A rectangle has one vertex at (0, 4) on
a coordinate plane. The rectangle has
at least one side with a length of
6 units. Which vertices could represent
the other three vertices of the
rectangle?
Select all the correct answers.
A (0, -2), (-2, -2), and (-2, 4)
B (3, 4), (3, 1), and (0, 1)
(6, 4), (0, 2), and (6, 2)
D(-6, 4), (0, 5), and (-6, 5)
E (0, 6), (2, 6), and (2, 4)

The weight of the ice after 10 hours is 1217 lb

A 523 lb mass of ice melts at 4.3 hours

The first step is to calculate the weight after one hour

523= 4.3

x= 1

cross multiply both sides

4.3x= 523

x= 523/4.3

x= 121.7

The weight after 10 hours can be calculated as follows

121.7= 1

y= 10

y= 121.7 × 10

y= 1217

Hence the weight after 10 hours is 1217 lb

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The sample standard deviation is approximately 15.29 miles.

Using the formula for sample standard deviation:

Find the mean of the data:

mean = (11 + 6 + 36 + 16 + 5 + 40) / 6 = 114 / 6 = 19

Subtract the mean from each data point, square the result, and sum the squares:

[tex](11 - 19)^2 + (6 - 19)^2 + (36 - 19)^2 + (16 - 19)^2 + (5 - 19)^2 + (40 - 19)^2\\= (-8)^2 + (-13)^2 + 17^2 + (-3)^2 + (-14)^2 + 21^264 + 169 + 289 + 9 + 196 + 441\\= 1168[/tex]

Divide the sum of squares by (n-1), where n is the sample size:

[tex]s^2 = 1168 / (6-1) = 233.6[/tex]

Take the square root of [tex]s^2[/tex] to find the sample standard deviation:

s = sqrt(233.6) ≈ 15.29

Therefore, the sample standard deviation is approximately 15.29 miles.

The deviation is a metric used in statistics and mathematics to determine how different a variable's observed value and predicted value are from one another. The deviation is the distance from the centre point, to put it simply.

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For equation A+C=B the matrix C is [tex]\left[\begin{array}{ccc}-2&-7\\-5&8\end{array}\right][/tex] and C-B=A then C is [tex]\left[\begin{array}{ccc}2&-9\\7&0\end{array}\right][/tex]

The given matrix A = [tex]\left[\begin{array}{ccc}2&-1\\6&-4\end{array}\right][/tex]

B=[tex]\left[\begin{array}{ccc}0&-8\\1&4\end{array}\right][/tex]

Now the equation is A+C=B

[tex]\left[\begin{array}{ccc}2&-1\\6&4\end{array}\right][/tex]+C  =[tex]\left[\begin{array}{ccc}0&-8\\1&4\end{array}\right][/tex]

C=[tex]\left[\begin{array}{ccc}0&-8\\1&4\end{array}\right][/tex]- [tex]\left[\begin{array}{ccc}2&-1\\6&-4\end{array}\right][/tex]

C=[tex]\left[\begin{array}{ccc}-2&-7\\-5&8\end{array}\right][/tex]

Now equation is C-B=A

C=A+B

= [tex]\left[\begin{array}{ccc}2&-1\\6&-4\end{array}\right][/tex]+[tex]\left[\begin{array}{ccc}0&-8\\1&4\end{array}\right][/tex]

C=[tex]\left[\begin{array}{ccc}2&-9\\7&0\end{array}\right][/tex]

Hence, for equation A+C=B the matrix C is [tex]\left[\begin{array}{ccc}-2&-7\\-5&8\end{array}\right][/tex] and C-B=A then C is [tex]\left[\begin{array}{ccc}2&-9\\7&0\end{array}\right][/tex]

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Answer:

[tex]SA=1657 cm^2[/tex]

Step-by-step explanation:

Surface Area Formula for Rectangular Prism.

[tex]SA=2*[ (l*h) + (w*h) + (l*w)][/tex]

Your l = 15 cm , w = 6.5 cm , and h = 34 cm.

Plug these values into the equation.

[tex]SA=2*[ (15*34) + (6.5*34) + (15*6.5)][/tex]

[tex]SA=2*[(510)+(221)+(97.5)][/tex]

[tex]SA=2*[510+221+97.5][/tex]

[tex]SA=2*(828.5)[/tex]

[tex]SA=1657 cm^2[/tex]

Boolean algebra is a type of algebra that deals with binary variables and logical operations. In this case, we're simplifying expressions using Boolean algebra.

1. ABC +(A+B+7)

First, let's simplify the expression in the parentheses:
A + B + 7 = 1 (since any input to a Boolean function that is not 0 is considered 1)
Now, let's substitute this value back into the original expression:
ABC + 1
This is the simplified expression.

2. (A+Ā)(AB+ ABC)

Using the identity A + Ā = 1, we can simplify the first set of parentheses:
(A + Ā)(AB + ABC) = AB + ABC

3. (B+BC)(B+ BC)(B+D)

Using the distributive property of Boolean algebra, we can simplify the first set of parentheses:
(B + BC)(B + BC)(B + D) = (B + BC)(B + D)
Using the distributive property again, we can simplify this further:
(B + BC)(B + D) = BB + BCD
Simplifying further, we know that B + BC = B and BB = B, so we can simplify to:
B + BCD

So, the simplified expressions are:
1. ABC + 1
2. AB + ABC
3. B + BCD

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we conclude that the helicopter will land in a finite amount of time if and only if p <= 0.

The differential equation dh/dt = [tex]-h^{p}[/tex], where h is the height of the helicopter, represents the rate of change of the height with respect to time. To find the value of p that results in predictions that the helicopter will land in a finite amount of time, we need to consider the behavior of the solution as h approaches zero.

If we assume that the helicopter will eventually land, then the height h will approach zero as time goes to infinity. Therefore, we can consider the behavior of the solution near the origin. To do this, we will use a technique called separation of variables.

Separation of variables involves writing the differential equation in the form dh/[tex]h^{p}[/tex] = -dt and then integrating both sides. This gives:

∫h_[tex]0^{h}[/tex] dh / [tex]h^{p}[/tex] = ∫0^t -dt

where h_0 is the initial height of the helicopter.

The left-hand side can be evaluated using the power rule of integration:

[tex][1/(1-p)] [h^{(1-p)}]_h_0^{h} = -t[/tex]

where [f(x)]_aᵇ denotes the value of f(x) evaluated at b minus the value of f(x) evaluated at a.

We can simplify this expression by using the fact that h_0 is nonzero, so h^(1-p)_0 approaches infinity as h approaches zero. Therefore, we can neglect the term h^(1-p)_0 and write:

[tex][1/(1-p)] h^{(1-p)} = -t[/tex]

If p > 1, then h^(1-p) approaches zero as h approaches zero. Therefore, the left-hand side of the equation approaches infinity as t approaches a finite value. This implies that the helicopter will never land, which contradicts our assumption that it will eventually land. Therefore, we must have p <= 1.

If p = 1, then the left-hand side of the equation becomes ln(h), which approaches negative infinity as h approaches zero. Therefore, the equation cannot hold for any finite value of t. This implies that the helicopter will not land in a finite amount of time.

If p < 1, then the left-hand side of the equation approaches infinity as h approaches zero. Therefore, the equation cannot hold for any finite value of t. This implies that the helicopter will not land in a finite amount of time.

Therefore, we conclude that the helicopter will land in a finite amount of time if and only if p <= 0.

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The account will be worth $39,847.15 in 40 years.

Given,

P = 8500 is the amount deposited

r = 0.039 is the decimal form of the 3.9% interest rate

n= 2 is the number of times the money is compounded per year

t = 40 is the number of years

We know that the amount  calculated semi-annually is:

[tex]A = 8500 (1 + \frac{0.039}{2})^{2*40}[/tex]

[tex]A = 8500( 1 + 0.0195)^{80}[/tex]

[tex]A = 8500 * 4.6875[/tex]

A = $39,847.15

As a result, The account will be worth $39,847.15 in 40 years.

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To answer your question about the inhomogeneous differential equation.

a) To guess a solution for the inhomogeneous equation f''(x) = e^x, we first observe that the right side of the equation is e^x. A function whose second derivative is e^x is a linear combination of e^x, x*e^x, and x^2*e^x. Therefore, we can guess the solution to be f(x) = A*x^2*e^x + B*x*e^x + C*e^x, where A, B, and C are constants to be determined.

b) To describe all solutions of the homogeneous equation f''(x) = 0, we can note that the second derivative of a linear function is zero. Therefore, the general solution of the homogeneous equation is f(x) = Ax + B, where A and B are constants.

c) To describe all solutions of the inhomogeneous equation f'''(x) = e^x, we combine the particular solution from part a) with the general solution from part b). This gives us the general solution for the inhomogeneous equation: f(x) = A_1*x^2*e^x + B_1*x*e^x + C_1*e^x + A_2*x + B_2, where A_1, B_1, C_1, A_2, and B_2 are constants.

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To find the standard normal random variable z transformed from x, we first need to calculate the z-score of x. The formula for the z-score is:

z = (x - mean) / standard deviation

Substituting the values given in the question, we get:

z = (x - 12) / 9

We can then transform this equation to solve for x in terms of z:

x = mean + z * standard deviation

Substituting the values for mean and standard deviation, we get:

x = 12 + z * 9

Therefore, the standard normal random variable z transformed from x is:

x = 12 + z * 9
To transform a given value (x) from a normal distribution with mean (μ) and standard deviation (σ) to a standard normal random variable (z), you can use the z-score formula:

z = (x - μ) / σ

In this case, the normal distribution has a mean (μ) of 12 and a standard deviation (σ) of 9. To transform any value x from this distribution to a standard normal random variable (z), you can follow these steps:

Step 1: Subtract the mean (μ) from the given value (x).
z = (x - 12)

Step 2: Divide the result by the standard deviation (σ).
z = (x - 12) / 9

Now you have the formula to transform any value x from the given normal distribution to a standard normal random variable (z).

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The probability that the point lies on the square is P = 0.498

to find that probability, we need to take the quotient between the area of the square and the area of the circle.

We can see that the square has a side length of 5 units, then its area is.

A = 5*5 = 25 square units.

The circle has a radius of 4 units, then its area is:

A' = 3.14*4^2 = 50.24 square units

Then the probability is:

P = 25/50.24 = 0.498

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The value of MAS to the nearest degree, given the radius of the circle, is 103 degrees.

Angle MAS can be found by the formula for calculating arc length which is:

L = ( n π r ) / 180

Further solving will give us:

n π r = 197 x 180 = 3, 546

n = 3, 546 / π r

n = 3, 546 / ( 3.14 x 110 )

n = 103 degrees

In conclusion, angle mAS can be found to be 103 degrees.

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The simplified expression is given by (x² - 3x - 3) / ((x + 3)(x - 2)(x - 4)).

To simplify this expression, we need to find a common denominator for the two fractions and then combine them. To do this, we need to factor the denominators of both fractions.

Let's start with the first fraction's denominator:

x² + x - 6

We need to find two numbers that multiply to -6 and add to +1. These numbers are +3 and -2. Therefore, we can write:

x² + x - 6 = (x + 3)(x - 2)

Now let's factor the second fraction's denominator:

x² - 6x + 8

We need to find two numbers that multiply to 8 and add to -6. These numbers are -2 and -4. Therefore, we can write:

x² - 6x + 8 = (x - 2)(x - 4)

Now we can rewrite the original expression with a common denominator:

(x(x - 2) - (1)(x + 3)) / ((x + 3)(x - 2)(x - 4))

Next, we can simplify the numerator:

(x² - 2x - x - 3) / ((x + 3)(x - 2)(x - 4))

(x² - 3x - 3) / ((x + 3)(x - 2)(x - 4))

Finally, we can't simplify this expression any further. Therefore, the simplified expression is:

(x² - 3x - 3) / ((x + 3)(x - 2)(x - 4))

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Answer:

The original cost of the apartment: was $1800

The increased price of the apartment per year: $90

To figure this problem out we need to calculate how much the apartment would cost around the 10th year. To do that, we would first need to multiply the $90 increase per year and the 10th years of living in the apartment.

$90 x 10 = $900

Now that we know how much it increased, we need to add that to our original cost. So, we add $1800 and $900.

$1800 + $900 = $27000

Yay! Know we know our monthly rent is $27000 during the 10th year of living there.

The cam lift is 3 1/2 inches, which is option D.

To determine the lift of the cam, we need to find the distance from the center of the disc to the highest point of the cam surface.

First, we can find the distance from the center of the disc to the edge of the 1/2" hole. Since the hole is drilled 1 1/2" off-center, this distance is:

(4"/2) - 1 1/2" = 1"

Next, we can find the radius of the cam surface by adding the radius of the shaft (1/2") to the distance from the center of the disc to the edge of the 1/2" hole (1"):

1/2" + 1" = 1 1/2"

Finally, we can find the distance from the center of the disc to the highest point of the cam surface by adding the radius of the disc (4"/2 = 2") to the radius of the cam surface (1 1/2"):

2" + 1 1/2" = 3 1/2"

Therefore, the lift of the cam is 3 1/2 inches, which is option D.

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Sam lives in either a small house or a yellow house.

Mutually exclusive:

Maya chooses either red or yellow when taking one crayon from a set of 16.

Raymond draws a 2 or a 3 when taking a single card from a deck.

Hannah gets either heads or tails when she flips a coin.

Not mutually exclusive:

Sally wears a blue shirt or blue pants.

Jack either goes to his friend's house or does his homework.

Sam lives in either a small house or a yellow house.

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Farah's team runs a total of 3.2 kilometers in the relay race.

We have,

First, let's start with the number of laps each student runs:

2 laps per student x 4 students

= 8 laps in total

Next, let's convert the number of laps to the total distance:

8 laps x 400 meters per lap = 3200 meters

Finally, let's convert the distance from meters to kilometers:

3200 meters ÷ 1000 meters per kilometer

= 3.2 kilometers

Thus,

Farah's team runs a total of 3.2 kilometers in the relay race.

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There are indeed 48 triangles that can be chosen from the rectangular array shown .

To find the 48 triangles, you should use the Combination formula which will show you the number of ways to pick 3 points when given 8 points.

C ( n, k ) = n! / ( k ! x ( n - k ) ! )

C ( 8 , 3 ) = 8 ! / (3 ! x ( 8 - 3 ) ! )

C ( 8, 3 ) = 336 / 6

C ( 8, 3) = 56

Now, there are technically 56 ways to pick the points but some of these ways are collinear and these cannot form triangles. Each row will have 4 such points so the number of ways to pick triangles is:

= 56 - ( 4 x 2 )

= 48 triangles

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The area of the parallelogram formed by the given vertices A(1, 0, -1), B(1, 7, 2), C(2, 4, -1), and D(0, 3, 2) is 2√21 square units.

To calculate the area of a parallelogram, we can use the cross product of two vectors formed by the sides of the parallelogram. The vectors AB and AD can be calculated by subtracting the coordinates of the initial and final points.

The cross product of these vectors gives us a vector representing the area of the parallelogram. Taking the magnitude of this vector gives us the area of the parallelogram. The magnitude of the cross product of AB and AD is 24, so the area of the parallelogram is 24 square units.

In this case, the vector AB is (-3, 7, 3), and the vector AD is (-1, 3, 3). Taking the cross product of these vectors gives us the vector (-12, 6, 24). The magnitude of this vector is √(12² + 6² + 24²) = √756 = 2√21. Therefore, the area of the parallelogram is 2√21 square units.

Complete Question:

Find the area of the parallelogram whose vertices are A(1, 0, −1), B(1, 7, 2), C(2, 4, −1), D(0, 3, 2).

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The correct answer is (a) $160.

The prediction equation, Spending = 60 + 5(Age of Consumer), suggests that there is a linear relationship between the age of a consumer and their spending. The equation can be interpreted as follows: for every additional year in age, the consumer's spending is expected to increase by $5, and the baseline spending for a consumer of any age is $60.

To find the model's prediction for spending when the consumer is 20 years old, we simply substitute 20 for Age of Consumer in the equation:

Spending = 60 + 5(20) = 60 + 100 = $160

Therefore, the correct answer is (a) $160.

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The probability that both event will occur is 1/6

A probability is a number that reflects the chance or likelihood that a particular event will occur. The certainty of an event to occur is 1 which is equivalent to 100%.

Probability = sample space/ total outcome

the probability that a coin will lands on head = 1/2 since a coin has two faces.

Also the probability that when a die is rolled , the probability of getting 5 or greater = 2/6, since die has 6 sides and 2 sides has 5 and greater.

Therefore the probability of getting both event = 1/2 × 2/6 = 1/6

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The equation of the ellipse is

Calculate the distance between the foci (2c):

2c = |8 - (-2)| = 10

c = 5

Determine the length of the semi-major axis (a):

a = 26 / 2 = 13

Solving for the center of the ellipse, denoted by (h, k):

h = (5 + 5)/2 = 5

k = (8 + -2)/2 = 3

hence, center of the ellipse is (5, 3).

solving for the length of the semi-minor axis denoted by (b):

a^2 = b^2 + c^2

knowing that the values of parameter a and c, so we can solve for b:

13^2 = b^2 + 5^2

169 = b^2 + 25

b^2 = 144

b = 12

equation of the ellipse:

((x - h)^2) / b^2 + ((y - k)^2) / a^2 = 1

Plugging in the values:

((x - 5)^2) / 12^2 + ((y - 3)^2) / 13^2 = 1

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The probability of a woman having four girls in a row is 6.25%.

To calculate the probability that if a woman has four children, they will all be girls, you should use the rule of multiplication. This rule states that to calculate the probability of two or more independent events occurring together, you multiply the probability of each individual event. In this case, the probability of each child being a girl is 0.5 (assuming an equal chance of having a boy or girl), so you would calculate the probability as 0.5 x 0.5 x 0.5 x 0.5 = 0.0625 or 6.25%. Therefore, the probability of a woman having four girls in a row is 6.25%.

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The Reinventing Government (REGO) approach to public administration and bureaucracy emphasizes a mission-driven approach to public service delivery, rather than a rule-driven approach. REGO advocates argue that this approach is beneficial to bureaucracies for several reasons.

Firstly, a mission-driven approach allows for improved budget management, as resources can be allocated more efficiently based on the organization's priorities and goals. This can lead to cost savings and better use of public funds.

Secondly, a mission-driven approach fosters greater employee creativity and flexibility. By focusing on the overarching objectives, employees have more freedom to innovate and develop new strategies to achieve those goals. This can lead to increased productivity and better outcomes for the public.

There may be certain types of agencies that are more open to adopting the REGO approach. Generally, agencies with a clear and well-defined mission, as well as those that can easily measure their performance, might be more receptive to the idea of a mission-driven approach. Additionally, agencies with a culture of innovation and adaptability might also be more inclined to embrace this approach.

In summary, the Reinventing Government approach aims to improve public service delivery by focusing on the mission of an organization. Its benefits include better budget management, increased employee creativity and flexibility, and overall improved outcomes for the public. The suitability of the REGO approach may vary depending on the agency's mission, performance measurement capabilities, and culture.

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a) The value of circumference of circle is,

C  = 219.8 Inches

b) The value of area of circle is,

A = 3,846.5 in²

Given that;

The diameter of circle is, 70 inches

a) Circumference of circle is,

C = π × diameter,

So, We get;

C = 3.14 × 70

C  = 219.8 Inches

b) The area of circle is,

A = πr²,

And, Radius is 1/2 of diameter,

Hence, Radius = 70/2 = 35 inches

Thus, We get;

A =  3.14 × 35²

A = 3.14 × 1225,

A = 3,846.5 in²

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The value of the integral ∫ [tex]cos^7(x) dx[/tex] is[tex](3\pi /32).[/tex]

Wallis's formulas are used to evaluate integrals of the form:

∫ [tex]sin^{n(x)} cos^{m(x)} dx[/tex]

where n and m are non-negative integers. We can use the trigonometric identity[tex]cos^{2(x)] + sin^{2(x)} = 1[/tex] to convert the powers of cosine to powers of sine.

Here, we have m = 7, so we can use the identity [tex]cos^{2(x)} = 1 - sin^{2(x)}[/tex] to write:

[tex]cos^{7(x)} = cos^{6(x)}[/tex] × [tex]cos(x)[/tex]

[tex]= (1 - sin^2(x))^3[/tex] ×[tex]cos(x)[/tex]

Now, we can use a substitution of [tex]u = sin(x), du = cos(x) dx[/tex]to convert the integral to a form that can be evaluated using Wallis's formulas:

∫ [tex]cos^7(x) dx =[/tex] ∫ [tex](1 - sin^2(x))^3[/tex] × [tex]cos(x) dx[/tex]

= ∫ [tex](1 - u^2)^3 du[/tex]

Using Wallis's formulas, we have:

∫ [tex](1 - u^2)^3 du = (1/8)[/tex]× β[tex](4, 4)[/tex]

[tex]= (1/8)[/tex] ×[tex][(3\pi /4) / sin(3\pi /4)][/tex]

[tex]= (3\pi /32)[/tex]

Substituting [tex]u = sin(x)[/tex], we have:

∫ [tex]cos^7(x) dx =[/tex] ∫ [tex](1 - u^2)^3 du = (3π/32)[/tex]

Therefore, the value of the integral ∫ [tex]cos^7(x) dx[/tex] is [tex](3\pi /32).[/tex]

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As we add more terms to the series, the plot approaches the original function f(x) = 1/x. Note that the series is only defined for x > 0, since f(x) is not defined at x = 0.

To sketch the Fourier cosine series of f(x) = 1/x, we need to first determine the Fourier coefficients. Recall that the Fourier cosine series is given by:

f(x) = a0/2 + ∑[n=1 to ∞] an cos(nπx/L)

where L is the period of the function (in this case, L = 2), and the Fourier coefficients are given by:

an = (2/L) ∫[0 to L] f(x) cos(nπx/L) dx

Using f(x) = 1/x, we can compute the Fourier coefficients as follows:

a0 = (2/L) ∫[0 to L] f(x) dx
  = (2/2) ∫[0 to 2] 1/x dx
  = ∞ (divergent)

an = (2/L) ∫[0 to L] f(x) cos(nπx/L) dx
  = (2/2) ∫[0 to 2] (1/x) cos(nπx/2) dx
  = (-1)^n π/2 (n ≠ 0)

Note that a0 is divergent, which means that the Fourier cosine series of f(x) will not have a constant term. Therefore, the Fourier cosine series of f(x) is given by:

f(x) = ∑[n=1 to ∞] (-1)^n π/2 cos(nπx/2)

To sketch this series, we can plot the partial sums of the series for a few values of n. For example, we can plot:

f1(x) = (-1)^1 π/2 cos(πx/2)
f2(x) = (-1)^1 π/2 cos(πx/2) + (-1)^2 π/2 cos(2πx/2)
f3(x) = (-1)^1 π/2 cos(πx/2) + (-1)^2 π/2 cos(2πx/2) + (-1)^3 π/2 cos(3πx/2)

and so on, up to some value of n. Here is what the plots look like for n = 1, 2, and 3:

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The probability that in a given week the airline will lose less than 20 suitcases is approximately 0.8186 or 81.86%.

We are given that the distribution of the number of suitcases that get lost each week on a certain route is approximately normal with a mean of [tex]$\mu = 16.9$[/tex] and standard deviation of [tex]$\sigma = 3.3$[/tex]. We need to find the probability that in a given week the airline will lose less than 20 suitcases.

Let X be the number of suitcases lost in a week. Then we need to find P(X < 20).

Using the Z-score formula, we can standardize the variable X as:

[tex]Z=\frac{X-\mu}{\sigma}[/tex]

Substituting the given values, we get:

[tex]Z=\frac{20-16.9}{3.3}=0.91[/tex]

Now, we need to find the probability that Z is less than 0.91. We can use a standard normal distribution table or calculator to find this probability, which is approximately 0.8186.

Therefore, the probability that in a given week the airline will lose less than 20 suitcases is approximately 0.8186 or 81.86%.

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5, instances of the number 5 will be stored in this 2D array. Option B.

To count the instances of the number 5 in the 2D array, we need to iterate through all the elements of the array and count the occurrences of 5.

Starting from the top-left element, we see that it is not a 5. Moving to the right, we find a 5 in the second column. Continuing to the right, we find another 5 in the same row. So far, we have counted 2 instances of 5.

Moving to the next row, we find no 5s in the first column. In the second column, we find two 5s in the same row. This brings our total count to 4. Moving to the last row, we find a single 5 in the second column, bringing our final count to 5.

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Answer:

75

Step-by-step explanation:

300/4 = 75

Change of y/ change of x = gradient

In the case of discrete random variables, the expectation of a function is defined as the sum of the function's values multiplied by their probabilities:

E(aX + bY + c) = ∑(aX + bY + c)P(X,Y)

We can break down the sum using properties of summation:

= a∑XP(X,Y) + b∑YP(X,Y) + c∑P(X,Y)

Since the sum of probabilities over all events equals 1:

= aE(X) + bE(Y) + c

For the continuous case, the expectation of a function is defined as the integral of the function's values multiplied by the joint probability density function (PDF):

E(aX + bY + c) = ∫∫(aX + bY + c)f(X,Y)dXdY

We can break down the integral using properties of integration:

= a∫∫Xf(X,Y)dXdY + b∫∫Yf(X,Y)dXdY + c∫∫f(X,Y)dXdY

Again, since the integral of the joint PDF over all events equals 1:

= aE(X) + bE(Y) + c

Thus, we have shown that for both discrete and continuous cases, the linearity of expectation holds:

E(aX + bY + c) = aE(X) + bE(Y) + c

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