Shaniece practices the piano 1610 minutes in 5 weeks. Assuming she practices the same amount every week, how many minutes would she practice in 4 weeks?

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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1. What is the probability a randomly selected bag will have a positive test? Give your answer to four decimal places is 0.1032

2. Given a randomly selected bag has a positive test, what is the probability it actually contains a large amount of liquid is 0.3780

3. Given a randomly selected bag has a positive test, what is the probability it does not contain a large amount of liquid is  0.6219

Let's characterize the taking after occasions:

A: The pack contains huge sums of fluid.

B: The test is positive.

We are given the taking after probabilities:

P(A) = 0.04

P(B | A) = 0.98

P(B | not A) = 0.07

1. To discover the likelihood of a positive test, we are able to utilize the law of adding up to likelihood:

P(B) = P(B | A) P(A) + P(B | not A) P(not A)

= 0.98 * 0.04 + 0.07 * 0.96

= 0.1032

So the likelihood of a haphazardly chosen pack having a positive test is 0.1032 (adjusted to four decimal places).

2. To discover the likelihood that a sack really contains large amounts of fluid given a positive test, we are able to utilize Bayes' hypothesis:

P(A | B) = P(B | A) P(A) / P(B)

= 0.98 * 0.04 / 0.1032

= 0.3780

So the likelihood that a pack really contains expansive sums of fluid given a positive test is 0.3780 (adjusted to four decimal places).

3. To discover the likelihood that a sack does not contain expansive sums of fluid given a positive test, ready to utilize Bayes' hypothesis again:

P(not A | B) = P(B | not A) P(not A) / P(B)

= 0.07 * 0.96 / 0.1032

= 0.6219

So the likelihood that a pack does not contain expansive sums of fluid given a positive test is 0.6219 (adjusted to four decimal places). 

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The exact values of the cosecant, secant, and cotangent ratios of 5π/6 are 2, -2/√3, and -√3, respectively.

To find the exact values of the cosecant, secant, and cotangent ratios of an angle of 5π/6, we need to use the definitions of these trigonometric functions and the values of the sine, cosine, and tangent of this angle.

First, we can find the sine and cosine of 5π/6 using the unit circle or reference angles:

sin(5π/6) = sin(π/6) = 1/2

cos(5π/6) = -cos(π/6) = -√3/2

Then, we can use the definitions of the cosecant, secant, and cotangent ratios:

cosec(5π/6) = 1/sin(5π/6) = 1/(1/2) = 2

sec(5π/6) = 1/cos(5π/6) = -2/√3

cot(5π/6) = cos(5π/6)/sin(5π/6) = (-√3/2)/(1/2) = -√3

Therefore, the exact values of the cosecant, secant, and cotangent ratios of 5π/6 are 2, -2/√3, and -√3, respectively.

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Answer: The box is large enough to ship the coat.

15000cm to the power of 3>10000cm to the power of 3

Step-by-step explanation:

V box=25x30x20

         =750+20

         =15000cm to the power of 3

So the box is large enough to ship the coat

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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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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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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a. At (0,2] f is uniformly continuous.

b. At (2,∞) f is uniformly continuous.

c. At (0,∞) f is uniformly continuous.

A function connects an input with an output. It is analogous to a machine with an input and an output. And the output is somehow related to the input. The standard manner of writing a function is f(x) "f(x) =... "

(a) Let f(x) = x³ on (0,2]. Let ε > 0 be given. We need to find a δ > 0 such that |x - y| < δ implies |f(x) - f(y)| < ε for all x,y in (0,2]. Note that |f(x) - f(y)| = |x³ - y³| = |x - y||x² + xy + y²|. Since x,y ∈ (0,2], we have x² + xy + y² ≤ 12. Thus, if we choose δ = ε/12, then for any x,y ∈ (0,2] such that |x - y| < δ, we have |f(x) - f(y)| < ε. Hence, f is uniformly continuous on (0,2].

(b) Let f(x) = 1/2 on (2,∞). Let ε > 0 be given. We can choose any δ > 0 since for any x,y ∈ (2,∞), we have |f(x) - f(y)| = 0 < ε. Thus, f is uniformly continuous on (2,∞).

(c) Let f(x) = (x-1)/(x+1) on (0,∞). Let ε > 0 be given. We need to find a δ > 0 such that |x - y| < δ implies |f(x) - f(y)| < ε for all x,y in (0,∞). Note that |f(x) - f(y)| = |(x-1)/(x+1) - (y-1)/(y+1)| = |(x-y)(2/(x+1)(y+1))|. Thus, if we choose δ = ε/2, then for any x,y in (0,∞) such that |x - y| < δ, we have |f(x) - f(y)| = |(x-y)(2/(x+1)(y+1))| < ε. Hence, f is uniformly continuous on (0,∞).

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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 result of 48 - 36 is 12. Then, if you divide 12 by 36, the result is 0.3333 or 1/3.

Step-by-step explanation:

The simplified difference is:

3y/8 - 5/6y = (-y)/24

Note that there are no non-permissible values in this case.

a. 4x/(2x+5) + 10/(2x+5)

To add these two expressions, we need to find a common denominator. In this case, the common denominator is (2x+5):

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

Now we can simplify the numerator by factoring out a 2:

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

And we can cancel out the common factor of (2x+5):

2(2x+5)/(2x+5) = 2

Therefore, the simplified sum is:

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

Note that the non-permissible value is x = -2.5, since this value would make the denominator equal to zero.

b. 3y/8 - 5/6y

To subtract these two expressions, we also need a common denominator. In this case, the common denominator is 24y:

3y/8 - 5/6y = (9y^2)/(24y) - (20y)/(24y)

We can simplify the first term in the numerator by canceling out a common factor of 3:

([tex]9y^2[/tex])/(24y) = (3y)/8

So the subtraction becomes:

3y/8 - 5/6y = (3y)/8 - (10y)/12

Now we can find a common denominator of 24:

(3y)/8 - (10y)/12 = (9y)/24 - (10y)/24

Simplifying the numerator gives:

(9y)/24 - (10y)/24 = (-y)/24

Therefore, the simplified difference is:

3y/8 - 5/6y = (-y)/24

Note that there are no non-permissible values in this case.

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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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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 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 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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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]

From multiplcation operation, Alex has enough paint to cover 1,000 ft² of his apartment. The true statement is 2 gallons of paint will cover 1068 ft², Alex have enough paint of quantity 2 gallons.

We have Mr. Alex painted his apartment. Area of his apartment'walls = 178 ft²

Quantity of paint used by him to paint his apartment'walls with area 178 ft² =[tex] \frac{1}{3} \: \: gallons[/tex]

Total quantity of paint used in all

= 2 gallons

We have to check the provide paint is enough or not to cover 1,000 ft² of his apartment. Let the required paint for 1000 ft² be x gallons. Using multiplcation, 1/3 gallons quantity of paint will cover the area of apartment = 178 ft², so, 1 gallons quantity of paint will cover the area of apartment = 178 ×3 ft²= 534 ft²

Now, 2 gallons quantity of paint will cover the area of apartment = 2× 534 ft² = 1068 ft²> 1000 ft²

But he wants to paint 1000 ft² of his apartment in 2 gallons quantity (x=1.9 gal ). So, he has enough paint to paint his apartment.

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Complete question:

The above figure complete the question.

Alex painted 178 ft2 of his apartment’s walls with 1/3 gallon of paint. He has 2 gallons of paint in all. If he wants to cover 1,000 ft2 of his apartment, does he have enough paint? Complete a true statement

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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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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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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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 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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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 given relationship in the task content is not a function as more than one output value is paired with the same input value.

Recall that a relationship is said to be a function only if one output value is attached to each input value of the relationship.

On this note, by observation; the pair of coordinates (6, 3) and (6, 9) implies that two output values are assigned to the same input value. Consequently, the given relationship is not a function.

The complete and correct sentence is therefore; Since one input value is paired with two output values; the relationship is not a function.

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The value of variable x and y in the right triangle are 4 and 4√3 units respectively.

A right angle triangle is a triangle that has one of its angles as 90 degrees.

The variable x and y can be as follows:

cos 60° = adjacent / hypotenuse

cos 60° = x / 8

cross multiply

x = 8 cos 60°

x = 8 × 0.5

x = 4 units

Let's find the value of y as follows:

sin 60° = opposite / hypotenuse

Therefore,

sin 60° =  y / 8

cross multiply

y = 8 sin 60

y = 8 × √3 / 2

y = 4√3 units  

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Answer:

Step-by-step explanation:

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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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 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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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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