Minimum radius of convergence R of power series solutions about the ordinary point x = 1 is 1.
The minimum radius of convergence R of power series solutions about the ordinary point x = 1 of the differential equation (x²-9)y''+3xy'+y=0 can be determined as follows:Let us first write the differential equation as: y''+ (3x/(x²-9)) y' + (1/(x²-9)) y= 0Therefore, we have the following properties: p(x) = (3x/(x²-9)), q(x) = (1/(x²-9)), and x0 = 1. Since p(x) and q(x) are both rational functions, and are defined for all x except ±3, the point x0 = 1 is an ordinary point.
To obtain power series solutions about the ordinary point x = 1, we should look for solutions of the form: y = (x - 1)²Σn≥0 an(x - 1)^n, where an's are constants to be determined. This is the power series solution about x0, expanded around x0, which is x = 1.Let us now plug in this series for y, y', and y'' into the differential equation:Σn≥0 an(n+1) (x - 1)^n + 3Σn≥0 an(x - 1)^(n+1) / (x - 1)² - Σn≥0 an(x - 1)^n / (x - 1)²= 0Multiplying everything by (x - 1)² and grouping together all coefficients of the same power, we have:Σn≥0 [(n+1)(n-2) an - 3an] (x - 1)^n= 0
Comparing the coefficients of the like powers of (x - 1), we get:2a1 = 0, and [(n+1)(n-2) an - 3an] = 0 for n ≥ 2.The solution for a1 = 0 does not affect the radius of convergence. Therefore, for n ≥ 2, we obtain an = 3 / [n(n-3)] by solving the quadratic equation, n² - n - 6 = 0.Therefore, the power series solution about x0 = 1 is: y = Σn≥2 [3 / (n(n-3))] (x - 1)² (x - 1)^nThe radius of convergence of this series is given by:R = limn→∞ |an / an+1|= limn→∞ |n(n-3) / (n+1)(n-2)|= 1Therefore, the minimum radius of convergence of the power series solutions about the ordinary point x = 1 is R = 1.
Hence, the answer is: Minimum radius of convergence R of power series solutions about the ordinary point x = 1 is 1.
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Please help physics due in 30 mins!!!!
The work done is 3750 Joules on the box.
What is the recipe for work completed?To quantitatively express this concept, the work W is equal to the force f times the distance d, or W = fd. If the force is applied at an angle to the displacement, the work is W = fd cos.t.
The equation W = F * d * cos(theta), where W is the work done, F is the force applied, d is the displacement of the item, and theta is the angle between the force and displacement vectors, can be used to solve this problem.
The force in this instance is 500 N, and the distance is provided as 15 m, and the 60 degree angle between the vectors of force and displacement.
So, by changing these numbers in the equation, we obtain:
W = 500 N x 15 m x cos (60 degrees)
We can simplify this to: Applying the trigonometric identity cos(60 degrees) = 1/2
W = (500 N) * (15 m) * (1/2)
W = 3750 J.
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ABCD is a parallelogram where AB = m and BĆ = n
В.
с
F is the midpoint of the line BD.
a) Express FD in terms of m and n.
FD =
m
A
D
E is the point such that ADE is a straight line with AD: DE = k:1
and
FÈ = = 2n m
b) Find the value of k.
Note: Please make sure your
final answer says k.
Mina Minute Harian
hand
ABCD is a parallelogram,
a) Expression of FD in terms of m and n is equals to vector FD = ( 1/2)(m + n).
b) The value of k is equals to the 2n/(n - 2m).
According to the Parallelogram law of vector addition, if two vectors vector a and vector b represent two sides of a parallelogram in magnitude and direction, then their sum vector a + vector b = the diagonal of the parallelogram through their common point in magnitude and direction. We have a parallelogram ABCD, where vector AB
= m and vector BC = n. Point F is the midpoint of the line BD. Using the Parallelogram law, vector AB + vector BC
= vector BD = m + n
a)As we know, mid point F divides the diagonal BD into two equal parts FD, and DB. So, the expression of vector FD is
vector FD = ( 1/2) vector BD
=> vector FD = ( 1/2)(m + n)
b)Now, ADE is a straight line with AD: DE = k : 1
and vector FE = 2n - (1/2)m
vector FA = vector FD = n/2 + m/2
vector AE + vector FA = vector FE
=> vector AE = 3n/2 - m
AD : DE = k : 1 so k = 2n/(n - 2m). Hence, required value is 2n/(n - 2m).
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Complete question:
ABCD is a parallelogram where AB = m and BĆ = n В. с F is the midpoint of the line BD. a) Express FD in terms of m and n. FD = m A D E is the point such that ADE is a straight line with AD: DE = k:1 and FÈ = = 2n m b) Find the value of k. Note: Please make sure your final answer says k... Mina Minute Harian hand Question Progress.
Y=5x+17 Y=-2x+4 solve with elimination
Answer:
x = -13/7
y = 54/7
Step-by-step explanation:
Y = 5x + 17 Y = -2x + 4
5x + 17 = -2x + 4
7x + 17 = 4
7x = -13
x = -13/7
Not put -13/7 in for x and solve for y
y = 5(-13/7) + 17
y = 54/7
So, the answer is x = -13/7 and y = 54/7
Answer: x = -13 / 7, y = 54/7
Step-by-step explanation:
To eliminate a variable, we can substitute y for 5x + 17
We get 5x + 17 = -2x + 4
7x = -13
x = -13 / 7
Substituting x into the 2nd equation y = 5 * -13 / 7 + 17
y = 119/7 - 65/7
y = 54/7
If the angles of a pentagon are xº, x°, 2xº, (2x +
40), (2x+10)º, find the value of the biggest
angle
Answer:
285°
Step-by-step explanation:
x + x + 2x + 2x + 40 + 2x + 10 = (5 - 2)180
8x + 50 = 540
8x = 490
x = 61.25
2x + 40 = 2(61.25) + 40 = 285
Answer:
Step-by-step explanation:
Interior angles in a pentagon equal 540°.
Simplify
x°,x°,2x°, (2x+40) and (2x+10) = 8x+50
Calculate x
8x + 50 = 540
8x = 540 - 50 = 490
x = 490/8
x = 61.25°
Calculate largest angle
2x + 40, where x = 61.25°
=162.5°
Can someone help me with this
Solving a system of equations we can see that he cost of a corn dog is $1.25 and the cost of the fries is $3.50
How to find the cost of each item?We can define two variables here:
x = cost of a corn dog.
y = cost of the fries.
With the information in the question we can write two equations, these two equations form a system of equations that we can solve, the system of equations is the one below:
2x + 3y = 13
4x + y = 8.50
We can isolate y in the second equation to get:
y = 8.50 - 4x
Replace it in the other one:
2x + 3*(8.5 - 4x) = 13
2x + 25.5 - 12x = 13
-10x = 13 - 25.5
x = -12.5/-10 = 1.25
Then:
y = 8.5 - 4*1.25 = 3.5
The cost of a corn dog is 1.25 dollars and the cost of the fries is 3.50 dollars.
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Find the value of x in the following diagram.
Round your answer to the nearest hundredth.
Answer:
Were sorry! Answer is not available right now check in later.
Step-by-step explanation:
solve for x, simplify your answer.
Answer:
x = 6
Step-by-step explanation:
given a line parallel to a side of a triangle and it intersects the other two sides, it divides those sides proportionally , that is
[tex]\frac{x}{x+3}[/tex] = [tex]\frac{8}{12}[/tex] = [tex]\frac{2}{3}[/tex] ( cross- multiply )
3x = 2(x + 3)
3x = 2x + 6 ( subtract 2x from both sides )
x = 6
A shadow 9 feet long is cast by a plum tree that is 12 feet tall. What is the height of a nearby lemon tree that casts a shadow 6 feet long?
The height of the nearby lemon tree is 8 feet using the property of similar triangles.
What are similar triangles?Triangles that are similar to one another in shape but differ in size are called similar triangles. Their respective sides are proportional to one another, and their corresponding angles are equal. The ratio of the corresponding sides of two triangles that are comparable is constant across all pairs of corresponding sides. Problems involving unknown lengths or heights in comparable forms can be resolved using this attribute.
The given situation represent two proportional triangles.
We know that the ratio of lengths of similar triangles are equal thus,
height of plum tree / length of plum tree's shadow = height of lemon tree / length of lemon tree's shadow
Substituting the values we have:
12 / 9 = height of lemon tree / 6
Using cross multiplication:
height of lemon tree = 12 * 6 / 9 = 8 feet
Hence, the height of the nearby lemon tree is 8 feet using the property of similar triangles.
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Express the following in exponential notation: 16384
The exponential notation of 16384 is 2^14
16384 can be expressed in exponential notation as 2^14, where 2 is the base and 14 is the exponent.
In exponential notation, a number is expressed as a base raised to an exponent, where the base is the number being multiplied repeatedly and the exponent represents the number of times the base is multiplied.
In this case, 2 is being multiplied 14 times to arrive at the value of 16384. Exponential notation is a useful way to represent very large or very small numbers in a concise and standardized format, making it easier to work with and compare values across different scales. It is commonly used in scientific and mathematical contexts.
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commuting times for employees of a local company have a mean of 63.6 minutes and astandard deviation of 2.5 minutes. what does chebyshev's theorem say about thepercentage of employees with commuting times between 58.6 minutes and 68.6 minutes?
According to Chebyshev's theorem, at least 75% of the employees will have commuting times that fall within 2 standard deviations of the mean, or between 58.6 minutes and 68.6 minutes.
Chebyshev's theorem states that for any set of data, regardless of its distribution, a certain percentage of the data lies within a certain number of standard deviations from the mean. Specifically, Chebyshev's theorem states that for any data set, at least 1 – 1/k² of the data values will lie within k standard deviations of the mean, where k is any number greater than 1. If k=2, at least 75% of the data values lie within 2 standard deviations of the mean. If k=3, at least 89% of the data values lie within 3 standard deviations of the mean.
Therefore, for a data set with a mean of 63.6 minutes and a standard deviation of 2.5 minutes, we can use Chebyshev's theorem to determine that at least 75% of the employees will have commuting times that fall within 2 standard deviations of the mean, or between 58.6 minutes and 68.6 minutes.
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The volume of a cylinder is 15, 919.8 cm³. If the height is 30 cm, what is the
radius? Use
The radius of the cylinder is r = 13 cm
What is the radius of a cylinder?The radius of a cylinder is the radius of the circular base of the cylinder.
Since the volume of a cylinder is 15, 919.8 cm³. If the height is 30 cm, we require it radius.
Using the formula for volume of a cylinder V = πr²h where
r = radius of cylinder and h = height of cylinderMaking r subject of the formula, we have that
r = √V/πh
Since
V = 15,919.8 cm³h = 30 cm and π = 3.142Substituting the values of the variables into the equation for the radius, we have that
r = √V/πh
r = √(15,919.8 cm³/[3.142 × 30 cm])
r = √(15,919.8 cm³/94.26 cm)
r = √168.8924 cm²
r = 12.995 cm
r ≅ 13 cm
So, the radius r = 13 cm
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What is the exact value of the trigonometric expression? State your answer in simplified radical form and include all work.
*Please reference included picture for problem. Thank you!
Answer:
The exact value of the given trigonometric expression is undefined.
Step-by-step explanation:
Given trigonometric expression:
[tex]\dfrac{\cos\left(\dfrac{2 \pi}{3}\right)}{\tan\left(-\dfrac{7 \pi}{4}\right)}+\csc(\pi)[/tex]
To find the exact value of the given trigonometric expression, begin by finding the exact values of each of the trigonometric functions in the expression
The exact value of cos(2π/3) is:
[tex]\implies \cos\left(\dfrac{2 \pi}{3}\right)=-\dfrac{1}{2}[/tex]
The exact value of tan(-7π/4) is:
[tex]\implies \tan\left(-\dfrac{7 \pi}{4}\right)=1[/tex]
Since the cosecant function is the reciprocal of the sine function, the exact value of csc(π) is:
[tex]\implies \csc (\pi)=\dfrac{1}{\sin(\pi)}=\dfrac{1}{0}=\textsf{unde\:\!fined}[/tex]
Therefore:
[tex]\begin{aligned}\implies \dfrac{\cos\left(\dfrac{2 \pi}{3}\right)}{\tan\left(-\dfrac{7 \pi}{4}\right)}+\csc(\pi)&=\dfrac{-\dfrac{1}{2}}{1}+\dfrac{1}{0}\\&=-\dfrac{1}{2}+\dfrac{1}{0}\\\\&=\textsf{unde\:\!fined}\end{aligned}[/tex]
You ate 4/12 of the pizza your family bought for dinner. Your brother ate 3/12 of the pizza. Which equation represents the fraction of pizza both you and your brother ate?
Answer:
7/12 pizzas have been eaten
Step-by-step explanation:
3 + 4= 7
7/12
7-12=5
5 pizzas left
Find an equation for the line of best fit for the table below.
X
Y
3
7
8
5
6
7
10 8
12
8
y = 0.4x +3.8
y = 3.8x + 0.4
y = 0.96x + 3.8
y = .08x - 3.8
The equation for the line of best fit for the table above is y = 0.96x + 3.8. This equation can be determined by using the least squares method, which is a mathematical procedure for finding the best fit line for a given set of data points.
What is determined ?The determination of something is the process of making a decision or arriving at a conclusion about it. Determining something can involve a variety of methods, such as research, observation, analysis, and experimentation. It can also involve the use of deductive reasoning, inductive reasoning, intuition, and intuition combined with logic.
This method involves finding the sum of the squares of the differences between the given y-values and the corresponding y-values of the line, and then minimizing this sum by varying the slope and y-intercept of the line. In this case, the best fit line is found to have a slope of 0.96 and a y-intercept of 3.8.
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The equation for the line of best fit for the table above is y = 0.4x + 3.8.
What is equation?An equation is a mathematical statement that describes the relationship between two or more variables. Equations are written using an equal sign and can be used to solve for unknown values. For example, the equation 2x + 5 = 15 can be used to solve for x; in this case, x = 5.
This equation can be determined by using the two-point form of a linear equation, which is y = mx + b, where m is the slope and b is the y-intercept. The slope of the line can be determined by calculating the rise over run between the two points, which in this case is (6-5)/(7-3)
= 0.4.
The y-intercept of the line can then be determined by plugging in one of the points into the equation and solving for b,
which in this case is (5-0.4*3)
= 3.8.
Therefore, the equation for the line of best fit is y
= 0.4x + 3.8.
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pls helllpppp i beg u i reallly need this tysmmm
In linear equation, value of expression is -5 and 0.
What in mathematics is a linear equation?
A linear equation is an algebraic equation of the form y=mx+b, where m is the slope and b is the y-intercept, and only a constant and a first-order (linear) term are included. The variables in the previous sentence, y and x, are referred to as a "linear equation with two variables" at times.
Equations with power 1 variables are known as linear equations. One example with only one variable is where ax+b = 0, where a and b are real values and x is the variable.
-6d + 4a + 5c/5
1)
d = 1 , a = 1 , c = 3
= -6d + 4a + 5c/5
= -6 * 1 + 4 * -1 + 5 * -3/5
= -6 -4 - 15/5
= -25/5
= -5
2)
d = -10, a = -5 , c = -8
= -6d + 4a + 5c/5
= -6 * -10 + 4 * -5 + 5 * -8/5
= 60 -20 - 40/5
= 60-60/5
= 0
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Lainey bought a set of
20
2020 markers for
$
6
$6dollar sign, 6. What is the cost of
1
11 marker?
If Lainey bought a set of 20 markers for $6, then the cost of one marker is $0.30.
To determine the cost of 1 marker, we need to divide the total cost of the set of markers by the number of markers in the set.
Lainey bought a set of 20 markers for $6, so the cost of the set is $6. To find the cost of 1 marker, we divide the cost of the set by the number of markers in the set:
Cost of 1 marker = Total cost of set / Number of markers in set
Cost of 1 marker = $6 / 20
Cost of 1 marker = $0.30
Therefore, the cost of 1 marker is $0.30.
In summary, to determine the cost of 1 item in a set, we divide the total cost of the set by the number of items in the set. In this case, Lainey bought a set of 20 markers for $6, so the cost of 1 marker is $0.30.
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Complete question is:
Lainey bought a set of 20 markers for $6. What is the cost of 1 marker?
Translate the sentence into an equation.
Seven more than the quotient of a number and 4 is equal 5to .
The following equation can be used to represent the sentence:
7 + (x/4) = 5 where x stands for a number.
What is a linear equation?A straight line on a graph is represented by a linear equation. It has a constant slope and y-intercept, one or more variables, typically expressed by x and y. Several different real-world situations can be represented by linear equations, such as estimating the cost of goods based on the quantity purchased or calculating a car's distance traveled based on speed and time. Finding the value of the variable that causes the equation to be true is the first step in solving a linear equation. In mathematics, science, engineering, and economics, linear equations are frequently employed.
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How do you calculate FV in Python?
In Python, to calculate FV (Future Value), you can use the following formula: "FV = PV * (1 + r) ^ n" Here, PV = Present Value, r = interest rate, and n = number of periods. There are different ways to implement this formula in Python, but one possible way is using a function.
Here is an example: function fv(pv, r, n): return pv * (1 + r) ** n
The function takes three arguments: pv, r, and n. Then it returns the future value computed with the formula. In your Python code, you can call the function and pass the required values. For example fv_result = fv(1000, 0.05, 10)print(fv_result)The output will be the future value of $1000 after 10 years with a 5% annual interest rate.
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Lesson: 3.03
Buffy and Addy launch their rockets at the same time. The height of Buffy's rocket, in
meters, is given by the function f x=-4.9x²+25x. where x is the number of seconds
after the launch. The height of Addy's rocket, in meters, is given by the function
g x=-4.9x² +16x+15 where x is the number of seconds after the launch. Algebraically
find the moment when the rockets are at the same height, and then use that to
calculate the height. Round to the nearest hundredth.
Answer:
21.93 meters.
Step-by-step explanation:
To find when the rockets are at the same height, we need to set the two functions equal to each other and solve for x:
-4.9x² + 25x = -4.9x² + 16x + 15
Simplifying this equation, we get:
9x = 15
x = 15/9
x ≈ 1.67
Therefore, the rockets are at the same height after approximately 1.67 seconds.
To find the height at which the rockets are at the same height, we can plug this value of x into either of the two functions. Let's use Buffy's function:
f(1.67) = -4.9(1.67)² + 25(1.67)
f(1.67) ≈ 21.93
Therefore, the height at which the rockets are at the same height is approximately 21.93 meters. Rounded to the nearest hundredth, this is 21.93 meters.
Find the mean, median, mode, and range of the data set after you perform the given operation on each data value.
9, 7, 12, 13, 9, 3; add 5
mean is average =9+7+12+13+9+3+5÷7=8.29
median is middle term =9
mode is frequented data=9
rang is the difference between maximum and minimum data=13_3=10
Solve for x to the nearest tenth.
2
X
4
10
Answer:
x ≈ 8.9
Step-by-step explanation:
using Pythagoras' identity.
the square on the hypotenuse of a right triangle is equal to the sum of the squares on the other 2 sides.
consider the right triangle on the left with hypotenuse h , then
h² = 2² + 4² = 4 + 16 = 20 ( take square root of both sides )
h = [tex]\sqrt{20}[/tex]
consider the right triangle on the right , then
x² + h² = 10²
x² + ( [tex]\sqrt{20}[/tex] )² = 10²
x² + 20 = 100 ( subtract 20 from both sides )
x² = 80 ( take square root of both sides )
x = [tex]\sqrt{80}[/tex] ≈ 8.9 ( to the nearest tenth )
For the value of the sum, enter an expression that gives the exact value, rather than entering an approximation.
A.-12 + 4 – 4/3 +4/9 – 4/27 +4/81 - … = -12/(1+1/3)
B.∑ (1/3)^n = 6*1/3^6(1/3^11-1)/(1/3-1)
a. The exact value of the sum of -12 + 4 – 4/3 +4/9 – 4/27 +4/81 - … = -12/(1+1/3) is 12/7.
b.The exact value of the sum of∑ (1/3)ⁿ = 6*1/3⁶(1/3¹¹-1)/(1/3-1) is 3/2.
A.-12 + 4 – 4/3 +4/9 – 4/27 +4/81 - …
This is an infinite geometric series with first term a = -12 and common ratio r = 4/(-3). The sum of an infinite geometric series is given by:
S = a / (1 - r)
Substituting the values of a and r, we get:
S = (-12) / [1 - (4/(-3))]
Simplify the denominator by multiplying both numerator and denominator by (-3):
S = (-12) / [-3 - 4]
S = (-12) / (-7)
S = 12/7
Therefore, the exact value of the sum is 12/7.
B. 6*1/3⁶(1/3¹¹-1)/(1/3-1)
This is a geometric series with first term a = 1 and common ratio r = 1/3. The sum of a geometric series with n terms is given by:
S = a (1 - rⁿ) / (1 - r)
As n approaches infinity, rⁿ approaches zero and the sum converges to:
S = a / (1 - r)
Substituting the values of a and r, we get:
S = 1 / (1 - 1/3)
S = 3/2
Therefore, an expression that gives the exact value of the sum is 3/2.
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Here is the question please help
1) With regard to the statistical diagram given:
The total number of women in the group = 180130 women intend to vote for party A.2)
24 Adults chose EnglishYes, an equal number of Adults and children chose Math because the portion of the pie chart that represents the choice of Children and adults are both identical and equal to 1/4 or 25%.What is a statistical diagram?A statistical diagram is a visual representation of data using graphs or charts, typically used to summarize and communicate information about the distribution, relationship, or comparison of variables.
1)
a) To derive the total number of women in the group, we must remove 200 from 380.
That is:
Total women in the group = 380 - 200
= 180 Women.
b) To get women who intend to vote for party A.
We must first derive the total who intend to vote for Party A.
the total who intend to vote for Party A = 380 - 130
= 250
Women who intend to vote for party A = Total who want to vote for Party A - Men who want to vote for Party A.
Women who intend to vote for party A = 250 - 120
= 130
2)
a) Given that 48 adults were evaluated, the the portion of the pie chart that shows their choice of English reads 50%, it means that
Adults who Chose English = 48 * 50%
= 24
b) an equal number of Adults and children chose Math because the portion of the pie chart that represents the choice of Children and adults are both identical and equal to 1/4 or 25%.
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please answer the question in the photo (will mark brainliest + 15p)
we have
13x+6y=−30------------- > 6y=-30-13x--------------- > y=(-30-13x)/6
x−2y=−4-- > 2y=x+4-------- > y=(x+4)/2
Using a graphing tool---------- > see attached figure
the solution of the system is the point (-2.625,0688)
the best estimate pair for the solution to the system is (−2.5, 0.75)
Please help, due very soon !!
Please urgent need the work and answer
X=3.2
Y=6.1
Z=0.2
XZ +Y2
Answer: 37.85
Step-by-step explanation:
Substitute: 3.2x0.2+6.1^2
Calculate the product or quotient: 0.64+6.1^2
Calculate the power: 0.64+37.21
Calculate the sum or difference: 37.85
Answer: 37.85
pls mark brainliest
Parallelogram RKMT has an area of 343 units^2 . RKMT is dialated by a scale factor of 2/7 . What is the area of the resulting image
Therefore , the solution of the given problem of area comes out to be the area of the resulting picture is therefore 28 units.
What is an area exactly?By calculating how much space would be needed to fully cover its exterior, its overall size can be calculated. The neighboring area is considered when selecting a comparable item for a rectangular design. The surface area of something determines its overall dimensions. The number of sides between a cuboid four trapezoidal curves determines how much water it can hold.
Here,
The following algorithm determines the area of a parallelogram:
=> Base area * height
We can write: given that the trapezoid RKMT has an area of 343 units2,
=> Base(RKMT) x Height(RKMT) = Area(RKMT) = 343
All of the parallelogram's linear measurements (lengths and widths) are multiplied by 2/7 when it is dilated by a scale factor of 2/7. As a result, the expanded parallelogram's base and height will be as follows:
=> base(dilated) equals base x (2/7)(RKMT)
=> Height(dilated) = Height * (2/7)(RKMT)
The dilated parallelogram's size will be as follows:
=> Area(dilated) = base(dilated) x height(dilated)
=> (2/7) x base(RKMT) x (2/7) x height(RKMT)
=> (4/49) x Area(RKMT)
=> (4/49) x 343
=> 28
The area of the resulting picture is therefore 28 units2.
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When testing for an association between class level (Freshman, Sophomore, Junior, Senior) and IQ score, which of the following is NOT an appropriate way to write the alternative hypothesis? There is an association between class level and IQ Not all the population 1Q means are the same. At least one of the population IQ means will be different
The alternative hypothesis states the presence of an association between class level and IQ score. The statement "Not all the population 1Q means are the same" is incorrect as it does not indicate any association between the two variables. A more appropriate alternative hypothesis would be "At least one of the population IQ means will be different" which suggests that the population IQ means are not all the same and there is an association between class level and IQ score.
In more technical terms, the alternative hypothesis can be expressed as:
H1: μ1 ≠ μ2 ≠ μ3 ≠ μ4
where μ
is the population mean IQ score for Seniors. This statement suggests that at least one of the population means will be different.
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A love expert carried out a study to quantify the effect of love songs on emotion. To do so, he used 30 volunteers. He random
Publishers
assigned the 30 volunteers to listen to either a love song or classical music. Then he asked them to draw a heart on a piece of paper. He measured the size of the heart drawn from bottom to top, in inches, for each person. The results are displayed in the stem and leaf plots.
The analysis of the data to obtain the confidence interval of the difference in the means indicates;
99% confidence interval for [tex]\bar x[/tex]₁ - [tex]\bar x[/tex]₂
The correct options are;
Name of Procedure
Two sample interval for [tex]\bar{x}[/tex]₁ - [tex]\bar{x}[/tex]₂Random
The volunteers are randomly selectedWe have a random sample of 15 subjects who listen to love songsWe have a random sample of 15 subjects who listen to classical music10%
The 10% condition is metNormal/Large Sample
The stemplot of the classical music sample data shows no strong skewness or outliersThe stemplot of the love song music sample data shows no strong skewness or outliers99% CI = (-0.302, 2.301)
Conclude;
We are 99% confident that the interval give in the previous step captures -0.301 to 2.301 = the true difference in mean heart height for all subject like these who listen to love songs versus classical music.
What is a confidence interval?A confidence interval is a range of value that is likely to contain the true value of a population parameter with a certain degree of confidence.
The two-sample t-test can be used to construct the 99% confidence interval as follows;
([tex]\bar x[/tex]₂ - [tex]\bar x[/tex]₁) ± t(α/2, df) × √(s₁²/n₁ + s₂²/n₂)
Where;
[tex]\bar x[/tex]₂ and [tex]\bar x[/tex]₁ = The sample means of the love song and classical music groups
s₁, and s₂ = The sample standard deviations
n₁ and n₂ = The sample sizes
df = The degrees of freedom
t(α/2, df) = The value from the t-distribution table with a significance level of 0.01 and df = n₁ + n₂ - 2
The data indicates;
n₁ = n₂ = 15
[tex]\bar x[/tex]₁ = 5.07, s₁ = 1.63
[tex]\bar x[/tex]₂ = 4.07, s₂ = 1.13
Therefore, we get;
([tex]\bar x[/tex]₂ - [tex]\bar x[/tex]₁) ± t(α/2, df) × √(s₁²/n₁ + s₂²/n₂)
= (5.07 - 4.07) ± t(0.005, 28) × √(1.62²/15 + 1.13²/15)
= 1 ± 2.763 × 0.469
= 1 ± 1.301
The 99% confidence interval for the difference in the true mean heart for subjects who listen to a love song versus classical music is; (-0.301, 2.301).
The correct statement is; 99% confidence interval for [tex]\bar x[/tex]₁ - [tex]\bar x[/tex]₂
The correct statements, placed in the box are;
Name of Procedure;
Two sample interval for [tex]\bar x[/tex]₁ - [tex]\bar x[/tex]₂
Random
The volunteers are randomly selected
The random condition is met
We have a random sample of 15 subjects who listen to a love song
We have a random sample of 15 subjects who listen to classical music
10%
The 10% condition is met
15 < 10% of all subjects like these who listen to love songs
15 < 10% of all subjects like these who listen to classical music
Normal/Large Sample
The Normal/Large condition is met
The stemplot of the classical music sample data shows no strong skewness or outliers
The stemplot of the love song music sample data shows no strong skewness or outliers
Therefore;
99% CI = (-0.301, 2.301)
Conclude;
We are 99% confident that the interval given in the previous step captures -0.301 to 2.301 = the true difference in mean heart height for all subject like these who listen to love songs versus classical music.
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Which of these following lies between 6. 3 and 6. 6
6. 2
6. 9
6. 05
6. 41