The inverse Laplace transform of F(s) = (s + 3)² (s + 5) is given by L⁻¹{F} = 1/2 * e^(-3t) - t * e^(-3t) + 1/2 * e^(-5t).
To determine the inverse Laplace transform of F(s) = (s + 3)² (s + 5), we can use the properties and formulas of Laplace transforms.
The inverse Laplace transform of F(s) can be obtained by applying partial fraction decomposition, followed by looking up the corresponding inverse Laplace transform in a table of Laplace transforms.
Let's first factorize F(s):
F(s) = (s + 3)² (s + 5)
Next, we perform partial fraction decomposition. We express F(s) as the sum of simpler fractions:
F(s) = A/(s + 3) + B/(s + 3)² + C/(s + 5)
To find the values of A, B, and C, we can equate the numerator of F(s) with the sum of the numerators in the partial fraction decomposition:
(s + 3)² (s + 5) = A(s + 3)(s + 5) + B(s + 5) + C(s + 3)²
Expanding the equations and collecting like terms, we get:
s² + 8s + 15 = (A + C)s² + (8A + 3C + B)s + (15A + 5B)
Equating the coefficients of the terms on both sides, we have the following system of equations:
A + C = 1
8A + 3C + B = 8
15A + 5B = 15
Solving this system of equations, we find A = 1/2, B = -1/2, and C = 1/2.
Now, we can rewrite F(s) in terms of the partial fractions:
F(s) = 1/2/(s + 3) + (-1/2)/(s + 3)² + 1/2/(s + 5)
Looking up the inverse Laplace transform of each term in the table, we find:
L⁻¹{1/2/(s + 3)} = 1/2 * e^(-3t)
L⁻¹{(-1/2)/(s + 3)²} = -t * e^(-3t)
L⁻¹{1/2/(s + 5)} = 1/2 * e^(-5t)
Therefore, the inverse Laplace transform of F(s) is:
L⁻¹{F} = 1/2 * e^(-3t) - t * e^(-3t) + 1/2 * e^(-5t)
This is the desired result for the inverse Laplace transform of F(s).
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Solve the system of equations graphically: x – y = 1 and 3x = 3y + 3
Answer:
x=y+1,y=y
Step-by-step explanation:
Graph the function. f(x) = x+2+1 Plot four points on the graph of the function: the leftmost point and three 12 х 5 FO 8 - 5 -2 6 8 ID 12 10 -12
The graph should pass through the points (-5, -2), (6, 8), (12, 10), and (-12, -12), forming a diagonal line from the bottom left to the top right of the graph.
To graph the function f(x) = x + 2 + 1, we will plot four points on the graph.
Given the points: (-5, -2), (6, 8), (12, 10), and (-12, -12).
Plotting the points on a graph:
(-5, -2):
Starting from the origin (0,0), move 5 units to the left along the x-axis and 2 units downward along the y-axis. Plot the point (-5, -2).
(6, 8):
From the origin, move 6 units to the right along the x-axis and 8 units upward along the y-axis. Plot the point (6, 8).
(12, 10):
Move 12 units to the right along the x-axis and 10 units upward along the y-axis from the origin. Plot the point (12, 10).
(-12, -12):
Move 12 units to the left along the x-axis and 12 units downward along the y-axis from the origin. Plot the point (-12, -12).
Connecting the plotted points, we get a straight line. This line represents the graph of the function f(x) = x + 2 + 1.
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Joel says to Kevin, "Give me $100, and I shall become twice as rich as you."
Kevin replies, "Give me $10, and I shall become six times as rich as you."
How many dollars do Joel and Kevin have together?
2. Find measure of arc AB.
Find mAB
A) 68
B) 78
C) 88
D) 98
Answer:
C) 88
Step-by-step explanation:
so we are given a circle, and m<ACB is 44°. We need to find the measure of arc AB
<ACB is an inscribed angle (it is inside the circle), and it intercepts the arc AB
Inscribed angle theorem is a theorem that states the measure of an inscribed angle is half of the measure of the arc it intercepts
which also means that the measure of the intercepted arc is twice the measure of the inscribed angle (this is because of how algebra works)
which means mAB=2m<ACB (inscribed angle theorem)
mAB=2*44° (substitution)
mAB=88° (algebra)
therefore, your answer is C
Hope this helps!
Luis paid a total of $72.00 for 3 pairs of shorts. If each pair of shorts costs the same amount, what is the price of one pair of shorts?
Answer:
24
Step-by-step explanation:
72/3
Find the interest on the loan, $35,000 at 6% for 9 months. A. $1,910 C. $1,575 B. $1,395 D. $1,465
The interest on the loan of $35,000 at a 6% interest rate for 9 months is $1,575.
To calculate the interest on a loan, you can use the formula:
Simple Interest = Principal x Rate x Time.
In this case, the principal is $35,000, the interest rate is 6%, and the time is 9 months.
First, convert the interest rate to a decimal by dividing it by 100: 6% = 0.06. Next, calculate the interest by multiplying the principal, rate, and time: Interest = $35,000 x 0.06 x 9/12.
Simplifying the calculation, we get: Interest = $35,000 x 0.06 x 0.75 = $1,575.
Therefore, the correct answer is option B: $1,575.
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in the case of a triangle with angle measures of 30°, 60°, and 90° and a hypotenuse length equal to x, what is the perimeter of the triangle in terms of x?
The perimeter of triangle is the sum of the lengths of its three sides. In the case of a triangle with angle measures of 30°, 60°, and 90° and a hypotenuse length equal to x, we can determine the perimeter in terms of x.
Let's consider the sides of the triangle:
The side opposite the 30° angle is x/2, which can be derived using the properties of a 30-60-90 triangle.
The side opposite the 60° angle is x√3/2, which can also be derived using the properties of a 30-60-90 triangle.
The hypotenuse, which is opposite the 90° angle, has a length of x.
To find the perimeter, we add up the lengths of these three sides:
Perimeter = x/2 + x√3/2 + x
Combining like terms, we can simplify the expression:
Perimeter = (x + x√3 + 2x)/2
Perimeter = (3x + x√3)/2
Perimeter = x(3 + √3)/2
Therefore, the perimeter of the triangle in terms of x is 2x + x√3.
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Micah has been learning about Scientific Notation in math class and is frustrated because he doesn’t understand why he needs to learn it. While working on his homework with his mother, she told Micah that Scientific Notation can be very useful in certain careers. What do you think she meant?
Answer:
Step-by-step explanation:
The master equation describing the evolution of the probability Pm (t) to find an electron on site m of a linear chain of molecules with lattice constant a = 1nm is given by: dt -2RP.m + RPm+1 + RPm-1 Assuming that R = 10125-1, find the diffusion constant in cm²/s. Estimate a characteristic time for the electron to move a distance of 1 micron.
The electron to move a distance of 1 micron is approximately 0.255 s.
The main equation describing the evolution of the probability Pm (t) of finding an electron at point m in a molecular chain with lattice constant a = 1 nm is given by the formula: dt = -2RP.m RPm 1 RPm- 1 Assuming that R = 10-125 , calculate the diffusion constant in units of cm²/s. The diffusion constant is given by D = a²v/2t where a = 1 nm, v = RPm = 2RP and R = 10-125D = a²v/2tD = (10-9)² x 2RP / 2t = 4.2 x 10 ⁻¹⁰ RPt
Evaluate the characteristic time for an electron to travel a distance of 1 micron. Room temperature can be assumed for the electron, T = 27°C = 300K. The diffusion constant is given by the formula D = kbT/6πηra = (1.38 x 10-23 J/K) (300K) / (6π(8.9 x 10-4) Nsm-² x 1 x 10-⁷ m)D = 1 .57 x 10-2 cm²/s Distance 1 micron = 10⁻⁴ cm So by the formula D = sqrt((2d²)/t) t = 2d²/Dt = 2 x (10⁻⁴)² / (1.57 x 10 ⁻⁵ ) = 0.255 s Therefore, the characteristic time of an electron to travel a distance of 1 micron is approximately 0.255 s.
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Shamma is working at an addition recovery center. She reads somewhere that one of the differences between casual drug use and addiction is despair or depression. She randomly gives a group of her patients the Beck's Depression Inventory (BDI). She knows from previous research that a group of local patients with Major Depressive Disorder had a mean BDI score of 24.
After analyzing the One Sample T-test from her study she writes:
We conducted a one-tailed, one-sample t-test comparing the sample mean BDI score (28.4) against a population mean of 24, t(14) = 2.25, p = 0.021.
How many people were in Shamma's sample?
Using the information from Shamma's study in Question 21 write a null hypothesis and an alternative hypothesis.
From Shamma's study, there are 14 people in her sample. How many people were in Shamma's sample? Shamma's sample consists of 14 people.
Explanation: We can find the answer in the given text, which is the number of people in Shamma's sample.
The sentence that holds the answer is: "We conducted a one-tailed, one-sample t-test comparing the sample mean BDI score (28.4) against a population mean of 24, t(14) = 2.25, p = 0.021." We see that the value of t is in brackets with a value of 14.
Therefore, there are 14 people in Shamma's sample. Now, let's write the null hypothesis and an alternative hypothesis.
Null Hypothesis, H0: H0: μ = 24, There is no significant difference between the sample mean BDI score and population mean. Alternative Hypothesis, Ha: Ha: μ > 24, There is a significant difference between the sample mean BDI score and population mean.
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The sample mean is greater than the population mean i.e., μ > 24.
There were 15 people in Shamma's sample.
Shamma randomly gives the Beck's Depression Inventory (BDI) to a group of her patients.
After analyzing the One Sample T-test from her study she writes:
We conducted a one-tailed, one-sample t-test comparing the sample mean BDI score (28.4) against a population mean of 24, t(14) = 2.25, p = 0.021.
The value given in the bracket is 14.
As we know that N-1 is used as the degrees of freedom, so the number of people in Shamma's sample is:
[tex]N-1 = 14N = 14 + 1 = 15[/tex]
Thus, the number of people in Shamma's sample is 15.
The null hypothesis for Shamma's study would be:
H0: The sample mean is equal to the population mean i.e., μ = 24.
The alternative hypothesis for Shamma's study would be:
HA: The sample mean is greater than the population mean i.e., μ > 24.
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in february 2002 the argentine peso lost 70% of its value compared to the united states dollar. this devaluation drastically raised the price of imported products. according to a survey conducted by ac nielsen in april 2002, 68% of the consumers in argentina were buying fewer products than before the devaluation, 24% were buying the same number of products, and 8% were buying more products. furthermore, in a trend toward purchasing less-expensive brands, 88% indicated that they had changed the brands they purchased. f an appetizer, entree, beverage, and dessert. you have a choice of five appetizers, ten entrees, three beverages, and six desserts. how many possible complete dinners are possible?
After considering the given data we conclude that the there are 900 possible complete dinners generated that will be satisfactory to the dedicated question.
To evaluate the number of possible complete dinners, we need to apply multiplication regarding the number of choices for each course.
The number of possible complete dinners is the product of the number of choices for the appetizer, entree, beverage, and dessert.
Its is known to us that we have a choice of five appetizers, ten entrees, three beverages, and six desserts, the number of possible complete dinners is:
Number of possible complete dinners = number of choices for appetizer × number of choices for entree × number of choices for beverage × number of choices for dessert
Number of possible complete dinners = 5 × 10 × 3 × 6
Number of possible complete dinners = 900
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The complete question is
A restaurant menu has a price-fixed complete dinner that consists of an appetizer, an entrée, a beverage, and a dessert. You have a choice of 5 appetizers, 10 entrées, 3 beverages, and 6 desserts. Determine the total number of possible dinners.
Problem 5 Let X₁.....Xy be iid according to EX= for all i=1,2,.... Define Y = 1 if X, > density function of Y₁. a continuous probability density fx. Suppose and Y₁ = 0 otherwise. Find the probability
Given that X1, X2, ..., Xy are i.i.d according to the probability density function fx such that EX = 2. Define Y = 1 if X > 2 and Y = 0 otherwise.
Find P(Y = 1).
Given that X1, X2, ..., Xy are i.i.d according to the probability density function fx such that EX = 2. Thus the probability density function is,fx = 1/2 for x ∈ (0, 2) fx = 0 elsewhere
Therefore, P(X > 2) = ∫2∞ fx dx= ∫2∞ 0 dx= 0Now, P(Y = 1) = P(X > 2) = 0
Hence, the required probability is 0. Therefore, the correct option is (D) 0.
Likelihood is a proportion of the probability of an occasion to happen. Numerous occurrences cannot be completely predicted. Using it, we can only predict the chance of an event happening, or how likely it is to happen.
The probability of an event occurring is determined by dividing the total number of possible outcomes by the number of favorable outcomes. A coin flip is the simplest illustration. There are only two outcomes that can occur when flipping a coin; either heads or tails is the outcome.
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Lind domain and Range for 9x) = 1-dx²_X+2 without graphing. Prove that [0, 1] & [11, 21]
The domain of the function is all real numbers, and the range is [0, 1] and [11, 21].
To find the domain and range of the function f(x) = 1 - dx² - x + 2, we can analyze its properties without graphing.
Domain:
The domain of a function consists of all the valid input values, or the values of x for which the function is defined. In this case, the only restriction we need to consider is the square root of a negative number.
In the given function, there are no square roots involved. Therefore, the function is defined for all real numbers. The domain is the set of all real numbers, which can be represented as (-∞, ∞).
Range:
The range of a function consists of all the possible output values, or the values that f(x) can take. To determine the range, we need to consider the behavior of the function as x approaches positive or negative infinity.
As x approaches positive or negative infinity, the dominant term in the function is -dx². If d is positive, as x gets larger, the term -dx² becomes more negative, approaching negative infinity. Similarly, if d is negative, as x gets larger, the term -dx² becomes more positive, approaching positive infinity.
Since the remaining terms in the function (1-x+2) are constants, they do not affect the behavior as x approaches infinity. Therefore, the range of the function depends on the value of d.
If d is positive, the range of the function is (-∞, 1+d). As x approaches negative infinity, the function approaches positive infinity, and as x approaches positive infinity, the function approaches 1+d.
If d is negative, the range of the function is (1+d, ∞). As x approaches negative infinity, the function approaches 1+d, and as x approaches positive infinity, the function approaches negative infinity.
Given that the range is [0, 1] & [11, 21], we can conclude that d is positive, and the range is [0, 1+d]. Since the range also includes [11, 21], we can infer that 1+d = 11, and solving for d gives d = 10.
Therefore, the domain of the function is (-∞, ∞), and the range is [0, 1] & [11, 21].
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Given the following perfect square trinomial, fill in the missing term. (2 points)
9x^2 +___+25
Answer:
30x
Step-by-step explanation:
missing must be 2×3x×5=30x
9x²+30x+25=(3x+5)²
evaluate plssssssssssssssssssss
Answer:
2/5!
Step-by-step explanation:
Which graph is a function of x?
Answer:
The first graph is the function of x
4. (15 points) One of the eigenvalues of A = independent eigenvectors corresponding to λ = 2. −1 1 −1 !. is λ = 2. Find two linearly
Two linearly independent eigenvectors corresponding to the eigenvalue λ = 2 are v₁ = [-1, 0, 3] and v₂ = [0, 1, 1].
Given the matrix A = [−1 1 −1], one of the eigenvalues is λ = 2. We need to find two linearly independent eigenvectors corresponding to this eigenvalue.
To find the eigenvectors, we solve the equation (A - λI)v = 0, where A is the matrix, λ is the eigenvalue, I is the identity matrix, and v is the eigenvector.
In this case, the equation becomes:
(A - 2I)v = 0
Substituting the values, we have:
[−1 1 −1] - 2[1 0 0] [x] [0]
[y] = [0]
[z] [0]
Simplifying further:
[−3 1 −1] [x] [0]
[y] = [0]
[z] [0]
This gives us the following system of equations:
-3x + y - z = 0
y = 0
z = 0
From the second equation, we get y = 0. Plugging this into the first equation, we have -3x - z = 0, which simplifies to -3x = z.
Choosing a value for z, let's set z = 3. Then, -3x = 3, and solving for x gives x = -1.
Therefore, one eigenvector corresponding to the eigenvalue λ = 2 is v₁ = [-1, 0, 3].
To find the second linearly independent eigenvector, we can choose a different value for z. Let's set z = 1.
Again, from the equation -3x + y - z = 0, we have -3x + y - 1 = 0. By choosing x = 0, we get y = 1.
Thus, another eigenvector corresponding to λ = 2 is v₂ = [0, 1, 1].
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Calculate the range, interquartile range, variance and standard deviation for the data of a set A and set B and answer each of the following question. = Set B= Set A = 1,2,3,4,5,6,7 1.2.3.4.5.6.50 (a) Which measure of dispersion for the data of set A and set B has significant difference? (b) Determine the most appropriate measure of dispersion to be used to measure the distribution of the data of set B
(a) The range has a significant difference between Set A and Set B.
(b) The interquartile range is a more appropriate measure for the data of Set B, considering the presence of outliers.
For Set A:
Range = 7 - 1 = 6
Interquartile Range = Q3 - Q1 = 5 - 2 = 3
Variance = 4.67
Standard Deviation = 2.16
For Set B:
Range = 50 - 1 = 49
Interquartile Range = Q3 - Q1 = 6 - 2 = 4
Variance = 205.14
Standard Deviation = 14.33
(a) The measure of dispersion that has a significant difference between Set A and Set B is the range.
(b) The most appropriate measure of dispersion to be used to measure the distribution of the data of Set B depends on the specific characteristics of the data. However, considering the presence of outliers (such as the value 50), a robust measure like the interquartile range may be more suitable for Set B.
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What is the answer. Please do NOT send me a link. I am severely struggling with this question.
Answer:
360 ft²
³
Step-by-step explanation:
The volume of the rectangular pyramid is calculated as
V = [tex]\frac{1}{3}[/tex] Ah ( A is the area of the base and h the perpendicular height )
Here A = 10 × 6 = 60 ft² and h = 6 , then
V = [tex]\frac{1}{3}[/tex] × 60 × 6 = 20 × 6 = 120 ft³
The volume of the rectangular prism is calculated as
V = 10 × 6 × 4 = 60 × 4 = 240 ft³
Total volume = 120 + 240 = 360 ft³
Select all the expressions that are equivalent to 4 – x. x – 4 4 + -x -x + 4 -4 + x 4 + x Use the distributive property to write an expression that is equivalent to 5(-2x – 3). If you get stuck, use the boxes to help organize your work.
Answer:
B
Step-by-step explanation:
Answer:
B
Step-by-step explanation:
When approximating S(x)dx using Romberg integration, R9,4 gives an approximation of order: h10 h8 h4 h6
R₄,₄ gives an approximation of order h⁸ when approximating ∫(a to b)f(x)dx using Romberg integration. Therefore second option is the correct answer.
When approximating the integral ∫(a to b) f(x) dx using Romberg integration, the term "R₄,₄" refers to the fourth row and fourth column of the Romberg matrix.
This specific entry represents the approximation obtained using the Romberg method with four iterations. The order of approximation is determined by the highest power of h in the error term of the approximation.
Since R₄,₄ has a subscript of 4, it indicates that the approximation is of order h⁸. This means that the error decreases at a rate of h⁸ as the step size h decreases, providing a more accurate estimation of the integral.
Therefore the correct answer is second option.
The question should be:
When approximating ∫(a to b)f(x)dx using Romberg integration, R₄,₄ gives an approximation of order:
h10
h8
h4
h6
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Find the volume of the cone.
please help me.
Answer:
46.08
Step-by-step explanation:
the formula for finding the volume of the cone is: V = π r^2 h/3
Insert the information provided (2 because half the diameter is the radius)
and solve. Hope this helps!
I REALLY NEED HELP IF YOU CAN PLEASE SHOW WORK SO I CAN UNDERSTAND
Answer:
G
Step-by-step explanation:
The number of total bottles sold on that day is 40
11 + 7 + 18 + 4 = 40
On Tuesday, theoretically, this is what will be sold:
22 bottles of apple juice
14 bottles of cranberry juice
36 bottles of orange juice
8 bottles of pineapple juice
We need to find out what doesn't fit the data.
F: 14 - 8 = 6, not correct answer
G: the number of cranberry juice will be 6 times the number of pineapple juice sold? It's not even twice the amount!
H: 36 + 14 = 50, not correct answer
J: 22 - 8 = 14, not correct.
From this, G is the correct answer as it is not even close to matching the data.
Let f (x) = x2 + 2x + 3 What is the average rate of change for the quadratic function from X = -2 to x= to x = 5?
Answer:
Answer: 5
Step-by-step explanation:
The average rate of change for the function f(x) from x=a to x=b is given by :-
The given function : (Its a quadratic function with degree 2.)
The average rate of change for the quadratic function from x=−2 to x = 5 will be :-
[Note (-)(-)=(+)]
Hence, the average rate of change for the quadratic function from x=−2 to x = 5 is 5 .
Step-by-step explanation:
How do you do this problem?
Answer:
Step-by-step explanation:
0.3a = -6
Divide 0.3 from both sides
a = -20
4,14,14, 14.4.4
Determine if the data set is unimodal, bimodal, multimodal, or has no mode. Identify the models). If any exist
The data set is bimodal with modes at 4 and 14.
To determine the mode(s) and the modality of a data set, we need to identify the values that occur most frequently.
The given data set is: 4, 14, 14, 14, 4
To find the mode(s), we can count the frequency of each value:
4 appears 2 times14 appears 3 times
The mode(s) are the value(s) that appear with the highest frequency. In this case, both 4 and 14 have the same frequency of occurrence, so this data set is bimodal, with modes at 4 and 14.
Therefore, the data set is bimodal with modes at 4 and 14.
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Mark hopes to one day earn $10,000. Estimate how
many clients Mark would need.
Find the value of the variables in the image above
Answer:
Step-by-step explanation:
Round off 3,489 to the nearest hundred.
Answer:
3,500
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Triangle DEF ~ Triangle BCA
Select ALL angles that have a sine of 12/37
- A
- B
- C
- D
- E
- F
The two triangles, Triangle DEF ~ Triangle BCA. answer that DE = 3.6, EF = 1.47 and FD = 2.8.
Given the two triangles, Triangle DEF ~ Triangle BCA. Let us understand this notation first. '~' means 'is similar to'. So, it says that Triangle DEF is similar to Triangle BCA. The meaning of similarity is that the two triangles have the same shape but different sizes,
Similarly, angles E and C, as well as angles F and A, are corresponding angles. Since corresponding angles of similar triangles are congruent, we can write:D = B, E = C and F = ABy the transitive property of equality, we can write:D = B, E = C, F = A => DE = BC, EF = AB and FD = CA (using corresponding sides of the triangles).So, the triangles DEF and BCA are similar and their corresponding sides are proportional. Therefore, we can write: DE/BC = EF/AB = FD/CA = k (some constant)If we consider DE/BC = k, we can write: DE/k = BC and DE/k + EF/k = AB and (DE + EF)/k = FDAnd, from the given information, we know that CD = 3, EA = 4 and FB = 6.
So, we can write:BC + CD = 5 (since AB = EA + FB = 4 + 6 = 10 and AB/BC = EA/CD => BC = AB * CD/EA = 10 * 3/4 = 7.5)FD + CD = 9 (since FD/CA = EF/EA => FD = EF * CA/EA = 6 * 9/4 = 13.5)So, we get the following system of equations:DE/k + EF/k = 10/k = 2.5 (from the given EF = 2.5)DE/k = 7.5 - CD = 4.5 (from the equation BC + CD = 5)EF/k = AB - DE/k = 5.5 (using the first equation)DE/k + FD/k = 13.5/k (from the equation FD + CD = 9)DE/k = CD = 3 (from the given information)FD/k = 10.5/k (using the third equation)Solving these equations, we get:k = 3.75, DE = 13.5/3.75 = 3.6, EF = 5.5/3.75 = 1.47 and FD = 10.5/3.75 = 2.8.
So,we get the answer that DE = 3.6, EF = 1.47 and FD = 2.8.
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