culate these. Increase $45 by 20%.​

The mass of the neptunium-230  is 6.25 g

The mass of the plutonium is 3 mg

The duration it takes for half of a substance to deteriorate or disappear is known as its half-life. This idea is frequently used to describe the breakdown of certain chemicals .

We know that;

N/No = (1/2)^t/t1/2

N = Mass left at time t

No = amount initially present

t = time taken

t1/2 = half life

Then;

N/100 = (1/2)^18.4/4.6

N = (1/2)^18.4/4.6 * 100

N = 6.25 g

The amount of the plutonium left is;

N/No = (1/2)^t/t1/2

N/12.8 =  (1/2)^250/120

N =  (1/2)^250/120 * 12.8

N = 3 mg

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The third, fourth, and fifth smallest eigenvalues and their corresponding eigenvectors for the Laplacian matrix constructed in exercise 10.4.1(c) are 0.753 and [-0.271, -0.090, 0.103, 0.248, 0.451, 0.506], 0.926 and [-0.186, -0.296, -0.107, 0.435, 0.518, -0.580], and 1.036 and [-0.126, -0.259, 0.309, 0.368, -0.783, 0.350], respectively.

The Laplacian matrix constructed in exercise 10.4.1(c) is a symmetric matrix with a size of 5 x 5. To find the eigenvalues and eigenvectors, we can use a linear algebra software package or a calculator that has this functionality.

The third smallest eigenvalue of this Laplacian matrix is approximately 0.2361, and its corresponding eigenvector is [0.4472, 0.3293, -0.7397, 0.2403, -0.3239].

The fourth smallest eigenvalue is approximately 0.5273, and its corresponding eigenvector is [0.5326, 0.5569, 0.3211, -0.0045, -0.5676].

The fifth smallest eigenvalue is approximately 1.0000, and its corresponding eigenvector is [-0.4418, 0.4418, -0.4418, 0.4418, -0.4418].

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--The complete question is,What are the third and subsequent smallest eigenvalues and their eigenvectors for the Laplacian matrix constructed in exercise 10.4.1(c)?--

Answer: D)

Step-by-step explanation:

In A) we have two of the same y-values, which means x is NOT a function of y, but y is still a function of x.

In B), the graph passes the vertical line test so it is a function!

In C), if we were to graph this function, it's a linear graph and will pass the vertical line test.

In D) we have an x-value (3) that is connected to two different y-values (1 and 6) so it is NOT a function of x. This goes against the idea that for every x-value, there is one y-value.

Hope this helps!

Answer:

answer d

Step-by-step explanation:

bye have a great day!!! :D please give brainliest if u want k bye

This is the desired expression for the total solution.

To solve the differential equation, we assume that the solution is of the form y(t) = c(t)eat. Then, we have:

y'(t) = c'(t)eat + aceat

y(0) = c(0)e0a = y

Substituting these into the differential equation, we get:

c'(t)eat + aceat = a(c(t)eat) + bu(t)

Simplifying this equation, we get:

c'(t) = bu(t)e-at

Integrating both sides from 0 to t, we get:

c(t) - c(0) = ∫0t bu(r)e-ar dr

Multiplying both sides by eat, we get:

y(t) - y = e-at(c(0) + ∫0t bu(r)e-ar dr)

Rewriting the right-hand side in terms of a constant c and a function h(t), we get:

y(t) = e-at y + c ∫0t bu(r)e-ar dr

where c = eay and h(t) = bu(t)e-at. This is the desired expression for the total solution.

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The statement "The second derivative test can always be used to determine whether a critical number is a relative extremum" is False.

The second derivative test is a useful method for determining if a critical number is a relative extremum (maximum or minimum).

However, it cannot always be used, as it is inconclusive when the second derivative is equal to zero or undefined. In these cases, other methods such as the first derivative test or analyzing the function's behavior around the critical number must be used.

To apply the second derivative test, follow these steps:

1. Find the first derivative (f') of the function.

2. Identify the critical numbers by setting f' equal to zero or where it's undefined.

3. Find the second derivative (f'') of the function.

4. Evaluate f'' at each critical number. If f'' > 0, it's a relative minimum; if f'' < 0, it's a relative maximum. If f'' = 0 or undefined, the test is inconclusive.

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The number of shirts on each rack is n = 24 shirts

Given data ,

Number of shirts = 97

Number of racks = 4

97 shirts / 4 racks = 24.25 shirts per rack

However, it's not possible to have a fraction of a shirt, as shirts are discrete items and cannot be divided into smaller parts. Therefore, we cannot evenly distribute 24.25 shirts on each rack.

On simplifying the equation , we get

In this instance, the remaining is 0.25 shirts, indicating that 0.25 shirts will remain after the shirts have been evenly distributed across the racks. We cannot utilize only a little portion of a garment, thus the rest is usually thrown away or not used.

Hence , it would be most practical to arrange the shirts as evenly as possible, giving each rack 24 shirts. The final shirt might then be put on any of the racks or set aside separately

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The required answer is the series is divergent  we cannot find its sum.

The given series is an alternating series with decreasing absolute values of its terms. Thus, we can use the Alternating Series Test to determine its convergence. According to this test, if the absolute values of the terms in an alternating series are decreasing and approach zero, then the series is convergent.

In this case, the absolute values of the terms are:

|4|, |-16/3|, |64/9|, |-256/27|, ...

which are decreasing and approach zero. Therefore, we can conclude that the series is convergent.
the property that different transformations of the same state have a transformation to the same end state. Convergent series, the process of some functions and sequences approaching a limit under certain conditions.
To find its sum, we can use the formula for the sum of an alternating series:

sum = a - a/2 + a/3 - a/4 + a/5 - ...

where a is the first term in the series. In this case, a = 4. Therefore, we have:

sum = 4 - 4/2 + 4/3 - 4/4 + 4/5 - ...

Simplifying this expression, we get:

sum = 4(1 - 1/2 + 1/3 - 1/4 + 1/5 - ...)
This is an alternating series, and to test its convergence, we can apply the Alternating Series Test.
which is a well-known series called the harmonic series. It is known to be divergent, which means that the sum of our alternating series is also divergent. Therefore, we cannot find its sum.

To determine whether the series 4 - (16/3) + (64/9) - (256/27) + ... is convergent or divergent, and if convergent, find its sum, we'll first identify the pattern and express it as a general series formula.
the property that different transformations of the same state have a transformation to the same end state. Convergent series, the process of some functions and sequences approaching a limit under certain conditions.

Notice that the terms are alternating in sign and the numerators are powers of 4, while the denominators are powers of 3. We can express the series as:

∑((-1)^(n+1) * (4^n) / (3^n)) for n = 1 to infinity.

This is an alternating series, and to test its convergence, we can apply the Alternating Series Test. The test has two conditions:

1. The absolute value of the terms must be decreasing: |a_(n+1)| <= |a_n|.
2. The limit of the absolute value of the terms must be 0: lim(n->infinity) |a_n| = 0.

For condition 1:
|a_(n+1)| = |(4^(n+1))/(3^(n+1))|
|a_n| = |(4^n)/(3^n)|

Since 4/3 > 1, the terms in absolute value will be decreasing.

For condition 2:
lim(n->infinity) |(4^n)/(3^n)| = lim(n->infinity) |(4/3)^n|

As n approaches infinity, (4/3)^n also approaches infinity, so the limit is not 0.

Since the second condition is not met, the Alternating Series Test fails, and the series is divergent. Therefore, we cannot find its sum.

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The P-value is approximately 0.003. The 95% upper confidence interval on the true process fraction nonconforming is (0.116, 1).

CI are typically estimated at a confidence level of 95% in accordance with the conventional acceptance of statistical significance at a P-value of 0.05 or 5%. In general, the null hypothesis shouldn't fall inside the 95% CI if an observed result is statistically significant at a P-value of 0.05.

We can use the following calculation to create a 95% upper confidence interval on the actual process percent nonconforming:

p+ z√(p(1-p)/n) ≤ p ≤ 1

Plugging in the values, we get:

0.13 + 1.96*√(0.13(1-0.13)/500) ≤ p ≤ 1

Simplifying, we get:

0.116 ≤ p ≤ 1

Therefore, the 95% upper confidence interval on the true process fraction nonconforming is (0.116, 1).

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Answer:

Pretty sure It's $1.40

Step-by-step explanation:

Sylas may not have enough money to buy the shoes with sales tax included. The sales tax on the shoes would be $18.60 (6% of $310), bringing the total cost to $328.60. This leaves Sylas with only $1.40 in his account. However, it is unclear if Sylas has any other sources of income or if he needs to pay for any other expenses. As for Tobias, his grandfather's account had earned $3,000 in simple interest. It is unclear if this is the full balance of the account or just the interest earned.

Answer:

1.40

Step-by-step explanatiom:

The value of dI/da can be obtained by using the chain rule, which states that dI/da = (dI/dt) / (da/dt).

Given that I_dt = Floo and y = cos(4t^2) and y_dt = -8t*sin(4t^2), we can solve for dI/da by finding da/dt when t = 5 and substituting the values in the formula.

Let's first apply the chain rule to the equation y = cos(4t^2) to find dy/dt:

dy/dt = -sin(4t^2) * d/dt (4t^2) = -8t*sin(4t^2)

Next, we can use the given value I_dt = Floo, which means that dI/dt = 1.

To find da/dt, we need to differentiate a with respect to t. However, the value of a is not given explicitly in the equation. Therefore, we need to use the given information that when t = 5, a = -7. This means that we can write the equation for a as follows:

a = 5t - 42

Taking the derivative of both sides with respect to t, we get:

da/dt = 5

Now we can substitute the values we have found into the formula for dI/da:

dI/da = (dI/dt) / (da/dt) = 1 / 5 = 1/5

Therefore, the exact value of dI/da is 1/5.

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The  f(x) = -3x + 4. 1 is a bijection as it is both one to one and onto.

We need to carry out two steps to check for the bijection-
1) To determine if f is one-to-one, we need to check if different inputs yield different outputs.

So, let's suppose f(a) = f(b) for some real numbers a and b.

Then, -3a + 4 = -3b + 4, which simplifies to -3a = -3b.

Dividing by -3 (which is allowed since -3 is not zero), we get a = b.

Therefore, f is one-to-one.

2) To determine if f is onto (also known as surjective), we need to check if every output has at least one corresponding input.

Let's suppose y is a real number.

Then, we need to solve the equation f(x) = y for x.

That is, -3x + 4 = y,  which gives us x = (4 - y)/3.

Therefore, every real number y has a preimage under f, and f is onto (surjective).

3) Since f is both one-to-one and onto, it is a bijection.
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The  f(x) = -3x + 4. 1 is a bijection as it is both one to one and onto.

We need to carry out two steps to check for the bijection-
1) To determine if f is one-to-one, we need to check if different inputs yield different outputs.

So, let's suppose f(a) = f(b) for some real numbers a and b.

Then, -3a + 4 = -3b + 4, which simplifies to -3a = -3b.

Dividing by -3 (which is allowed since -3 is not zero), we get a = b.

Therefore, f is one-to-one.

2) To determine if f is onto (also known as surjective), we need to check if every output has at least one corresponding input.

Let's suppose y is a real number.

Then, we need to solve the equation f(x) = y for x.

That is, -3x + 4 = y,  which gives us x = (4 - y)/3.

Therefore, every real number y has a preimage under f, and f is onto (surjective).

3) Since f is both one-to-one and onto, it is a bijection.
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The optimal solution is (b) and occurs where x = 1 and y = 0

The given linear program is:

Max Z = 5x + 3y
Subject To: x - y ≤ 6, x ≤ 1

Let's analyze the possible solutions:

a) If the linear program is infeasible, it means that there is no feasible region satisfying all constraints. However, there is a feasible region in this case, so this option is incorrect.

b) If the optimal solution occurs where x = 1 and y = 0, the function value would be Z = 5(1) + 3(0) = 5, and it satisfies the constraints. So, this option is possible.

c) If the objective function value is 5, this corresponds to the result in option b, where x = 1 and y = 0.

d) If the optimal solution occurs where x = 0 and y = 1, the function value would be Z = 5(0) + 3(1) = 3. This solution also satisfies the constraints, but it does not yield the maximum value for the objective function.

Thus, the optimal solution is (b) and occurs where x = 1 and y = 0

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We need at least n = 127 terms to ensure the remainder is less than 0.0005 for infinite series.

We can use the formula for the remainder of a convergent geometric series:

R = a(1 - [tex]r^n[/tex])/(1 - r)

where R is the remainder, a is the first term, r is the common ratio, and n is the number of terms.

In this case, a = 2/5, r = (1/5)² = 1/25, and we want R < 0.0005. Substituting these values and solving for n, we get:

0.0005 > (2/5)(1 - [tex](1/25)^n[/tex])/(1 - 1/25)

0.0005(24/25) > 1 - [tex](1/25)^n[/tex]

[tex](1/25)^n[/tex] > 0.9992

n log(1/25) > log(0.9992)

n < log(0.9992)/log(1/25)

n > 126.06

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Answer:2.82

Step-by-step explanation:

Using the formula

A=πr2

Solving forr

r=A

π=25

π≈2.82095

r≈2.82

The radius of the circle is approximately 2.82 meters.

To find the radius of a circle with a given area, you can use the formula for the area of a circle:

Area = π * r^2

In this case, the area is 25 square meters:

25 = π * r^2

To solve for the radius (r), first divide both sides of the equation by π:

25/π = r^2

Now, take the square root of both sides to find the value of r:

r = √(25/π)

r ≈ √(25/3.1416)

r ≈ √(7.9577)

r ≈ 2.82

The linear function graphed is defined as follows:

y = -2x + 5.

The slope-intercept representation of a linear function is given by the equation presented as follows:

y = mx + b

The coefficients of the function and their meaning are described as follows:

The graph crosses the y-axis at y = 5, hence the intercept b is given as follows:

b = 5.

When x increases by 1, y decays by 2, hence the slope m is given as follows:

m = -2.

Thus the equation of the line is given as follows:

y = -2x + 5.

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Answer:

Yes

Step-by-step explanation:

Dance party playlist:

She has 15 rock songs, and 25 country songs, meaning the ratio is:

15:25

Summer barbecue playlist:

She has 9 rock songs, and 15 country songs, meaning the ratio is:

9:15

Having a ratio is the same as dividing, so for the dance party playlist we can divide 15/25, which is 0.6, and do the same for the summer BBQ playlist, 9/15, which is also 0.6

So, yes, both playlists have the same ratio of both songs.

The company's mean annual growth rate over the past six years is 12.9%

To calculate the mean annual growth rate of the company over the past six years, we can use the following formula:

mean annual growth rate = (current value / past value) ^ (1/number of years) - 1

Substituting the given values, we get:

mean annual growth rate = (10 - 4) ^ (1/6) - 1
mean annual growth rate = 0.129 or 12.9% (rounded to one decimal place)

Therefore, the company's mean annual growth rate over the past six years is 12.9%.

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The Jeremy spends $22.27 per day, rounded to 2 decimal places.

the P(A and B) is 0.2500, rounded to 4 decimal places.

the probability of drawing at least one red card is 23/35.

Amount Jeremy spends per day = Mean + (4 * Standard Deviation)

[tex]= $14.27 + (4 * $2.25)[/tex]

= [tex]$22.27[/tex] (rounded to 2 decimal places)

P(A and B) = P(A) * P(B|A)

where P(B|A) is the conditional probability of B given A.

From the information given, we know that P(A) = 0.4000, P(B) = 0.2000, and P(AB) = 0.2500. To find P(B|A), we can use the formula:

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

Substituting the values we have, we get:

P(B|A) = 0.2500 / 0.4000

= 0.6250

Now, we can calculate P(A and B) using the formula above:

P(A and B) = P(A) * P(B|A)

= 0.4000 * 0.6250

= 0.2500

P(at least one red) = 1 - P(no red)

To calculate P(no red), we need to find the probability of drawing two yellow cards in a row, since there are no red cards left after the first card is drawn.

The probability of drawing a yellow card on the first draw is 9/15, or 3/5. The probability of drawing a yellow card on the second draw, given that the first card was yellow, is 8/14, or 4/7

(since there are now 8 yellow cards and 14 cards total remaining). Therefore, the probability of drawing two yellow cards in a row is:

P(no red) = (9/15) * (8/14)

= 24/70

= 12/35

Using the complement rule, we can now find the probability of drawing at least one red card:

P(at least one red) = 1 - P(no red)

= 1 - 12/35

= 23/35

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Therefore, the probability of selecting two red marbles from the jar is 21/496.

The total number of marbles in the jar is 7 + 25 = 32.

The probability of selecting a red marble on the first draw is 7/32.

Since the marble is not replaced, there are only 31 marbles left, including 6 red marbles.

Therefore, the probability of selecting a red marble on the second draw, given that the first marble was red, is 6/31.

To find the probability of both events happening (selecting 2 red marbles), we multiply the probabilities:

(7/32) * (6/31) = 42/992 = 21/496

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Yhey have infinitely many solutions, since any point on the line satisfies both equations.

Given the system of equation in the question;

y = -2x + 2

2y + 4x = 4

To find the number of solutions, we can solve for y in the first equation and substitute it into the second equation:

y = -2x + 2

Plug y = -2x + 2 into the second equation.

2( -2x + 2 ) + 4x = 4

Simplify and solve for x.

-4x + 4 + 4x = 4

4 = 4

Since this equation is always true.

Option C) infinitely many solutions is the correct answer.

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The function contains a natural logarithm, which is only defined for positive values of t. Therefore, the domain of r(t) is t ∈ (0, ∞).

The given vector function is r(t) = (9 - t^2, e^(-3t), ln(t+1)).

To find the domain, we need to determine the range of values of t for which the function is valid.

1. For the first component, 9 - t^2, there is no restriction on t. It can be any real number.

2. For the second component, e^(-3t), there is also no restriction on t. The exponential function is defined for all real numbers.

3. For the third component, in (t+1), the natural logarithm function is defined only for positive values inside the parentheses. So, we must have t + 1 > 0, which implies t > -1.

Considering all the components, the domain of the vector function r(t) is the intersection of their individual domains. In interval notation, the domain of r(t) is (-1, ∞).

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Answer:

True

Step-by-step explanation:

The given statement can be translated into the following algebraic equation:

7x+2=12

Where x is the unknown number.

With the help of the given graph, we know that Juan has read 10 books by day 12.

A graph is a visual representation or diagram that displays facts or values in an organized manner in mathematics.

The points on a graph are typically used to depict the relationships between two or more things.

Making a diagram that shows the link between two or more objects is known as developing a graph.

Create a graph by placing a series of bars on graph paper.

So, for path:
The curve line: y = kx + 5 which passes (4, 10)

10 = 4k + 5, k = 5/4

For Juan:

His reading rate same as Patti's.

Curve line: y = ax + b

a = 5/4

He has not read any book before 4th day.

So, when x = 4, y = 0.

0 = 4a + 5, 4 * 5/4 + b

b = -5

y = 5/4 * -5 (x > 4)

When x = 12, y = 5/4 * 12 - 5 = 15 - 5 = 10

Juan has read 10 books by day 12.

Therefore, with the help of the given graph, we know that Juan has read 10 books by day 12.

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With the help of the given graph, we know that Juan has read 10 books by day 12.

A graph is a visual representation or diagram that displays facts or values in an organized manner in mathematics.

The points on a graph are typically used to depict the relationships between two or more things.

Making a diagram that shows the link between two or more objects is known as developing a graph.

Create a graph by placing a series of bars on graph paper.

So, for path:
The curve line: y = kx + 5 which passes (4, 10)

10 = 4k + 5, k = 5/4

For Juan:

His reading rate same as Patti's.

Curve line: y = ax + b

a = 5/4

He has not read any book before 4th day.

So, when x = 4, y = 0.

0 = 4a + 5, 4 * 5/4 + b

b = -5

y = 5/4 * -5 (x > 4)

When x = 12, y = 5/4 * 12 - 5 = 15 - 5 = 10

Juan has read 10 books by day 12.

Therefore, with the help of the given graph, we know that Juan has read 10 books by day 12.

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Suppose we fix a tree 1. The descendent relation on the nodes of T is a strict partial order. So the option d is correct.

The descendent relation on the nodes of a tree T is a strict partial order if, for any two nodes x and y in T, either x is a descendent of y, y is a descendent of x, or neither x nor y is a descendent of the other.

This means that for any two nodes x and y, either x is a parent node, a grandparent node, etc. of y, or y is a parent node, a grandparent node, etc. of x, or neither x nor y is related in any way.

The descendent relation is a strict partial order because it is a partial order (it is reflexive, antisymmetric, and transitive) and it is also strict (neither x nor y can be a descendent of the other). So the option d is correct.

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The probability of observing no births in a given hour at the hospital is approximately 0.165, or 16.5% when rounded to 3 decimal places.

To find the probability of observing no births in a given hour at the hospital, we'll use the Poisson distribution formula:

P(X = k) = (e^(-λ) * λ^k) / k!

Here, λ (lambda) is the average rate of births per hour (1.8), k is the number of births we want to find the probability for (0 in this case), and e is the base of the natural logarithm (approximately 2.71828).

Step 1: Plug in the values: P(X = 0) = (e^(-1.8) * 1.8^0) / 0!
Step 2: Calculate e^(-1.8) ≈ 0.1653
Step 3: Calculate 1.8^0 = 1
Step 4: Calculate 0! (0 factorial) = 1
Step 5: Substitute the calculated values back into the formula: P(X = 0) = (0.1653 * 1) / 1
Step 6: Simplify the equation: P(X = 0) = 0.1653

The probability of observing no births in a given hour at the hospital is approximately 0.165, or 16.5% when rounded to 3 decimal places.

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Answer:

Hence, the answer is option C. 0.7.

Step-by-step explanation:

To find P(A or B), we need to find the probability of the event that at least one of the 10 students plays basketball or soccer or both.

We can see from the Venn diagram that there are 3 students who play only basketball (Jada, Gabby, and Fran), 1 student who plays only soccer (Mickey), and 3 students who play both basketball and soccer (lan, Juan, and Ella).

Therefore, the total number of students who play basketball or soccer or both is 3 + 1 + 3 = 7.

So, P(A or B) = probability of the event that at least one of the 10 students plays basketball or soccer or both = 7/10 = 0.7.

Hence, the answer is option C. 0.7.

The value of P(A/B)=1/3 from the venn diagram.

It is a branch of mathematics that deals with the occurrence of a random event.

Let A represents a set of students playing Basketball and B represents a set of students playing Soccer

So, A = {Fran, Ian, Juan, Ella}

⇒ n(A) = 4

B = {Mickey, Marcus, Ella}

⇒ n(B) = 3

(A ∩ B) = {Ella}

⇒ n(A ∩ B) = 1

sample space: S = {Fran, Ian, Juan, Ella, Mickey, Marcus, Mai, Karl, Jada, Gabby}

⇒ n(S) = 10

Now, we calculate probability of B and A∩B.

P(B) = n(B)/n(S)

⇒ P(B) = 3/10

Also, P(A∩B) = n(A∩B) / n(S)

⇒ P(A∩B) = 1/10

So, the required probability would be,

⇒ P(A|B) = P(A∩B) ÷ P(B)

⇒ P(A|B) = (1/10) ÷ (3/10)

⇒ P(A|B) = 1/3

Therefore, the value of P(A|B) = 1/3

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To prove that n^3 + 3n^4 is θ (2n^3), we need to show that there exist positive constants c1, c2, and n0 such that

c1 * 2n^3 ≤ n^3 + 3n^4 ≤ c2 * 2n^3 for all n ≥ n0.

First, we will show that n^3 + 3n^4 ≤ c2 * 2n^3 for all n ≥ 1, where c2 = 4. For n ≥ 1, we have

n^3 + 3n^4 ≤ 4n^4 = 2(2n^3) ≤ 2n^3 * c2

Therefore, n^3 + 3n^4 is O(n^3), and hence it is also O(2n^3).

Next, we will show that n^3 + 3n^4 ≥ c1 * 2n^3 for all n ≥ 1, where c1 = 1/4. For n ≥ 1, we have

n^3 + 3n^4 ≥ (1/4) * 3n^4 = (3/4) * 4n^4/4 = (3/4) * 2n^3

Therefore, n^3 + 3n^4 is Ω(n^3), and hence it is also Ω(2n^3).

Since n^3 + 3n^4 is both O(2n^3) and Ω(2n^3), we conclude that n^3 + 3n^4 is θ(2n^3).

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Answer:

Let's start with the formula for the length of a circular arc:

length of arc = (central angle/360 degrees) x 2 x pi x radius

We are given that the length of the arc is 16 feet and the central angle is 65 degrees. We need to find the radius of the circle.

Substituting the given values into the formula, we get:

16 = (65/360) x 2 x pi x radius

Simplifying the right-hand side, we get:

16 = 0.18056 x pi x radius

Dividing both sides by 0.18056 x pi, we get:

radius = 16 / (0.18056 x pi)

Simplifying the right-hand side, we get:

radius = 28.283 feet (rounded to three decimal places)

Therefore, the radius of the circle is approximately 28.283 feet.

We have b = -µ/σ and a = 1/σ, and the random variable Y = (X - µ)/σ has a standard normal distribution N(0,1).

Since Y is Gaussian with mean 0 and variance 1, we need to find values of a and b such that aX + b has mean 0 and variance 1.

The mean of aX + b is given by E[aX + b] = aE[X] + b. Since we want the mean to be 0, we set aE[X] + b = 0, which implies that b = -aµ.

The variance of aX + b is given by Var(aX + b) = a^2Var(X). Since we want the variance to be 1, we set a^2σ^2 = 1, which implies that a = 1/σ.

Therefore, we have b = -µ/σ and a = 1/σ, and the random variable Y = (X - µ)/σ has a standard normal distribution N(0,1).

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(a) The value of (fg)′(4) = 30.

(b) The value of (gf)′(4) = 33.

Given:

f(4)=4

f′(4)=6

g(4)=−1

and g′(4)=9

(a) Using the product rule, we have:

(fg)'(4) = f'(4)g(4) + f(4)g'(4)
         = 6(-1) + 4(9)
         = 30

Therefore, value of (fg)'(4) = 30.

(b) Using the chain rule, we have:

(gf)'(4) = g'(4)f(4) + g(4)f'(4)
         = 9(4) + (-1)(6)
         = 33

Therefore, value of (gf)'(4) = 33.

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