When designing a roller coaster, engineers need to know about geometry and
how to use angles that will support the ride. Engineers take into account the
materials used, the height of the roller coaster, and whether or not there are
inversions, or loops, in the roller coaster when deciding the angle measures
needed to support the coaster. They may use different combinations of vertical
and adjacent angles to ensure the safety of the ride.
Explain the difference between vertical angles and adjacent angles.

We can conclude that for any natural number n,[tex]n^2[/tex]= 1 (mod 4) depending on the remainder of n when divided by 4.

The statement "For any natural number n, it is true that in=1,i,â1, depending on the remainder of n when divided by 4" is not true.

In fact, the statement is not well-defined because it is unclear what "in" refers to.

However, if the statement is intended to be "For any natural number n, it is true that [tex]n^2[/tex]=1 (mod 4) depending on the remainder of n when divided by 4," then this statement is true.

To see why, note that any natural number can be written as 4k, 4k+1, 4k+2, or 4k+3 for some integer k.

If n = 4k, then [tex]n^2 = (4k)^2 = 16k^2[/tex], which is divisible by 4 and hence is congruent to 0 (mod 4). Therefore, [tex]n^2[/tex] = 1 (mod 4).

If n = 4k + 1, then [tex]n^2 = (4k + 1)^2 = 16k^2 + 8k + 1 = 4(4k^2 + 2k) + 1[/tex], which is congruent to 1 (mod 4). Therefore, [tex]n^2[/tex] = 1 (mod 4).

If n = 4k + 2, then [tex]n^2 = (4k + 2)^2 = 16k^2 + 16k + 4 = 4(4k^2 + 4k + 1)[/tex], which is congruent to 0 (mod 4). Therefore, n^2 = 0 (mod 4), which is not equal to 1 (mod 4).

If n = 4k + 3, then[tex]n^2 = (4k + 3)^2 = 16k^2 + 24k + 9 = 4(4k^2 + 6k + 2)[/tex] + 1, which is congruent to 1 (mod 4). Therefore, [tex]n^2 = 1[/tex] (mod 4).

Therefore, we can conclude that for any natural number n,[tex]n^2 =[/tex]1 (mod 4) depending on the remainder of n when divided by 4.

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It will take the plant 65 days to grow to a height of 5 ft.

A relationship between two variables in which one variable is a fixed multiple of the other is known as direct variation. Accordingly, as one variable changes, the other changes proportionally as well. Likewise, as one variable declines, the other variable changes similarly.

If two variables x and y vary directly, we can say in mathematical terms:

y = kx

where k is the variational constant. This indicates that the y/x ratio is constant and equal to k. The initial conditions of the issue, such as the values of y and x at a specific moment, determine the constant k.

The height of the plant varies directly this is given as:

h = k t

Now, the plant is 2 ft tall after 26 days:

2 = k × 26

k = 2/26

k = 1/13

Now, for h = 5 ft we have:

5 = (1/13) t

t = 65

Hence, it will take the plant 65 days to grow to a height of 5 ft.

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The fourth derivative of f(x) over the interval [2π/4,2π/2] is -3/8.

The given function is f(x) = 6cos(x/2) over the interval [2π/4,2π/2]. To find the fourth derivative of this function, we need to apply the chain rule and the product rule repeatedly.

First, let's find the first derivative of f(x):

f'(x) = -3sin(x/2)

Next, let's find the second derivative of f(x):

f''(x) = -3/2cos(x/2)

Now, let's find the third derivative of f(x):

f'''(x) = 3/4sin(x/2)

Finally, let's find the fourth derivative of f(x):

f''''(x) = 3/8cos(x/2)

Now that we have the fourth derivative of the function, we can evaluate it over the interval [2π/4,2π/2] to get the value of L4. To do this, we simply substitute the upper limit of the interval (2π/2) and the lower limit of the interval (2π/4) into the fourth derivative expression and subtract the results. This gives us:

L4 = f''''(2π/2) - f''''(2π/4)

= (3/8)cos(π) - (3/8)cos(π/2)

= -(3/8)

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The instantaneous rate of change of f equal the average rate of change of f on that interval is approximately 2.769

The given function is f(x) = ∫0 to 2x (sin(t) dt), where f(x) represents the area under the curve of sin(t) from 0 to 2x. To find the average rate of change of this function on the interval [0, π], we can use the formula:

average rate of change = [f(π) - f(0)] / (π - 0)

We can simplify this expression by evaluating f(π) and f(0) using the given function:

f(π) = ∫ (sin(t) dt)

= ∫ (sin(t) dt) + ∫(sin(t) dt)

= 2

f(0) = ∫ (sin(t) dt)

= 0

Substituting these values into the formula for the average rate of change, we get:

average rate of change = (2 - 0) / π

= 2/π

We can use the fundamental theorem of calculus to evaluate this derivative:

f'(x) = d/dx [F(2x) - F(0)]

= 2sin(2x)

where F(x) is an antiderivative of sin(x).

Now we can compare the instantaneous rate of change of f(x) at any point x in [0, π] to the average rate of change of f(x) over the entire interval [0, π]. We want to find the values of x for which these rates of change are equal:

2sin(2x) = 2/π

Simplifying this expression, we get:

sin(2x) = 1/π

We know that sin(x) is a periodic function with period 2π. So, we can find the solutions to this equation by finding all values of 2x that satisfy sin(2x) = 1/π within the interval [0, 2π], and then dividing these solutions by 2 to get the corresponding values of x in [0, π].

Using a trigonometric identity, sin(2x) = 2sin(x)cos(x), we can rewrite the equation as:

2sin(x)cos(x) = 1/π

Squaring both sides, we get:

4sin²(x)cos²(x) = 1/π²

Using another trigonometric identity, 2sin(x)cos(x) = sin(2x), we can rewrite the left-hand side as:

sin²(2x) = 1/π²

Taking the square root of both sides, we get:

sin(2x) = ±1/π

So, the solutions to the equation sin(2x) = 1/π are:

2x = sin⁻¹(1/π) + 2kπ or 2x = π - sin⁻¹(1/π) + 2kπ

where k is an integer. To get the solutions for x, we divide both sides by 2, which gives:

x = (1/2)sin⁻¹(1/π) + kπ or x = (1/2)(π - sin⁻¹(1/π)) + kπ

where k is an integer.

Now we need to find which of these solutions lie in the interval [0, π]. To do this, we can check if each solution satisfies 0 ≤ x ≤ π. Since sin⁻¹(1/π) is positive, we only need to check the first solution:

x = (1/2)sin⁻¹(1/π) + kπ

For k = 0, we get:

x = (1/2)sin⁻¹(1/π) ≈ 0.372

For k = 1, we get:

x = (1/2)sin⁻¹(1/π) + π ≈ 2.769

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Answer:

1, 2, 4, 8, 16, 32, 64, 128, 256, and 512

Step-by-step explanation:

Answer:

answers are on picture

Step-by-step explanation:

please mark mine brainliest. answrs on picture

Answer:

25.4 x 17.78 x 12.7 cm

Step-by-step explanation:

The x values of the inflection points of f(0) = 4x² + 8 are 'NONE'.

To find inflection points, we first need to find the second derivative of the function. The original function is f(x) = 4x² + 8. The first derivative, f'(x), is the derivative of 4x² + 8 with respect to x, which is 8x.

Now, find the second derivative, f''(x), by taking the derivative of 8x with respect to x, which is 8. Since the second derivative is a constant value (8) and does not change with x, there are no inflection points. Inflection points occur when the second derivative changes sign, but in this case, it remains constant.

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For f(x) to be a valid probability mass function, it must satisfy the following conditions:

f(x) ≥ 0 for all x

Σ f(x) = 1 over all possible values of x

Let's check these conditions:

For x = 1, 2, 3, ..., we have (1/3)^X ≥ 0, so f(x) ≥ 0 for all x.

Σ f(x) = a Σ (1/3)^X = a (1 + 1/3 + 1/9 + ...) = a (3/2) (geometric series with r = 1/3 and a = 1), which converges to a (3/2)/(1-1/3) = a (3/2)/(2/3) = a (9/4). For this to equal 1, we need:

a (9/4) = 1

a = 4/9

Therefore, the value of a for f(x) to be a valid probability function is 4/9.

To be a valid probability function, the sum of probabilities for all possible values of X should be equal to 1. So, we need to find the value of a such that the sum of probabilities is equal to 1.

Let's first find the sum of probabilities for all possible values of X:

∑f(x) = ∑a(1/3)^X = a(1/3)^1 + a(1/3)^2 + a(1/3)^3 + ...

This is an infinite geometric series with first term a(1/3)^1 and common ratio (1/3). The sum of an infinite geometric series with first term a and common ratio r is given by:

sum = a / (1 - r)

So, for our series, we have:

∑f(x) = a(1/3)^1 + a(1/3)^2 + a(1/3)^3 + ... = a / (1 - 1/3) = a / (2/3) = (3/2)a

Now, we want this sum to be equal to 1, so:

(3/2)a = 1

a = 2/3

Therefore, the value of a for this to be a valid probability function is 2/3.

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The dimensions that minimize the material costs are 6.0 m by 6.0 m by 2.0 m. The minimum material cost is C=65(6.0)²+192(6.0)(2.0)=$1248.

Let's assume that the dimensions of the square base are x by x, and the height of the container is h. Therefore, the volume of the container can be expressed as V=x²h=92.0 m².

To minimize the material costs, we need to find the dimensions that minimize the cost of the base and the cost of the sides and top. The cost of the base can be expressed as C₁=65x², and the cost of the sides and top can be expressed as C₂=4(48xh)=192xh.

To minimize the total cost, we need to minimize the sum of the costs of the base and the sides/top, which can be expressed as C=C₁+C₂=65x^2+192xh.

Using the volume equation, we can solve for h in terms of x: h=92/x^2. Substituting this into the total cost equation, we get C=65x²+192(92/x²)=65x²+17664/x².

To find the minimum cost, we need to find the critical points of C. Taking the derivative with respect to x, we get dC/dx=130x-35328/x³=0. Solving for x, we get x=6.0 m. Substituting this into the volume equation, we get h=2.0 m.

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(a) Probability is 0.755 (b) Probability is 0.0322 (c) Probability is 0.5871 (d) Percentile rank is 3.22%

(a) To find the probability that a randomly selected bag contains between 1100 and 1500 chocolate chips, inclusive, we need to find the area under the normal curve between the values of 1100 and 1500.

Using a z-score formula, we can standardize the values:

z1 = (1100 - 1252) / 123 = -1.24
z2 = (1500 - 1252) / 123 = 2.09

Then, we can use a standard normal distribution table or calculator to find the area under the curve between these z-scores:

P(-1.24 < Z < 2.09) = 0.755

Therefore, the probability that a randomly selected bag contains between 1100 and 1500 chocolate chips, inclusive, is 0.755.

(b) To find the probability that a randomly selected bag contains fewer than 1025 chocolate chips, we need to find the area under the normal curve to the left of 1025.

Again, we can standardize the value using a z-score formula:

z = (1025 - 1252) / 123 = -1.85

Then, we can use a standard normal distribution table or calculator to find the area under the curve to the left of this z-score:

P(Z < -1.85) = 0.0322

Therefore, the probability that a randomly selected bag contains fewer than 1025 chocolate chips is 0.0322.

(c) To find the proportion of bags that contains more than 1225 chocolate chips, we need to find the area under the normal curve to the right of 1225.

Again, we can standardize the value using a z-score formula:

z = (1225 - 1252) / 123 = -0.22

Then, we can use a standard normal distribution table or calculator to find the area under the curve to the right of this z-score:

P(Z > -0.22) = 0.5871

Therefore, the proportion of bags that contains more than 1225 chocolate chips is 0.5871.

(d) To find the percentile rank of a bag that contains 1025 chocolate chips, we need to find the percentage of bags that contain fewer chips than this bag.

We can use the same z-score formula to standardize the value:

z = (1025 - 1252) / 123 = -1.85

Then, we can use a standard normal distribution table or calculator to find the area under the curve to the left of this z-score:

P(Z < -1.85) = 0.0322

This means that approximately 3.22% of bags contain fewer than 1025 chocolate chips. Therefore, the percentile rank of a bag that contains 1025 chocolate chips is approximately 3.22%.

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There is actually a decrease in the mean for the second week compared to the first, and it is $7.58 less.

To find the mean for the first week, we add up all the lunch expenses and divide by the number of lunches:

(12.99 + 10.50 + 9.89 + 6.90 + 7.58) / 5 = 9.97

So the mean for the first week is $9.97.

In the second week, Devon spent $2 more in total for the 5 lunches than the first week, which means he spent:

9.97 + 2 = $11.97

To find the mean for the second week, we divide the total spent by the number of lunches:

11.97 / 5 = $2.39

The increase in the mean for the second week compared to the first is:

2.39 - 9.97 = -$7.58

So there is actually a decrease in the mean for the second week compared to the first, and it is $7.58 less.

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The characteristic equation has a repeated root of r=1.

The characteristic equation for the associated homogeneous equation

y''-2y'+y=0 can be found by substituting [tex]y=e^{(rt)[/tex]and solving for r:

[tex]r^2-2r+1=0[/tex]

In this problem you will use variation of parameters to solve the

nonhomogeneous equation y″+2y′+y=−2e−t

Given equation is,

y″+2y′+y=−2e−t

The characteristic equation associated with the homogeneous equation

is,r2+2r+1=0Upon solving, (r+1)2=0 (r+1) (r+1)=0

This is a quadratic equation that can be factored as:

[tex](r-1)^2=0[/tex]

Thus, the characteristic equation has a repeated root of r=1.

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Let’s call the number we’re looking for “x”. If x is rounded to 400,000 when rounded to the nearest 100,000 and rounded to 350,000 when rounded to the nearest 10,000, then we know that x must be between 375,000 and 424,999.

This is because if we round x down to the nearest 100,000, we get 300,000 (since it rounds down to the nearest hundred thousand), and if we round x up to the nearest 100,000, we get 500,000 (since it rounds up to the nearest hundred thousand). Therefore, x must be between these two numbers.

Similarly, if we round x down to the nearest 10,000, we get 340,000 (since it rounds down to the nearest ten thousand), and if we round x up to the nearest 10,000, we get 359,999 (since it rounds up to the nearest ten thousand). Therefore, x must be between these two numbers as well.

Therefore, a possible number that satisfies these conditions is any number between 375,000 and 424,999 that rounds to 400,000 when rounded to the nearest hundred thousand and 350,000 when rounded to the nearest ten thousand.

I hope that helps!

The approximate probability that the average salary of the 100 players exceeded $1.1 million is 0.0026 or 0.26%.

To unravel this issue, we are able to utilize the central restrain hypothesis, which states that the dissemination of test implies will be around typical, notwithstanding the basic populace dispersion, as long as the test measure is huge sufficient.

In this case, the test measure is 100, which is considered large and sufficient to apply the central restrain hypothesis. Ready to discover the z-score compared to the test cruel of $1.1 million by utilizing the equation:

z = (X bar - μ) / (σ / √n)

where X bar is the test cruel(mean),

μ is the populace cruel (given as $1.5 million),

σ is the populace standard deviation (given as $0.7 million),

and n is the test estimate (given as 100).

Stopping within the values, we get:

z = (1.1 - 1.5)  / (0.7 / √100) = -2.86

Employing a standard typical conveyance table or calculator, ready to discover that the likelihood of getting a z-score more noteworthy than -2.86 is around 0.9974. Hence, the inexact likelihood that the normal compensation of the 100 players surpassed $1.1 million is 1 - 0.9974 = 0.0026 or 0.26%.

This means that it is exceptionally improbable for the test cruel to be that low, given the populace cruel and standard deviation. 

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Answer:

[tex]x = 0\\\\y = 1[/tex]

Step-by-step explanation:

We have the equations

[tex]-3x - 9y = - 9[/tex]

[tex]3x - 3y = - 3[/tex]

Add the equations

[tex]\begin{aligned}3x-3y& =-3\\+\\\underline{-3x-9y&=-9}\\-12y&=-12\\\end{aligned}\\\\\\y = \dfrac{-12}{-12} = 1\\\\[/tex]

Substitute y = 1 in the first equation:
[tex]-3x - 9 \cdot 1 = -9\\ \\-3x - 9 = -9\\\\-3x = -9 + 9 \text{ (add -9 to both sides)}\\\\-3x = 0\\\\x = 0\\[/tex]

The result of 3.95 x 10² ÷ 1.5 x 10⁶ is 0.00263 which when converted in proper decimal notation is written as 2.63 x [tex]10^{-3[/tex]

To solve the given equation, we convert the given decimal notation into normal numerals:

To do this we multiply the number by 10 the times the power of 10 or shift the decimal to the right by the power of 10 if the sign of exponent is positive or to the left if the sign of exponent is negative

and we get the equation as 395 ÷ 150000

By solving the above equation, we get:

= 0.00263

In proper decimal notation, we get 2.63 x [tex]10^{-3[/tex]

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For a 90 degrees clockwise rotation,

A' = (y, -x) = (1, -2)

B' = (y, -x) = (3, -3)

C' = (y, -x) = (2, -5)

To plot a triangle onto a coordinate plane, these instructions must be followed:

Begin by drawing x and y axes to establish the necessary framework.

Choose three points upon which to place the vertices of the triangle on the graph.

Then connect the selected points through straight lines, thus resulting in the appearance of three sides; representing each point's distance from one another respectively.

Subsequently attach letter designations such as A, B, and C to each vertex.

Lastly, inspect the measurements of the sides and angles between them to confirm that they correspond with requisites specific to your chosen triangle type (such as an equilateral or Isosceles shape).

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The first step in finding the maximum likelihood estimator for this distribution is to write the likelihood function, which is the joint probability density function of the sample. For a random sample of size n, this is given by:

L(θ | y1, y2, ..., yn) = f(y1 | θ) × f(y2 | θ) × ... × f(yn | θ)

where θ is the parameter(s) of the distribution.

In this case, the parameter of interest is not explicitly stated, but based on the given probability density function f(y), we can identify that it is the probability of success p, where success is defined as the event that Y takes on a value between 0 and 1. This probability is given by:

p = P(0 ≤ Y ≤ 1) = ∫₀¹ f(y) dy

We can simplify this integral by using the Beta function, which is defined as:

B(a, b) = ∫₀¹ x^(a-1) (1-x)^(b-1) dx

Substituting in the values of a and b, we get:

B(5, 4) = ∫₀¹ y^4 (1-y)^3 dy

Therefore, we can express the probability of success as:

p = B(5, 4) = 280/429

Now we can write the likelihood function as:

L(p | y1, y2, ..., yn) = ∏ᵢ f(yᵢ | p) = ∏ᵢ (280yᵢ^4(1 - yᵢ)^3)

Taking the natural logarithm of the likelihood function, we get:

log L(p | y1, y2, ..., yn) = ∑ᵢ [log 280 + 4 log yᵢ + 3 log(1 - yᵢ)]

To find the maximum likelihood estimator for p, we need to differentiate the log likelihood function with respect to p and set the result equal to zero:

d/dp log L(p | y1, y2, ..., yn) = 0

Since p appears only in the expression B(5, 4), we can substitute in the value we previously derived:

log L(p | y1, y2, ..., yn) = ∑ᵢ [log(280/429) + 4 log yᵢ + 3 log(1 - yᵢ)]

d/dp log L(p | y1, y2, ..., yn) = 0

Simplifying this expression, we get:

∑ᵢ [(4/yᵢ) - (3/(1-yᵢ))] = 0

Multiplying both sides by p = 280/429, we get:

∑ᵢ [(4p/yᵢ) - (3p/(1-yᵢ))] = 0

This equation does not have a closed-form solution for p, so we need to use numerical methods to find an approximate solution. One common method is to use an iterative algorithm, such as Newton-Raphson, to update our estimate of p based on the derivative of the log likelihood function. We start with an initial guess for p, and then repeat the following steps until convergence:

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Answer:

There are 8 apples in basket

The minimum value of the function f(x,y) subject to the constraint 2x + 10y = 5 is:

f(5/6, 1/12) = 4(5/6)^2 + 5(1/12)^2 = 13/9

So, fmin = 13/9.

We can use the method of Lagrange multipliers to find the minimum value of the function f(x,y) subject to the constraint 2x + 10y = 5.

First, we define the Lagrangian function L(x,y,λ) as:

L(x,y,λ) = f(x,y) - λ(g(x,y))

where g(x,y) is the constraint equation and λ is the Lagrange multiplier.

In this case, we have:

f(x,y) = 4x^2 + 5y^2

g(x,y) = 2x + 10y - 5

So, the Lagrangian function becomes:

L(x,y,λ) = 4x^2 + 5y^2 - λ(2x + 10y - 5)

Next, we find the partial derivatives of L with respect to x, y, and λ, and set them equal to zero to find the critical points:

∂L/∂x = 8x - 2λ = 0

∂L/∂y = 10y - 10λ = 0

∂L/∂λ = 5 - 2x - 10y = 0

Solving these equations simultaneously, we get:

x = 5/6

y = 1/12

λ = 5/12

These values represent a critical point of the Lagrangian function, and we need to determine whether this critical point corresponds to a minimum, maximum, or saddle point.

To do this, we need to find the second partial derivatives of L with respect to x and y:

∂^2L/∂x^2 = 8

∂^2L/∂y^2 = 10

The determinant of the Hessian matrix is:

∂^2L/∂x^2 * ∂^2L/∂y^2 - (∂^2L/∂x∂y)^2 = (8)(10) - (0)^2 = 80

Since the determinant is positive and ∂^2L/∂x^2 is positive, we can conclude that the critical point (5/6, 1/12) corresponds to a minimum of the Lagrangian function.

Therefore, the minimum value of the function f(x,y) subject to the constraint 2x + 10y = 5 is:

f(5/6, 1/12) = 4(5/6)^2 + 5(1/12)^2 = 13/9

So, fmin = 13/9.

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Answer:

D

Step-by-step explanation:

it should be D because the others seem too big or too small

The perimeter of the shaded region is 39.71 units approximately to the nearest hundredth using the arc length of each sector

Arc length = (central angle / 360) x (2 x π x radius)

central angle = 120°

radius = 5/2 = 2.5

Arc length of a sector = (120°/360º) × 2 × 22/7 × 2.5

Arc length of a sector = 5.2381

Arc length of the three sector = 3 × 5.2381

Arc length of the three sector = 15.7143

perimeter of the shaded region = (3 ×5) + 15.7143

perimeter of the shaded region = 30.7143

Therefore, perimeter of the shaded region is 39.71 units approximately to the nearest hundredth using the arc length of each sector

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After answering the presented question, we may conclude that  So, the expressions correct answer is option B: -2x + 1.

In mathematics, you can multiply, divide, add, or subtract. An expression is constructed as follows: Number, expression, and mathematical operator A mathematical expression (such as addition, subtraction, multiplication, or division) is made up of numbers, variables, and functions. It is possible to contrast expressions and phrases. An expression or algebraic expression is any mathematical statement that has variables, integers, and an arithmetic operation between them. For example, the phrase 4m + 5 has the terms 4m and 5, as well as the provided expression's variable m, all separated by the arithmetic sign +.

expression that is equivalent to 1/3 (9-6x+12),

9 - 6x + 12 = 21 - 6x

1/3 (21 - 6x) = (1/3) * 21 - (1/3) * 6x = 7 - 2x

Therefore, the expression that is equivalent to 1/3 (9-6x+12) is:

7 - 2x

So, the correct answer is option B: -2x + 1.

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The amount of salt in the tank at any time is given by the expression S(t) = 6(1 - [tex](\frac{1}{2} )^t[/tex] ), where t is time in minutes. After 56 minutes, there are approximately 5.18 grams of salt present.

To find an expression for the amount of salt in the tank at any time, we need to consider the rate at which the salt concentration is changing. Since pure alcohol is flowing in at 2 liters/min and the solution is flowing out at 1 liter/min, the tank's volume remains constant at 112 liters.

The concentration of salt decreases by half for every doubling of the volume of the solution. Thus, the amount of salt after t minutes is given by:

S(t) = 6(1 -  [tex](\frac{1}{2} )^t[/tex] )

To find the amount of salt present after 56 minutes, plug t = 56 into the equation:

S(56) = 6(1 - (1/2)⁵⁶)

S(56) ≈ 5.18 grams of salt

Therefore, after 56 minutes, there are approximately 5.18 grams of salt in the tank.

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There are 216 possible outcomes when you roll 3 dice in this game of chance.

In this game, players win if they roll triples, meaning all three dice display the same number. To find out how many possible outcomes there are when you roll 3 dice, follow these steps:
Step:1. Determine the number of sides on a die. A standard die has 6 sides, each with a different number (1-6).
Step:2. Calculate the total possible outcomes for each die. Since there are 6 sides on a die, there are 6 possible outcomes for each die.
Step:3. Multiply the possible outcomes of each die together. In this case, that would be 6 (for the first die) * 6 (for the second die) * 6 (for the third die). 6 * 6 * 6 = 216
So, there are 216 possible outcomes when you roll 3 dice in this game of chance.

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For squares, a quadrilateral, the line segments connecting the halfway points of opposite sides are perpendicular.

Line segments refer to lines joining two endpoints. It has a fixed length and a definite length, unlike ray and line.

A line is said to be perpendicular to another line if the two lines intersect at a right angle. It is represented by ⊥.

A quadrilateral is a 2-dimensional shape that has four sides and four angles. Examples include squares, rectangles, and so on.

The quadrilaterals Square and Rectangle are such that the line segments connecting the halfway points of opposite sides are perpendicular that is the angle of intersection is of the magnitude of 90°

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The speed s(t) of the particle with the given trajectory at t = 14 is 166.13 m/s.

A trajectory is a path or an orbit that an object follows. It is the path that a moving object follows through space and time.

The speed s(t) of a particle with a given trajectory at a given time t is equal to the magnitude of the velocity vector. The velocity vector can be calculated by taking the first derivative of the position vector c(t).

Taking the derivative of c(t) with respect to t yields:

c'(t) = (2t / (t² + 1), 3t²).

The magnitude of c'(t) is equal to the speed of the particle at time t and is given by the following equation:

s(t) = √(4t² / (t² + 1) + 9t⁴).

Substituting t = 14 into the equation above yields:

s(14) = √(4*14² / (14² + 1) + 9*14⁴)

   = √(2176 / 15 + 27456)

   = √(27601)

   = 166.13 m/s.

Therefore, the speed s(t) of the particle with the given trajectory at t = 14 is 166.13 m/s.

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