let f (x) = 9sin(x) for 0 ≤ x ≤ 2 . find lf (p) and uf (p) (to the nearest thousandth) for f and the partition p = 0, 6 , 4 , 3 , 2 .

Answer:

562 bags.

Step-by-step explanation:

8,430 divided by 15 is 562.

In triangle ΔGHI, with ∠I measuring 90° and ∠G measuring 82°, and GH measuring 3.4 feet, the length of HI is 24.2 feet.

To find the length of HI, we can use the trigonometric function tangent (tan). In a right triangle, the tangent of an angle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to it. In this case, the side opposite ∠G is HI, and the side adjacent to ∠G is GH. Therefore, we can set up the equation: tan(82°) = HI / GH.

Rearranging the equation to solve for HI, we have: HI = GH * tan(82°). Plugging in the given values, we get: HI = 3.4 * tan(82°). Using a calculator, we find that tan(82°) is approximately 7.115. Multiplying 3.4 by 7.115, we find that HI is approximately 24.161 feet. Rounded to the nearest tenth of a foot, the length of HI is 24.2 feet.

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Answer:

P(divisor of 70) = 1/2

Step-by-step explanation:

P(divisor of 70) means what is the probability that the role results in a divisor of 70.

The divisors of 70 are: 1, 2, 5, 7, 10, 14, 35, 70

Since 1,2, and 5 are the only ones that can actually be rolled on a 6-sided die, there is a [tex]\frac{3}{6}[/tex]  or [tex]\frac{1}{2}[/tex] chance to roll a divisor of 70.

Answer:  5%

Step-by-step explanation:

(1), (2), 3, 4, (5), 6,

70/1 = 70.                 ( these are integers)

70/2 = 35

70/5 = 14

3 over 6 = 1 over 2 = 50%

hope this helps!!                                                                                                                                                                                      

Factors related to cause and effect is not one of the things the relative frequency (Rf) of z-scores allows us to calculate for corresponding raw scores. The correct answer is a) Factors related to cause and effect.

The relative frequency (Rf) of z-scores is a statistical tool that calculates the probability of obtaining a certain raw score or a score more extreme than that. It allows for inferences to be made about the population from which the sample was drawn and for comparisons to be made between variables. However, Rf does not provide information on factors related to cause and effect, as it cannot establish cause-and-effect relationships between variables. It is useful in analyzing data in the context of a normal distribution and calculating the frequency of occurrence of certain scores in a given population. Therefore the correct answer is a) Factors related to cause and effect.

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We get two different answers for fg depending on the method used to compute the convolution. Using a change of variables, we get fg = 1/√(2π), while using integration by parts, we get f°g = ∞.

Since both functions f!(t) and g(t) are equal to h(t), their convolution f°g can be computed as follows:

f°g = ∫[0,∞] f(τ)g(t-τ) dτ

= ∫[0,∞] h(τ)h(t-τ) dτ

Method 1: Change of Variables

To compute the convolution using a change of variables, let u = t' and v = t - t'. Then, τ = u and t = u + v, and we have:

f°g = ∫∫[D] h(u)h(u+v) dudv

where D is the region of integration corresponding to the domain of u and v. Since the limits of integration are 0 to ∞ for both u and v, we can write:

f°g = ∫[0,∞] ∫[0,∞] h(u)h(u+v) dudv

Using the convolution theorem, we know that f°g is equal to the Fourier transform of H(f), where H(f) is the Fourier transform of h(t). Since h(t) is a constant function, H(f) is a Dirac delta function, given by:

H(f) = 1/√(2π) δ(f)

where δ(f) is the Dirac delta function. Therefore, we have:

f°g = Fourier^-1{H(f)} = Fourier^-1{1/√(2π) δ(f)} = 1/√(2π)

Method 2: Integration by Parts

To compute the convolution using integration by parts, we have:

f°g = ∫[0,∞] h(τ)h(t-τ) dτ

= h(t) ∫[0,∞] h(τ-t) dτ (using a change of variables)

= h(t) ∫[0,∞] h(u) du (since h is a constant function)

= h(t) [u]0^∞

= h(t) [∞ - 0]

= ∞

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To compute the approximate elasticity of demand, we can use the formula:

Elasticity of Demand = [(Q2 - Q1) / ((Q2 + Q1) / 2)] / [(P2 - P1) / ((P2 + P1) / 2)]

Given the following data:

Initial situation:

Price (P1) = $23

Quantity (Q1) = 11.5

New situation:

Price (P2) = $20

Quantity (Q2) = 13.5

Using the formula, we can calculate the approximate elasticity of demand:

Elasticity of Demand = [(13.5 - 11.5) / ((13.5 + 11.5) / 2)] / [(20 - 23) / ((20 + 23) / 2)]

Elasticity of Demand = [(2) / (12.5)] / [(-3) / (21.5)]

Elasticity of Demand = (2/12.5) * (-21.5/3)

Elasticity of Demand = -0.34

Therefore, the approximate elasticity of demand is approximately -0.34.

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The equilibrium point (1,1) in the system x′ = 2(x − y)y, y′ = xy - 2 is a(n) stable spiral.

To determine the type of equilibrium point, we first linearize the system around the point (1,1) by finding the Jacobian matrix:

J(x,y) = | ∂x′/∂x  ∂x′/∂y | = |  2y     -2y  |
        | ∂y′/∂x  ∂y′/∂y |    |  y      x   |

Evaluate the Jacobian at the equilibrium point (1,1):

J(1,1) = |  2  -2 |
        |  1   1  |

Next, find the eigenvalues of the Jacobian matrix. The characteristic equation is:

(2 - λ)(1 - λ) - (-2)(1) = λ² - 3λ + 4 = 0

Solve for the eigenvalues:

λ₁ = (3 + √7i)/2, λ₂ = (3 - √7i)/2

Since the eigenvalues have positive real parts and nonzero imaginary parts, the equilibrium point at (1,1) is a stable spiral. This means that trajectories near the point spiral towards it over time.

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With a mean of 20 inches and a standard deviation of 2.6 inches, the probability can be calculated as P(z > 0), which is approximately 0.5.

To find the probability that a given infant is longer than 20 inches, we need to use the normal distribution. The given information provides the mean (20 inches) and the standard deviation (2.6 inches) of the length of newborn babies.

In order to calculate the probability, we need to convert the value of 20 inches into a standardized z-score. The z-score formula is given by (x - μ) / σ, where x is the observed value, μ is the mean, and σ is the standard deviation.

Substituting the given values, we get (20 - 20) / 2.6 = 0.

Next, we find the area under the normal curve to the right of the z-score of 0. This represents the probability that a given infant is longer than 20 inches.

Using a standard normal distribution table or a calculator, we find that the area to the right of 0 is approximately 0.5.

Therefore, the probability that a given infant is longer than 20 inches is approximately 0.5, or 50%.

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To derive the trigonometric Fourier series from the complex exponential series, we can start with the complex exponential Fourier series: The trigonometric Fourier series is f(x) = a0/2 + Σ[cn e^(inx)]

where cn = (an - ibn)/2.

f(x) = a0/2 + Σ(an cos(nx) + bn sin(nx))

where a0/2 is the average value of f(x), and an and bn are the Fourier coefficients given by:

an = (1/π) ∫f(x)cos(nx)dx

bn = (1/π) ∫f(x)sin(nx)dx

We can rewrite the trigonometric terms in terms of complex exponentials as follows:

cos(nx) = (e^(inx) + e^(-inx))/2

sin(nx) = (e^(inx) - e^(-inx))/(2i)

Substituting these expressions into the complex exponential Fourier series, we get:

f(x) = a0/2 + Σ[(an + ibn)(e^(inx) + e^(-inx))/2]

where ibn = bn/i.

We can simplify this expression as follows:

f(x) = a0/2 + Σ[cn e^(inx)]

where cn = (an - ibn)/2.

This is the trigonometric Fourier series, which expresses the function f(x) as a sum of complex exponential terms with real coefficients. We can write this more explicitly as:

f(x) = a0/2 + Σ[cn (cos(nx) + i sin(nx))]

which is the same as:

f(x) = a0/2 + Σ[cn cos(nx)] + i Σ[cn sin(nx)]

So, to derive the trigonometric Fourier series from the complex exponential series, we simply substitute the complex exponential expressions for cos(nx) and sin(nx), and simplify the resulting expression to obtain the coefficients cn.

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there is no equations

Hey There!

Step-by-step explanation:

Which of the matrices are equal?

Two matrices are said to be equal if: Both the matrices are of the same order i.e., they have the same number of rows and columns A m × n = B m × n .

Rounding to three decimal places, the probability is approximately 0.007.

To find the probability that the first two adults get news from television and the third gets news from the newspaper, we need to use the multiplication rule for independent events.
The probability of selecting an adult who gets news from television on the first draw is 13/75, since there are 13 adults who get news from television out of a total of 75 adults.
Assuming the first draw is an adult who gets news from television, there are now 12 adults who get news from television out of a total of 74 adults.

So the probability of selecting another adult who gets news from television on the second draw, given that the first draw was an adult who gets news from television, is 12/74.
Assuming the first two draws are adults who get news from television, there are now 14 adults who get news from a newspaper out of a total of 73 adults.

So the probability of selecting an adult who gets news from a newspaper on the third draw, given that the first two draws were adults who get news from television, is 14/73.
Therefore, the probability that the first two adults get news from television and the third gets news from the newspaper is:
(13/75) * (12/74) * (14/73) = 0.0067
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The true inequalities in the number line are:

0 < √0.9, √0.9 < 0.9

√0.9 < 1, 0.9 > √0.9 < 1

To plot √0.9 on the number line, we need to find its approximate value.

√0.9 is between 0 and 1 because 0.9 is greater than 0 but less than 1. However, it is closer to 1 than 0.

So, we can represent √0.9 as a point on the number line between 0 and 1, closer to 1.

Now let's analyze the given inequalities:

0 < √0.9: This inequality is true because √0.9 is greater than 0.

√0.9 < 0.9: This inequality is true because √0.9 is less than 0.9.

√0.9 < 1: This inequality is true because √0.9 is less than 1.

√0.9 > √1: This inequality is false because √0.9 is less than √1.

0.9 > √0.9 < 1: This inequality is true because √0.9 is less than 1 and greater than 0.9.

So, the true inequalities are:

0 < √0.9

√0.9 < 0.9

√0.9 < 1

0.9 > √0.9 < 1

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Answer:

v

Step-by-step explanation:

Any permutation can be achieved by performing a series of transpositions, where you repeatedly swap elements until all objects are in their correct positions.

A permutation is an arrangement of n objects in a specific order, while a transposition is a simple swap of two elements in a permutation.

To show that any permutation can be achieved by a product of transpositions, let's follow these steps:

Step 1: Consider a permutation of n objects, where at least one element is not in its desired position.

Step 2: Identify the first element that is not in its correct position. This element should be at position i but should be in position j.

Step 3: Perform a transposition by swapping the element in position i with the element in position j. Now, the element that was originally in position i is in its correct position.

Step 4: Repeat steps 2 and 3 for the remaining n-1 objects, excluding the element that has been placed in its correct position.

Step 5: Continue this process until all elements are in their correct positions. At this point, you have achieved the desired permutation by performing a series of transpositions (swaps).

In summary, any permutation can be achieved by performing a series of transpositions, where you repeatedly swap elements until all objects are in their correct positions.

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The limit exists, and the limit of the function as (x, y)→(0, 0) is 0.

To find the limit of the given function as (x, y)→(0, 0), we need to consider the function and the terms you mentioned, "limit" and "exists."

The given function is:

f(x, y) = [tex](x^2 * y^2) / (x^2 * y^2 + 16 - 4)[/tex]

We want to find the limit as (x, y)→(0, 0):

lim (x, y)→(0, 0) f(x, y)

Step 1: Check if the function is continuous at (0,0)

When x = 0 and y = 0:

f(0, 0) = [tex](0^2 * 0^2) / (0^2 * 0^2 + 16 - 4)[/tex]

f(0, 0) = 0 / (0 + 12)

f(0, 0) = 0

Since the function is defined at (0, 0), it is continuous at this point.

Step 2: Analyze the limit

As (x, y) approach (0, 0), the numerator [tex](x^2 * y^2)[/tex] also approaches 0. The denominator [tex](x^2 * y^2 + 16 - 4)[/tex]approaches 12. Thus, we have:

lim (x, y)→(0, 0) f(x, y) = 0 / 12 = 0

So, the limit exists, and the limit of the function as (x, y)→(0, 0) is 0.

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A pile of 55 keyboard boxes will be approximately 1983.08 cm tall.

To determine the total height of a pile of 55 keyboard boxes, we need to first calculate the height of a single box and then multiply it by 55.

Given that a single box is 3/2 cm thick, we need to know the dimensions of the box to calculate its height. If we assume that the box has a standard width and length of, say, 30 cm and 20 cm respectively, we can calculate its height using the Pythagorean theorem.

The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. In this case, the hypotenuse is the height of the box, and the other two sides are the width and length.

So, we have:

height^2 = width^2 + length^2

height^2 = 30^2 + 20^2

height^2 = 900 + 400

height^2 = 1300

height = sqrt(1300)

height = 36.0555... cm (rounded to 3 decimal places)

Therefore, the height of a single keyboard box is approximately 36.056 cm.

To find the height of a pile of 55 keyboard boxes, we can simply multiply the height of a single box by 55:

height of pile = 36.056 cm x 55

height of pile = 1983.08 cm

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Imani's net pay decreased by approximately 7.7% when she increased her 401k contributions, resulting in a decrease of $49.00 from her initial net pay of $637.00.

To determine the percent that Imani's net pay was decreased, we need to find the difference between her initial net pay and her net pay after increasing her 401k contributions, and then calculate that difference as a percentage of her initial net pay.

Let's denote the initial net pay as A and the net pay after increasing the 401k contributions as B.

A = $637.00 (initial net pay)

B = $588.00 (net pay after increasing 401k contributions)

The decrease in net pay can be calculated by subtracting B from A:

Decrease = A - B = $637.00 - $588.00 = $49.00

Now, to find the percentage decrease, we divide the decrease by the initial net pay (A) and multiply by 100:

Percentage Decrease = (Decrease / A) * 100 = ($49.00 / $637.00) * 100 ≈ 7.68%

Therefore, the percent that Imani's net pay was decreased, rounded to the nearest tenth of a percent, is approximately 7.7%.

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Therefore, x and y for the indefinite integral are not independent, even though they are uncorrelated.

To show that x and y are uncorrelated, we need to compute their indefinite integraland show that it is zero:

Cov(x, y) = E(xy) - E(x)E(y)

We can compute E(x) and E(y) as follows:

E(x) = E(cos) = ∫(cos*f( )d ) = ∫(cos(1/2π)*d ) = 0

E(y) = E(sin) = ∫(sin*f( )d ) = ∫(sin(1/2π)*d ) = 0

where f( ) is the probability density function of , which is a uniform distribution over the range (0, 2π).

Next, we compute E(xy):

E(xy) = E(cossin) = ∫(cossinf( )d ) = ∫(cossin(1/2π)*d )

Since cos*sin is an odd function, we have:

∫(cossin(1/2π)*d ) = 0

Therefore, Cov(x, y) = E(xy) - E(x)E(y) = 0 - 0*0 = 0.

Hence, x and y are uncorrelated.

To show that x and y are not independent, we need to find P(x, y) and show that it does not factorize into P(x)P(y):

P(x, y) = P(cos, sin) = P( ) = (1/2π)

Since P(x, y) is constant over the entire range of (cos, sin), we can see that P(x, y) does not depend on either x or y, i.e., it does not factorize into P(x)P(y).

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The optimal solution of the given linear program is (x, y) = (2, 1).

How to solve linear programming problems?

To solve the linear program, we first plot the feasible region determined by the constraints:

We can rewrite the second constraint as y >= (4 - Ax)/2.

Next, we plot the lines 3x - 4y = 12 and Ax + 2y = 4 - 2x and shade the appropriate regions:

We can see that the feasible region is bounded, so we can find the optimal solution by evaluating the objective function C at each of the corner points of the feasible region.

The corner points are:

Evaluating C at each corner point, we get:

Thus, the optimal solution is at (2, 1) with a minimum value of C = 11.

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x = 5pi/6 is not in the interval [0, pi/2]

Starting with the given equation:

4 cos x - 8 sin x cos x = 0

We can factor out 4 cos x:

4 cos x (1 - 2 sin x) = 0

So either cos x = 0 or (1 - 2 sin x) = 0.

If cos x = 0, then x = pi/2 since we're only considering the interval [0, pi/2].

If 1 - 2 sin x = 0, then sin x = 1/2, which means x = pi/6 or x = 5pi/6 in the interval [0, pi/2].

So the two solutions in the interval [0, pi/2] are x = pi/2 and x = pi/6.

That x = 5pi/6 is not in the interval [0, pi/2].

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The given equation is 4 cos x - 8 sin x cos x = 0. To find the solutions in the interval [0, pi/2], we need to solve for x.
Find the solutions within the given interval. Equation: 4 cos x - 8 sin x cos x = 0

First, let's factor out the common term, which is cos x:

cos x (4 - 8 sin x) = 0

Now, we have two cases to find the solutions:

Case 1: cos x = 0
In the interval [0, π/2], cos x is never equal to 0, so there is no solution for this case.

Case 2: 4 - 8 sin x = 0
Now, we'll solve for sin x:

8 sin x = 4
sin x = 4/8
sin x = 1/2

We know that in the interval [0, π/2], sin x = 1/2 has one solution, which is x = π/6.

So, in the given interval [0, π/2], the equation has only one solution: x = π/6.

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Therefore, the MLE of θ is θ = -(1/n) ∑ln(x_i). Therefore, θ is an unbiased estimator of θ.

(a) To find the maximum likelihood estimator (MLE) of θ, we first write the likelihood function as follows:

L(θ|x_1, x_2, ..., x_n) = ∏(i=1 to n) f(x_i; θ)

= ∏(i=1 to n) [(1/θ)x_i(1-θ)/θ]

= (1/θ^n) ∏(i=1 to n) x_i(1-θ)

Taking the natural logarithm of L(θ|x_1, x_2, ..., x_n), we have:

ln(L(θ|x_1, x_2, ..., x_n)) = -n ln(θ) + (1-θ) ∑ln(x_i)

To find the MLE of θ, we differentiate ln(L(θ|x_1, x_2, ..., x_n)) with respect to θ and set the derivative to zero:

d/dθ ln(L(θ|x_1, x_2, ..., x_n)) = -n/θ + ∑ln(x_i) = 0

Solving for θ, we get:

θ = -(1/n) ∑ln(x_i)

(b) To show that θ is an unbiased estimator of θ, we need to find its expected value:

E(θ) = E[-(1/n) ∑ln(x_i)]

= -(1/n) ∑E[ln(x_i)]

= -(1/n) ∑[∫0^1 ln(x_i) (1/θ)x_i(1-θ)/θ dx_i]

= -(1/n) ∑[∫0^1 (1/θ)ln(x_i)x_i(1-θ) d(x_i)]

= -(1/n) ∑[θ(-1/(θ^2))(1/2)ln(x_i)^2|0^1 + (1/θ)(1/2)x_i^2(1-θ)|0^1]

= -(1/n) ∑[(1/2θ)ln(x_i)^2 - (1/2θ)x_i^2(θ-1)]

= -(1/n) [(1/2θ)∑ln(x_i)^2 - (1/2θ)(θ-1)∑x_i^2]

Note that ∑ln(x_i)^2 and ∑x_i^2 are constants with respect to θ. Therefore, we have:

E(θ) = -(1/n) [(1/2θ)∑ln(x_i)^2 - (1/2θ)(θ-1)∑x_i^2]

= (1/2) - (1/2nθ)

Since E(θ) = θ, we have:

θ = (1/2) - (1/2nθ)

Solving for θ, we get:

θ = 1/(n+2)

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To set up a triple integral for the volume of the solid in the first octant bounded by the coordinate planes and the plane z = 8 − x − y, we need to break down the solid into its boundaries and express them in terms of the limits of integration for the triple integral.

Since the solid is in the first octant, all three coordinates (x, y, z) are positive. Therefore, the boundaries for the solid are: 0 ≤ x ≤ ∞ (bounded by the x-axis and the plane x = ∞)
0 ≤ y ≤ ∞ (bounded by the y-axis and the plane y = ∞)
0 ≤ z ≤ 8 − x − y (bounded by the plane z = 8 − x − y)
Thus, the triple integral for the volume of the solid can be expressed as:
∫∫∫ E dz dy dx
where E is the region in xyz-space defined by the boundaries above.
Therefore, ∫∫∫ E dz dy dx = ∫0^∞ ∫0^(∞-x) ∫0^(8-x-y) dz dy dx
This triple integral represents the volume of the solid in the first octant bounded by the coordinate planes and the plane z = 8 − x − y. However, we have not evaluated the integral yet, so we cannot find the actual value of the volume.

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Answer: Oui, vous pouvez le factoriser parfaitement

Step-by-step explanation:

x^2 + 8x + 64

ajouter et soustraire (b/2a)^2

x^2+8x+64+16-16

factoriser le trinôme carré parfait : x^2 + 8x + 16

(x+4)^2 + 64 - 16

réponse finale:

48 + (x+4)^2

The value of g of the given triangular prism is: 3.5 meters

The formula for calculating the Volume of a triangular prism is expressed as the area of the base times it's height. Thus:

Volume = Base area * height

We are given that the volume is 140 cubic meters.

Thus,

140 = (10 * g) * 4

because we are given one of the base length as 10 and the height as 4 m. Thus:

40g = 140

g = 140/40

g = 3.5 meters

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The steps are explained below.

Factorization of the given polynomial to find the product is as follows;

(x² + 3x + 2)/(x² + 6x + 5) = (x + 1)(x + 2)/(x + 1)(x + 5)

(x² + 7x + 10)/(x² + 4x + 4) = (x + 2)(x + 5)/(x + 2)(x + 2)

Expressing the product in terms of the factors

(x² + 3x + 2)/(x ^ 2 + 6x + 5) × (x² + 7x + 10)/(x² + 4x + 4) = (x + 2)(x + 5)/(x + 2)(x + 2) × (x + 1)(x + 2)/(x + 1)(x + 5)

The steps arranged in the order in which they would be performed are;

First step;

(x² + 3x + 2)/(x² + 6x + 5) × (x² + 7x + 10)/(x² + 4x + 4)

Second step (factorizing) =

(x + 1)(x + 2)/(x + 1)(x + 5) × (x + 2)(x + 5)/(x + 2)(x + 2)

Third step (dividing out common terms) =

(x+5)/(x+2) × (x+2)/(x+5)

Fourth step (rearranging and removing terms that cancel each other)

= 1

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Answer:

Step-by-step explanation:

32.00 increased by 25% is 40.00

The increase is 8.00

Using the formula of conditional probability, the probability that a student buys lunch given that they ride the bus is approximately 81.25%. So,  81.25% is the right answer.

To find the probability that a student buys lunch given that they ride the bus, we can use conditional probability.

Let's denote the following events:

A: Student buys lunch

B: Student rides the bus

We are given:

P(B) = 80% = 0.80 (probability that a student rides the bus)

P(A) = 75% = 0.75 (probability that a student buys lunch)

P(A|B) = 65% = 0.65 (probability that a student buys lunch given that they ride the bus)

Using the concept of conditional probability

Probability of a student buying lunch and riding the bus = 65%

Probability of a student riding the bus = 80%

Probability of a student buying lunch given that they ride the bus = (Probability of a student buying lunch and riding the bus) / (Probability of a student riding the bus) = 65% / 80% = 0.8125 = 81.25%

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The integral ∫csc^2(x) dx can be rewritten in terms of g(x) as F(g(x)) - 2/7 ∫csc(g(x)/7) cot(g(x)/7) dx, where F(g(x)) is the antiderivative of csc^2(g(x)/7).

Let's start by substituting g(x) into the function f(x):

f(g(x)) = csc^2(g(x)/7)

Next, we can use the chain rule to find the derivative of f(g(x)):

f'(g(x)) = -2csc(g(x)/7) cot(g(x)/7) / 7

Using the substitution u = g(x), we can rewrite the integral in terms of g(x) as follows:

∫csc^2(x) dx = ∫f(g(x)) dx = ∫f(u) du = F(u) + C

Substituting back in for u, we get:

∫csc^2(x) dx = F(g(x)) + C

Using the derivative of f(g(x)) that we found earlier, we can substitute it into the integral:

∫csc^2(x) dx = -2/7 ∫csc(g(x)/7) cot(g(x)/7) dx

Therefore, the integral in terms of g(x) and the antiderivative F(g(x)) is:

∫csc^2(x) dx = F(g(x)) - 2/7 ∫csc(g(x)/7) cot(g(x)/7) dx

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By substituting u = x²   and using the appropriate differential, the integral can be transformed into 3∫(669 sin(u)) du, which can be further evaluated.

The given expression, 6∫(669x sin(x²)) dx, can be simplified using the substitution u = x² and 2x dx = du, which implies that x dx = (1/2) du. By substituting these values, the integral becomes 6∫(1/2)(669 sin(u)) du.

When x = 0, u = 0, and when x = 4, u = 16.

Thus, the integral can be rewritten as 6(1/2) ∫(669 sin(u)) du from 0 to 16.

Simplifying further, we get 3∫(669 sin(u)) du from 0 to 16, which evaluates to 3[-669 cos(u)] from 0 to 16, resulting in a final answer of -669[cos(16) - cos(0)].

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The correct equation is,

⇒ 8(x + 6) = 144

Now, We can start building this equation by making everything equal to 144 since the problem is representing the total number of apples:

? = 144

Next, we don't know how many apples are in each basket, so we can represent it with a variable, x.

Since 6 apples are added to each basket we will simply add 6 to the "x" amount of apples in each basket:

x + 6 = 144

Lastly, according to the scenario, we have 8 baskets, each holding "x" amount of apples plus the extra 6 that was added, so it will be multiplied:

8(x + 6) = 144

Thus, The correct equation is,

8(x + 6) = 144

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