Solve the system of equations x' 2x – 3y + 2 sin(2t) y' = x – 2y — 2 cos(2t)
Answers
Upon solving the given system of equations:
\(x(t) = (c_1 + e^{-t}) + (3/2) * (c_2 + e^{-t}) * cos(2t) + (1/2) * (c_1 + e^{-t}) * sin(2t),\\y(t) = (c_1 + e^{-t}) + (3/2) * (c_2 + e^{-t}) * sin(2t) - (1/4) * (c_1 + e^{-t}) * cos(2t)\)
To solve the system of equations:
x' = 2x - 3y + 2sin(2t)
y' = x - 2y - 2cos(2t)
We can use the method of undetermined coefficients to find the particular solution. Assuming the particular solution takes the form:
\(x_p(t) = A sin(2t) + B cos(2t)\\y_p(t) = C sin(2t) + D cos(2t)\)
Substituting these expressions into the original equations, we get:
2(A sin(2t) + B cos(2t)) - 3(C sin(2t) + D cos(2t)) + 2sin(2t) = 2sin(2t)
(A sin(2t) + B cos(2t)) - 2(C sin(2t) + D cos(2t)) - 2cos(2t) = cos(2t)
(2A - 3C + 2)sin(2t) + (2B - 3D)cos(2t) = 2sin(2t)
(A - 2C)sin(2t) + (B - 2D - 2)cos(2t) = cos(2t)
By comparing the coefficients of sine and cosine on both sides, we can equate them separately:
2A - 3C + 2 = 2
2B - 3D = 0
A - 2C = 0
B - 2D - 2 = 1
Solving these equations, we find:
A = 1
B = 3/2
C = 1/2
D = -1/4
So the particular solution is:
\(x_p(t)\) = sin(2t) + (3/2)cos(2t)
\(y_p(t)\) = (1/2)sin(2t) - (1/4)cos(2t)
To find the complementary solution, we solve the homogeneous system:
x' = 2x - 3y
y' = x - 2y
We can rewrite this system as a matrix equation:
X' = AX
where \(X = [x, y]^T\) and
\(A = \left[\begin{array}{ccc}2&-3\\1&-2\end{array}\right]\)
The characteristic equation is:
det(A - λI) = 0, where I is the identity matrix. Solving this equation, we find the eigenvalues:
\(\lambda_1 = -1\\\lambda_2 = -1\)
For each eigenvalue, we solve the corresponding eigenvector equation:
(A - λI)V = 0
For \(\lambda_1 = -1\), we have:
\(\left[\begin{array}{ccc}3&-3\\1&-1\end{array}\right] * V_1 = 0\)
Solving this system, we find the eigenvector:
\(V_1 = [1\ \ 1]\)
For \(\lambda_2 = -1\), we have:
\(\left[\begin{array}{ccc}3&-3\\1&-1\end{array}\right] * V_2= 0\)
Solving this system, we find the eigenvector:
\(V_2 = [3\ \ 1]\)
So the complementary solution is:
\(x_c(t) = c_1 * e^{-t} * [1\ \ 1]^T + c_2 * e^{-t} * [3\ \ 1]^T\\y_c(t) = c_1 * e^{-t} * [1\ \1]^T + c_2 * e^{-t} * [3\ \ 1]^T\)
where
\(c_1\ and\ c_2\) are arbitrary constants.
The general solution is the sum of the particular and complementary solutions:
\(x(t) = x_p(t) + x_c(t)\\y(t) = y_p(t) + y_c(t)\)
Simplifying and combining terms, we get:
\(x(t) = (c_1 + e^{-t}) + (3/2) * (c_2 + e^{-t}) * cos(2t) + (1/2) * (c_1 + e^{-t}) * sin(2t)\\y(t) = (c_1 + e^{-t}) + (3/2) * (c_2 + e^{-t}) * sin(2t) - (1/4) * (c_1 + e^{-t}) * cos(2t)\)
where \(c_1\ and\ c_2\) are arbitrary constants.
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Related Questions
Perform the indicated operation, if possible.
\(\ \textless \ br /\ \textgreater \
\left[\(\(\(\begin{array}{rrrr}\ \textless \ br /\ \textgreater \
2 & 8 & 13 & 0 \\\ \textless \ br /\ \textgreater \
7 & 4 & -2 & 5 \\\ \textless \ br /\ \textgreater \
1 & 2 & 1 & 10\ \textless \ br /\ \textgreater \
\end{array}\right]-\left[\begin{array}{rrrr}\ \textless \ br /\ \textgreater \
2 & 3 & 6 & 10 \\\ \textless \ br /\ \textgreater \
3 & -4 & -4 & 4 \\\ \textless \ br /\ \textgreater \
9 & 0 & -2 & 17\ \textless \ br /\ \textgreater \
\end{array}\right]\)\)\)
\)
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
A. The resulting matrix is (Simplify your answer.)
B. The matrices cannot be subtracted.
Answers
The correct choice is A. The resulting matrix is
\(\[\begin{array}{rrrr}0 & 5 & 7 & -10 \\4 & 8 & 2 & 1 \\-8 & 2 & 3 & -7 \\\end{array}\]\)
To perform the indicated operation, we need to subtract the second matrix from the first matrix. The matrices must have the same dimensions to be subtracted.
Given matrices:
\(\[ \begin{array}{rrrr}2 & 8 & 13 & 0 \\7 & 4 & -2 & 5 \\1 & 2 & 1 & 10 \\\end{array}\]\)
and
\(\[ \begin{array}{rrrr}2 & 3 & 6 & 10 \\3 & -4 & -4 & 4 \\9 & 0 & -2 & 17 \\\end{array}\]\)
These matrices have the same dimensions, so we can subtract them element by element.
Subtracting the corresponding elements, we get:
\(\[ \begin{array}{rrrr}2-2 & 8-3 & 13-6 & 0-10 \\7-3 & 4-(-4) & -2-(-4) & 5-4 \\1-9 & 2-0 & 1-(-2) & 10-17 \\\end{array}\]\)
Simplifying the subtraction, we have:
\(\[ \begin{array}{rrrr}0 & 5 & 7 & -10 \\4 & 8 & 2 & 1 \\-8 & 2 & 3 & -7 \\\end{array}\]\)
Therefore, the resulting matrix is:
\(\[ \begin{array}{rrrr}0 & 5 & 7 & -10 \\4 & 8 & 2 & 1 \\-8 & 2 & 3 & -7 \\\end{array}\]\)
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Which choice is equivalent to the expression below?
Answers
Answer:
C. \(4\sqrt{7} -4x\sqrt{7}\) is correct
Steps:
\(4\sqrt{7} -3\sqrt{7}x-x\sqrt{7} \\\\4\sqrt{7} -3\sqrt{7} x-\sqrt{7} x\\\\4\sqrt{7} +(-3\sqrt{7} x-\sqrt{7} x)\\\\4\sqrt{7} -4\sqrt{7} x\)
Answer:
C. 4√7 - 4x√7
Step-by-step explanation:
4√7 - 3x√7 - x√7
combine like terms
4√7 - 4x√7
7. IfQ, and Q2 are orthogonal 1 X matrices, show that the product QO2 is orthogonal.
Answers
The product of the two matrices Q₁Q₂ is orthogonal
What i orthogonal matrix?In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors. ... {\displaystyle Q^{\mathrm {T} }Q=QQ^{\mathrm {T} }=I,} where QT is the transpose of Q and I is the identity matrix.
It is said to be an orthogonal matrix if its transpose is equal to its inverse matrix, or when the product of a square matrix and its transpose gives an identity matrix of the same order.
If A is an n*n orthogonal matric, then A*A¹ = A¹*A
Therefore A*A¹ = A¹*A = 1
This implies that the product Q₁O₂ is orthogonal.
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is to bee $4.00 on the culcome 7 . For this bee, the player wins $1600 if the 18sut of the roll is 7 and lases $4.000 oiterwise. Complate pants fa) through (o) Click the reen to vew a tatie of all possitele oinsomes of a two rice rol. (Type an axact aremer in tirrolfiod form.)
Answers
In this scenario, a player can place a bet of $4.00 on the outcome of rolling two dice. If the sum of the roll is 7, the player wins $1600, otherwise, they lose $4.00.
To analyze the possible outcomes of rolling two dice, we need to consider all the combinations of numbers that can appear on each die. Each die has six sides, numbered from 1 to 6. When rolling two dice, we can have a total of 36 different outcomes (6 possibilities for the first die multiplied by 6 possibilities for the second die).
Now, we need to determine the sum of each pair of numbers. For example, if the first die shows a 1 and the second die shows a 6, the sum is 7. We can list all the possible outcomes and their corresponding sums:
(1, 1) - Sum: 2
(1, 2) - Sum: 3
(1, 3) - Sum: 4
(1, 4) - Sum: 5
(1, 5) - Sum: 6
(1, 6) - Sum: 7
(2, 1) - Sum: 3
(2, 2) - Sum: 4
(6, 6) - Sum: 12
Out of these 36 outcomes, there are six combinations that result in a sum of 7. These combinations are: (1, 6), (2, 5), (3, 4), (4, 3), (5, 2), and (6, 1). If the sum is 7, the player wins $1600, and if the sum is any other number, the player loses $4.00.
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which number is the next logical number in the following sequence of numbers: 2, 6, 14, 30,
Answers
The next logical number in the sequence is 50
How to find the next logical number in the given sequence (2, 6, 14, 30)?To find the next logical number in the given sequence (2, 6, 14, 30), we need to observe the pattern or rule governing the sequence. Let's analyze the differences between consecutive terms:
6 - 2 = 4
14 - 6 = 8
30 - 14 = 16
By looking at the differences, we can see that they are increasing by 4 each time. Therefore, it appears that the sequence is based on adding the successive odd numbers: 1, 3, 5, 7, and so on.
Now, let's calculate the next difference:
16 + 4 = 20
To find the next number in the sequence, we add this difference to the last term:
30 + 20 = 50
Hence, the next logical number in the sequence is 50.
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Find the exact length of the curve. y = In(sec(x)), 0≤x≤ Need Help? Read It π 4 Watch It
Answers
The curve is y = In(sec(x)) and we have to find its length. We are given the range as 0 ≤ x ≤ π/4. So, the formula for the length of the curve is given as:
To solve for the length of the curve of y = In(sec(x)), we use the formula,
`L = ∫[a,b] √[1+(f′(x))^2] dx`.Where, `a = 0` and `b = π/4`. And `f′(x)` is the derivative of `In(sec(x))`.
We know that:`f′(x) = d/dx[In(sec(x))]`
Using the formula of logarithm differentiation, we can write the above equation as:
`f′(x) = d/dx[In(1/cos(x))]`
So,`f′(x) = -d/dx[In(cos(x))]`
Therefore,`f′(x) = -sin(x)/cos(x)`
Substituting the values, we get:
`L = ∫[a,b] √[1+(f′(x))^2] dx`
`L = ∫[0,π/4] √[1+(-sin(x)/cos(x))^2] dx`
`L = ∫[0,π/4] √[(cos^2(x)+sin^2(x))/(cos^2(x))] dx`
`L = ∫[0,π/4] sec(x) dx`
Now, `L = ln(sec(x) + tan(x)) + C` where `C` is a constant.
We calculate the constant by substituting the values of `a = 0` and `b = π/4`:
`L = ln(sec(π/4) + tan(π/4)) - ln(sec(0) + tan(0))`
`L = ln(√2 + 1) - ln(1 + 0)`
`L = ln(√2 + 1)`
Thus, the exact length of the curve is `ln(√2 + 1)` units.
Thus, the exact length of the curve of y = In(sec(x)), 0≤x≤π/4 is `ln(√2 + 1)` units.
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May someone help me on this question please
Answers
Answer:
y = 225x + 625
Step-by-step explanation:
slope (m) = rise/run = Δy/Δx = (1750-625) / (5-0) = 1125/5 = 225
It means for one additional child (Run) will charge $225 (Rise) per month.
Basic fee is $625 (y intercept: b)
y = mx + b
Equation: y = 225x + 625
Select the correct answer.
The class sizes of the Introductory Psychology courses at a college are shown below.
121, 134, 106, 93, 149, 130, 119, 128
The college adds a new Honors Introductory Psychology course with 45 students. What effect does the new class size have on the center and spread of the class sizes of the Introductory Psychology courses at the college?
A. The value of the center increases and the distribution has a larger spread.
B. The value of the center decreases and the distribution has a smaller spread.
C. The value of the center decreases and the distribution has a larger spread.
D. The value of the center increases and the distribution has a smaller spread.
Answers
Center : Mean Before the introduction of the new course, center = average(121,134,106,93,149,130,119,128) = 122.5 After the introduction of the new course, center = average(121,134,106,93,149,130,119,128,45) = 113.9 The center has moved to the left (if plotted in a graph) because of the low intake for the new course. Spread before introduction of the new course : Arrange the numbers in ascending order: (93, 106,119, 121), (128, 130,134, 149) Q1=median(93,106,119,121) = 112.5 Q3=median(128,130,134,149) = 132 Spread = Interquartile range = Q3-Q1 = 19.5 After addition of the new course,
(45,93, 106,119,) 121, (128, 130,134, 149)
Q1=median(45,93,106,119)=99.5
Q3=median (128, 130,134, 149)= 132
Spread = Interquartile range = 132-99.5 =32.5
We see that the spread has increased after the addition of the new course.
What is 4 1/2÷(−2 2/3)?
Write the answer as a mixed number in simplest form.
help plssss
Answers
Answer:
Step-by-step explanation:
you cant divide fractions so you multiply the reciprocal so your answer would be
1 11/16
identify the similar triangles in the diagram, then sketch them so the corresponding sides and angles have the same orientation.
Answers
The triangle KLM and triangle JKM are congruent triangle
In geometry, two figures or objects are said to be congruent if their shapes and sizes match, or if one is the mirror image of the other.
Triangles that are identical in size and shape are said to be congruent triangles. Inferred from this is that the matching sides and angles are equal. Without checking each of the triangles' sides and angles, we may determine whether two triangles are congruent.
Given a triangle LKJ
We have to find similar triangles in the diagram so the corresponding sides and angles have the same orientation.
By the figure given we can write following things:
LK/LJ = MK/JK
LK/JK = MK/MJ
So the triangle KLM ≅ triangle JKM ≅ triangle JLK
Hence the triangle KLM and triangle JKM are congruent triangle
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The second of two numbers is two more than the first.
The sum is 40. Find the numbers.
Answers
Answer:
19 + 21 = 40.
Step-by-step explanation:
a number is at least 6 and less than 15. Write as inequality and graph.
Answers
The inequality expression 6 <= x < 15 to represent the statement "a number is at least 6 and less than 15"
What are expressions?Expressions are mathematical statements that are represented by variables, coefficients and operators
How to determine the expression and the inequality?The mathematical statement is given as
a number is at least 6 and less than 15.
Represent the number with x
So, we have
x is at least 6 and less than 15.
At least 6 means greater than or equal 6
So, we have
x >= 6
Less than 15
So, we have
x < 15
When the above expressions are combined, we have
6 <= x < 15
Next, we plot the inequality expression 6 <= x < 15 to represent the statement "a number is at least 6 and less than 15"
See attachment for the inequality expression
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 GIVING BRAINLIEST TO THE PERSON THAT CAN EXPLAIN THIS THE BEST (:
Answers
Answer:
the given picture is not clear
mark has 100$in his savings account. He earns 25$ a week mowing lawns. If mark puts all of his earnings from mowing lawns into a savings account, how many weeks will it take for him to have 500$?
Answers
400+ 100 = 500
Therefore the answer is 16
Si A es tu edad, la tasa de pulsos máxima que deberías mantener durante actividades aeróbicas es de 0.88 (220-A). ¿Cúal es la tasa de pulsos máxima que deberías mantener si tú estuvieras en el rango de los 20 años?
Answers
Respuesta:
176
Explicación paso a paso:
Dado :
La frecuencia máxima del pulso que se debe mantener durante las actividades aeróbicas viene dada por:
0,88 (220-A); donde A = edad
Para una persona de 20 años
Supongamos que la edad es de 20 años; A = 20
Poniendo A = 20 en la ecuación;
0,88 (220 - A); A = 20
Frecuencia de pulso máxima:
0,88 (220 - 20)
0,88 (200)
= 176
La frecuencia de pulso máxima es 176
Find the supplemet of x
Answers
Answer:
the answer is 180 degrees
pls mark me as the brainliest
Evaluate the following logical expressions for all combinations of variables. (a) F1 = A + B + C (b) F2 (B) (C) (c) F3 = A +B +C (d) F4 = ABC (e) Fs ABC+(B+C)
Answers
There seems to be an incomplete question as there are missing logical expressions for (b), (c), and (e). Could you please provide the missing information?
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Determine the period
Answers
Answer:
10
Step-by-step explanation:
acellus
The period of the given graph is 10.
Use the concept of a period of function defined as:
One full wave is set up in a medium for the length of time it takes a particle of that medium to complete one vibration or one wave period.
In the given wave,
X-axis represents time
The Y- axis represents the amplitude of the curve
As we can see,
The first peak of this graph is located at x = 1 and the second peak is at,
x = 13
Since the period is the time interval between two consecutive peaks in a graph.
In this case, the time interval is 10.
Hence,
The period of the given graph is 10.
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Tyrone ran 7 laps on he quarter mile track during practice. Which subset of real numbers would include the distance Tyrone ran at practice?
Answers
Answer:
Step-by-step explanation:
7 quarter mile laps totals 7/4 miles, or 1 3/4 miles.
1 3/4 miles is between 1 and 2, a subset of real numbers.
a mathematics teacher gives her class a two-question clicker quiz at the end of each class period and tabulates their answers according to their mathematical understanding, misconceptions, and error patterns. if her goal is improvement in her students' mathematical proficiency, her best use of the data would be to use it to:
Answers
The mathematics teacher should use the data from the clicker quiz to identify the areas where her students have misconceptions or errors in their understanding of mathematical concepts.
She can then adjust her lesson plans to focus on these areas and provide additional instruction or resources to help her students improve their understanding. By analyzing the data and using it to inform her teaching strategies, the teacher can help her students develop better mathematical proficiency and achieve better results on future assessments.
Based on the given scenario, if the mathematics teacher's goal is to improve her students' mathematical proficiency, her best use of the data from the two-question clicker quiz would be to:
1. Identify areas of misunderstanding and error patterns: By analyzing the students' responses, the teacher can pinpoint specific concepts or problem-solving strategies that are causing difficulties.
2. Tailor instruction accordingly: Once the teacher has identified areas of weakness, she can adapt her lessons and teaching methods to address these issues more effectively, ensuring that students receive targeted support to improve their understanding.
3. Monitor progress over time: Regularly collecting and analyzing data from the quizzes allows the teacher to track the progress of her students and determine if her instructional adjustments are resulting in improved mathematical proficiency.
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The mathematics teacher can use the data from the clicker quiz to identify the areas where her students are struggling the most and focus her teaching on those topics.
She can also use the data to provide individualized feedback to each student, addressing their specific misconceptions and errors. By analyzing the patterns in the data, the teacher can modify her teaching strategies and methods to better suit the learning needs of her students.
In short, the data from the clicker quiz can be used to inform and improve the teacher's instruction and enhance her students' mathematical proficiency.
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If you have the option of choosing a loan that will accumulate 4%/ a interest compounded semi-annually compared to a loan that will accumulate 4% /a interest compounded monthly, which one would you choose
Answers
Answer:
4%/ a interest compounded semi-annually
Step-by-step explanation:
The option that gives the lower interest rate payment would be more appropriate. to determine this calculate the effective interest rate
Effective annual rate = (1 + APR / m ) ^m - 1
M = number of compounding
(1 + 0.04/2)^2 - 1 = 4.04%
(1 + 0.04/12)^12 - 1 = 4.07%
the choice should be 4%/ a interest compounded semi-annually
Please solve this question.
Answers
Answer: 2
Step-by-step explanation:
Current Age Age in future (x years)
Brother 1: 15 x + 15
Brother 2: 22 x + 22
The product of their ages in the future is 408:
(x + 15)(x + 22) = 408
x² + 37x + 330 = 408 Expanded
x² + 37x - 78 = 0 Subtracted 408 from both sides
(x + 39)(x - 2) = 0 Factored
x + 39 = 0 x - 2 = 0 Applied Zero Product Property
x = -39 x = 2 Solved for x
↓
Disregard since "future" years cannot be negative
The segment AC is?
A- the adjacent
B- the hypotenuse
C- The opposite
D- The tangent
Answers
Answer:
B
Step-by-step explanation:
the hypotenuse........
Answer:
B
Step-by-step explanation:
The hypotenus is always opposite to the right and angle and always the longest side
Solve this problem... Really urgent
Answers
Answer:
\( \boxed{\sf Time \ taken = 15 \ minutes} \)
Given:
Initial speed (u) = 65 km/h
Final speed (v) = 85 km/h
Acceleration (a) = 80 km/h²
To Find:
Time taken for car to achieve a speed of 85 km/h in minutes
Step-by-step explanation:
\(\sf From \ equation \ of \ motion:\)
\( \boxed{ \bold{v = u + at}}\)
By substituting value of v, u & a we get:
\( \sf \implies 85 = 65 + 80t\)
Substract 65 from both sides:
\( \sf \implies 85 - 65 = 65 - 65 + 80t\)
\( \sf \implies 20 = 80t\)
\( \sf \implies 80t = 20\)
Dividing both sides by 80:
\( \sf \implies \frac{ \cancel{80}t}{ \cancel{80}} = \frac{20}{80} \)
\( \sf \implies t = \frac{2 \cancel{0}}{8 \cancel{0}} \)
\( \sf \implies t = \frac{ \cancel{2}}{ \cancel{2} \times 4} \)
\( \sf \implies t = \frac{1}{4} \: h\)
\( \sf \implies t = \frac{1}{4} \times 60 \: minutes\)
\( \sf \implies t = 15 \: minutes\)
So,
Time taken for car to achieve a speed of 85 km/h in minutes = 15 minutes
Calculate the work done in lifting a 15-lb flower pot to a height of 4 ft above the ground.
Answers
Answer:
A. 60 ft·lb
Step-by-step explanation:
You want the work done lifting a 15-lb flower pot to a height of 4 ft.
WorkWork is the product of force and distance. When the pot is raised 4 ft, the work done is ...
W = F·d
W = (15 lb)(4 ft) = 60 ft·lb
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Round 685,135 to the nearest ten thousand.
Answers
Answer:
690,000 I think
Step-by-step explanation:
if pumpkin weights are normally distributed with a mean of 10 pounds and a standard deviation of 2 pounds, what is the probability that a randomly selected sample of 16 pumpkins has a mean weight greater than 11.1 pounds?
Answers
The probability of a selected sample is 0.0001.
What is a normal distribution?
It is also known as Gaussian distribution. It shows the data near the more. This appears as bell curve in graphical form. It is a bell shaped frequency distribution curve of a continuous random variable.
Given that,
μ= 10 pounds
σ=2 pounds
n=16
Sampling distribution of mean is
μₓ⁻ =μ=10
Standard error of mean is
σₓ⁻ =σ/√n
=2/√16
=0.5
p(μₓ⁻ >12.5)
=1-p((x⁻-μₓ⁻/σₓ⁻ )≤ ((12.5-10)/0.5)
=1-p(z≤5)
using standard normal table,
=1-0.9999
=0.0001
The probability of a selected sample is 0.0001.
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What is the value of x?
Answers
Answer: 5
Step-by-step explanation:
By the inscribed angle theorem.
\(\frac{9}{15}=\frac{2x-1}{3x}\\\\27x=30x-15\\\\-3x=-15\\\\x=5\)
radius is half of a diameter true or false
Answers
The relation between the radius and the diameter of a circle is given by:
r = d/2
where r is the radius and d the diameter.
Hence, radius is half of a diameter
Mary, who is married and the mother of three, is 35 years and expects to work until 75. She earns $55,000 per year. Mary expects inflation to be 3% over her working life, and the appropriate risk-free discount rate is 5%. Her personal consumption is equal to 27% of her after-tax earnings, and her combined federal and state marginal tax bracket is 15%. What is the amount of life insurance necessary for Mary using the Human Life Value method?
What if Mary works until 70? What would be the amount of life insurance necessary for her?
What is the amount of life insurance necessary for Mary using the Capitalization of Earnings method if she works until 75?
What is the amount of life insurance necessary for Mary using the Capitalization of Earnings method if she works until 70?
Answers
Mary's HLV is $2,215,773.95.
Mary's HLV if she works until 70 is $1,611,008.23.
Mary's COE if she works until 75 is $5,130,372.40.
Mary's COE if she works until 70 is $2,985,308.60.
How to find amount of life insurance?To calculate the Human Life Value insurance (HLV) of Mary, we need to determine the present value of her future income stream.
Annual after-tax earnings = $55,000 x (1 - 0.15) = $46,750
Total after-tax earnings over working life = $46,750 x (1 + 0.03)^(75-35) = $3,382,820.33
Total personal consumption = 27% x $3,382,820.33 = $914,073.89
Calculate Mary's HLV:
= $3,382,820.33 - (1 + 0.05)^(-1) x $914,073.89
= $2,215,773.95
Therefore, Mary's HLV is $2,215,773.95.
If Mary works until 70 total after-tax = $46,750 x (1 + 0.03)^(70-35) = $2,124,182.09
Mary's HLV if she works until 70 is $1,611,008.23.
To calculate the Capitalization of Earnings (COE) method. Here's how we can calculate it:
Calculate Mary's expected earnings in her final year of work:
Expected earnings in final year = $55,000 x (1 + 0.03)^(75-35) = $256,518.62
Apply a capitalization rate to the expected earnings:
COE = expected earnings in final year / capitalization rate
= $256,518.62 / 0.05
= $5,130,372.40
Therefore, Mary's COE if she works until 75 is $5,130,372.40.
If Mary works until 70, her expected earnings in her final year of work would be:
Expected earnings in final year = $55,000 x (1 + 0.03)^(70-35) = $149,265.43
Mary's COE if she works until 70 is $2,985,308.60.
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