consider the equation
d2t/dx2 - 10^-7(t-273)4+4(150-t)=0
subject to boundary conditions T(0) = 200 and T(0.5) = 100. This equation represents the
temperature distribution T along a rod of length 0.5 subject to convective and radiative heat
transfer. Solve this equation using the shooting method and plot the results (Temperature vs x)

Two events are

Whether the page number was even.

Whether the page number was odd.

A number that can be divided by two without leaving a remainder is an even number. Even numbers, in the context of book pages, are those that begin with 0, 2, 4, 6, or 8. For instance, page numbers 10, 12, and 14 are even numbers.

An odd number, on the other hand, is one that cannot be divided by 2 without leaving a remainder. Odd numbers, in the context of book pages, are those that begin with 1, 3, 5, 7, or 9. Page numbers 9, 11, and 13 are examples of odd numbers.

Mina is gathering information regarding the frequency of each event by noting whether the page numbers are even or odd. After that, this data can be used to figure out probabilities and make predictions about what will happen in the future, like whether the next page will be even or odd.

Mina opened the book 15 times to determine whether the page number was even or odd, so she conducted 15 trials.

She documented the following two events:

Whether the page number was even.

Whether the page number was odd.

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The arc length by the given data is 4π cm.

We are given that;

AC≅AD,  ∠C =∠D = 30°

Now,

The arc length formula is:

s = rθ

where s is the arc length, r is the radius, and θ is the central angle in radians.

To use this formula, we need to convert the angle of 120° to radians. We can use the fact that 180° = π radians, so:

120° × π/180° = 2π/3 radians

Then we can plug in the values of r = 6 cm and θ = 2π/3 radians into the formula:

s = 6 × 2π/3 s = 4π cm

Therefore, by the given angle the answer will be 4π cm.

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The slope of the line tangent to the polar curve r = 9 + 3cosθ at (12, 0) is 0 (choice A).
The slope of the line tangent to the polar curve r = 9 + 3cosθ at (6, π) is 0 (choice A).

To find the slope of the line tangent to the polar curve r = 9 + 3cosθ at the given points (12, 0) and (6, π):

We'll first find the derivative dr/dθ and then use the formula for the slope of a tangent line in polar coordinates:

dy/dx = (r(dr/dθ) + dr/dθcosθ)/(r - dr/dθsinθ).
Step 1: Find dr/dθ.
r = 9 + 3cosθ
dr/dθ = -3sinθ
Step 2: Compute dy/dx for each point.
For (12, 0):
r = 12, θ = 0
dy/dx = (12(-3sin0) + (-3sin0)cos0)/(12 - (-3sin0)sin0)
dy/dx = (0 + 0)/(12 - 0) = 0
So the slope at (12, 0) is 0, which corresponds to choice A in your question.
For (6, π):
r = 6, θ = π
dy/dx = (6(-3sinπ) + (-3sinπ)cosπ)/(6 - (-3sinπ)sinπ)
dy/dx = (0 - 0)/(6 - 0) = 0
So the slope at (6, π) is 0, which corresponds to choice A in your question.
In summary:
The slope of the line tangent to the polar curve r = 9 + 3cosθ at (12, 0) is 0 (choice A).
The slope of the line tangent to the polar curve r = 9 + 3cosθ at (6, π) is 0 (choice A).

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The volume of his barn is calculated as:

= 40,000 cubic feet.

The barn as seen in the image attached below comprises of a rectangular prism and a triangular prism. Therefore:

Volume of the barn = (volume of rectangular prism) + (volume of triangular prism).

Volume of triangular prism = 1/2(base * height) * length of prism

= 1/2(40 * 10) * 50

= 10,000 cubic feet.

Volume of the rectangular prism = length * width * height

= 50 * 40 * 15

= 30,000 cubic feet.

Therefore, volume of his barn = 30,000 + 10,000 = 40,000 cubic feet.

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The derivative of y√(1/(t(8t+1))) is (dy/dt √(1/(t(8t+1))) - y * (4t√t+1 + √(8t+1))/(2t√(t(8t+1)))).

To use logarithmic differentiation to find the derivative of y√(1/(t(8t+1))), we can follow these steps:

ln(y√(1/(t(8t+1)))) = ln(y) + 1/2 ln(1/(t(8t+1)))

d/dt ln(y√(1/(t(8t+1)))) = d/dt [ln(y) + 1/2 ln(1/(t(8t+1)))]

d/dt ln(y√(1/(t(8t+1)))) = (dy/dt √(1/(t(8t+1))) - y * (4t√t+1 + √(8t+1))/(2t√(t(8t+1))))

So, the final expression y√(1/(t(8t+1))) is

(dy/dt √(1/(t(8t+1))) - y * (4t√t+1 + √(8t+1))/(2t√(t(8t+1)))).

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The basis for the orthogonal complement of the row space of the given matrix is (-2, -2, 1).

To find the basis for the orthogonal complement of the row space of matrix A, we can use the fact that the row space and the nullspace of a matrix are orthogonal complements of each other.

So, we first need to find the null space of A. To do this, we can row-reduce A to echelon form and solve the corresponding homogeneous system of linear equations.

RREF(A) =
1 1 4
0 2 -7

This gives us the homogeneous system:

x + y + 4z = 0
2y - 7z = 0

Solving for the free variables, we get:

x = -y - 4z
y = (7/2)z

So, the nullspace of A is spanned by the vector:

v = [-1/2, 7/2, 1]

Now, we can find a basis for the orthogonal complement of the row space of A by taking the orthogonal complement of the span of the rows of A.

The rows of A are:

[1 0 2]
[1 1 4]

We can take the cross-product of these two vectors to get a vector that is orthogonal to both of them:

[0 -2 1]

This vector is also in the orthogonal complement of the row space of A.

Therefore, a basis for the orthogonal complement of the row space of A is [-1/2, 7/2, 1], [0, -2, 1].
Hi! I'd be happy to help you find a basis for the orthogonal complement of the row space of the given matrix. Here's a step-by-step explanation:

1. Write down the given matrix A:
  A = | 1 0 2 |
      | 1 1 4 |

2. To find the orthogonal complement, we first need to find the row space of matrix A. Since there are two linearly independent rows, the row space is spanned by these two rows:

  Row space of A = span{ (1, 0, 2), (1, 1, 4) }

3. Now we need to find a vector that is orthogonal to both of these rows. To do this, we can take the cross-product of two-row row vectors:

  Cross product: (1, 0, 2) x (1, 1, 4) = (-2, -2, 1)

4. The cross product gives us a vector that is orthogonal to both of the rows and therefore lies in the orthogonal complement of the row space of matrix A.

  Orthogonal complement of row space of A = span{ (-2, -2, 1) }

So, the basis for the orthogonal complement of the row space of the given matrix is (-2, -2, 1).

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Answer:

c

Step-by-step explanation:

tysm

a) Poppy's petrol was better value for money as it cost less per liter. b) 20 liters of petrol from the cheaper petrol station (Poppy's petrol station) would cost £26.00.

a) To determine which petrol was better value for money, we need to calculate the price per liter for each petrol station:

Jacob's petrol: £18.90 / 14 litres = £1.35 per litre

Poppy's petrol: £22.10 / 17 litres = £1.30 per litre

Therefore, Poppy's petrol was better value for money as it cost less per litre.

b) To calculate the cost of 20 litres of petrol from the cheaper petrol station, we need to determine which petrol station was cheaper:

Jacob's petrol: £1.35 per litre x 20 litres = £27.00

Poppy's petrol: £1.30 per litre x 20 litres = £26.00

Therefore, 20 litres of petrol from the cheaper petrol station (Poppy's petrol station) would cost £26.00.

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Answer:

  0.0137

Step-by-step explanation:

You want the area under a standard normal probability distribution curve that is not between z = -2.59 and z = 2.37.

The desired area is the complement of the area between the limits -2.59 and 2.37. The value of the desired area is shown by the attached calculator to be about 0.0137.

Part A:

The new deck will be a 4:5 scaled version of the original deck. This means that every dimension of the new deck will be 4/5 times the corresponding dimension of the original deck.

The original deck has a base of 15 feet and a height of 9 feet.

The new deck will have a base of (4/5) * 15 = 12 feet and a height of (4/5) * 9 = 7.2 feet.

Therefore, the dimensions of the new deck are 12 feet for the base and 7.2 feet for the height.

Part B:

To find the area of the original deck, we use the formula for the area of a triangle:

Area = (1/2) * base * height = (1/2) * 15 * 9 = 67.5 square feet.

To find the area of the new deck, we use the same formula with the new dimensions:

Area = (1/2) * 12 * 7.2 = 43.2 square feet.

Therefore, the area of the original deck is 67.5 square feet, and the area of the new deck is 43.2 square feet.

Part C:

The ratio of the areas is:

Area of new deck / Area of original deck = 43.2 / 67.5

Simplifying this fraction, we get:

Area of new deck / Area of original deck = 8 / 15

The scale factor is 4/5, which simplifies to 8/10 or 4/5.

Comparing the ratio of the areas to the scale factor, we see that:

Area ratio / Scale factor = (8/15) / (4/5) = (8/15) * (5/4) = 1

Therefore, the ratio of the areas is equal to the scale factor. This makes sense since the area of a triangle is proportional to the square of its dimensions. In this case, the scale factor is applied to both the base and the height, so the area ratio is equal to the scale factor squared, which is 16/25.

Answer:

Step-by-step explanation:

Part A: To find the dimensions of the new deck, we need to scale the base and height of the original deck by a factor of 4:5.

Scaling factor = 4/5

New base = 15 * (4/5) = 12 feet

New height = 9 * (4/5) = 7.2 feet

Therefore, the dimensions of the new deck are 12 feet for the base and 7.2 feet for the height.

Part B: The area of the original deck can be found by using the formula for the area of a triangle:

Area = (1/2) * base * height = (1/2) * 15 * 9 = 67.5 square feet.

The area of the new deck can also be found using the same formula:

Area = (1/2) * base * height = (1/2) * 12 * 7.2 = 43.2 square feet.

Part C: The ratio of the areas of the two decks can be found by dividing the area of the new deck by the area of the original deck:

Ratio of areas = (43.2 / 67.5) ≈ 0.64

The scale factor is 4:5 or 0.8.

Comparing the ratio of areas to the scale factor:

Ratio of areas / scale factor = (0.64 / 0.8) = 0.8

The ratio of the areas divided by the scale factor is equal to 0.8, which makes sense since the scale factor is the factor by which the dimensions were scaled up, and the ratio of areas tells us how much the area was scaled up.

Answer:

11.2

Step-by-step explanation:

Vector that is perpendicular to the level curve at the point (a, b) is the opposite of the gradient vector:

-∇f(a, b) = ([tex]-e^a[/tex] + 1, a sin b)

We can use the gradient of f(x, y) at that point.

The gradient of f(x, y) is given by:

∇f(x, y) = ( ∂f/∂x , ∂f/∂y )

So, we have:

∂f/∂x = [tex]e^x[/tex] - 1

∂f/∂y = -x sin y

At the point (a, b), the gradient vector is:

∇f(a, b) = ( [tex]e^a[/tex] - 1 , -a sin b )

The level curve of f(x, y) is the set of points (x, y) where f(x, y) = k for some constant k. In other words, the level curve is the curve where the surface s intersects the plane z = k.

Let (a, b, c) be a point on the surface s that lies on the level curve at the point (a, b). Then, we have:

f(a, b) = c

Differentiating both sides with respect to x and y, we get:

∂f/∂x dx + ∂f/∂y dy = 0

This equation says that the gradient vector of f(x, y) is orthogonal to the tangent vector of the level curve at the point (a, b).

Therefore, the vector that is perpendicular to the level curve at the point (a, b) is the opposite of the gradient vector:

-∇f(a, b) = ([tex]-e^a[/tex] + 1, a sin b)

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Vector that is perpendicular to the level curve at the point (a, b) is the opposite of the gradient vector:

-∇f(a, b) = ([tex]-e^a[/tex] + 1, a sin b)

We can use the gradient of f(x, y) at that point.

The gradient of f(x, y) is given by:

∇f(x, y) = ( ∂f/∂x , ∂f/∂y )

So, we have:

∂f/∂x = [tex]e^x[/tex] - 1

∂f/∂y = -x sin y

At the point (a, b), the gradient vector is:

∇f(a, b) = ( [tex]e^a[/tex] - 1 , -a sin b )

The level curve of f(x, y) is the set of points (x, y) where f(x, y) = k for some constant k. In other words, the level curve is the curve where the surface s intersects the plane z = k.

Let (a, b, c) be a point on the surface s that lies on the level curve at the point (a, b). Then, we have:

f(a, b) = c

Differentiating both sides with respect to x and y, we get:

∂f/∂x dx + ∂f/∂y dy = 0

This equation says that the gradient vector of f(x, y) is orthogonal to the tangent vector of the level curve at the point (a, b).

Therefore, the vector that is perpendicular to the level curve at the point (a, b) is the opposite of the gradient vector:

-∇f(a, b) = ([tex]-e^a[/tex] + 1, a sin b)

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Answer:

  18 square yards

Step-by-step explanation:

You want the area of a garden that fills a back yard that is 4 yd by 6 yd except for a patio that is 3 yd by 2 yd.

The area of the backyard is ...

  A = LW = (6 yd)(4 yd) = 24 yd²

The area of the patio is ...

  A = LW = (3 yd)(2 yd) = 6 yd²

Garden area

The garden area is the area of the backyard that is not taken up by the patio:

  24 yd² -6 yd² = 18 yd²

The garden covers 18 square yards.

__

Additional comment

You can compute this many ways. You can divide the garden area into rectangles or trapezoids, or you can recognize that the garden is 3/4 of the area of the back yard.

(You get two trapezoids by cutting the garden along a line between the upper left corner of the yard and the upper left corner of the patio.)

The exposure variable in the cohort study was consumption of artificially sweetened beverages.

The exposure variable in a cohort study refers to the factor that researchers are interested in studying to determine its potential association with an outcome. In this case, the exposure variable was consumption of artificially sweetened beverages.

The researchers looked at how often individuals consumed these beverages, and the amount or frequency of consumption may have been measured to assess the exposure. The researchers aimed to investigate whether there was a relationship between consumption of artificially sweetened beverages and the outcomes of incident stroke and dementia.

Therefore, the exposure variable in the cohort study was artificially sweetened beverage consumption.

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The z-value for a score of 16.5 in this normal distribution is -2.5.

To find the z-value for a score of 16.5 in a normal distribution with a mean of 24 and a variance of 9, we use the formula:

z = (x - μ) / σ

where x is the score we're interested in, μ is the mean, and σ is the standard deviation (which is the square root of the variance). Plugging in the values we have:

z = (16.5 - 24) / √9
z = -7.5 / 3
z = -2.5

Therefore, the z-value for a score of 16.5 in this normal distribution is -2.5.

To find the z-value for a score of 16.5 in a normal distribution with a mean (µ) of 24 and a variance of 9, you'll need to use the z-score formula:

z = (X - µ) / σ

Where X is the given score (16.5), µ is the mean (24), and σ is the standard deviation. Since the variance is 9, the standard deviation (σ) is the square root of 9, which is 3. Plug these values into the formula:

z = (16.5 - 24) / 3
z = -7.5 / 3
z = -2.5

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The probability of not selecting a person with blood type B+ is 90/100.

a) To find the probability of selecting a person with blood type A+ or A- (P(A+ or A-)), first count the number of people with each blood type, then divide the sum of those counts by the total number of people (100).

Number of people with blood type A+ = 34
Number of people with blood type A- = 6

P(A+ or A-) = (34 + 6) / 100 = 40/100

So, the probability of selecting a person with blood type A+ or A- is 40/100.

b) To find the probability of not selecting a person with blood type B+ (P(not B+)), first count the number of people without blood type B+ and then divide that count by the total number of people (100).

Number of people with blood type B+ = 10
Number of people without blood type B+ = 100 - 10 = 90

P(not B+) = 90 / 100 = 90/100

So, the probability of not selecting a person with blood type B+ is 90/100.

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Answer:

area of triangle GHI =16.5 unit ^2

Step-by-step explanation:

triangle B =3 unit^2

triangle A = 9unit^2

triangle C = 7.5 unit^2

so

area of rectangle = 6 unit × 6 unit

= 36 unit^2

area of triangle GHI = 36 unit^2 - ( 3+9+7.5) unit^2

= 36unit^2 - 19.5unit^2

= 16.5 unit ^2

The final location of all relative and absolute extrema of the function H(x) is, H has an absolute minimum at (0, 5) and an absolute maximum at (2, 5). no relative extrema.

To find the exact location of all the relative and absolute extrema of the function h(x) = 5(x-1)^(2/3) with domain [0, 2], follow these steps:

1. Find the first derivative of h(x) with respect to x:
h'(x) = d/dx [5(x-1)^(2/3)]
h'(x) = (2/3) * 5(x-1)^(-1/3)

2. Set the first derivative to 0 to find critical points:
(2/3) * 5(x-1)^(-1/3) = 0
No real solutions exist for x.

3. Check the endpoints of the domain for absolute extrema:
h(0) = 5(0-1)^(2/3) = 5(-1)^(2/3) = 5
h(2) = 5(2-1)^(2/3) = 5(1)^(2/3) = 5

4. Compare the function values at the endpoints:
Since h(0) = h(2) = 5, and there are no critical points within the domain, h(x) has an absolute minimum at (0,5) and an absolute maximum at (2,5). There are no relative extrema in this case.

Your answer:
h has an absolute minimum at (0, 5).
h has an absolute maximum at (2, 5).
h has no relative extrema.

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Step-by-step explanation:

Area B is 4 times Area A

This implies that B = 4 × A

So you take the measurements given and replace it, A is x and B is (x+3) so

(x+3) = 4x

3 = 4x - x

3x = 3

x = 1

Answer:

63.4 square feet

Step-by-step explanation:

area of sector= (60/360)× 3.143×11²=

1/6 × ~380

= 63.38~63.4

Answer:

i think it might be 25350¹⁷

or 2⁵x3⁴x5⁴x13⁵

Subtract the at vector from the acceleration vector a(t) and simplify the result.

To find the tangential and normal components of the acceleration vector for the given function [tex]r(t) = ti + t^2j + 5tk[/tex], we first need to find the velocity and acceleration vectors.

Velocity vector v(t) is the first derivative of r(t):
[tex]v(t) = dr/dt = (1)i + (2t)j + (5)k[/tex]

Acceleration vector a(t) is the second derivative of r(t) or the first derivative of v(t):
[tex]a(t) = dv/dt = (0)i + (2)j + (0)k[/tex]
Now, we need to find the tangential and normal components of the acceleration vector.

Tangential component (at) is the projection of the acceleration vector onto the velocity vector:
[tex]at = (a(t) • v(t)) / ||v(t)||^2 * v(t)[/tex]Dot product[tex]a(t) • v(t) = (0*1) + (2*2t) + (0*5) = 4t[/tex]Magnitude of v(t) squared = (1^2 + (2t)^2 + 5^2) = 1 + 4t^2 + 25

Thus, at =[tex](4t / (1 + 4t^2 + 25)) * (i + 2tj + 5k)[/tex]

Normal component (an) is given by:
an = a(t) - at

To find an, subtract the at vector from the acceleration vector a(t) and simplify the result.

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The area of the triangle with vertices (-4,-2), (2,0), and (-2,8) in r2 is 20 square units.

To find the area of a triangle in R² with vertices A(-4, -2), B(2, 0), and C(-2, 8), you can use the determinant method. The formula is:
Area = (1/2) * | det(A, B, C) |

where det(A, B, C) is the determinant of the matrix formed by the coordinates of the vertices. Arrange the coordinates in a matrix like this:
| -4  -2  1 |
|  2   0  1 |
| -2   8  1 |

To find the area of the triangle with vertices (-4,-2), (2,0), and (-2,8) in r2 using a determinant. To calculate the determinant, we can expand along the first row:
det = -4 * det(0 8; 1 1) - 2 * det(2 -2; 1 1) + (-2) * det(2 -2; -2 0)
det = -4 * (0 - 8) - 2 * (2 + 2) + (-2) * (-4 - 4)
det = 32 - 8 + 16
det = 40
The absolute value of the determinant gives us the area of the triangle, which is:
area = |det|/2
area = 20

The area of the triangle with vertices (-4, -2), (2, 0), and (-2, 8) is 20 square units.

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The values of probability are a. P(X < 71) = 0.90, b. P(X ≤ 24) = 0.73, and c. P(X ≤ 71) = 0.90

We need to calculate the probabilities for a continuous random variable X,

given that P(24 ≤ X ≤ 71) = 0.17 and P(X > 71) = 0.10.

a. P(X < 71)
To find P(X < 71), we can use the fact that P(X < 71) = 1 - P(X ≥ 71).

Since P(X > 71) = 0.10, we know that P(X ≥ 71) = P(X > 71) = 0.10. Thus, P(X < 71) = 1 - 0.10 = 0.90.

b. P(X ≤ 24)
We can use the given information P(24 ≤ X ≤ 71) = 0.17 and P(X < 71) = 0.90 to find P(X ≤ 24).

We know that P(X ≤ 24) = P(X < 71) - P(24 ≤ X ≤ 71) = 0.90 - 0.17 = 0.73.

c. P(X ≤ 71)
To find P(X ≤ 71), we can use the fact that P(X ≤ 71) = P(X < 71) + P(X = 71).

Since X is a continuous random variable, the probability of it taking any specific value, such as 71, is 0.

Therefore, P(X ≤ 71) = P(X < 71) = 0.90.

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10 cartons of eggs will contain 8 more broken eggs than 4 cartons if a grocery store worker is checking for broken egg cartons. Each carton of eggs contains 12 eggs. He checks 4 cartons and finds 8 broken eggs.

Assuming that the proportion of broken eggs in the sampled cartons is representative of the entire batch of cartons, the worker can use this information to predict the number of broken eggs in the rest of the cartons.

The worker checked 4 cartons of eggs, each with 12 eggs, for a total of 4 x 12 = 48 eggs. Out of these 48 eggs, 8 were found to be broken.

To estimate the number of broken eggs in the rest of the cartons, the worker can use proportionality. The proportion of broken eggs in the sample is 8/48 = 1/6. Therefore, the worker can predict that out of the marginal cost remaining cartons of eggs, 1/6 of the eggs will be broken.

The closest option is C, which predicts that there will be 16 broken eggs in 10 cartons, which is 8 more broken eggs than in the 4 cartons the worker checked. This implies an average of 2 broken eggs per carton, which is consistent with the prediction of 2x broken eggs in the remaining cartons. Therefore, option C is the best answer.

The median of complaints received per week is 2.

To find the median of complaints received per week, we need to arrange the probabilities in ascending order and then find the probability at the middle position.

Arranging the probabilities in ascending order, we get:

Number of complaints 0 1 2 3 4 5
Probability 0.05 0.07 0.09 0.18 0.26 0.35

The median position is (n+1)/2, where n is the total number of probabilities. In this case, n=6, so the median position is (6+1)/2=3.5.

The probability at position 3 is 0.09 and the probability at position 4 is 0.18. Therefore, the median probability is the average of these two probabilities, which is (0.09+0.18)/2=0.135.

To find the median number of complaints, we need to find the number of complaints that corresponds to the median probability. Starting from the first probability, we add up the probabilities until we reach a total of at least 0.135.

0.05 + 0.07 + 0.09 = 0.21

Therefore, the median number of complaints is 2.
So, the answer is 2 (rounded to the nearest integer).

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The median of complaints received per week is 2.

To find the median of complaints received per week, we need to arrange the probabilities in ascending order and then find the probability at the middle position.

Arranging the probabilities in ascending order, we get:

Number of complaints 0 1 2 3 4 5
Probability 0.05 0.07 0.09 0.18 0.26 0.35

The median position is (n+1)/2, where n is the total number of probabilities. In this case, n=6, so the median position is (6+1)/2=3.5.

The probability at position 3 is 0.09 and the probability at position 4 is 0.18. Therefore, the median probability is the average of these two probabilities, which is (0.09+0.18)/2=0.135.

To find the median number of complaints, we need to find the number of complaints that corresponds to the median probability. Starting from the first probability, we add up the probabilities until we reach a total of at least 0.135.

0.05 + 0.07 + 0.09 = 0.21

Therefore, the median number of complaints is 2.
So, the answer is 2 (rounded to the nearest integer).

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Answer:

3

Step-by-step explanation:

since there are 6 sides, you do 18 divide 6 which is 3

Answer:

The length of the hexagon= 3 because perimeter= the distance around that particular polygon and 3 multiplied by the number of sides of the regular polygon which is 6 to get 18 therefore 3 becomes the length of each side of the hexagon

The general solution is y(x) = yc(x) + yp(x) y(x) = c1 cos(x) + c2 sin(x) - 5/24 + (1/8)cos(2x).

For the first part of the question, we need to find the complementary function of the differential equation y'' + y = sec(θ)tan(θ).

The characteristic equation is r^2 + 1 = 0, which has roots r = ±i.

So the complementary function is yc(θ) = c1 cos(θ) + c2 sin(θ).

For the second part of the question, we need to find the general solution of the differential equation y'' + y = sec(θ)tan(θ).

To find the particular solution, we need to use variation of parameters. Let's assume the particular solution has the form yp(θ) = u(θ)cos(θ) + v(θ)sin(θ).

Then, we can find the derivatives: yp'(θ) = u'(θ)cos(θ) - u(θ)sin(θ) + v'(θ)sin(θ) + v(θ)cos(θ), and

yp''(θ) = -u(θ)cos(θ) - u'(θ)sin(θ) + v(θ)sin(θ) + v'(θ)cos(θ).

Substituting these into the differential equation, we get

(-u(θ)cos(θ) - u'(θ)sin(θ) + v(θ)sin(θ) + v'(θ)cos(θ)) + (u(θ)cos(θ) + v(θ)sin(θ)) = sec(θ)tan(θ).

Simplifying and grouping terms, we get

u'(θ)sin(θ) + v'(θ)cos(θ) = sec(θ)tan(θ).

To solve for u'(θ) and v'(θ), we need to use the trig identity sec(θ)tan(θ) = sin(θ)/cos(θ).

So, we have u'(θ)sin(θ) = sin(θ), and v'(θ)cos(θ) = 1.

Integrating both sides, we get

u(θ) = -cos(θ) + c1, and v(θ) = ln|sec(θ)| + c2.

Therefore, the particular solution is

yp(θ) = (-cos(θ) + c1)cos(θ) + (ln|sec(θ)| + c2)sin(θ).

Thus, the general solution is

y(θ) = yc(θ) + yp(θ)

y(θ) = c1 cos(θ) + c2 sin(θ) - cos(θ)cos(θ) + (c1ln|sec(θ)| + c2)sin(θ)

y(θ) = c1 cos(θ) - cos^2(θ) + c2 sin(θ) + c1 sin(θ)ln|sec(θ)|.

For the second part of the question, we need to solve the differential equation y'' + y = sin^2(x).

The characteristic equation is r² + 1 = 0, which has roots r = ±i.

So the complementary function is yc(x) = c1 cos(x) + c2 sin(x).

To find the particular solution, we can use the method of undetermined coefficients. Since sin²(x) = (1/2) - (1/2)cos(2x), we can guess a particular solution of the form yp(x) = a + bcos(2x).

Then, yp'(x) = -2bsin(2x) and yp''(x) = -4bcos(2x).

Substituting these into the differential equation, we get

-4bcos(2x) + a + bcos(2x) = (1/2) - (1/2)cos(2x).

Equating coefficients of cos(2x) and the constant term, we get the system of equations

a - 3b = 1/2

-4b = -1/2

Solving for a and b, we get a = -5/24 and b = 1/8.

Therefore, the particular solution is

yp(x) = -5/24 + (1/8)cos(2x).

Thus, the general solution is

y(x) = yc(x) + yp(x)

y(x) = c1 cos(x) + c2 sin(x) - 5/24 + (1/8)cos(2x).

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