The margin of error for the duration of telephone calls directed by a local telephone company is approximately 0.511 minutes or 30.66 seconds, with a 98% confidence level.
Given the provided information, we can use the following terms to calculate the margin of error:
1. Standard deviation (s) = 4.5 minutes
2. Sample size (n) = 420 calls
3. Confidence level = 98%
First, let's find the critical value (z-score) for a 98% confidence level. Using a standard normal distribution table, we can determine that the critical value is approximately 2.33.
Next, we need to find the standard error. The standard error (SE) is calculated as follows:
SE = s / √n
SE = 4.5 / √420
SE ≈ 0.2195
Now that we have the critical value and standard error, we can calculate the margin of error (ME) using the formula:
ME = z-score * SE
ME = 2.33 * 0.2195
ME ≈ 0.511
Thus, the margin of error for the duration of telephone calls directed by a local telephone company is approximately 0.511 minutes or 30.66 seconds, with a 98% confidence level. This means that the true mean duration of telephone calls is likely to be within 0.511 minutes (or 30.66 seconds) of the sample mean.
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Write the transformation matrix and the resultant matrix that will translate the triangle, (-2, 4), given the triangles vertices are at A(-1, 3), B(0, -4) and C(3, 3).
The resultant matrix represents the vertices of the translated triangle, with A' at (-3, 7), B' at (-2, 0), and C' at (1, 7).
Define the term transformation matrix?A transformation matrix is a mathematical matrix that represents a geometric transformation of space.
To translate the triangle by (-2, 4), we need to add -2 to the x-coordinates and add 4 to the y-coordinates of each vertex. The transformation matrix for this translation is:
[tex]\left[\begin{array}{ccc}1&0&-2\end{array}\right][/tex]
[tex]\left[\begin{array}{ccc}0&1&4\end{array}\right][/tex]
[tex]\left[\begin{array}{ccc}0&0&1\end{array}\right][/tex]
To apply this transformation to the vertices of the triangle, we can represent each vertex as a column vector with a third coordinate of 1:
[tex]A=\left[\begin{array}{ccc}-1&3&1\end{array}\right][/tex]
[tex]B=\left[\begin{array}{ccc}0&-4&1\end{array}\right][/tex]
[tex]C=\left[\begin{array}{ccc}3&3&1\end{array}\right][/tex]
To apply the translation, we multiply each vertex by the transformation matrix:
[tex]A'=\left[\begin{array}{ccc}1&0&-2\end{array}\right] \left[\begin{array}{ccc}-1&3&1\end{array}\right] \left[\begin{array}{ccc}-3&7&1\end{array}\right][/tex]
[0 1 4] × [ 0 -4 1] = [-2 0 1]
[0 0 1] [ 1 3 1] [-1 7 1]
[tex]B'=\left[\begin{array}{ccc}1&0&-2\end{array}\right] \left[\begin{array}{ccc}0&-4&1\end{array}\right] \left[\begin{array}{ccc}-2&-4&1\end{array}\right][/tex]
[0 1 4] × [ 0 -4 1] = [-2 0 1]
[0 0 1] [ 0 3 1] [-2 7 1]
[tex]C'=\left[\begin{array}{ccc}1&0&-2\end{array}\right] \left[\begin{array}{ccc}3&3&1\end{array}\right] \left[\begin{array}{ccc}1&7&1\end{array}\right][/tex]
[0 1 4] × [ 3 3 1] = [ 1 7 1]
[0 0 1] [ 3 3 1] [ 1 7 1]
The resultant matrix represents the vertices of the translated triangle, with A' at (-3, 7), B' at (-2, 0), and C' at (1, 7).
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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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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.
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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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
which is the crrect answer?
Answer:
Step-by-step explanation:
An object is placed 16.2 cm from a first converging lens of focal length 11.6 cm. A second converging lens with focal length 5.00 cm is placed 10.0 cm to the right of the first converging lens.(a) Find the position q1 of the image formed by the first converging lens. (Enter your answer to at least two decimal places.)cm(b) How far from the second lens is the image of the first lens? (Enter your answer to at least two decimal places.)cm beyond the second lens(c) What is the value of p2, the object position for the second lens? (Enter your answer to at least two decimal places.)cm(d) Find the position q2 of the image formed by the second lens. (Enter your answer to at least two decimal places.)cm(e) Calculate the magnification of the first lens.(f) Calculate the magnification of the second lens.(g) What is the total magnification for the system?(h) Is the final image real or virtual (compared to the original object for the lens system)?realvirtualIs it upright or inverted (compared to the original object for the lens system)? upright
The final image is real since it is located to the right of the second lens. It is also upright since the magnification is positive.
(a) Using the thin lens formula, 1/f = 1/p + 1/q, where f is the focal length, p is the object distance, and q is the image distance, we have:
1/f = 1/p - 1/q1
Substituting the given values, we get:
1/11.6 = 1/16.2 - 1/q1
Solving for q1, we get:
q1 = 6.97 cm
Therefore, the image formed by the first lens is located 6.97 cm to the right of the lens.
(b) The image formed by the first lens acts as an object for the second lens. Using the thin lens formula again, we have:
1/f = 1/p2 + 1/q1
Substituting the given values, we get:
1/5 = 1/p2 + 1/6.97
Solving for p2, we get:
p2 = 3.32 cm
The distance from the second lens to the image of the first lens is then:
q2 = p2 + q1 = 3.32 + 6.97 = 10.29 cm
Therefore, the image of the first lens is located 10.29 cm to the right of the second lens.
(c) The object distance for the second lens is simply the image distance of the first lens:
p2 = q1 = 6.97 cm
(d) Using the thin lens formula again, we have:
1/f = 1/p2 + 1/q2
Substituting the given values, we get:
1/5 = 1/6.97 + 1/q2
Solving for q2, we get:
q2 = 13.95 cm
Therefore, the final image is located 13.95 cm to the right of the second lens.
(e) The magnification of the first lens is given by:
m1 = -q1/p = -6.97/16.2 ≈ -0.43
(f) The magnification of the second lens is given by:
m2 = -q2/p2 = -13.95/3.32 ≈ -4.20
(g) The total magnification of the system is given by the product of the magnifications of the two lenses:
m = m1 × m2 ≈ 1.81
(h) The final image is real since it is located to the right of the second lens. It is also upright since the magnification is positive.
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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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help quick/100 points Select all of the following statements that are true. If 6 > 10, then 8 · 3 = 24. 6 + 3 = 9 and 4 · 4 = 16 If 6 · 3 = 18, then 4 + 8 = 20 5 · 3 = 15 or 7 + 5 = 20
Answer:
The statements that are true are:
6 + 3 = 9 and 4 · 4 = 16 (both are true statements)
5 · 3 = 15 or 7 + 5 = 20 (at least one of these statements is true, since the word "or" means that only one of the two statements needs to be true for the entire statement to be true)
The other two statements are false:
If 6 > 10, then 8 · 3 = 24 (this statement is false, because the premise "6 > 10" is false, and a false premise can never imply a true conclusion)
If 6 · 3 = 18, then 4 + 8 = 20 (this statement is false, because the conclusion "4 + 8 = 20" does not follow logically from the premise "6 · 3 = 18")
Please help!!!
006
A survey was conducted at a local mall in which 100 customers were asked what flavor of soft drink they preferred. The results of the survey are in the chart. Based on this survey if 300 customers were asked their preference how many would you expect to select cola as their favorite flavor? Answer in units of customers.
Answer: 87
Step-by-step explanation: 3 x 29=87 because you are multiplying the amount of people in the survey by 3
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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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
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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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
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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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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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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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!
Find the point on y=x3+6x2-15x+1 at which the gradient is zero
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
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.
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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√3x + √2x-1/√3x -√2x-1 = 5
prove that x = 3/2
Answer:
this is a correct answer 5√6/2
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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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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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.......................
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:
which equation represents the relationship show in the graph?
let's firstly get the EQUATion, of the graph before we get the inequality.
so we have a quadratic with two zeros, at -6 and 8, hmmm and we also know that it passes through (-2 , 10)
[tex]\begin{cases} x = -6 &\implies x +6=0\\ x = 8 &\implies x -8=0\\ \end{cases} \\\\[-0.35em] ~\dotfill\\\\ \stackrel{original~polynomial}{a ( x +6 )( x -8 ) = \stackrel{0}{y}}\hspace{5em}\textit{we also know that } \begin{cases} x=-2\\ y=10 \end{cases}[/tex]
[tex]a ( -2 +6 )( -2 -8 ) = 10\implies a(4)(-10)=10\implies -40a=10 \\\\\\ a=\cfrac{10}{-40}\implies a=-\cfrac{1}{4} \\\\[-0.35em] ~\dotfill\\\\ -\cfrac{1}{4}(x+6)(x-8)=y\implies -\cfrac{1}{4}(x^2-2x-48)=y \\\\\\ ~\hfill {\Large \begin{array}{llll} -\cfrac{x^2}{4}+\cfrac{x}{2}+12=y \end{array}}~\hfill[/tex]
now, hmmm let's notice something, the line of the graph is a solid line, that means the borderline is included in the inequality, so we'll have either ⩾ or ⩽.
so hmmm we could do a true/false region check by choosing a point and shade accordingly, or we can just settle with that, since the bottom is shaded, we're looking at "less than or equal" type, or namely ⩽, so that's our inequality
[tex]{\Large \begin{array}{llll} -\cfrac{x^2}{4}+\cfrac{x}{2}+12\geqslant y \end{array}}[/tex]
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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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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