18.2 mandadjacent side is [tex]27.8 mθ = tan⁻¹ (18.2/27.8)θ = 35.44°[/tex] (rounded to 2 decimal places)Thus, the vector angle which describes the person's direction from the forward direction (east) is 35.44° (rounded to 2 decimal places).

Hence, the correct option is 35.44°.

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The equation of the line in the graph is

From the graph the line passed through points (4,-1) and (0,5)

using the slope-intercept form of a line, which is y = mx + b,

where

m is the slope and

b is the y-intercept.

the slope of the line

m = (5 - (-1)) / (0 - 4) = 6 / -4 = -3/2

form the points the y intercept is 5

Therefore, the equation of the line is y = -3/2 x + 5.

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the series ∑(n=1 to infinity) [tex]n^{6}[/tex] / n! is convergent by using ratio test.

To apply the Ratio Test, we need to evaluate the limit of the ratio of consecutive terms, lim(n→∞) (a(n+1) / a(n)).

In this case, a(n) = [tex]n^{6}[/tex] / n! and a(n+1) =[tex](n+1)^{6}[/tex] / (n+1)!.

Taking the limit, we have:

lim(n→∞) [[tex](n+1)^{6}[/tex] / (n+1)!] / [[tex]n^{6}[/tex] / n!]

= lim(n→∞) [[tex](n+1)^{6}[/tex] / [tex]n^{6}[/tex]] * [n! / (n+1)!]

= lim(n→∞) [[tex](n+1)^{6}[/tex] / [tex]n^{6}[/tex]] * [1 / (n+1)]

= 1 * 0 = 0.

Since the limit of the ratio of consecutive terms is 0, which is less than 1, the series converges by the Ratio Test.

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Answer:

a=4, b=11

Step-by-step explanation:

You have to complete the square.

x²-8x+5 = (x-4)²-16 +5 = (x-4)² - 11

The net force acting on the object is 10N up

When multiple forces act on an object, the net force is the total force acting on the object. It determines the object's motion, including its direction and speed.

To calculate the net force, we need to add all the forces acting on the object. If the net force is zero, the object will remain at rest or move with a constant velocity, while if it is non-zero, the object's velocity will change, and it will accelerate in the direction of the net force.

In this scenario, there are four forces acting on the object, two pointing up and two pointing down. To calculate the net force, we need to add all the forces together, taking into account their direction and magnitude.

Since the forces pointing up and down are opposite in direction, we subtract the smaller force from the larger one to get the resultant force. In other words, we can cancel out the forces pointing in opposite directions, leaving us with a single net force acting on the object.

So, in this case, we have a 30N force pointing up, a 25N force pointing down, a 5N force pointing down, and a 5N force pointing up.

First, we'll cancel out the 5N force pointing down with the 5N force pointing up.

30N up - 25N down - 5N down + 5N up

= 30N - 25N - 5N + 5N

= 30N - 20N

= 10N up

Therefore, the net force acting on the object is 10N up. This means that the object will accelerate in the upward direction with a force of 10N

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Complete Question

Calculate the net force these forces acts on a single object, 30N [up],

25N [down], 5N [down] and 5N [up]

When government spending increases by $5 billion and the MPC = .8, in the first round of the spending multiplier process, spending increases by $20 billion.


The spending multiplier is the amount by which GDP will increase for each unit increase in government spending. It is calculated as 1/(1-MPC), where MPC is the marginal propensity to consume. In this case, MPC = .8, so the spending multiplier is 1/(1-.8) = 5.

Therefore, when government spending increases by $5 billion, the total increase in spending in the economy will be $5 billion multiplied by the spending multiplier of 5, which equals $25 billion. However, the initial increase in spending is only $5 billion, hence the increase in the first round of the spending multiplier process is $20 billion.

In summary, when government spending increases by $5 billion and the MPC = .8, the initial increase in spending is $5 billion, but the total increase in the first round of the spending multiplier process is $20 billion.

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So there are Output pairs that cannot be obtained for any input pair.

(a) T(x, y) = (2x, y)

This function is one-to-one but not onto. It is one-to-one because different input pairs (x1, y1) and (x2, y2) will always result in different output pairs (2x1, y1) and (2x2, y2). However, it is not onto because for any y ≠ 0, there is no input pair (x, y) that maps to the output pair (0, y).

(b) T(x, y) = (x^4, y)

This function is onto but not one-to-one. It is onto because for any given output pair (a, b), we can find an input pair (x, y) such that T(x, y) = (a, b) by taking the fourth root of a for x and setting y to b. However, it is not one-to-one because different input pairs can result in the same output pair. For example, T(1, 2) = T(-1, 2) = (1, 2).

(c) T(x, y) = (sin(x), cos(y))

This function is neither one-to-one nor onto. It is not one-to-one because different input pairs can result in the same output pair due to the periodic nature of sine and cosine functions. For example, T(0, 0) = T(2π, 0) = (0, 1). It is also not onto because the range of the function is limited to the interval [-1, 1] for both x and y, so there are output pairs that cannot be obtained for any input pair.

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T(x, y) = (2x, y) is one-to-one and onto.

To show one-to-one, assume T(a, b) = T(c, d). Then we have (2a, b) = (2c, d), which implies a = c and b = d.

To show onto, we need to show that for any (x, y) in R2, there exists (a, b) in R2 such that T(a, b) = (x, y). If we take (a, b) = (x/2, y), then T(a, b) = (x, y).

(b) T(x, y) = (x^4, y) is one-to-one but not onto.

To show one-to-one, assume T(a, b) = T(c, d). Then we have (a^4, b) = (c^4, d), which implies a = c and b = d.

To show not onto, note that there is no (a, b) in R2 such that T(a, b) = (-1, 0), since x^4 is always non-negative.

(d) T(x, y) = (sin(x), cos(y)) is neither one-to-one nor onto.

To show not one-to-one, note that T(0, 0) = T(2π, 0), but (0, 0) ≠ (2π, 0).

To show not onto, note that there is no (x, y) in R2 such that T(x, y) = (0, 1), since sin(x) is always between -1 and 1.

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A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. It can be used to represent systems of linear equations, transformations in geometry, and a wide range of other mathematical concepts in a compact and organized form.

When you multiply a matrix A (n × n) by a vector x (in R^n), the resulting product Ax is a linear combination of the columns of matrix A.

Here's a step-by-step explanation:

1. Let A be an n × n matrix with columns C₁, C₂, ..., Cₙ, and x be a vector in R^n with elements [x₁, x₂, ..., xₙ]^T (transpose).

2. When you multiply the matrix A by the vector x, the resulting vector Ax can be represented as:

  Ax = A * x = [C₁ C₂ ... Cₙ] * [x₁, x₂, ..., xₙ]^T

3. The multiplication of A and x results in a new vector, where each element is formed by taking the dot product of the corresponding row of A with the vector x:

  Ax = [x₁*C₁ + x₂*C₂ + ... + xₙ*Cₙ]

4. In the resulting vector Ax, you can see that each column of matrix A is multiplied by its corresponding scalar from the vector x, forming a linear combination of the columns of matrix A.

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The profit function is 9x - 7000 and the revenue function is 12x.

Given that the cost function for a company to produce a lunch box is c(x)= 3x+7000 where x is the number of lunch boxes and the company sells the lunch boxes for $12 each.

To write a profit function, the revenue function is required to calculate the profit earned by the company.

The revenue function is given as:

Revenue = Selling Price × Quantity Sold

Price is $12 for each lunch box, therefore

Revenue = $12 × Quantity sold

Quantity sold is represented as x, therefore,

Revenue = 12x

The profit function is given as:

Profit = Revenue - Cost

The cost function is given as c(x)= 3x+7000

Therefore,

Profit = 12x - (3x + 7000)

Profit = 9x - 7000

Hence, the profit function is 9x - 7000 and the revenue function is 12x.

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The mechanistic basis for linear polystyrene having phenyl groups attached to alternate carbons of the polymer chain is due to the nature of the polymerization reaction, specifically free-radical polymerization.

1. Free-radical polymerization of styrene starts with the initiation step, where a free radical initiator generates a reactive radical site.
2. The reactive radical site reacts with the double bond of the styrene monomer, forming a new radical site on the styrene molecule.
3. This new radical site on the styrene molecule can now react with another styrene monomer, effectively joining them together.
4. As the radical site is always at the end of the growing polymer chain, the phenyl groups of each added styrene monomer will be attached to alternate carbons. This occurs because the reactive site is situated between the phenyl group and the double bond in the monomer, creating a zigzag pattern as the chain grows.

Conclusion:
The attachment of phenyl groups to alternate carbons of the polymer chain in linear polystyrene can be attributed to the free-radical polymerization mechanism. The reactive radical site, created during the polymerization, allows the phenyl groups to be connected in an alternating pattern along the chain.

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The total cost of transporting the raw materials is $890, which is less than the $450 budget of the factory manager.

The best way to maximize the transportation of raw materials from a factory to its storage area using Big Red and Common Joe trucking firms while ensuring the factory manager does not spend more than $450 is to use 5 Big Red trucks and 5 Common Joe trucks.In order to get the best result from the two trucking firms, the following steps should be followed.

Step 1: Determine the number of trucks that can be transported using Big Red's heavy-duty trucks.

$200 per truck is the cost of transporting 200 tons by Big Red.

The formula for calculating the number of trucks that can be used is as follows:

$450/$50 = 9 truckloads

Step 2: Determine the number of trucks that can be transported using Common Joe trucks.

$20 per truck is the cost of transporting 90 tons by Common Joe.

The formula for calculating the number of trucks that can be used is as follows:

$450/$20 = 22.5 truckloads

The number of trucks that can be used is 22, but since it is not an integer, it will be rounded down to 22.The total number of tons that can be transported using the two trucking firms is calculated as follows:

5 * 200 = 1000 tons of raw materials can be transported by Big Red

5 * 90 = 450 tons of raw materials can be transported by Common Joe

The total tons of raw materials that can be transported is therefore 1,450 tons.

Therefore, to transport a total of 1,500 tons of raw materials, 50 more tons need to be transported. 10 more truckloads of Big Red will transport these additional tons.

Therefore, 15 truckloads will be transported by Big Red (5 + 10 = 15), and the remaining 7 truckloads will be transported by Common Joe. (22 - 15 = 7).

As a result, the total cost of transporting the raw materials is:

$50 * 15 + $20 * 7 = $750 + $140

= $890, which is less than the $450 budget of the factory manager.

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The value of the given integral is (1/18) + (17/ln(17)).

The integral is evaluated using the sum rule of integration, which states that the integral of the sum of two functions is equal to the sum of their integrals. Therefore, we can evaluate the integral of each term separately and then add them together.

For the first term, we use the power rule of integration, which states that the integral of [tex]x^n[/tex]is equal to[tex](x^(n+1))/(n+1)[/tex]. Therefore, the integral of [tex]x^17[/tex]is [tex](x^18)/18.[/tex]

For the second term, we use the exponential rule of integration, which states that the integral of [tex]a^x[/tex]is equal to [tex](a^x)/(ln(a))[/tex]. Therefore, the integral of [tex]17^x is (17^x)/(ln(17)).[/tex]

Adding these two integrals together gives us the final answer of (1/18) + [tex](17/ln(17)[/tex]).

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Sam the snail's rate is approximately 0.03 miles per hour.

To find Sam's rate in miles per hour, we need to convert his speed from feet per minute to miles per hour.

We know that 1 mile is equal to 5280 feet. First, we can convert Sam's speed from feet per minute to feet per hour by multiplying it by 60 since there are 60 minutes in an hour.

Therefore, Sam's speed in feet per hour is 2.64 ft/min * 60 min/hr = 158.4 ft/hr.

Next, we can convert Sam's speed from feet per hour to miles per hour. Since 1 mile is equal to 5280 feet, we can divide Sam's speed in feet per hour by 5280 to get his speed in miles per hour.

Therefore, Sam's speed in miles per hour is 158.4 ft/hr / 5280 ft/mi = 0.03 mi/hr.

Therefore, Sam the snail crawls at a rate of approximately 0.03 miles per hour.

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The polar equation r = 3+6cosθ is symmetric to the polar axis with respect to the polar axis.

To test the polar equation r = 3 + 6 cos(θ) for symmetry, we will consider each type of symmetry one by one:

1. Polar axis symmetry: Replace θ with -θ and check if the equation remains the same.
r = 3 + 6 cos(-θ) = 3 + 6 cos(θ) (since cosine is an even function)
Since the equation remains the same, the curve is symmetric with respect to the polar axis.

2. Pole symmetry: Replace r with -r and check if the equation remains the same.
-r = 3 + 6 cos(θ)
This equation is not equivalent to the original equation, so the curve is not symmetric with respect to the pole.

3. Line θ = π/2 symmetry: Replace θ with (π - θ) and check if the equation remains the same.
r = 3 + 6 cos(π - θ) = 3 - 6 cos(θ) (since cos(π - θ) = -cos(θ))
This equation is not equivalent to the original equation, so the curve is not symmetric with respect to the line θ = π/2.

In conclusion, the polar equation r = 3 + 6 cos(θ) is symmetric with respect to the polar axis, but not with respect to the pole or the line θ = π/2.

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The given simultaneous equations are x² + y² = 9  ...............(1)

x + y = 2 ...............(2)

Equation (2) is solved for y by taking x as the subject:

y = 2 - x

Substitute this value of y in the equation (1):

x² + y² = 9x² + (2 - x)² = 9x² + 4 - 4x + x² = 9

Rearrange the above equation in the standard quadratic form by bringing all terms to one side of the equation:

x² + x² - 4x - 5 = 02

x² - 4x - 5 = 0

This equation is a quadratic equation and can be solved by using the quadratic formula:

x = [-(-4) ± √(-4)² - 4(2)(-5)]/2(2)

x = [4 ± √56]/4

x = [4 ± 2√14]/4

x = [2 ± √14]/2

Substitute these values of x in equation (2) to find the corresponding values of y:

For x = [2 + √14]/2,

y = 2 - [2 + √14]/2

y = (4 - [2 + √14])/2

y = (2 - √14)/2

For x = [2 - √14]/2,

y = 2 - [2 - √14]/2

y = (4 - [2 - √14])/2

y = (2 + √14)/2

Therefore, the solution of the given simultaneous equations is

x = [2 + √14]/2,

y = (2 - √14)/2

OR

x = [2 - √14]/2,

y = (2 + √14)/2

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The formula for the indicated rate of change dc/dk is dc/dk = 32c.

To find the indicated rate of change, we need to calculate dc/dk, which represents the partial derivative of the function s(c, k) = c(32k) with respect to k while treating c as a constant.

To calculate dc/dk, we differentiate the function s(c, k) with respect to k while considering c as a constant:

dc/dk = d/dk (c * (32k))

Applying the product rule of differentiation, we have:

dc/dk = c * d/dk (32k) + (32k) * d/dk (c)

The derivative of 32k with respect to k is 32, as it is a constant multiple of k. The derivative of c with respect to k is zero since c is treated as a constant.

Therefore, dc/dk simplifies to:

dc/dk = c * 32 + 0

dc/dk = 32c

So, the formula for the indicated rate of change dc/dk is dc/dk = 32c.

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Answer:

y = (3/4)x - 7/4

Step-by-step explanation:

y – y1 = m (x – x1), where y1 and x1 are the coordinates of a given point.

4x + 3y = 15

3y = -4x + 15

y = -(4/3)x + 5.

the slope of this line is -4/3.

the slope of the perpendicular line is -1 / (-4/3) = +3/4.

equation of perpendicular line through (5, 2) is:

y - 2 = (3/4) (x -5)  = (3/4)x - (15/4)

y =  (3/4)x - (15/4) + 2

y = (3/4)x - 7/4

Answer:

a³²

Step-by-step explanation:

the law of exponents states that when raising a power to another power multiply the exponents

our answer will go like this:

(a⁸)⁴

a⁸*⁴

a³²

Answer:

The formatting is a bit off but assuming that -x + 2y = 6 and -3x + y = -2 are the two separate equations, the solution to your system of equations is (2,4) or x = 2, y = 4.

Step-by-step explanation:

Here is how you could solve this system of equations using the elimination method:

1. The first step is to find a variable you can eliminate, such as y.
-x+2y=6
-3x+y=-2

(multiply the second equation by -2)

−x+2y=6
6x-2y=4
This is your new set

2. Next, "add" your set together by lining it up and combining like terms.
   -x+2y=6
+. 6x-2y=4
——————
    5x = 10

3. Solve for x by dividing by 5
5x=10
10÷5=2
x=2

4. Now that you have your x, find y by substituting 2 for x in any of your original set's equations. We'll do the first equation, −x+2y=6.
−x+2y=6
-2+2y=6 ---> add 2 on both sides to remove -2
2y=8 ---> divide by 2 on both sides to remove the 2 from y
y=4

5. Set your answers up as an ordered pair like this ( ___ , ___ )
x=2 , y=4
(2, 4)

Hope this helps!

Answer:

a = 3, -17

Step-by-step explanation:

a ² +  14a  - 51 = 0

1) put the a, not a ², in parenthesis.

2) half the coefficient (14) of a. that is 7. Put that into same parenthesis.

3) we have (a + 7)

4) square this and multiply out. (a + 7) ² = a ² + 7a + 7a +49 = a ² +14a + 49

5) this looks just like the original equation except for +49. What do we have to do to get back to original? 49 – (-51) = 49 + 51 = 100. We have to subtract 100

6) now we have (a + 7) ² – 100 =0

7) (a + 7) ² = 100

8) (a + 7) = ± √100

9) a = ± √100  - 7

a = ±10 - 7

= -17 and 3

Option A is correct, 57.17% is the probability that he hits the target at least four out of next six attempts.

Let's calculate the probability of hitting the target exactly four times out of six attempts:

P(4 hits) = C(6, 4) × (0.65)⁴ ×  (1 - 0.65)⁶⁻⁴

The probability of hitting the target exactly five times out of six attempts:

P(5 hits) = C(6, 5) × (0.65)⁵ × (1 - 0.65)⁶⁻⁵

Now calculate the probability of hitting the target all six times:

P(6 hits) = (0.65)⁶

Now, we can find the probability that Tom hits the target at least four times by summing up the individual probabilities:

P(at least 4 hits) = P(4 hits) + P(5 hits) + P(6 hits)

P(at least 4 hits) = C(6, 4) × (0.65)⁴ ×  (1 - 0.65)⁶⁻⁴ + C(6, 5) × (0.65)⁵ × (1 - 0.65)⁶⁻⁵ +  (0.65)⁶

=57.17%

Hence,  57.17% is the probability that he hits the target at least four out of next six attempts.

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Main answer: Transformations of basic functions depend on the changes made to their variables.

Supporting answer :Functions can be transformed in different ways. The variable a modifies the vertical stretch or compression of a function. A negative value of a produces a reflection over the x-axis. The variable b is used to modify the horizontal stretch or compression of the function. A negative value of b produces a reflection over the y-axis. The variable h translates the graph to the left (h > 0) or to the right (h < 0). Lastly, the variable k translates the graph up (k > 0) or down (k < 0).

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The Reparametrized curve with respect to arc length is:

r(s) = (1/2) * sqrt(2) * e^(4t) cos(4t) i + 3 j + (1/2) * sqrt(2) * e^(4t) sin(4t) k

To reparametrize the curve with respect to arc length, we need to find the expression for the curve in terms of the arc length parameter s.

The arc length parameter s is given by the integral of the speed function |r'(t)| with respect to t:

s = ∫|r'(t)| dt

Let's calculate the speed function |r'(t)| first:

r(t) = e^(4t) cos(4t) i + 3 j + e^(4t) sin(4t) k

r'(t) = (4e^(4t) cos(4t) - 4e^(4t) sin(4t)) i + 0 j + (4e^(4t) sin(4t) + 4e^(4t) cos(4t)) k

|r'(t)| = sqrt((4e^(4t) cos(4t) - 4e^(4t) sin(4t))^2 + (4e^(4t) sin(4t) + 4e^(4t) cos(4t))^2)

= sqrt(16e^(8t) cos^2(4t) - 32e^(8t) cos(4t) sin(4t) + 16e^(8t) sin^2(4t) + 16e^(8t) sin^2(4t) + 32e^(8t) cos(4t) sin(4t) + 16e^(8t) cos^2(4t))

= sqrt(32e^(8t))

Now, we can express s in terms of t by integrating |r'(t)|:

s = ∫sqrt(32e^(8t)) dt

To find the integral, we can make a substitution u = 8t, du = 8 dt:

s = (1/8) ∫sqrt(32e^u) du

= (1/8) ∫2sqrt(2e^u) du

= (1/8) * 2 * sqrt(2) ∫e^(u/2) du

= (1/4) * sqrt(2) * ∫e^(u/2) du

= (1/4) * sqrt(2) * 2e^(u/2) + C

= (1/2) * sqrt(2) * e^(4t) + C

Therefore, the reparametrized curve with respect to arc length is:

r(s) = (1/2) * sqrt(2) * e^(4t) cos(4t) i + 3 j + (1/2) * sqrt(2) * e^(4t) sin(4t) k

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The curve reparametrized with respect to arc length is:

r(u) = e^(2u/√2) cos(2u/√2) i + 3j + e^(2u/√2) sin(2u/√2) k

We have the curve given by:

r(t) = e^(4t) cos(4t) i + 3j + e^(4t) sin(4t) k

The speed of the curve is:

|v(t)| = √( (dx/dt)^2 + (dy/dt)^2 + (dz/dt)^2 )

= √( 16e^(8t) + 16e^(8t) )

= 4e^(4t) √2

Thus, the arc length from t = 0 to t = s is:

s = ∫0s |v(t)| dt

= ∫0s 4e^(4t) √2 dt

= √2 e^(4t) |_0^s

= √2 ( e^(4s) - 1 )

Solving for s, we get:

s = (1/4) ln( (s/√2) + 1 )

Let u be the parameter with respect to arc length, then we have:

u = ∫0t |v(t)| dt

= ∫0t 4e^(4t) √2 dt

= √2 e^(4t) |_0^t

= √2 ( e^(4t) - 1 )

Solving for t, we get:

t = (1/4) ln( (u/√2) + 1 )

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