In a carnival game, the player selects two coins from a bag containing two silver dollars and six slugs. Write down the probability distribution for winnings and determine how much the player would have to pay so that he would break even, on the average, over many repetitions of the game.

Trace(QpP) is a non-negative integer. In other words, trace(QpP) ≥ 0.

Since trace(PQ) = trace(QpP), we have shown that trace(PQ) ≥ 0, as required.

Since p and q are orthogonal projections, we have p^2 = p and q^2 = q, and p and q are also orthogonal in the sense that p q = q p = 0.

We want to show that trace (PQ) ≥ 0. Let's first expand the expression for the trace of PQ:

trace(PQ) = trace(PQP) = trace(QPP)

Here, we have used the fact that trace(AB) = trace(BA) for any matrices A and B.

Next, let's consider the product QP. Since p and q are orthogonal, their sum p + q is also an orthogonal projection, and we have:

QP = Q(p + q)P = QpP + QqP

Note that QpP is a product of two orthogonal projections, so it is itself an orthogonal projection. Furthermore, since p and q are orthogonal, we have QqP = 0. Therefore, we have:

QP = QpP

Substituting this into the expression for trace(PQ), we get:

trace(PQ) = trace(QpP)

Now, let's consider the eigenvalues of the product Qp. Since p is a projection, its eigenvalues are either 0 or 1. Therefore, the eigenvalues of QpP are also either 0 or 1.

Let λ1, λ2, ..., λn be the eigenvalues of QpP (where n is the dimension of V). Then we have:

trace(QpP) = λ1 + λ2 + ... + λn

Since each eigenvalue is either 0 or 1, we have:

trace(QpP) = number of 1's in the list λ1, λ2, ..., λn

Therefore, trace(QpP) is a non-negative integer. In other words, trace(QpP) ≥ 0.

Since trace(PQ) = trace(QpP), we have shown that trace(PQ) ≥ 0, as required.

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Answer: The third image.

When a template is created, dummy data is used to verify the formulas in the template.

A template is a pre-designed format or layout that can be used for various purposes, such as creating reports, invoices, or budgets. One of the most important aspects of creating a template is ensuring that all the formulas and calculations used in it are accurate and produce the desired results.

To verify the formulas in a template, dummy data is often used. Dummy data is a set of fictitious or meaningless data that is used to test and verify the functionality of a system or application. In the context of templates, dummy data is used to ensure that the formulas and calculations used in the template are working correctly.

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Answer:

  5√30 ≈ 27.386 square units

Step-by-step explanation:

You want the area of the portion of the plane 5x +2y +z = 10 that lies in the first octant.

The axis-intercepts are found by setting the other variables to zero.

  x-intercept: 5x = 10   ⇒   x = 2

  y-intercept: 2y = 10   ⇒   y = 5

  z-intercept: z = 10

The boundaries of the triangular first-octant portion of the plane will be the lines between these intercepts. The length of each boundary can be found using the distance formula. For example, the length in the X-Y plane will be ...

  d = √((x2 -x1)² +(y2 -y1)² +(z2 -z1)²)

  d = √((0 -2)² +(5 -0)² +(0 -0)²) = √(4+25) = √29

The first attachment shows the other side lengths to be ...

  Y-Z plane: 5√5

  X-Z plane: 2√26

The area of the triangular portion of the plane can be found using Heron's formula. For semi-perimeter s and side lengths a, b, c, the area is ...

  A = √(s(s -a)(s -b)(s -c)) . . . . . . where s = (a+b+c)/2

The second attachment shows the area to be 5√30 ≈ 27.386 square units.

#95141404393

The approximate value of the surface area is 4.32 square units.

To find the area of the surface, we need to first find the equation of the plane and then determine the portion of the plane that lies in the first octant.

The equation of the plane can be written as:

z = 10 - 5x - 2y

To determine the portion of the plane that lies in the first octant, we need to find the points where the plane intersects the x, y, and z axes. Setting x = 0, y = 0, and z = 0 in the equation of the plane, we get:

z = 10 (when x = 0 and y = 0)

y = 5x (when z = 0 and y = 0)

x = 2 (when z = 0 and x = 0)

The portion of the plane that lies in the first octant is bounded by the x-axis, the y-axis, and the line y = 5x. To find the area of this surface, we can use a double integral:

∬R √(1+f_x^2+f_y^2) dA

where R is the region bounded by the x-axis, the y-axis, and the line y = 5x, and f(x,y) = 10 - 5x - 2y.

Converting to polar coordinates, we have:

x = r cosθ

y = r sinθ

The line y = 5x becomes y = 5r cosθ, and the region R is described by:

0 ≤ r ≤ 2sinθ

0 ≤ θ ≤ π/4

The surface area is then:

A = ∫(0 to π/4) ∫(0 to 2sinθ) √(1+f_r^2+f_θ^2) r dr dθ

Using f(x,y) = 10 - 5x - 2y, we can find:

f_r = -5

f_θ = -2r

So we have:

A = ∫(0 to π/4) ∫(0 to 2sinθ) √(1+25+4r^2) r dr dθ

= ∫(0 to π/4) ∫(0 to 2sinθ) √(29+4r^2) r dr dθ

This integral is difficult to evaluate analytically, but it can be approximated using numerical methods or a computer algebra system.

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The correct equation that states the relationship between the dot distances and speeds is dot speed = dot distance / 5 hr. The correct option is (b).

This is because the dots are observed to be moving away from the observer at a constant rate, which is given in units of centimeters per hour.

This equation correctly represents the observed relationship because it implies that the speed at which a dot is moving away from you is directly proportional to its distance from you.

The coefficient of proportionality in this case is 1 cm/hr per 5 cm of distance.

This means that for every 5 centimeters of distance, the dot's speed increases by 1 centimeter per hour. As the distance increases, the speed also increases proportionally.

This is consistent with the information where dots at greater distances have higher speeds.

Therefore, to relate the dot distance (in centimeters) to the dot speed (in centimeters per hour), we need to divide the dot distance by the time taken for the dot to move that distance, which is 5 hours for all dots.

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a. The coefficient of x1 (-8.69) suggests that for every one-year increase in length of service, the job satisfaction test score decreases by 8.69 points, holding all other variables constant. The coefficient of x2 (13.5) suggests that for every one-dollar increase in wage rate, the job satisfaction test score increases by 13.5 points, holding all other variables constant.

b. To estimate the job satisfaction test score for an employee with four years of service and a wage rate of $6.50 per hour, we plug in the values for x1 and x2 in the regression equation:

^y = 14.4 – 8.69x1 + 13.5x2

^y = 14.4 – 8.69(4) + 13.5(6.5)

^y = 14.4 – 34.76 + 87.75

^y = 67.39

Therefore, the estimated job satisfaction test score for an employee with four years of service and a wage rate of $6.50 per hour is 67.39.

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Null hypothesis: µ ≤ 4.2

Alternative hypothesis: µ > 4.2

The null hypothesis (H₀) states that the average monthly amount spent on office equipment and supplies is less than or equal to 4.2 hundred dollars. The alternative hypothesis (H₁) suggests that the average monthly amount spent is greater than 4.2 hundred dollars.

In this study, the accountant wants to test whether there is sufficient evidence to support the claim that the average monthly amount spent on office equipment and supplies exceeds 4.2 hundred dollars. The accountant collects data from a sample of 33 months, with a sample mean of 4.65 hundred dollars. The population standard deviation, based on past data, is known to be 0.9 hundred dollars.

By setting the significance level at 10%, the accountant will conduct a hypothesis test to evaluate whether the sample evidence supports the alternative hypothesis that the true population mean (µ) is greater than 4.2 hundred dollars. If the test provides strong evidence against the null hypothesis, the accountant may conclude that there is sufficient support for the claim that the average monthly amount s

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Without solving for x the exact value of sin(2x) is √99 / 50.

To find the exact value of sin(2x) given that sin(x) = 1/10 and x is in quadrant I, we can use the double-angle identity for sine. The double-angle identity for sine states that sin(2x) = 2sin(x)cos(x).

Given that sin(x) = 1/10, we need to find the value of cos(x) to determine sin(2x). To do this, we can use the Pythagorean identity for cosine, which states that cos^2(x) + sin^2(x) = 1.

Substituting sin(x) = 1/10, we have:

cos^2(x) + (1/10)^2 = 1

cos^2(x) + 1/100 = 1

cos^2(x) = 1 - 1/100

cos^2(x) = 99/100

Taking the square root of both sides (remembering that cos(x) is positive in quadrant I), we have:

cos(x) = √(99/100)

cos(x) = √99 / 10

Now we have the values of sin(x) and cos(x) in quadrant I. We can use the double-angle identity for sine to find sin(2x):

sin(2x) = 2sin(x)cos(x)

sin(2x) = 2 * (1/10) * (√99 / 10)

sin(2x) = √99 / 50

Therefore, the exact value of sin(2x) is √99 / 50.

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The mean number of televisions per household is approximately 1.93, the variance is approximately 0.7387, and the standard deviation is approximately 0.859.

To find the mean, variance, and standard deviation of the number of televisions per household, we use the formulas:

Mean = Σ(X * P(X))

Variance = Σ[(X - Mean)^2 * P(X)]

Standard deviation = sqrt(Variance)

Using the given probabilities, we can calculate the mean as follows:

Mean = (10.32) + (20.51) + (30.12) + (40.05) = 1.93

To calculate the variance, we first need to calculate the deviations from the mean:

1 - 1.93 = -0.93

2 - 1.93 = 0.07

3 - 1.93 = 1.07

4 - 1.93 = 2.07

Using these deviations and the given probabilities, we can calculate the variance as follows:

Variance = (-0.93^2 * 0.32) + (0.07^2 * 0.51) + (1.07^2 * 0.12) + (2.07^2 * 0.05) = 0.7387

Finally, we can calculate the standard deviation as the square root of the variance:

Standard deviation = sqrt(0.7387) = 0.859

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Answer:

23.25

Step-by-step explanation:

Calculate the discount amount: 25% of $31

Discount = (25 / 100) * $31

Discount = 0.25 * $31

Discount = $7.75

Subtract the discount amount from the regular price:

Sale price = Regular price - Discount

Sale price = $31 - $7.75

Sale price = $23.25

sale price of the jeans is $23.25.

Their bill will increase by $32.99 each month when they make the switch to the unlimited data plan.

To calculate the increase in their monthly bill when switching to the unlimited data plan, we need to compare the costs before and after the switch. Before the switch: Price for 30GB data + unlimited talk and text: $55, Line costs: $19 per line, Taxes: $9.50 per month, Total cost before the switch: $55 + $19 (2 lines) + $9.50 = $102.50 per month.

After the switch: Price for unlimited data + unlimited talk and text: $85, Line costs: $19 per line, Taxes: $12.49 per month, Total cost after the switch: $85 + $19 (2 lines) + $12.49 = $135.49 per month. To determine the increase in their bill, we subtract the total cost before the switch from the total cost after the switch: Increase in bill = Total cost after the switch - Total cost before the switch, Increase in bill = $135.49 - $102.50, Increase in bill = $32.99 per month.

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Answer:

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Point A is the center and both B and D are on the circle.

As per definition, the length between the center and any point on the circle is its radius.

Therefore both AB and AD have same length of 12 cm as radius.

The absolute minimum value of 4 - 6x on an interval, we need to take the derivative of the function does not have an absolute minimum value on the interval (-∞, ∞).

Then, we can plug those critical points and the endpoints of the interval into the original function to find the absolute minimum value.
The derivative of 4 - 6x is -6, which is a constant. Setting it equal to zero, we get -6 = 0, which has no solution. Therefore, there are no critical points.

Next, we need to evaluate the function at the endpoints of the interval. Since no interval is given in the question, we will assume that the interval is the set of all real numbers, denoted by (-∞, ∞).
When x = -∞, we have 4 - 6x = 4 - 6(-∞) = ∞, which is not a finite value. Similarly, when x = ∞, we have 4 - 6x = 4 - 6(∞) = -∞, which is also not a finite value.

Therefore, the function does not have an absolute minimum value on the interval (-∞, ∞).

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The given integral can be rewritten as [tex]e^-7* (-7) dx[/tex] and used the general exponential rule to evaluate it as (-1/7) e^-7x + C.

Now, we rewrite the given integral as follows:

−7e−7x dx = e−7x (−7) dx

Next, we can use the general exponential rule, which states that ∫ [tex]e^u du = e^u + C,[/tex] where C is the constant of integration.

So, substituting u = -7x, we get:

e−7x (−7) dx = (-1/7) ∫ [tex]-7e^-7x d(-7x)[/tex]

Using integration by substitution, where v = -7x and dv/dx = -7, we get:

(-1/7) ∫ [tex]-7e^-7x  d(-7x)[/tex] = (-1/7) ∫ [tex]e^v dv[/tex]

Simplifying this, we get:

(-1/7) ∫ [tex]e^v dv = (-1/7) e^v + C[/tex]

Substituting back for v = -7x, we get:

[tex](-1/7) e^{(-7x)} + C[/tex]

Therefore, we have rewritten the given integral as [tex]e^-7x (-7) dx[/tex]  and used the general exponential rule to evaluate it as [tex](-1/7) e^{-7x} + C.[/tex]

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Answer:

First 3 positions can be filled in 7C3 ways; last 2 positions can be filled in 6C2 ways.

Evaluate each of these two numbers, and MULTIPLY them to produce the answer.

Step-by-step explanation:

Answer:

The answer is 615.75in .

Answer:

A≈615.75in²

Step-by-step explanation:

A=πr2=π·142≈615.75216in²

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The girls did break the school's record, and they were 78/5 minutes slower than the record time.

To determine whether the girls broke the school's record, we need to calculate their total time and compare it to the record time.

From the chart, we can see that Barbara's time is 3 minutes, Donna's time is 10 minutes, Cindy's time is 2 minutes, and Nicole's time is 5 minutes.

To calculate the total time, we sum up the individual times:

Total time = Barbara's time + Donna's time + Cindy's time + Nicole's time

= 3 + 10 + 2 + 5

= 20 minutes

The school's record time is given as 2 12/5 minutes. To compare the two times, we need to convert the record time to a common denominator.

[tex]2 12/5 minutes = (2 \times 5 + 12)/5 minutes = 22/5 minutes[/tex]

Now we can compare the total time with the record time:

Total time = 20 minutes

Record time = 22/5 minutes

Since the girls' total time of 20 minutes is less than the record time of 22/5 minutes, we can conclude that they did break the school's record.

To determine how much faster they were, we subtract the total time from the record time:

Record time - Total time = 22/5 - 20

= 22/5 - 100/5

= (22 - 100)/5

= -78/5

The result is -78/5, which means they were 78/5 minutes slower than the record time.

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The solution is c = 12.1, A = 28°, and B = 46°, which corresponds to option B.

To solve the triangle with given information a = 6, b = 9, and C = 106°, we will use the Law of Sines. We need to find side c and angles A and B.

1. First, we find angle A:

(sin A) / a = (sin C) / b

(sin A) / 6 = (sin 106°) / 9

(sin A) = (6 * sin 106°) / 9

A = sin^(-1)((6 * sin 106°) / 9)

A ≈ 28° (rounded to the nearest degree)

2. Next, we find angle B using the fact that the sum of the angles in a triangle is 180°:

B = 180° - A - C

B = 180° - 28° - 106°

B ≈ 46° (rounded to the nearest degree)

3. Finally, we find side c using the Law of Sines again:

(sin C) / c = (sin A) / a

(sin 106°) / c = (sin 28°) / 6

c = (sin 106°) * 6 / (sin 28°)

c ≈ 12.1 (rounded to the nearest tenth)

So, the solution is c = 12.1, A = 28°, and B = 46°, which corresponds to option B.

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The correct sequence that shows the increasing density of the three phases of water is C) liquid>solid>gas.


- Liquid: In the liquid phase, water molecules are closely packed, but they can still move and slide past each other, resulting in a relatively high density.
- Solid: In the solid phase (ice), water molecules are arranged in a hexagonal lattice structure. This structure has open spaces between the molecules, making it less dense than liquid water.
- Gas: In the gas phase (water vapor), water molecules are widely spaced and move freely, resulting in the lowest density among the three phases.

The increasing density of the three phases of water follows the sequence liquid>solid>gas, which corresponds to option C.

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The linear programming problem can be formulated as follows:

Maximize: P = 25x + 40y (profit function)

Subject to:

90x + 160y ≤ 600,000 (manufacturing cost constraint)

x + y ≤ 2,400 (demand constraint)

x, y ≥ 0 (non-negativity constraint)

Maximize: P = 25x + 40y (profit function) - This objective function represents the total profit, which is the sum of the profits from selling model A (25x) and model B (40y).

Subject to: 90x + 160y ≤ 600,000 (manufacturing cost constraint) - This constraint ensures that the total cost of manufacturing model A and model B does not exceed $600,000.

Subject to: x + y ≤ 2,400 (demand constraint) - This constraint ensures that the total number of portable printers produced (model A + model B) does not exceed 2,400 units, which represents the total demand.

Subject to: x, y ≥ 0 (non-negativity constraint) - This constraint ensures that the number of units produced for each model cannot be negative; they must be non-negative values.

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To make the given system dependent, we need to find a value of c that makes the two equations proportional or equivalent. This occurs when the coefficients of x and y in both equations are proportional to each other.

Let's compare the coefficients of x and y in both equations:

For the first equation, 10x + 5y = -5, the ratio of the coefficients is 10/5 = 2.

For the second equation, 6x + 3y = c, the ratio of the coefficients is 6/3 = 2.

To make the system dependent, the ratio of the coefficients must be the same for both equations. Therefore, we need to find a value of c that makes the ratio of coefficients in the second equation equal to 2.

Setting 6/3 equal to 2, we have:

6/3 = 2

Dividing both sides by 2:

2 = 2

The equation 2 = 2 is always true, regardless of the value of c. Therefore, any value of c will make the system dependent, as the lines will coincide.

In conclusion, there is no specific value of c that makes the system dependent. Any value of c will result in a dependent system with coinciding lines.

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Answer:

Money left over = -$388,600

Step-by-step explanation:

To calculate how much money you have left over from the selling of your home, we need to first calculate the profit you made from selling your home, and then subtract the cost of the new home and the down payment.

Profit from selling the home = selling price - purchase price

Profit = $389,000 - $264,000

Profit = $125,000

Next, we need to subtract the cost of the new home and the down payment:

Cost of new home = $428,000

Down payment = 20% of $428,000 = $85,600

Total cost of new home and down payment = $428,000 + $85,600 = $513,600

Finally, we can calculate how much money you have left over:

Money left over = Profit - (Cost of new home + Down payment)

Money left over = $125,000 - $513,600

Money left over = -$388,600

Since the result is negative, it means you do not have any money left over and you need to cover the difference of $388,600.

Answer:

$39,400

Step-by-step explanation:

Calculate the profit from the sale: Selling price - Original purchase price = $389,000 - $264,000 = $125,000.

Calculate the deposit paid for the new home: 20% of the purchase price = 0.2 x $428,000 = $85,600.

Subtract the deposit paid from the net proceeds of the sale: $125,000 - $85,600 = $39,400 (money left over).

Bassam's uncle eats 4 figs when he visits.

Let's assume the number of figs that Bassam's uncle eats is "x". According to the given information, Bassam eats 8 figs, which is twice the number his uncle eats. So, we can set up the equation:

8 = 2x

To find the value of x, we can divide both sides of the equation by 2:

8/2 = x

x = 4

Therefore, Bassam's uncle eats 4 figs when he visits.

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There is a one-to-one correspondence between the even and odd divisors, we can conclude that an even integer k, not divisible by 4, has the same number of even and odd divisors.

To prove that an even integer k, not divisible by 4, has the same number of even and odd divisors, we can establish a bijection between them.

Let's consider an even integer k. Since k is even, it can be expressed as k = 2n, where n is an integer.

First, let's consider the even divisors of k. An even divisor of k is of the form 2m, where m is an integer. Since k = 2n, we can write the even divisor as 2m = 2n. Simplifying this equation, we get m = n.

Now, let's consider the odd divisors of k. An odd divisor of k is of the form 2m + 1, where m is an integer. Again, we can write this odd divisor as 2m + 1 = 2n. Simplifying this equation, we get m = n - 1/2.

Now, let's establish a bijection between the even and odd divisors. For every even divisor 2m, we can find a corresponding odd divisor 2m + 1, and vice versa. The correspondence is given by the equation m = n for even divisors and m = n - 1/2 for odd divisors.

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The simplified expression is 1+14i.

Given that an expression including the complex numbers (-2 + 3i) - (-3 - 11i)

We need to solve the expression,

= (-2 + 3i) - (-3 - 11i)

= -2 + 3i + 3 + 11i

= 1 + 14i

Hence the simplified expression is 1+14i.

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A power series expansions for 1/(1-x)², 1/(1-x)³, 1/(1-x)⁴ are:

1/(1-x)² = 1 + 2x + 3x² + 4x³ + ...

1/(1-x)³ = 2 + 6x + 12x² + ...

1/(1-x)⁴ = 6 + 24x + ...

The radius of convergence for these power series is 1.

To find the power series expansions for the given functions, we can differentiate the power series expansion for 1/(1-x) term by term. Let's start with 1/(1-x).

Power series expansion for 1/(1-x):

1/(1-x) = 1 + x + x² + x³ + ...

Differentiating both sides with respect to x:

d/dx (1/(1-x)) = d/dx (1 + x + x² + x³ + ...)

1/(1-x)² = 1 + 2x + 3x² + 4x³ + ...

Power series expansion for 1/(1-x)²:

1/(1-x)² = 1 + 2x + 3x² + 4x³ + ...

Now, let's differentiate the power series expansion for 1/(1-x)² to find the power series expansion for 1/(1-x)³.

Differentiating both sides with respect to x:

d/dx (1/(1-x)²) = d/dx (1 + 2x + 3x² + 4x³ + ...)

1/(1-x)³ = 2 + 6x + 12x² + ...

Power series expansion for 1/(1-x)³:

1/(1-x)³ = 2 + 6x + 12x² + ...

Similarly, we can differentiate the power series expansion for 1/(1-x)³ to find the power series expansion for 1/(1-x)⁴.

Differentiating both sides with respect to x:

d/dx (1/(1-x)³) = d/dx (2 + 6x + 12x² + ...)

1/(1-x)⁴ = 6 + 24x + ...

Power series expansion for 1/(1-x)⁴:

1/(1-x)⁴ = 6 + 24x + ...

The radius of convergence for each of these power series expansions is 1 since they are centered around x = 0.

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Check the picture below.

so the parabola looks more or less like so, with a vertex at (-3 , 10) and the directrix above it at -79/8 or namely -9⅞, now, the directrix is just 1/8 of a unit above the vertex, that's our "p" distance, and since the directrix is above the vertex, the parabola is opening downwards and "p" is negative.

[tex]\textit{vertical parabola vertex form with focus point distance} \\\\ 4p(y- k)=(x- h)^2 \qquad \begin{cases} \stackrel{vertex}{(h,k)}\qquad \stackrel{focus~point}{(h,k+p)}\qquad \stackrel{directrix}{y=k-p}\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix}\\\\ \stackrel{p~is~negative}{op ens~\cap}\qquad \stackrel{p~is~positive}{op ens~\cup} \end{cases} \\\\[-0.35em] ~\dotfill[/tex]

[tex]\begin{cases} h=-3\\ k=-10\\ p=-\frac{1}{8} \end{cases}\implies 4(-\frac{1}{8})(~~y-(-10)~~) = (~~x-(-3)~~)^2 \\\\\\ -\cfrac{1}{2}(y+10)=(x+3)^2\implies y+10=-2(x+3)^2\implies \boxed{y=-2(x+3)^2-10}[/tex]

The answer is ⟨y(x) = (-1/27)(5 - 3x)^9 + C, where C is an arbitrary constant⟩.

The given differential equation is:

y' = (5 - 3x)^8

We can separate the variables and integrate both sides:

dy/dx = (5 - 3x)^8

dy = (5 - 3x)^8 dx

Integrating both sides, we get:

∫dy = ∫(5 - 3x)^8 dx

y = (-1/27)(5 - 3x)^9 + C

where C is an arbitrary constant of integration.

Therefore, the general solution of the differential equation is:

y(x) = (-1/27)(5 - 3x)^9 + C

where C is an arbitrary constant.

Hence, the answer is ⟨y(x) = (-1/27)(5 - 3x)^9 + C, where C is an arbitrary constant⟩.

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The variance of a binomial distribution is np(1-p), so probability of success is 0.9798 (rounded to 4 decimal places).

Using the hypergeometric distribution, the probability of selecting exactly 3 rods that meet the specification is:

P(X=3) = (12 choose 3) * (3 choose 3) / (15 choose 6) ≈ 0.2727

To find the standard deviation of X, we use the formula:

σX = sqrt(np(1-p)/N)

where n is the sample size (6), p is the probability of success (12/15), and N is the population size (15).

σX = sqrt(6 * (12/15) * (3/15)) ≈ 0.925

Using the binomial distribution formula with n = 6 and p = 12/15 = 0.8:

P(X = 3) = (6 choose 3) * (0.8)^3 * (0.2)^3 = 0.27648

The variance of a binomial distribution is np(1-p), so:

σX = sqrt(np(1-p)) = sqrt(60.80.2) = 0.9798 (rounded to 4 decimal places)

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The measure of ∠A is 37 degrees, ∠B is 90 degrees and ∠C is 53 degrees in the triangle ABC.

Let us find ∠A by using sine function.

We know that sine function is a ratio of opposite side and hypotenuse.

The opposite side of ∠A is BC, which is 21.

The hypotenuse side of ∠A is AC, which is 35.

Sin∠A =21/35

Sin∠A =0.6

∠A =Sin⁻¹(0.6)

∠A =36.87 degrees.

∠B is 90 degrees.

By angle sum property the sum of three angles is 180 degrees in triangle.

∠A+∠B+∠C=180

37+90+∠C=180

127+∠C=180

Subtract 127 from both sides:

∠C=180-127

∠C=53 degrees.

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