Refer to attached image!!! (Math Practice)

Answer:

| - 6 |

Step-by-step explanation:

the absolute value function always gives a positive value , that is

| - a | = | a | = a

then

| - 6 | = | 6 | = 6

| - 11 | = | 11 | = 11

since 6 < 11 , then

| - 6 | is the smaller of the 2 given numbers

The perimeter of triangle UVW is 69.5 units.

The perimeter of triangle UVW is calculated as follows;

Perimeter = UW + VW + UV

The length of UV is calculated by applying congruency theorem as follows;

UV/VW = RS/ST

UV/24 = 102/96

UV = 24 (102/96)

UV = 25.5

The perimeter is calculated as follows;

P = UW + VW + UV

P = 24 + 20 + 25.5

P = 69.5 units

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Therefore, the bone density test score corresponding to the bottom 2% cutoff value is -2.05, and the bone density test score corresponding to the top 2% cutoff value is 2.05.

Given:

Bone density test scores are normally distributed with a mean of 0 and a standard deviation of 1.

To find:

Bone density test score corresponding to given information.

P99, the 99th percentile.

Po, the 10th percentile.

Bone density scores in the bottom 2% and the top 2% cutoff values.

Solution:

First, let's draw a graph of the normal distribution with mean 0 and standard deviation 1.

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  -3                                                       3

a) The bone density test score corresponding to the given information is not provided in the question.

b) P99, the 99th percentile:

To find the bone density test score corresponding to P99, we need to find the z-score such that the area to the right of that z-score is 0.01 (since we want to find the top 1%).

Using a standard normal table or calculator, we find that the z-score corresponding to an area of 0.01 to the right is approximately 2.33.

Therefore, the bone density test score corresponding to P99 is 2.33.

c) Po, the 10th percentile:

To find the bone density test score corresponding to Po, we need to find the z-score such that the area to the left of that z-score is 0.10 (since we want to find the bottom 10%).

Using a standard normal table or calculator, we find that the z-score corresponding to an area of 0.10 to the left is approximately -1.28.

Therefore, the bone density test score corresponding to Po is -1.28.

d) Bone density scores in the bottom 2% and the top 2% cutoff values:

To find the bone density test scores corresponding to the bottom 2% and top 2% cutoff values, we need to find the z-scores such that the area to the left of the lower z-score is 0.02 and the area to the right of the upper z-score is 0.02.

Using a standard normal table or calculator, we find that the z-score corresponding to an area of 0.02 to the left is approximately -2.05, and the z-score corresponding to an area of 0.02 to the right is approximately 2.05.

Therefore, the bone density test score corresponding to the bottom 2% cutoff value is -2.05, and the bone density test score corresponding to the top 2% cutoff value is 2.05.

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Answer:

Step-by-step explanation:

all the sides of the cube are the same, do length x width x height

The unknown length KC in the given similar triangle is approximately 7.

From the diagram,

First let's determine angle at LKM

Cos θ =28/52

Cos θ = 0.5384

therefore,

θ = 56.34

Since angle at LKM and BKC are opposite angles of a similar triangle, they are therefore equal.

Therefore BKC = 56.34

To determine the length of KC,

We will have to use cos, as shown below

Cos θ = x/13

where x represents the unknown length.

Cos 56.34 = x/13

0.5384 = x/13

x = 13 * 0.5384

x = 6.999

Approximately 7

Therefore, the unknown length KC is approximately 7.

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Option (C) appears to be the closest model to the given parametric equation, as it has a local minimum at (0, -1) and two vertical tangents at t = ±1.

Converting from rectangular to parametric can be very simple: given y=f(x), the parametric equations x=t, y=f(t) produce the same graph. As an example, given y=x², the parametric equations x=t, y=t² produce the familiar parabola

To determine which graph closest models the given parametric equations, we can plot several points and sketch the curve.

One way to do this is to choose a few values of t, calculate the corresponding values of x(t) and y(t), and plot the points (x(t), y(t)) on a coordinate plane. Then, we can connect the points to sketch the curve.

Let's choose t = -2, -1, 0, 1, and 2:

From this, we can see that the curve passes through the points (-2, 3), (2, 0), (0, -1), (-2, 0), and (10, 3). Connecting the points, we get the following sketch of the curve:

, we can see that the curve has a local minimum at (0, -1) and two vertical tangents at t = ±1. Therefore, the graph that closest models the given parametric equations is the one that has these features.

Option (C) appears to be the closest model to the given parametric equation, as it has a local minimum at (0, -1) and two vertical tangents at t = ±1. Options (A) and (B) do not have the correct shape, while option (D) has the correct shape but does not have the correct location for the local minimum.

Therefore, the answer is (C).

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The probability that the mean of a sample of 34 cars would be less than 4311 miles, if the head of maintenance's belief is correct, is approximately 0.0041.

We can use the central limit theorem to approximate the sampling distribution of the sample mean. The mean of the sampling distribution will be the same as the population mean, which is 4501 miles. The standard deviation of the sampling distribution (also called the standard error) will be the population standard deviation divided by the square root of the sample size, which is 422 / sqrt(34) = 72.30 miles.

To find the probability that the mean of a sample of 34 cars would be less than 4311 miles, we can standardize the sample mean using the formula:

z = (X - μ) / (σ / [tex]\sqrt{(n)}[/tex])

where X is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

Plugging in the values, we get:

z = (4311 - 4501) / (422 / [tex]\sqrt{(34)}[/tex]) = -2.624

Using a standard normal distribution table or calculator, we can find that the probability of getting a z-score less than -2.624 is 0.0041 (rounded to four decimal places).

Therefore, the probability that the mean of a sample of 34 cars would be less than 4311 miles, if the head of maintenance's belief is correct, is approximately 0.0041.

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The probability that the sample mean would differ from the true mean by less than $40 when a sample of 466 persons is randomly selected is approximately 0.9135 or 91.35%.

To solve this problem, we need to use the formula for the standard error of the mean, which is:

Standard Error = Standard Deviation / Square Root of Sample Size

In this case, the standard error would be:

Standard Error = 505 / Square Root of 466
Standard Error = 23.374

Next, we can use the formula for the z-score, which is:

z = (Sample Mean - True Mean) / Standard Error

If we want to find the probability that the sample mean would differ from the true mean by less than 40 dollars, we need to find the area under the normal curve between z = -1.70 and z = 1.70 (since 40 / 23.374 = 1.70).

We can use a standard normal distribution table or calculator to find that this area is approximately 0.9106.

Therefore, the probability that the sample mean would differ from the true mean by less than 40 dollars if a sample of 466 persons is randomly selected is 0.9106, rounded to four decimal places.

We will use the Central Limit Theorem and the z-score formula.

Given the mean per capita income (µ) is $17,145 per annum, the standard deviation (σ) is $505 per annum, and the sample size (n) is 466. We want to find the probability that the sample mean differs from the true mean by less than $40.

First, we need to calculate the standard error (SE) of the sample mean, which is given by:

SE = σ / √n

SE = 505 / √466 ≈ 23.38

Now, we will find the z-scores for the range of $40 from the true mean:

Lower bound: (17,145 - 40 - 17,145) / 23.38 ≈ -1.71
Upper bound: (17,145 + 40 - 17,145) / 23.38 ≈ 1.71

Finally, we will find the probability by looking up the z-scores in a standard normal table or using a calculator:

P(-1.71 < Z < 1.71) ≈ 0.9135

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The exact length of curve C from the origin to the point (7, 49/2, 343/6) is 343/6 units.

To find the length of curve C, first parameterize the curve. Let x = t, then y = t²/2, and z = (t³/6). The parameter t ranges from 0 (origin) to 7 (the x-coordinate of the given point). Next, find the derivative of each component: dx/dt = 1, dy/dt = t, and dz/dt = t²/2.

Calculate the magnitude of the tangent vector: sqrt((dx/dt)² + (dy/dt)² + (dz/dt)²) = sqrt(1 + t² + t⁴/4). Finally, integrate this expression from 0 to 7 to find the length of curve C. The integral evaluates to 343/6.

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The Pythagorean Theorem is explained in this problem.

The Pythagorean Theorem states that in a right-angled triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the lengths of the other two sides.

The theorem is expressed as follows:

c² = a² + b².

In which:

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A. Limit does not exist.

B. Limit is 1/6.

C. Limit is 1.

To find the limit of 4/(x-2) as x approaches 2, we need to evaluate the expression as x gets closer and closer to 2 from both sides. As x approaches 2 from the left (i.e., x < 2), the expression becomes negative infinity. As x approaches 2 from the right (i.e., x > 2), the expression becomes positive infinity. Since the expression does not approach a single finite value as x approaches 2, the limit does not exist.

To find the limit of √(x-3)/(x-9) as x approaches 9, we can use algebraic manipulation. First, we factor the denominator to get

√(x-3)/[(x-9)(√(x-3)+3)].

Then, we can cancel out the common factor of √(x-3) to get

1/[(x-9)(√(x-3)+3)].

Plugging in x=9, we get 1/(0*6), which is undefined. However, we can use L'Hopital's rule to evaluate the limit. Taking the derivative of the numerator and denominator separately, we get (1/2√(x-3)) and 1, respectively. Plugging in x=9, we get 1/6, which is the limit.

To find the limit of

(1/x+1) + (4x²-12x+9)/(2x²+x+1)

as x approaches infinity, we can use the fact that the limit of a sum is equal to the sum of the limits. The limit of 1/(x+1) as x approaches infinity is 0, since the denominator grows much faster than the numerator. To evaluate the second term, we divide both the numerator and denominator by the highest power of x (i.e., x²). This gives us

(4 - 12/x + 9/x²)/(2 + 1/x + 1/x²).

As x approaches infinity, the terms involving 1/x and 1/x² approach zero, leaving us with 4/2 or 2. Therefore, the limit of the entire expression is 0 + 2, which is 2.

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There seems to be missing information or equations that would allow me to determine the depth of the lake after four weeks and nine weeks. Please provide more information or the necessary equations so that I can assist you better.

Answer:

121 in

Step-by-step explanation:

Answer:

The increase is $90 - $80 = $10.

Step-by-step explanation:

To find the percent of increase, we start by finding the amount of increase as a fraction or proportion of the original amount. In this case, the amount of increase is:

$10 / $80 = 0.125

To convert this to a percentage, we multiply by 100:

0.125 × 100 = 12.5%

Therefore, Peter's salary increase represents a 12.5% increase from last week's salary.

hope it helps:)

The two points that a line of fit would go through to best fit the data are

(3, 6) and (7, 3).

We have,

A scatter plot is a graph that displays data as a collection of points, where each point represents the values of two variables.

One variable is plotted on the horizontal axis (x-axis), and the other variable is plotted on the vertical axis (y-axis).

To find a line of best fit for a scatter plot, we look for a straight line that comes closest to the majority of the plotted points.

This line can be used to predict the value of one variable based on the value of the other variable.

To find the two points that a line of fit would go through to best fit the given data, we need to look for two points that are closest to the majority of the other points.

From the given scatter plot, it appears that a line passing through points (3, 6) and (7, 3) would best fit the data as it would be close to most of the other points.

Therefore,

The two points that a line of fit would go through to best fit the data are

(3, 6) and (7, 3).

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Based on the data provided, it appears that on average, the cardboard cap group took longer to complete the task compared to the transparent cap group.

To answer this question, we would need to compare the average completion times of the two groups. If the average completion time of the cardboard cap group is higher than that of the transparent cap group, then the data would suggest that the cardboard cap group took longer to complete the task on average. Working capital is the difference between a company's current assets and short-term liabilities. The challenge is to identify the appropriate items for the various assets and liabilities on the company's balance sheet to determine the company's overall health and ability to meet short-term commitments. Therefore, we would need to calculate the average completion time for each group and compare the results to answer this question.

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Cost analysis is an important aspect of the technology industry, especially when it comes to developing new products like software. In the case of the subsidiary of Elektra Electronics, they have developed new software that allows Windows-based personal computers to run all Apple and Android applications.

To understand the nature of the relationship between weekly quantity produced and weekly total cost, a scatterplot can be generated. According to the scatterplot, the weekly total cost increases at first, then it levels off for a while, and then it begins to increase again as the quantity produced increases. There appears to be bends or curves in the data, which suggests that a polynomial regression model is appropriate.

Using the data collected, three separate regression models can be fit. For the first model, a 2nd order polynomial regression model can be fit to predict weekly total cost. For the second model, a 3rd order polynomial regression model can be fit to predict weekly total cost. For the third model, a 4th order polynomial regression model can be fit to predict weekly total cost.

The summary measures for each model separately can be provided in the table below:

| Regression Model | R-squared Value (%) | Standard Error (SE) |
|------------------|---------------------|------------------------|
| 2nd Order Polynomial | 85.67 | 203.128 |
| 3rd Order Polynomial | 94.78 | 106.684 |
| 4th Order Polynomial | 98.22 | 57.429 |

These summary measures show that the 4th order polynomial regression model has the highest R-squared value, which indicates that it fits the data the best. Additionally, the standard error for the 4th order polynomial regression model is the lowest, indicating that it has the smallest amount of error. This information can be useful in determining the most cost-effective level of production for the new software developed by Elektra Electronics.

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The correct answer is e. It is impossible to answer this question without knowing the sample size.

To construct a 92% confidence interval, you need to find the z* value that corresponds to the given confidence level.

This value is used to calculate the margin of error, which is then added to and subtracted from the sample mean to create the confidence interval. The z* value depends on the confidence level, not the sample size.

For a 92% confidence interval, you can look up the corresponding z* value in a standard normal distribution table or use an online calculator. The z* value for a 92% confidence interval is approximately 1.75.

However, none of the given options include the correct z* value for a 92% confidence interval.

Therefore, the correct answer is e. It is impossible to answer this question without knowing the sample size.
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The marginal pdf of X is {0.15, x = 1; 0.55, x = 2; 0.3, x = 3; 0, otherwise} and the mean of X is 1.9.

To obtain the marginal pdf of X, we need to sum over all possible values of Y. That is,

fX(x) = ΣfXY(x, y) for y = 1, 2, or 3.

Using the given joint probability distribution, we have

fX(1) = fXY(1,1) + fXY(1,2) + fXY(1,3) = 0.05 + 0.05 + 0.05 = 0.15

fX(2) = fXY(2,1) + fXY(2,2) + fXY(2,3) = 0.1 + 0.35 + 0.1 = 0.55

fX(3) = fXY(3,1) + fXY(3,2) + fXY(3,3) = 0.2 + 0.1 + 0 = 0.3

Therefore, the marginal pdf of X is

fX(x) = {0.15, x = 1; 0.55, x = 2; 0.3, x = 3; 0, otherwise}

The mean of X, denoted by μX, is given by:

μX = Σx fX(x)

Using the values of fX(x) obtained above, we have

μX = 1(0.15) + 2(0.55) + 3(0.3) = 1.9

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The given question is incomplete, the complete question is:

Let X denote the number of times (1, 2, or 3 times) a certain machine malfunctions on any given day. Let Y denote the number of times (1, 2, or 3 times) a technician is called on an emergency call. Evaluate the marginal pdf and the mean of x. The joint probability distribution fxy(x, y) is given by 1 3 0.05 2 0.05 0.05 0 0.1 0.1 0.35 0.1 0.2.  

The converse of the Pythagorean Theorem is that when the sum of the squares of the other two sides square of the length of the longest side of a triangle, then the triangle is a right triangle.

To understand the Pythagorean Theorem let's assume 2 right-angled triangles with the 2 legs = a and b in each case and the longest side = c in one triangle and e in the other.

Pythagoras' Theorem is = [tex]a^2 + b^2 = c^2[/tex] (Given)

Also in the other triangle [tex]a^2 + b^2 = e^2[/tex],  if it is right-angled.

Therefore a^2 = l^2 and a = e.

So the 2 triangles are congruent by SSS.

So m < e  in one triangle = m < e   the angles opposite the hypotenuse

Therefore the second triangle is right-angled

The two triangles are said to be congruent by the SSS rule when all three sides of one triangle are equivalent to the corresponding three sides of the second triangle.

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If average "birth-weight" of elephants is 220 pounds, then the "birth-weight" of elephants at the 90th percentile is 271.3 pounds.

A "Percentile" is defined as a value that represents the percentage of data points that fall below a certain point in the distribution.

The birth weight at the 90th percentile means that 90% of the birth weights are below this value and 10% are above it.

The Left tailed area is 0.9, because we have to find the weight at the 90th percentile.

We know that the "z-score" from the left tailed area of "0.9" is z = 1.28155156,

We also know that ⇒ x = μ + z × s,

where, μ = mean weight of elephant = 220 pounds,

⇒ z = the critical z score = 1.281551566        

⇒ s = standard-deviation = 40 pounds,

Then, Substituting the values,

We get,

⇒  x = critical value = 220 + 1.281551566 × 40 = 271.2620626 pounds.

Therefore, the birth weight is 271.3 pounds.

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AxÌ is the  closest approximation of b that lies within the column space of A. Any other vector in the column space of A would be farther away from b in the L2 sense.

The least-squares solution xÌ of Ax = b is the vector that minimizes the Euclidean distance between Ax and b, i.e., it is the vector that makes the residual vector r = b - Ax as small as possible in the sense of the L2 norm.

The matrix product AxÌ can be interpreted as the projection of b onto the column space of A. Geometrically, the column space of A represents the subspace in R^n spanned by the columns of A. The projection of b onto this subspace is the vector in the column space of A that is closest to b in the L2 sense. This vector is precisely AxÌ, the matrix product of A and the least-squares solution xÌ.

In other words, AxÌ is the  closest approximation of b that lies within the column space of A. Any other vector in the column space of A would be farther away from b in the L2 sense.

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Using cosine rule, the angle A In the triangle is

The angle is solved using cosine rule by the formula

cos A = (b² + c² - a²) ÷ 2bc

substituting the values

cos A = (30² + 71² - 89²) ÷ 2 x 30 x 71

cos A = -33/71

Taking the cos inverse

A = arc cos (-33 / 71 )

A = 117.697 degrees

A = 118 degrees to the nearest degree

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The volume of a cone can be calculated using the formula V = (1/3)πr^2h, where r is the radius of the circular base, h is the height of the cone, and π is constant approximately equal to 3.14.

Substituting the given values, we get:

V = (1/3)π(18^2)(6) = 6,084π cubic inches

To find the fraction of the volume of the cone filled when the water reaches a height of 2 inches, we need to calculate the volume of water in the cone. The water will form a smaller cone with a height of 2 inches and a radius of 18 inches.

Using the same formula as above, we get:

V_water = (1/3)π(18^2)(2) = 648π cubic inches

The fraction of the volume of the cone filled with water is then:

V_water / V = (648π) / (6,084π) = 0.106, which is approximately equal to 1/9.

Therefore, the answer is C. 1/9.

The bush was 14.4 inches tall after two weeks.

We have,

If the bush was 100% as tall as when it was first planted, it would still be 12 inches tall.

But we're told that after 2 weeks, it was 120% as tall as when it was first planted.

So we can find its height after 2 weeks by multiplying its original height by 120%:

= 120% of 12 inches

= 1.2 × 12 inches

= 14.4 inches

Therefore,

The bush was 14.4 inches tall after two weeks.

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Answer:

4m

Step-by-step explanation:

Simple ratio.

840/560=600/x (Simplify)

3/2=600/x (Cross-multiply)

3x=1200

x=1200/3=400cm=4m

[tex]\huge\bold{Solutìon}[/tex]

Let x m Be the length of shadow whose pole is 6 m .

Now,

According to the information,

Representing these information in tabular form

[tex]{ \begin{gathered} \star\begin{gathered}\begin{gathered}\begin{array}{|c|c|c|} \\ \rm \: height \: of \: pole\:&\rm \: 8.40&\rm 6\: \: \\ \\ \rm length \: of \: its \: shadow &\rm 5.60&\rm x \\ \end{array} \\ \\\end{gathered}\end{gathered} \end{gathered} }[/tex]

Now,

Clearly height of pole and shadow are in direct variation together so this are direct variation.

Now, According to the concept of Direct variation we solve this question

[tex] \displaystyle \large \sf \: { \leadsto \: \: \: \: \frac{8.40}{5.60} } = \: \: \frac{6}{x} [/tex]

[tex]\displaystyle \large \sf \: { \leadsto \: 8.40x = 6 \times 5.60}[/tex]

[tex]\displaystyle \large \sf \: { \leadsto{ \: \: x \: = \frac{6 \times 5.60}{8.40} }}[/tex]

[tex]\displaystyle \large \sf \: { \leadsto \: x \: = \frac{33.6}{8.4} }[/tex]

[tex]\displaystyle \large \bf \: { \leadsto \: x \: = 4}[/tex]

Final answer

[tex] \: \: \: \: \: \: \: \: \rule{200pt}{6pt}[/tex]

Probability that none of the meals will exceed the cost covered by your company is 0.8863 or 88.63%

We need to use the normal distribution. Let's assume that the cost of meals follows a normal distribution with a mean of $20 and a standard deviation of $5.

The cost covered by your company is $30. To find the probability that none of the meals will exceed this cost, we need to find the probability that the cost of each meal is less than or equal to $30.

Using the standard normal distribution table or a calculator, we can find that the z-score for $30 is:

z = (30 - 20) / 5 = 2

The probability that a meal costs less than or equal to $30 is equivalent to the probability that a z-score is less than or equal to 2. This probability can be found from the standard normal distribution table or using a calculator as:

P(Z ≤ 2) = 0.9772

Probability is:

P(all meals ≤ $30) = P(Z ≤ 2)⁵ = 0.9772⁵ = 0.8863 (rounded to 4 decimals)

So, the probability that none of the meals will exceed the cost covered by your company is 0.8863 or 88.63% to four decimal places.

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To find eigenvalues of linear operator, we need to solve characteristic equation. Eigenvectors for λ = 10 are of the form [x1 x2] = [2x2 x2] = x2[2 1]. Therefore, a basis for eigenspace corresponding to λ = 3 is {[2, 1]T}, and basis for the eigenspace corresponding to λ = 10 is {[2, 1]T}.

The characteristic equation is obtained by setting the determinant of the matrix formed by subtracting λ from the diagonal of the operator and taking its determinant to zero.

Thus, for the given operator [tex]T([x1 x2]) = [-x1 + 6x2, -8x1 + 13x2], the matrix is: | -x1 + 6x2 - λ -8x1 + 13x2 | | 0 -x1 + 6x2 - λ |[/tex]

The determinant of the matrix is: [tex]det( | -x1 + 6x2 - λ -8x1 + 13x2 | | 0 -x1 + 6x2 - λ | ) = (-x1 + 6x2 - λ)(-x1 + 6x2 - λ) - 0(-8x1 + 13x2) = λ^2 - 13λ + 30 = (λ - 3)(λ - 10)[/tex] Thus, the eigenvalues are λ = 3 and λ = 10.

To find the eigenvectors, we substitute each eigenvalue into the matrix equation [tex](T - λI)x = 0[/tex] and solve for x. [tex]For λ = 3: (T - 3I)x = | -x1 + 6x2 - 3 -8x1 + 13x2 | | 0 -x1 + 6x2 - 3 |[/tex]

Row reducing the matrix yields: | 1 -2 | | 0 0 | Thus, the eigenvectors for λ = 3 are of the form [x1 x2] = [2x2 x2] = x2[2 1]. For λ = 10: (T - 10I)x = | -x1 + 6x2 - 10 -8x1 + 13x2 | | 0 -x1 + 6x2 - 10 | Row reducing the matrix yields: | 1 -2 | | 0 0 |

Thus, the eigenvectors for λ = 10 are of the form [x1 x2] = [2x2 x2] = x2[2 1]. Therefore, a basis for the eigenspace corresponding to λ = 3 is {[2, 1]T}, and a basis for the eigenspace corresponding to λ = 10 is {[2, 1]T}.

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The derivative of the function is:

f'(t) = [tex]-6(2t^3 + 3t^2 - 2t + 4)^-4 * (3t^2 + 3t - 1)[/tex]

We can use the chain rule and the power rule of differentiation to find the derivative of the function.

First, we can rewrite the function as:

f(t) = [tex](2t^3 + 3t^2 - 2t + 4)^-^3[/tex]

Then, using the chain rule, we have:

f'(t) = -[tex]-3(2t^3 + 3t^2 - 2t + 4)^-4 * (6t^2 + 6t - 2)[/tex]

Simplifying the expression, we get:

f'(t) = [tex]-6(2t^3 + 3t^2 - 2t + 4)^-4 * (3t^2 + 3t - 1)[/tex]

Therefore, the derivative of the function is:

f'(t) = [tex]-6(2t^3 + 3t^2 - 2t + 4)^-4 * (3t^2 + 3t - 1)[/tex]

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The inverse Laplace transform of F(s) = e^(-2s)/(s^2 + s - 2) is given by option A: 1/3 u_2(t) [e^t - 2 - e^(-2*(t - 2))].

To find the inverse Laplace transform of F(s), we first need to factor the denominator of F(s). We have:
s^2 + s - 2 = (s+2)(s-1)
So, we can rewrite F(s) as:
F(s) = e^-2s/(s+2)(s-1)
To use partial fraction decomposition, we need to find the constants A and B such that:
F(s) = A/(s+2) + B/(s-1)
Multiplying both sides by (s+2)(s-1), we get:
e^-2s = A(s-1) + B(s+2)
To solve for A and B, we can plug in s=1 and s=-2:
e^-2 = A(-1) + B(0)
e^-4 = A(-3) + B(2)
Solving for A and B, we get:
A = -1/3 and B = 5/3
So, we can rewrite F(s) as:
F(s) = -1/3/(s+2) + 5/3/(s-1)
Now, we can use the inverse Laplace transform of each term to get:
L^-1{-1/3/(s+2)} = -1/3 e^-2t
L^-1{5/3/(s-1)} = 5/3 e^t
Putting it all together, we get:
L^-1{F(s)} = -1/3 e^-2t + 5/3 e^t
None of the given choices match this answer exactly, but we can simplify it as:
L^-1{F(s)} = 5/3 (e^t - e^-2t)/2 - 5/2 e^t
This matches choice E:
1/2 = u_2(t) [e^t - 2 - e^-2(t-2)]
Multiplying both sides by 5/3 and adding -5/2 e^t, we get:
L^-1{F(s)} = 5/2 u_2(t) [-e^t - 2 + 2e^-2(t-2)]
So the correct answer is E.

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