Answer:
Step-by-step explanation:
To find the x-intercepts of a quadratic function, we set y = 0 and solve for x. So, for the given function:
-1/2x^2 + x + 5/2 = 0
Multiplying both sides by -2 to eliminate the fraction:
x^2 - 2x - 5 = 0
We can solve for x using the quadratic formula:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
where a = 1, b = -2, and c = -5:
x = (2 ± sqrt(4 + 20)) / 2
x = (2 ± 2sqrt(6)) / 2
x = 1 ± sqrt(6)
Therefore, the x-intercepts of the quadratic function y = -1/2x^2 + x + 5/2 are
x = 1 + sqrt(6)
and
x = 1 - sqrt(6).
Triangles KLO and MNO are similar. Which proportion could be used to find LO? Select all that apply.
Answer:
LO=6
Step-by-step explanation:
MO/KO=NO/LO
5/10=3/LO
5LO=30
LO=6cm
Is W a subspace of the vector space? If not, state why. (Select all that apply.) W is the set of all vectors in R whose components are Pythagorean triples. (Assume all components of a Pythagorean triple are positive integers.) O W is a subspace of R3. W is not a subspace of R because it is not closed under addition W is not a subspace of R because it is not closed under scalar multiplication
No, W is not a subspace of R3 because it is not closed under vector addition and scalar multiplication, even though it contains the zero vector.
A set must meet three requirements to be a subspace of a vector space: (1) it must include the zero vector, (2) it must be closed under vector addition, and (3) it must be closed under scalar multiplication.
While W includes the zero vector (0, 0, 0), vector addition does not close it. For example, the triples (3, 4, 5) and (5, 12, 13) are both Pythagorean, but their addition (8, 16, 18) is not. As a result, W does not meet the second requirement and is not a subspace of R3.
Under scalar multiplication, W is likewise not closed. When we multiply the Pythagorean triple (3, 4, 5) by -1, we obtain (-3, -4, -5), which is not a Pythagorean triple. Therefore, W does not satisfy the third condition and is not a subspace of R.
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which is the crrect answer?
Answer:
Step-by-step explanation:
6 soccer players share 5 oranges as fraction
Answer:5/6
Step-by-step explanation: Theres 6 people that you are splitting 5 oranges among. so you do the equation 5÷6 or 5/6 which is the answer your looking for
Answer:
5/6
Step-by-step explanation:
=0.8333.......................
Consider the curve x2/3+y2/3=4.
Let L be the tangent line to this curve at the point (1,3√3), and let A and B be the x- and y-intercepts of L.
What is the length of the line segment AB?
(a) 8
(b) √38
(c) 8√3
(d) √3
(e) 31/38
If the curve x2/3+y2/3=4 and Let L be the tangent line to this curve at the point (1,3√3), and let A and B be the x- and y-intercepts of L, then the length of line segment AB is √38 (option b).
Explanation:
To find the length of line segment AB, follow these steps:
Step 1: We need to first find the equation of tangent line L at the given point (1, 3√3) on the curve x^(2/3) + y^(2/3) = 4.
Step 1. Find the derivative dy/dx using implicit differentiation:
(2/3)x^(-1/3) + (2/3)y^(-1/3)(dy/dx) = 0
Step 2. Solve for dy/dx (the slope of tangent line L) at point (1, 3√3):
(2/3)(1)^(-1/3) + (2/3)(3√3)^(-1/3)(dy/dx) = 0
dy/dx = -1/2
Step 3. Use the point-slope form to find the equation of tangent line L:
y - 3√3 = -1/2(x - 1)
Step 4. Find the x- and y-intercepts (A and B) of line L:
x-intercept (A): y = 0
0 - 3√3 = -1/2(x - 1)
x = 2 + 6√3
y-intercept (B): x = 0
y - 3√3 = -1/2(0 - 1)
y = 3√3 + 1/2
Step 5. Calculate the length of the line segment AB using the distance formula:
AB = √[(2 + 6√3 - 0)^2 + (3√3 + 1/2 - 0)^2] = √38
The length of line segment AB is √38 (option b).
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A Nyquist plot of a unity-feedback system with the feedforward transfer function G(s) is shown in Figure. If G(s) has one pole in the right-half s plane, is the system stable? If G(s) has no pole in the right-half s plane, but has one zero in the right-half s plane, is the system stable?
In a Nyquist plot, If G(s) has one pole in the right-half s plane, the system is marginally stable.
If G(s) has no pole in the right-half s plane, but has one zero in the right-half s plane, the system is stable.
In a Nyquist plot, the stability of a system can be determined by examining the number of encirclements of the -1 point. If the number of encirclements is equal to the number of right-half plane poles, then the system is unstable. If the number of encirclements is less than the number of right-half plane poles, then the system is marginally stable. If the number of encirclements is greater than the number of right-half plane poles, then the system is stable.
In the first case where G(s) has one pole in the right-half s plane, the Nyquist plot will encircle the -1 point once in the clockwise direction. Therefore, the number of encirclements is less than the number of right-half plane poles, which means the system is marginally stable.
In the second case where G(s) has one zero in the right-half s plane, the Nyquist plot will not encircle the -1 point at all. Therefore, the number of encirclements is less than the number of right-half plane poles, which means the system is stable.
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assume the random variable x is normal distributed with a mean of 395 and a standard deviation of 23. if x = 35, find the corresponding z-score.
The corresponding z-score for a random variable x that is normally distributed with a mean of 395 and a standard deviation of 23, when x = 35 is: -15.65
To find the z-score, you can use the following formula:
z = (x - μ) / σ
where z is the z-score, x is the value of the random variable, μ is the mean, and σ is the standard deviation.
Step 1: Identify the values.
x = 35, μ = 395, and σ = 23.
Step 2: Substitute the values into the formula.
z = (35 - 395) / 23
Step 3: Calculate the z-score.
z = (-360) / 23 = -15.65
So, the corresponding z-score for x = 35 is approximately -15.65.
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Find the point on y=x3+6x2-15x+1 at which the gradient is zero
Write the complex number e^i pi/3 in the form a + bi. a = and b =. Write the complex number e^i 4 pi/3 in the form a + bi. a = and b = Write the complex number z = 6 - 4i in polar form: z = r(cos theta + i sin theta) where r = and theta = The angle should satisfy 0 lessthanorequalto theta < 2 pi
z = 2√13(cos 5.49 + i sin 5.49) in polar form.
We can use Euler's formula, e^(iθ) = cos(θ) + i sin(θ), to write complex numbers in polar form.
1. For e^(iπ/3), we have:
e^(iπ/3) = cos(π/3) + i sin(π/3) = 1/2 + i√3/2
Therefore, a = 1/2 and b = √3/2.
2. For e^(i4π/3), we have:
e^(i4π/3) = cos(4π/3) + i sin(4π/3) = -1/2 - i√3/2
Therefore, a = -1/2 and b = -√3/2.
3. For z = 6 - 4i, we can find the magnitude (or modulus) of z, r, using the Pythagorean theorem:
|r| = √(6^2 + (-4)^2) = √52 = 2√13
To find the angle, theta, we can use the inverse tangent function:
tan⁻¹(-4/6) = -tan⁻¹(2/3) ≈ -0.93 radians
However, since we want the angle to satisfy 0 ≤ θ < 2π, we need to add 2π to the angle if it is negative:
θ = 2π - 0.93 ≈ 5.49 radians
Therefore, z = 2√13(cos 5.49 + i sin 5.49) in polar form.
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Let adj A 1 2 1 where adj A is the adjugate of matrix A, 1 1 2 with det A >0 and AX = A + X. Find det A and matrix X.
det(A) = ad - bc.
Matrix X is X = [8/5 -6/5, -2, 2/5 2/5]
What method is used to calculate det A and matrix X?We know that the adjugate of a 2x2 matrix is:
adj(A) = [d -b, -c a]
A = [a b, c d]
det(A) = ad - bc.
So, from the given adj(A), we have:
d - b = 1
-c = 2
-a = 1
d - c = 1
We have c = -2. Then, from the fourth equation, we have d = 1 - c = 3. Substituting these values into the first and third equations, we get:
b = 2
a = -1
So, the matrix A is:
A = [-1 2, -2 3]
We are given that AX = A + X. Substituting the matrix A and simplifying, we get:
AX = A + X
=> A(X - I) = X - A
=> (X - I)(-A) = X - A
=> X - I = (-A)⁻¹ (X - A)
=> X - I = A⁻¹ (X - A)
=> X = A⁻¹ X - A⁻¹ A + I
=> X = A⁻¹ (X - A) + I
Since we know A, we can find its inverse:
A⁻¹ = 1/(ad - bc) [d -b, -c a] = 1/5 [3 -2, 2 -1]
Substituting this into the above equation, we get:
X = 1/5 [3 -2, 2 -1] (X - [-1 2, -2 3]) + I
Simplifying this, we get:
X = 1/5 [4X + 5, -6X - 10, -2X - 10, 3X + 5]
Equating the corresponding elements on both sides, we get the following system of equations:
4x + 5 = x
-6x - 10 = 2
-2x - 10 = 1
3x + 5 = 3
Solving this system of equations, we get:
x = -1
Therefore, det(A) = ad - bc = (-1)(3) - (2)(-2) = -1 + 4 = 3.
And, matrix X is:
X = [8/5 -6/5, -2, 2/5 2/5]
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e. if a quantity increases exponentially, the time required to increase by a factor of 10 remains constant for all time. is this statement true or false?
The statement is false because in exponential growth, the time required to increase by a factor of 10 actually increases as the quantity gets larger.
We have,
In exponential growth, the rate of increase becomes progressively faster as the quantity grows.
This means that the time it takes to increase by a factor of 10 will also increase. For example, if it takes 1 year to increase from 1 to 10, it may take 2 years to increase from 10 to 100, and 3 years to increase from 100 to 1000.
The time required to achieve a factor of 10 increase will depend on the specific growth rate and initial quantity.
Therefore,
The time required to increase by a factor of 10 does not remain constant for all time in exponential growth. It increases as the quantity grows larger.
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Solve the expression 9 + (20 x one fourth) − 6 ÷ 2 using PEMDAS. (1 point)
8
11
13
16
Answer:
B) 11
Step-by-step explanation:
PEMDAS=parenthesis, exponents, multiplication, division, addition, and subtraction.
First start with the parenthesis:
20 x 1/4=5
9 + (5) - 6 ÷ 2
We don't have exponents or multiplication, so go onto division:
-6 ÷ 2 = -3
9+5-3
Finally addition and subtraction:
9+5-3=11
Hope this helps!
AABC is translated 4 units to the left and 8 units up. Answer the questions to find the coordinates of A after the translation.
1. Give the rule for translating a point 4 units left and 8 units up.
2. After the translation, where is A located
Now reflect the figure over the y-axis. Answer to find the coordinates of A after the reflection
3. Give the rule for reflecting a point over the y-axis
4. What are the coordinates of A after the reflection?
5. is the final figure congruent to the original figure? How do you know?
The shape and size of the figure are unchanged. and other soltuions are below
The rule for translating a point 4 units left and 8 units up.The rule for translating a point 4 units left and 8 units up is (x, y) → (x - 4, y + 8).
After the translation, where is A locatedUsing the above rule
After the translation, the new coordinates of A are (3, 13).
The rule for reflecting a point over the y-axisThe rule for reflecting a point over the y-axis is (x, y) → (-x, y).
The coordinates of A after the reflection?Using the above rule
After the reflection, the coordinates of A become (-7, 5).
Is the final figure congruent to the original figure?The final figure is congruent to the original figure because translation and reflection are both rigid transformations, which preserve distance and angles between points.
Therefore, the shape and size of the figure are unchanged.
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The student performed a different single transformation on PQR to create JKL. The coordinates of vertex K are (4,1). What could be the single transformation the student performed?
The single transformation which this student performed include the following: a reflection over the y-axis.
What is a reflection over the y-axis?In Geometry, a reflection over or across the y-axis or line x = 0 is represented and modeled by this transformation rule (x, y) → (-x, y).
By applying a reflection over the y-axis to the coordinate of the given triangle PQR, we have the following coordinates:
(x, y) → (-x, y).
Coordinate P = (-1, 1) → Coordinate J' = (-(-1), 1) = (1, 1).
Coordinate Q = (-4, 1) → Coordinate K' = (-(-4), 1) = (4, 1).
Coordinate R = (-4, 4) → Coordinate L' = (-(-4), 4) = (4, 4).
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√3x + √2x-1/√3x -√2x-1 = 5
prove that x = 3/2
Answer:
this is a correct answer 5√6/2
if the population standard deviation of aaa batteries (with a population mean of 9 hours) is 0.5 hours, the margin of error for a 95onfidence interval for μ (n = 25 batteries) would be
Therefore, the margin of error for a 95% confidence interval for the population mean of AAA batteries (with a population standard deviation of 0.5 hours and a population mean of 9 hours) based on a sample size of 25 batteries is 0.196 hours.
What is margin error?
Margin of error is a statistical term that refers to the amount of error that is allowed in a survey or poll's results. It is a measure of the precision of the results and indicates the range within which the true value is likely to fall.
In statistical terms, the margin of error is calculated as a percentage of the total number of respondents in a survey, and it takes into account factors such as the size of the sample, the level of confidence desired, and the variability in the responses.
For example, if a poll has a margin of error of +/- 3%, it means that the actual value of the result could be up to 3% higher or 3% lower than the reported value in the survey.
The formula for the margin of error for a 95% confidence interval is:
Margin of error = z * (σ / [tex]\sqrt(n)[/tex])
Where:
z = the z-score for the desired confidence level (95% confidence level corresponds to z = 1.96)
σ = the population standard deviation
n = the sample size
Substituting the given values into the formula:
Margin of error = 1.96 * (0.5 /[tex]\sqrt(25)[/tex])
Margin of error = 1.96 * (0.5 / 5)
Margin of error = 1.96 * 0.1
Margin of error = 0.196
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At the same time a 70 feet building casts a 50 foot shadow, a nearby pillar casts a 10 foot shadow. Which proportion could you use to solve for the height of the pillar?
The height of the pillar is 14 feet.
How to find height ?We can use the ratio of the building's height to the length of its shadow to solve for the pillar's height and apply it to the pillar as well. This is because similar-shaped objects will produce shadows that are proportional to their size.
Let h represent the pillar's height, and let's establish a proportion:
height of building / length of building's shadow = height of pillar / length of pillar's shadow
Substituting the given values:
70 / 50 = h / 10
We can simplify this proportion by cross-multiplying:
70 x 10 = 50h
700 = 50h
And solving for h:
h = 700 / 50 = 14
Therefore, the height of the pillar is 14 feet.
In conclusion, we can solve for the pillar's height by establishing a ratio between the building's height and the length of its shadow and applying that ratio to the pillar.
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find the slope of the parametric curve x=-4t^2-4, y=6t^3, for , at the point corresponding to t.
The slope of the parametric curve x=-4t^2-4, y=6t^3 at the point corresponding to t is -9t/4.
To find the slope of the parametric curve x=-4t^2-4, y=6t^3 at the point corresponding to t, follow these steps:
1. Find the derivatives of both x and y with respect to t:
dx/dt = -8t
dy/dt = 18t^2
2. The slope of the parametric curve is the ratio of the derivatives, dy/dx.
To find this, divide dy/dt by dx/dt:
dy/dx = (dy/dt) / (dx/dt)
= (18t^2) / (-8t)
3. Simplify the expression:
dy/dx = -9t / 4
So, the slope of the parametric curve x=-4t^2-4, y=6t^3 at the point corresponding to t is -9t/4.
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Solve the following inequality algebraically:
Negative 2 less-than StartFraction x Over 3 EndFraction + 1 less-than 5
a.
Negative 9 greater-than x greater-than 12
b.
Negative 9 less-than x less-than 12
c.
Negative 2 less-than x less-than 5
d.
Negative 2 greater-than x greater-than 5
The solution of inequality is -9 < x < 12.
The correct option is B.
We have,
-2 < x/3 + 1 < 5
Now, solving the inequation in parts as
-2 < x/3 + 1
-2 - 1 < x/3
-3 < x/3
-9 < x
and, x/3 + 1 < 5
x/3 < 4
x <12
Thus, the solution of inequality is -9 < x < 12.
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Find the volume of the rectangular prism
Answer:
2 1/3
Step-by-step explanation:
V = LWH
V = 1 3/4 ft × 2/3 ft × 2 ft
V = 7/4 × 2/3 × 2/1 ft³
V = 28/12 ft³
V = 7/3 ft³
V = 2 1/3 ft³
Right triangle ABC is inscribed in circle E. Find the area of the shaded region. Round your answer to the nearest tenth if necessary. C 8 A 15 E B
A New York City hotel surveyed its visitors to determine which type of transportation they used to get around the city. The hotel created a table of the data it gathered.
Type of Transportation Number of Visitors
Walk 120
Bicycle 24
Car Service 45
Bus 30
Subway 81
Which of the following circle graphs correctly represents the data in the table?
circle graph titled New York City visitor's transportation, with five sections labeled walk 80 percent, bus 16 percent, car service 30 percent, bicycle 20 percent, and subway 54 percent
circle graph titled New York City visitor's transportation, with five sections labeled walk 40 percent, bicycle 8 percent, car service 15 percent, bus 10 percent, and subway 27 percent
circle graph titled New York City visitor's transportation, with five sections labeled subway 40 percent, bus 8 percent, car service 15 percent, bicycle 10 percent, and walk 27 percent
circle graph titled New York City visitor's transportation, with five sections labeled subway 80 percent, bicycle 20 percent, car service 30 percent, bus 16 percent, and walk 54 percent
The option to the above question is a circle graph titled New York City visitor's transportation, with five sections labelled walk 40 per cent, bicycle 8 per cent, car service 15 per cent, bus 10 per cent, and subway 27 per cent.
What is a circle?A circle is a geometric shape that consists of a set of points in a plane that are equidistant from a fixed point called the centre. The distance from the centre to any point on the circle is called the radius, and the distance across the circle passing through the centre is called the diameter.
According to the given information:
The correct circle graph that represents the data in the table would be:
circle graph titled New York City visitor's transportation, with five sections labelled walk 40 per cent, bicycle 8 per cent, car service 15 per cent, bus 10 per cent, and subway 27 per cent.
This is because the percentages shown in this circle graph match the data given in the table for each type of transportation. Specifically, the circle graph shows that 40% of visitors walked, 8% used bicycles, 15% used car service, 10% used the bus, and 27% used the subway, which aligns with the numbers provided in the table.
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Answer:It’s B i got it correct on the quiz
Step-by-step explanation:
ILL GIVE BRAINLEIST THIS WAS DUE YESTERDAY!! 5. Use the following information to answer the questions.
.
A survey asked 75 people if they wanted a later school day start time.
.
45 people were students, and the rest were teachers.
.
50 people voted yes for the later start.
• 30 students voted yes for the later start.
.
a) Use this information to complete the frequency table. (5 points: 1 point for
each cell that was not given above)
Students
Teachers
Total
Vote YES for
later start
Vote NO for later
start
Total
b) Use the completed table from Part a. What percentage of the people surveyed
were teachers? (2 points)
Answer:
a) Yes No Total
Students 30 15 45
Teachers 20 10 30
Total 50 25 75
b) 30/75 = 2/5 = 40% of the people surveyed were teachers.
c) 20/75 = 4/15 = 26.7% of the people surveyed were teachers who wanted a later start time.
a faulty watch gains 10 seconds an hour if it is correctly set to 8 p.m. one evening what time will it show when the correct time is 8 p.m. the following evening
The watch gains 4 min till 8:00 PM in the next evening and show 8:04 pm the next evening.
What does it gain?Considering that,
A broken watch adds ten seconds per hour.
Find the number of hours between 8:00 PM this evening and 8:00 PM the following evening.
There are 24 hours in a day.
number of seconds the defective watch gained.
1 hour equals 10 seconds
24 hours ÷ by 10
24 * 10 is 240 seconds.
Now figure out how many minutes your defective watch has gained.
60 s = 1 minute
240 sec = 240/60
= 4 min
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Missing parts;
A faulty watch gains 10 seconds an hour. If it is set correctly at 8:00 pm one evening, what time will it show when the correct time is 8:00 pm the following evening
Give the correct singular, affirmative, formal command of each of the following verbs. 1. tener: 2. conocer: 3. buscar: 4. ir: 5. ser:
The singular, affirmative and formal command for each of the following are as follows: 1.tener: tenga, 2.conocer: conozca 3.buscar:busque 4.ir:vaya 5.ser:sea
What Spanish Affirmative and Negative commands?
The indicative, subjunctive, and imperative verb moods are the three primary categories of verb moods in Spanish.
When discussing actual actions, events, conditions, and facts, the indicative mood is utilized.The subjunctive mood, which denotes subjectivity, is typically employed to express a personal assertion or query.The imperative mood is used to issue clear instructions or directives. In other words, the imperative mood is employed to direct others as to what they should or should not do. As a result, Spanish has two command forms: Affirmative commands, also known as the positive imperative tense, are used to give specific instructions for something to occur. Spanish command phrases known as "negative commands” are used to issue clear directives against actions that should not happen(i.e., to tell people what not to do).To know more about Verb visit:
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Which equation shows the identity property of multiplication?
b.ama b
Submit
a b c d
0.a=0
· 1 = a
Answer:
Step-by-step explanation:
1=a
Answer:
Step-by-step explanation:
Identity means to get itself
For multiplication,
if you multiplied a number by 0, you would not get that number again, you would get 0. so multiplying by 0 is not and identity
ex. (a)(0)=0 does not = a so this is not an identity
if you multiplied a number by 1, yes that would be an identity because any number times 1 is itself
ex. a(1)=a multiplied by 1 the number is itself, so yes this is an identity.
convert the binary expansion of each of the following integers to a hexadecimal expansion. the hexadecimal notation of (0111 0111 0111 0111)2
Each group of four binary digits can be represented by a single hexadecimal digit. The hexadecimal notation of (0111 0111 0111 0111)₂ is (7777)₁₆.
To convert the binary expansion of (0111 0111 0111 0111)2 to hexadecimal, we first group the digits into groups of four starting from the right:
(0111 0111 0111 0111)2 = (7 7 7 7)16
Each group of four binary digits can be represented by a single hexadecimal digit. In this case, each group of four binary digits represents the hexadecimal digit 7. Therefore, the hexadecimal notation of (0111 0111 0111 0111)2 is (7777)16.
To convert the binary expansion (0111 0111 0111 0111)₂ to a hexadecimal expansion, you can group the binary digits into sets of four starting from the right and then convert each group to its corresponding hexadecimal value. Here's the process:
1. Group the binary digits: (0111) (0111) (0111) (0111)
2. Convert each group to hexadecimal:
- (0111)₂ = 7₁₆
- (0111)₂ = 7₁₆
- (0111)₂ = 7₁₆
- (0111)₂ = 7₁₆
Your answer: The hexadecimal notation of (0111 0111 0111 0111)₂ is (7777)₁₆.
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You want to buy a new cell phone. The sale price is $149
. The sign says that this is $35
less than the original cost. What is the original cost of the phone?
Answer:
In this problem it is saying that it is $35 less than the original cost so to find this you need to add. This should be just $149 + $35 to solve it which is a total of $184 which is the original cost for the phone.
$184 is your answer
Step-by-step explanation:
$114
as the question says the price is $35 less than the original cost, which means 149-35 which equal 144.
Can someone help me please? I've been trying to solve this for a while now, please help. Thank you
Answer:
-1(-4)=-5
Step-by-step explanation:
as h = -1
A viral linear DNA molecule of length, say, 1 is often known to contain a certain "marked position," with the exact location of this mark being unknown. One approach to locating the marked position is to cut the molecule by agents that break it at points chosen according to a Poisson process with rate λ. It is then possible to determine the fragment that contains the marked position. For instance, letting m denote the location on the line of the marked position, then if denotes the last Poisson event time before m (or 0 if there are no Poisson events in [0, m]), and R1 denotes the first Poisson event time after m (or 1 if there are no Poisson events in [m, 1]), then it would be learned that the marked position lies between L1 and R1. Find(a) P{L1 = 0}(b) P{L1 < x}, 0 < x < m, (c) P{R1 = 1}(d) P{R1 = x}, m < x < 1By repeating the preceding process on identical copies of the DNA molecule, we are able to zero in on the location of the marked position. If the cutting procedure is utilized on n identical copies of the molecule, yielding the data Li, Ri, i = 1, . . , n, then it follows that the marked position lies between L and R, where(e) Find E[R−L], and in doing so, show that E[R−L] ~
(a) P{L1 = 0}: Since L1 is the last Poisson event time before m, and m is uniformly distributed on [0, 1], we have:
P{L1 = 0} = P{no Poisson events in [0, m]} = e^(-λm)
(b) P{L1 < x}, 0 < x < m: We have:
P{L1 < x} = P{there is at least one Poisson event in [0, x]} = 1 - P{no Poisson events in [0, x]} = 1 - e^(-λx)
(c) P{R1 = 1}: Since R1 is the first Poisson event time after m, we have:
P{R1 = 1} = P{no Poisson events in [m, 1]} = e^(-λ(1-m))
(d) P{R1 = x}, m < x < 1: We have:
P{R1 = x} = P{there is no Poisson event in [m, x]} * P{there is at least one Poisson event in [x, 1]}
= e^(-λ(x-m)) * (1 - e^(-λ(1-x)))
(e) E[R-L]: We have:
E[R-L] = E[(R1-L1) + (R2-L2) + ... + (Rn-Ln)]
= E[R1-L1] + E[R2-L2] + ... + E[Rn-Ln] (by linearity of expectation)
= nE[R1-L1] (since the n copies are identical)
To find E[R1-L1], note that R1-L1 represents the length of the fragment that contains the marked position. This length is distributed as an exponential random variable with parameter λ, since it is the time until the next Poisson event in a Poisson process with rate λ. Therefore:
E[R1-L1] = 1/λ
Thus, we have:
E[R-L] = n/λ
This means that as n (the number of copies) becomes large, the expected length of the interval containing the marked position becomes smaller and smaller, converging to 0 as n approaches infinity.
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