Lambert invests $20,000 for a 1/3 interest in a partnership in which the other partners have capital totaling $34,000 before admitting Lambert. After distribution of the bonus, what is Lambert's capital?

If Pepe and Leo continue to deposit the same amount of money every month, their balances will be the same and continue to grow at the same rate i.e. Pepe's balance = $3,600 and Leo's balance = $3,600.

If we assume that Pepe and Leo continue to deposit the same amount of money every month and that the interest rate remains constant, we can use a formula to calculate the future value of their savings. The formula for future value is:

FV = PV x (1 + r)n

Where:

FV stands for the savings account's future value.

PV stands for the savings account's initial balance's present value.

The interest rate, r, is considered to be zero in this instance.

The number of months is n.

If we assume that Pepe and Leo deposit $100 each per month, we can use this formula to calculate the future value of their savings after a certain number of months. For example:

After 12 months:

Pepe's balance = $1,200

Leo's balance = $1,200

After 24 months:

Pepe's balance = $2,400

Leo's balance = $2,400

After 36 months:

Pepe's balance = $3,600

Leo's balance = $3,600

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Linear functions are widely used in various fields including business, economics, and science. In a linear function, the relationship between two variables, usually denoted by x and y, can be represented by a straight line on a graph. The equation of a linear function is y = mx + b, where m is the slope and b is the y-intercept. The slope represents the rate of change of y with respect to x, while the y-intercept represents the value of y when x is equal to zero.

In the given word problem, we are asked to find a linear function that represents the relationship between the number of shirts sold and the price charged per shirt. Historical data shows that at a price of $30, 2,000 shirts can be sold, while at a price of $25, 3,000 shirts can be sold. Using this information, we can find the slope of the linear function as follows:

slope (m) = (change in y)/(change in x) = (25-30)/(3000-2000) = -0.005

The negative value of the slope indicates that the price per shirt decreases as the number of shirts sold increases. To find the y-intercept (b), we can use either of the two data points. Let's use the first data point (2000, 30):

30 = -0.005(2000) + b

b = 40

Therefore, the linear function that represents the relationship between the number of shirts sold (n) and the price charged per shirt (p) is:

p(n) = -0.005n + 40

To find the output variable in a word problem that uses a linear function, we need to identify the input variable and substitute it into the equation of the linear function. In the given word problem, the input variable is the number of shirts sold (n), and the output variable is the price charged per shirt (p). To find the price charged for, say, 2500 shirts, we can substitute n = 2500 into the equation of the linear function:

p(2500) = -0.005(2500) + 40 = $27.50

Therefore, the price charged for 2500 shirts is $27.50.

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Given that event A is known to occur and both events A and B have nonzero probabilities, we can conclude that the probability of event B occurring is zero. So, the correct answer to your question is: d. zero

If events A and B are mutually exclusive, it means that they cannot occur at the same time. So, if we know that event A has occurred, we can safely say that event B cannot occur. Therefore, the probability of the occurrence of event B given that event A has occurred is zero. Therefore, the correct answer is d) zero.

Mutually exclusive events are a fundamental concept in probability theory. It means that the occurrence of one event excludes the occurrence of another event. For example, when flipping a coin, the event of getting heads is mutually exclusive with the event of getting tails. It is impossible to get both heads and tails at the same time.

Understanding mutually exclusive events is important because it helps us to calculate the probability of combined events. For mutually exclusive events, we can simply add their probabilities to get the probability of their union. However, if events are not mutually exclusive, we need to subtract the probability of their intersection to avoid counting the same outcome twice.

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A disc with a radius of 2 cm has a density of 14 g/cm2 in the center and 0 at its edge. The density increases linearly with distance from the center. The mass of the disk is 7π g.

To find the mass of the disk, we need to integrate the density function over the area of the disk. The density is a linear function of the distance from the center, which means it can be written as:

ρ(r) = Ar + B

where A and B are constants that we need to determine. We know that the density at the center of the disk, where r=0, is 14 g/cm2. Therefore,

ρ(0) = A(0) + B = 14

So we have B = 14.

We also know that the density at the edge of the disk, where r=2 cm, is 0 g/cm2. Therefore,

ρ(2) = A(2) + 14 = 0

So we have A = -7.

Now we can write the density function as:

ρ(r) = -7r + 14

To find the mass of the disk, we need to integrate the density function over the area of the disk:

m = ∫∫ ρ(r) dA

We can use polar coordinates to integrate over the disk. The area element in polar coordinates is:

dA = r dr dθ

The limits of integration are 0 ≤ r ≤ 2 and 0 ≤ θ ≤ 2π. Therefore,

[tex]$ m = \iint \rho(r) r dr d\theta[/tex]

[tex]= \int_0^2 \int_0^{2\pi} (-7r + 14) r dr d\theta[/tex]

[tex]= \int_0^2 (-\frac{7}{2} r^3 + 7r^2) d\theta[/tex]

[tex]= 2\pi [ -\frac{7}{8} r^4 + \frac{7}{3} r^3 ]\bigg\rvert_0^2[/tex]

[tex]= 2\pi [-(\frac{7}{8})(2^4) + (\frac{7}{3})(2^3)][/tex]

[tex]= \frac{4\pi}{3} (28 - 7)[/tex]

[tex]= \frac{21\pi}{3} $[/tex]

= 7π

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The simplified expression is 40xy.

To simplify (5x)² (2y)³/10x⁴y², we can first simplify the numerator by using the power of a power rule, which states that when we raise an exponent to another exponent, we multiply the exponents.

So, (5x)² can be simplified as 25x², and (2y)³ can be simplified as 8y³.

The expression now becomes:

(25x²)(8y³) / 10x⁴y²

We can simplify this further by canceling out common factors. We can divide both the numerator and denominator by 5x²y²:

(25x²)(8y³) / (10x⁴y²) = (5x²y³)(8) / (2x²y²)

Simplifying this further, we can cancel out the x² in the numerator and denominator:

(5xy³)(8) / y²

Finally, we can simplify by multiplying 5 and 8:

40xy³ / y²

This can be simplified further by dividing y³ by y², which gives us:

40xy

So, the simplified expression is 40xy.

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Answer:

56 degrees

Step-by-step explanation:

the total sum of the angles in a triangle is 180

90+34+b=180

b=180-124

=56

Answer:

56°

Step-by-step explanation:

The sum of interior angles in a triangle is equal to 180°.

The triangle shown in the image is a right triangle so one of the angle measure is 90°.

Given, the other angle is 34°, we can find the value of missing angle with the following equation:

x + 90° + 34° = 180°

x + 124° = 180°

x = 56°

The answers to the questions are:
(a)(i) Events X and Y are not independent.
(a)(ii) Events X and Y are not mutually exclusive.
(b)(i) P(X) = 2/3
(b)(ii) P(X U Y) = 3/5
(b)(iii) P(X|Y) = 6/5.

(a) (i) To determine if events X and Y are independent, we need to see if P(X|Y) = P(X).
P(Y|X) = P(XY)/P(X)
1/3 = 2/5 / P(X)
P(X) = (2/5)/(1/3)
P(X) = 6/5

Therefore, since P(X|Y) ≠ P(X), events X and Y are not independent.

(ii) To determine if events X and Y are mutually exclusive, we need to see if P(XY) = 0.
P(XY) = 2/5 ≠ 0

Therefore, events X and Y are not mutually exclusive.

(b)
(i) To find P(X), we can use the formula P(X) = P(XY) + P(XY').
P(Y') = 1 - P(Y) = 1 - 1/3 = 2/3

P(X) = P(XY) + P(XY')
P(X) = 2/5 + (2/3)(1 - 2/5)
P(X) = 2/5 + 4/15
P(X) = 10/15
P(X) = 2/3

(ii) To find P(X U Y), we can use the formula P(X U Y) = P(X) + P(Y) - P(XY).
P(X U Y) = 2/3 + 1/3 - 2/5
P(X U Y) = 10/15 + 5/15 - 6/15
P(X U Y) = 9/15
P(X U Y) = 3/5

(iii) To find P(X|Y), we can use the formula P(X|Y) = P(XY)/P(Y).
P(X|Y) = P(XY)/P(Y)
P(X|Y) = 2/5 / 1/3
P(X|Y) = (2/5)(3/1)
P(X|Y) = 6/5

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The 90% confidence interval for the proportion of all adults in the United States who own an e-reader is 0.32 ± 1.645 * √(0.32 * 0.68 / 1,005). This corresponds to option (B) in your list of choices.

The appropriate formula for calculating the confidence interval for a proportion is:

sample proportion ± z*standard error

where z is the z-score corresponding to the desired level of confidence (90% in this case) and the standard error is:

sqrt((sample proportion*(1-sample proportion))/sample size)

Plugging in the values given in the question, we get:

sample proportion = 0.32
sample size = 1005
z-score for 90% confidence = 1.645 (option B)
standard error = sqrt((0.32*(1-0.32))/1005) = 0.015

So the appropriate 90% confidence interval is:

0.32 ± 1.645(0.015)
= 0.32 ± 0.025
= (0.295, 0.345)

Therefore, the correct answer is option B: 0.32 +1.645,10.3270.68

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This is the minimum variance of 0, and we can see that it decreases as n increases.

What is the standard deviation?

A measure of a group of values' variance or dispersion in statistics is called the standard deviation. When the standard deviation is low, the values are more likely to fall within a narrow range, also known as the expected value, whereas when the standard deviation is high, the values tend to be closer to the mean. The lowercase Greek letter sigma, which stands for the population standard deviation, or the Latin letter s, which stands for the sample standard deviation, are most frequently used in mathematical equations and texts to represent standard deviation. Standard deviation is also sometimes referred to as SD.

To find the minimum variance unbiased estimator of 0, we first take the natural logarithm of the likelihood function:

ln [tex]L(0; Y_1, Y_2, ..., Y_n) = -n*0 - (Y_1+Y_2+...+Y_n)[/tex]

Taking the derivative with respect to 0 and setting it equal to zero, we get:

d/d0 ln [tex]L(0; Y_1, Y_2, ..., Y_n) = -n + 0 = 0[/tex]

Therefore, the maximum likelihood estimator of 0 is:

[tex]0 = ΣY_i / n[/tex]

To show that it is unbiased, we take the expected value of 0:

[tex]E(0) = E(ΣYi / n) = (1/n) E(ΣYi) = (1/n) nE(Y1) = (1/n) n(0+1) = 1[/tex]

Since E(0) = 1, we can see that 0 is an unbiased estimator of 0.

To find the variance of 0, we use the fact that [tex]Var(Yi) = E(Yi^2) - [E(Yi)]^2 = 1 - 0^2 = 1 - (ΣYi / n)^2.[/tex] Therefore:

[tex]Var(0^) = Var(ΣYi / n) = (1/n^2) Var(ΣYi) = (1/n^2) nVar(Y1) = 1/n - (ΣYi / n)^2[/tex]

This is the minimum variance of 0, and we can see that it decreases as n increases.

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The smallest number of properties that Terry could have had is 7 properties.

Let's assume that Terry had x properties. Then, we know that Emily had x + 6 properties. Together, they owned at least 20 properties,

so:x + (x + 6) ≥ 20

2x + 6 ≥ 20

2x ≥ 14

x ≥ 7

Hence, Terry must have had at least 7 properties.

To understand why, we can think of it this way: if Terry had fewer than 7 properties, then Emily would have had even fewer than Terry (since she has 6 fewer properties than him).

If their combined total is at least 20, and Emily has fewer than Terry, then there's no way they could have reached a total of 20 or more properties.

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Answer:

9

Step-by-step explanation:

i can not really tell what letters are on the picture but i think it is 9

The area of the rhombus is 17.5 square inches.

The formula to find the area of a rhombus is:

Area = (diagonal1 x diagonal2) / 2

where diagonal1 and diagonal2 are the lengths of the diagonals.

diagonal1 = 7 inches and diagonal2 = 5 inches.

we can plug these values into the formula:

Area = (7 x 5) / 2

Area = 35 / 2

Area = 17.5 square inches

Therefore, the area of the rhombus is 17.5 square inches.

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In a case whereby A mathematics student decides to use an approximation of π of 355/113 his percentage error in using this value,  can  be expressed as

Percent error (percentage error)  can be described as the difference that can be established between  experimental  as well as theoretical value,  which is been divided by theoretical value then find the multiplication by  100

The error can be calculad as (355/113 -  π) /  π

=8.5*10^-5

The percentage error = 8.5*10^-5*100 = 8.5*10^-6

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The unit vector along the direction from Q to P is:

(-6 - m)/√(m² + 10m + 45) i + m/√(m² + 10m + 45) j - 3/√(m² + 10m + 45) k

What is unit vector?

A unit vector is a vector that has a magnitude of 1 and is usually used to indicate a direction in a vector space.

To find the unit vector along the direction of a vector from point Q to P, we first need to find the vector from Q to P.

The vector from Q to P is given by:

P - Q = [-1 - (5 + m)]i + [4 + m - 4]j + [-3 - (0 + m)]k

= [-6 - m]i + m j - 3k

To find the unit vector along this direction, we need to divide this vector by its magnitude:

|P - Q| = √[(-6 - m)² + m² + (-3)²] = √(m² + 10m + 45)

So, the unit vector along the direction from Q to P is:

(-6 - m)/√(m² + 10m + 45) i + m/√(m² + 10m + 45) j - 3/√(m² + 10m + 45) k

(Note that we can simplify this expression by factoring out the common factor of √(m² + 10m + 45) from the numerator.)

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The A 5 gallons of water will pour through the hose in 8 minutes.

The formula to be used for calculation of amount of water pouring through hose :

Total amount of water = amount of water pouring per minute × amount of time (in minutes)

Keep the values in formula to find the total amount of water

Total amount of water = 2.5 × 8

Performing multiplication on Right Hand Side of the equation

Total amount of water = 20 quarts

Now performing unit conversion

Amount of water in gallon = amount of water in quarts × 0.25

Amount of water in gallon = 20 × 0.25

Amount of water = 5 gallon

Hence, the correct answer is A 5.

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Wilcoxon signed rank test - A non-parametric test used to compare two related samples or repeated measures.
Friedman block test - A non-parametric test used to compare three or more related samples or repeated measures.
Kendall's tau - A non-parametric test used to measure the strength of association between two variables that are ordinal or ranked.
Spearman's rho - A non-parametric test used to measure the strength of association between two variables that are measured on an ordinal or continuous scale.

1. Wilcoxon Signed Rank: A non-parametric test used to compare two related samples, matched samples, or repeated measurements on a single sample to assess whether their population mean ranks differ.

2. Friedman Block Test: A non-parametric test used to determine if there are any significant differences between the means of three or more paired groups by comparing the rankings of the data.

3. Kendall's Tau: A non-parametric measure of correlation that evaluates the strength and direction of association between two ordinal variables.

4. Spearman's Rho: A non-parametric measure of rank correlation that assesses the strength and direction of association between two ranked variables.

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If we roll a single die twice, what is the probability that the sum of the dots showing on the two rolls equals four (4)The probability that the sum of the dots showing on the two rolls equals four (4) is 1/12.

Explanation:
1. Identify the possible outcomes that result in a sum of 4: (1, 3), (2, 2), and (3, 1).
2. Calculate the probability of each outcome:
  - P(1, 3) = 1/6 (for the first roll) * 1/6 (for the second roll) = 1/36
  - P(2, 2) = 1/6 * 1/6 = 1/36
  - P(3, 1) = 1/6 * 1/6 = 1/36
3. Add the probabilities of each outcome to find the total probability: 1/36 + 1/36 + 1/36 = 3/36 = 1/12.

11 outcomes (of the total of 36 outcomes) which give us the desired output. Hence the probability of getting a sum of 6 or 7 is 1136

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The description of the statement is (b) inclusive.

Here, we have,

to describe the events:

The events are given as:

Drawing a red card

A card with a number

A red card can have a number and a numbered card can be a red card.

This means that

n(Red and Numbered) ≠ 0

Events that have similar features are referred to as inclusive events

Hence, the description of the statement is (b) inclusive

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The solid's volume is (79/15)π cubic units.

To find the volume of the solid obtained by rotating the region bounded by the curves y = x² and y = 1 about the line y = 5, use the method of cylindrical shells.

The solid we're interested in is a cylindrical shell with an outer radius of (5 - y), an inner radius of (5 - 1), and a height of (x² - 1).

The volume of this shell can be expressed as:

dV = 2πr × h × dx

where r = average radius of the shell, h = height, and dx = infinitesimal width of the shell.

To find the limits of integration, solve for x in terms of y for the equation y = x²:

x² = y

x = ±√y

Rotating about the line y = 5, the limits of integration will be from y = 1 to y = 5.

Volume of solid can be obtained by integrating the expression for dV from y = 1 to y = 5:

V = ∫1⁵ 2π(5 - y)(5 - 1)(√y - 1) dy

= 2π ∫1⁵ (4 - y)(√y - 1) dy

= 2π [4/3 y^(3/2) - 2/5 y^(5/2) - y^(3/2) + 2 y]1⁵

= 2π (79/15)

Therefore, the volume of the solid is (79/15)π cubic units.

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The first-rate is 4 times larger than the second rate, so we can say that the first-rate is 4 times the second rate.

To compare these two rates, we need to convert them to the same unit. Let's convert the first rate to dollars per ounce:

$2.56 per 1/2 pound = $2.56 / (1/2 lb) = $2.56 / 8 oz = $0.32 per oz

So the first rate is $0.32 per ounce.

Now, let's convert the second rate to dollars per ounce:

$0.48 per 6 ounces = $0.48 / 6 oz = $0.08 per oz

So the second rate is $0.08 per ounce.

Therefore, the equivalent rates are:

$0.32 per oz and $0.08 per oz

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The particular solution is y = 2√(1 + x^2) - x + 2.

To solve the differential equation dy/dx = (1 + x^2)^(1/2), we can separate variables and integrate both sides:

∫1/(1 + x^2)^(1/2) dy = ∫dx

Using the substitution u = x^2 + 1, du/dx = 2x, we can simplify the integral on the left:

∫1/(1 + x^2)^(1/2) dy = ∫1/u^(1/2) * (1/2x) dy
= ∫1/u^(1/2) du
= 2√(1 + x^2)

Therefore, we have:

2√(1 + x^2) = x + C

where C is the constant of integration. To find the particular solution that satisfies y(0) = 2, we substitute x = 0 and y = 2 into the equation:

2√(1 + 0^2) = 0 + C

C = 2

So the particular solution is:

2√(1 + x^2) = x + 2

To check, we can verify that y(0) = 2 by substituting x = 0:

2√(1 + 0^2) = 0 + 2

2 = 2, which is true. Therefore, the particular solution is y = 2√(1 + x^2) - x + 2.

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The basic eigenvector corresponding to λ₁ = 4 is v₁ = [3.75, 4], and the basic eigenvector corresponding to λ₂ = -3 is v₂ = [5, 6].

To find the eigenvalues of matrix A, we need to solve the characteristic equation:

|A - λI| = 0

where I is the identity matrix of the same size as A, and λ is the eigenvalue we are trying to find.

For matrix A given as:

a=[8 -15]

[6 -10]

we have:

|A - λI| =

|8 - λ -15 |

|6 -10- λ |

Expanding the determinant, we get:

(8 - λ)(-10 - λ) - (-15)(6) = 0

Simplifying the expression, we get:

λ² - 2λ - 12 = 0

Using the quadratic formula, we get:

λ₁ = 4

λ₂ = -3

Therefore, the distinct eigenvalues of A are λ₁ = 4 and λ₂ = -3.

Next, we find the eigenvectors corresponding to each eigenvalue. We do this by solving the system of equations:

(A - λI)x = 0

For λ₁ = 4:

A - λ₁I =

|8 - 4 -15 |

|6 -10 - 4 |

=

|4 -15 |

|6 -14|

RREF:

|1 -3.75|

|0 0 |

Thus, we have a free variable x₂. Setting x₂ = 4, we get the basic eigenvector:

v₁ = [3.75, 4]

Therefore, there is one basic eigenvector corresponding to eigenvalue λ₁ = 4.

For λ₂ = -3:

A - λ₂I =

|8 15 |

|6 7 |

RREF:

|1 -5/6|

|0 0 |

Thus, we have a free variable x₂. Setting x₂ = 6, we get the basic eigenvector:

v₂ = [5, 6]

Therefore, there is one basic eigenvector corresponding to eigenvalue λ₂ = -3.

In summary, the distinct eigenvalues of matrix A are λ₁ = 4 and λ₂ = -3. There is one basic eigenvector corresponding to each eigenvalue. The basic eigenvector corresponding to λ₁ = 4 is v₁ = [3.75, 4], and the basic eigenvector corresponding to λ₂ = -3 is v₂ = [5, 6].

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Answer:

Step-by-step explanation:

I used to do these as a kid! theyre pretty fun :)

for the first one:

the sum has to be 9. (as we can see from the first row).

the middle box in the last row will be -1. (since two boxes fill to be 10, you subtract 1 to get to 9)

and so on. it solves itself. use similar tactics for all others.

1:

0 7 2

5 3 1

4 -1 6

2:

1 2 6

8 3 -2

0 4 5

3:

3 -2 5

4 2 0

-1 6 1

The sequence given is {1,1/3,1/5,1/7,1/9,...} and we are asked to find a formula for the general term an of this sequence. Specifically, the nth term in the sequence is the reciprocal of the (2n - 1)th odd number. Thus, the formula for the general term an of the sequence is given by:

an = (-1)^(n+1) / (2n - 1)

This formula can be derived by noting that the signs of the terms alternate between positive and negative, with the first term being positive. Therefore, we introduce a factor of (-1)^(n+1) to account for the sign of each term. Additionally, we observe that the denominator of each term is an odd number of the form 2n - 1, where n is the position of the term in the sequence. Thus, we express the general term as the reciprocal of the denominator with the appropriate sign.

In summary, the formula for the general term an of the sequence {1,1/3,1/5,1/7,1/9,...} is an = (-1)^(n+1) / (2n - 1), where n is the position of the term in the sequence. This formula gives us a way to find any term in the sequence by plugging in its position for n.

To further explain, we can consider the first few terms of the sequence and see how the formula applies. The first term corresponds to n = 1, so we have a1 = (-1)^(1+1) / (2(1) - 1) = 1/1 = 1. The second term corresponds to n = 2, so we have a2 = (-1)^(2+1) / (2(2) - 1) = -1/3. Similarly, the third term corresponds to n = 3, so we have a3 = (-1)^(3+1) / (2(3) - 1) = 1/5. We can continue in this way to find any term in the sequence using the formula for the general term.

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We are similar to gram-negative. It must be noted that we are neither and have different cell characteristics compared to bacteria.

The bacterial cells are classified as gram postive and gram negative depending on their cell membrane structure. The gram negative bacteria are rich in lipid layer and thin peptidoglycan while gram postive have more peptidoglycan content.

Now, peptidoglycan are responsible for gram staining. Human epithelial cells do not have peptidoglycan which do not let them take up the stain. Hence, humans will be considered gram negative while noting the identity will be completely different.

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(a) The growth rate in the percentage of Microsoft Word users is not explicitly provided in the given equation. To find the growth rate, we need the derivative of the function P(t) with respect to time (t).

(b) Graphing the function P(t) requires a graphing utility. The graph will show the percentage of Microsoft Word users over time.

(c) To find the percentage of Microsoft Word users in 1990, substitute t = 1990 into the equation P(t) = 99.774/(1 + 3.014e).

(d) To determine the year when the percentage of Microsoft Word users reached 90%, we need to solve the equation P(t) = 90 for t.

(e) The numerator in the model, 99.774, represents the initial percentage of companies reporting usage of Microsoft Word in 1984. It implies that almost 100% of the surveyed companies were already using Microsoft Word at that time. The model assumes a high initial adoption rate for the software.

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The indefinite integral of [tex]1/(16-x^2)^{(3/2)}[/tex] dx using the substitution x = 4 sin(t) is (1/8) arcsin(x/4) - [tex](1/16)x(16-x^2)^{0.5}[/tex] + C here C is the constant of integration.
Let us take x = 4 sin(t), then dx/dt = 4 cos(t), and x² = 16 sin²(t). Substituting these values in the integral, we get:

[tex]\int\limits 1/(16-x^{2})^{(3/2}) dx[/tex] =[tex]\int\limits1/(16-16sin^{2}(t))^{(3/2)} * 4cos(t) dt[/tex]

= ∫1/16cos³(t) dt

= (1/16) ∫sec³(t) dt

= (1/16) (1/2 sec(t) tan(t) + 1/2 ln|sec(t)+tan(t)| + C)

Substituting back x = 4 sin(t), we get:

[tex]\int\limits1/(16-x^2)^{(3/2)}[/tex] dx = (1/8) [tex]arcsin(x/4) - (1/16)x(16-x^{2})^{0.5}[/tex] + C

where C is the constant of integration.

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Answer:

no because it put your neighbors at risk and you might get sued

Step-by-step explanation:

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N is closed under taking inverses.

To show that N is a subgroup of G, we need to show that it satisfies the three subgroup criteria:

N is non-empty: N contains 1 and x², so it is non-empty.

N is closed under multiplication: Since G is a group, we know that 1 and x² are both in G, and we have:

1 * 1 = 1, 1 * x² = x², x² * 1 = x², x² * x² = 1.

Therefore, N is closed under multiplication.

N is closed under taking inverses: Again, since G is a group, we know that 1 and x² have inverses in G, and we have:

1⁻¹ = 1, (x²)⁻¹ = x².

Therefore, N is closed under taking inverses.

Thus, N satisfies all three subgroup criteria, so it is a subgroup of G.

To show that N is a normal subgroup of G, we need to show that for any g in G and any n in N, we have gng⁻¹ in N. We can list the cosets of N in G to show this:

1N = {1, x²}

iN = {i, ix²}

jN = {j, jx²}

kN = {k, kx²}

We can see that each coset is of the form gN, where g is one of the elements in G. Since the left cosets and right cosets are the same in this case, it suffices to check whether gn and ng are in the same coset for each g in G and each n in N. We can do this by calculating gn and ng for each g and n:

1n = n1 = n

x²n = nx⁻² = n

in = ni = iN

ix²n = nx⁻²i = iN

jn = nj = jN

jx²n = nx⁻²j = jN

kn = nk = kN

kx²n = nx⁻²k = kN

Since gn and ng are in the same coset for every g in G and every n in N, we can conclude that N is a normal subgroup of G.

To determine the multiplication table for G/N, we need to calculate the cosets gN for each element g in G. We can do this by multiplying each element of G by the elements of N:

1N = {1, x²}

iN = {i, ix²}

jN = {j, jx²}

kN = {k, kx²}

To compute the multiplication table for G/N, we need to calculate the product of each coset with each other coset. Since the multiplication of cosets is defined by the product of their representatives, we can use the multiplication table for G to compute the products. Here is the multiplication table for G/N:

| 1N | iN | jN | kN |

1N| 1N | iN | jN | kN |

iN| iN | 1N | kN | jN |

jN| jN | kN | 1N | iN |

kN| kN | jN | iN | 1N |

We can see that G/N is isomorphic to the Klein four-group, which is the only group of order four up to isomorphism.

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