A histogram of the weights of all 2,000 athletes would be a graph with two normal distributions, one for offense and one for defense.
The offense distribution would have a bell shaped curve centered at 200 pounds, with a range of 150-250 pounds. The defense distribution would have a bell shaped curve centered at 300 pounds, with a range of 250-350 pounds. Each of the distributions would have the same area, representing 1,000 athletes in each group. The total area of the histogram would represent the combined 2,000 athletes with weights ranging from 150-350 pounds. It is important to note that the histogram would not represent the exact weights of the athletes, but rather the relative frequency of each weight.
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The number of tricycles in the preschool play yard was?
2
4
No answer text provided
3
The number of tricycles in the preschool play yard is given as follows:
10.
How to obtain the number of tricycles?The variables for the system of equations are given as follows:
Variable x: number of bicycles.Variable y: number of tricycles.There are 30 seats, hence:
x + y = 30
x = 30 - y.
There are 70 wheels, hence:
2x + 3y = 70.
Replacing x = 30 - y on the second equation, the number of tricycles is obtained as follows:
2(30 - y) + 3y = 70.
y = 10.
Missing InformationA preschool playground has both bicycles and tricycles. There is a total of 30 seats and 70 wheels. What is the number of tricycles?
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Julia sorted candies and found there were
10 each of these colors: green, red, and
pink and 20 blue in the package. Which
answer matches the percentage, decimal
and fraction of the blue candies in the
package?
F 20%, .2, 1/5
G 30%, .3, 3/10
H 40%, .4, 2/5
J 50%, .5, 1/2
The percentage is 40%, decimal point of blue candies is 0.4, and the fraction of blue candies is 2/5. The correct option is H. 40%, 0.4, 2/5.
How to calculate the percentage and decimal? Percentage is a unit of measurement used to express a proportion of a whole number in terms of hundredths. To calculate a percentage, divide the number of items in question by the total number of items and then multiply the result by 100. For instance, if you were to calculate the percentage of apples in a basket that contained 12 apples and 8 oranges, you would divide 12 by 20 (12 + 8) and then multiply the result by 100. The answer would be 60 percent.Decimals are numbers that are expressed in tenths, hundredths, or thousandths of a unit. To calculate a decimal, divide the number of items in question by the total number of items. For instance, if you were to calculate the decimal amount of apples in a basket that contained 12 apples and 8 oranges, you would divide 12 by 20 (12 + 8). The answer would be 0.6.Given,
Julia has 10 candies in green, 10 candies in red, 10 candies in pink and 20 candies in blue. There are 50 candies in Julia's hand.
We have to find percentage, decimal and fraction of blue candies = ?
Percentage = 20/50 x 100 = 40%
Decimal = 20/50 = 2/5 = 0.4
Fraction = 20/50 = 2/5.
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Sample Questions Chapters 3, 12, 13, 14. All of the following contribute to the success of incumbent members of Congress n election campaigns EXCEPT Incumbents.
All of the following contribute to the success of incumbent members of Congress n election campaigns EXCEPT Incumbents tend to understand national issues better than do their challengers.
What does the term incumbent mean?The term "incumbent" has several meanings depending on the context, but it generally refers to someone who holds a particular position or office. The most common usage of this term is in the context of politics, where an incumbent is the current holder of a political office, such as a president, governor, or member of Congress. An incumbent candidate is the candidate who is currently holding a political office and is running for re-election.
In other fields, the term incumbent is used to describe someone who holds a particular job, position or role and is responsible for certain tasks or duties, for example an incumbent employee, an incumbent contractor, an incumbent provider, etc.
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Complete question not found
If PQRS is a rectangle,
find the measure of ZQPT.
P
(16x - 17)
(10x + 1)°
T
R
S
Answer:
∠ QPT = 59°
Step-by-step explanation:
• the opposite sides are parallel
then
∠ SPT = ∠ QRT ( alternate angles ), that is
16x - 17 = 10x + 1 ( subtract 10x from both sides )
6x - 17 = 1 ( add 17 to both sides )
6x = 18 ( divide both sides by 6 )
x = 3
then
∠ SPT = 16x - 17 = 16(3) - 17 = 48 - 17 = 31°
• the vertices of a rectangle are right
∠ QPS = 90° , so
∠ QPT = 90° - ∠ SPT = 90° - 31° = 59°
Joe’s garden hose fills the pool in hours. His neighbor has a thinner hose that fills the pool in hours. How long will it
take to fill the pool using both hoses?
The length of time it would have taken for both garden hoses to fill the pool would be 6.67 hours.
How to find the time taken ?To solve this problem, we can use the formula:
1/rate = time
In the case of Joe's hose, the rate is 1/12 of the pool per hour. For his neighbor's hose, the rate is 1/15 of the pool per hour.
So, the combined rate is:
= (1/12) + (1/15)
= (5/60) + (4/60)
= 9/60 of the pool per hour
So, the time it will take to fill the pool using both hoses is:
= 1 / (9/60)
= 60/9 hours
= 6.7 hours
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The full question is:
Joe's garden hose fills the pool in 12 hours. His neighbor has thinner hose that fills the pool in 15 hours. How long will it take to fill the pool using both hoses?
At a family reunion, each of Shamar's aunts and uncles is getting photographed once. The aunts are taking
pictures in groups of 5 and the uncles are taking pictures in groups of 10. If Shamar has the same total number
of aunts and uncles, what is the minimum number of aunts that Shamar must have?
The minimum number of aunts that Shamar must have based on the information provided is 10.
What are the possible number of uncles and aunts?It is known that uncles will take a photograph in groups of 10, while aunts will take a photograph in groups of 5. Based on this, the possible number for each are:
Uncles: 10, 20, 30, 40, 50
Aunts: 5, 10, 15, 20, 25
How to find the minimum number of relatives?In this case, the minimum number is the LCM or Least Common Multiple and based on the previous list, this number is 10.
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steven is cutting a board to use for a project.
the board is 10 feet long. he first has to cut off 1 1/2 feet from one end of the board. he has to cut the remaining section board into 3 pieces.
what is the length of each equal piece?
write and solve an equation to represent this situation. write answer as a mixed number
The length of each equal piece is 17/6 feet, the mixed fraction is 2 5/6.
What are proper and improper fractions and how to convert mixed fraction to simple fraction?There is a fraction, containing numerator(upper value) and denominator(lower value).
When the numerator is less than the denominator, the fraction is called proper fraction, otherwise it is called improper fraction.
A proper fraction is also called fraction which is less than 1.
And an improper fraction is ≥ 1
A mixed fraction contains a sum of whole number and a proper fraction.
Given;
The board is 10 feet long
First he has to cut off 1 1/2 feet from one end of the board
Now, after the first cut remaining board will be;
=10-1 1/2=17/2
After cutting it in 3 equal pieces
=17/2/3
=17/6
=2 5/6
Therefore, the answer in mixed fraction will be 2 5/6
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20 people fit comfortably in a 10 feet by 10 feet area. Use this value to estimate the size of a
crowd that is 7 feet deep on ONE side of the street along a 1 mile section of a parade route.
The size of the crowd is 7392 people.
What is ratio?The ratio is a numerical relationship between two values that demonstrates how frequently one value contains or is contained within another.
20 people fit comfortably in a 10 feet by 10 feet area.
So,
the ratio of the number of people per unit area = 20/(10 × 10) = 20/100 = 1/5
The area, A, occupied by the crowd = 7 feet × 1 mile × 1
1 mile = 5280 feet
A = 7 feet × 5280 feet × 1 = 36960 feet².
Now, number of people = 36960 x 1/5
= 7392 people.
Therefore, the required value is 7392 people.
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estimate the distance (in mi) the race car traveled during this time period using the velocities at the beginning of the time intervals
The distance the race car traveled during this time period using the velocities at the beginning of the time intervals is 23.886 mi
Use areas of rectangles to approximate the area under the curve. The width of each rectangle is 10 sec = 1/36 hrs and the lengths are the velocities.
LH = 1/36 ( 183.9 + 167 + 107.6 + 98.8 + 125.5 + 177.1)
LH=1/36(862.9)
LH≈ 23.886 mi
To find the right-hand approximation:
RH = 1/36 (167 + 107.6 + 98.8 +125.5 +177.1 + 176.6)
RH=1/36(852)
RH≈ 23.683 mi
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The length of a new rectangular playing field is 4 yards longer than triple the width. If the perimeter of the rectangular playing field is 416 yards, what are its dimensions?
Answer: Let's call the width of the rectangular playing field "w" and the length "l". From the problem, we know that:
l = 3w + 4 (the length is 4 yards longer than triple the width)
We also know that the perimeter of a rectangle is the sum of the lengths of all four sides, so:
P = 2l + 2w
We are given that the perimeter is 416 yards, so we can substitute that into the perimeter equation:
416 = 2(3w + 4) + 2w
Simplifying and solving for w:
16 = 6w + 8
8 = 6w
w = 8/6 = 4/3
We can now substitute that back into the equation for the length:
l = 3w + 4
l = 3(4/3) + 4
l = 4 + 4
l = 8
So the dimensions of the rectangular playing field are 4/3 yards wide and 8 yards long.
Step-by-step explanation:
the question is in the picture
The positive interval and the negative interval for the function given is (-∞, 3) and (3, ∞).
What is Function?A function is a relation from a set A to a set B where the elements in set A only maps to one and only one image in set B. No elements in set A has more than one image in set B.
Given function is f(x) = [tex]-2\sqrt{x+1} +4[/tex]
Positive interval of the function is the values of x when function > 0.
To get the function > 0, the value [tex]2\sqrt{x+1}[/tex] < 4.
[tex]\sqrt{x+1}[/tex] < 2
x + 1 < 2²
x + 1 < 4
x < 3
Positive interval is (-∞, 3).
Negative interval of the function is the values of x when function < 0.
To get the function value < 0
[tex]-2\sqrt{x+1} +4[/tex] < 0
[tex]2\sqrt{x+1}[/tex] > 4
[tex]\sqrt{x+1}[/tex] > 2
x + 1 > 2²
x + 1 > 4
x > 3
Negative interval is (3, ∞).
Hence the positive interval of the function is (-∞, 3) and the negative interval is (3, ∞).
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A candle is 18 inches tall before it's lit. Let x represent the number of inches burned from the candle. Let y represent the number of inches of candle remaining a) what does the expression 18-x represent? length burned varying varying b) Label x,y,18, and 18-x on the diagram above. c) Write a formula that represents y in terms of X. d) Is it possible to write your formula in part (c) without using a constant quantity? What role do constant quantities play in helping you define a formula that describes how two varying quantities change together?
The expression 18-x represents the length of the candle that has been burned. As x increases (representing more inches being burned), 18-x decreases (representing less inches remaining).
b) x represents the inches burned from the candle, y represents the inches of candle remaining, 18 represents the initial height of the candle before it is lit, and 18-x represents the length of the candle that has been burned.
c) y = 18 - x (The height of the candle remaining is 18 inches minus the inches that have been burned, represented by x)
d) It is not possible to write the formula in part (c) without using a constant quantity. Constant quantities, such as the initial height of the candle (18 inches), help to define the relationship between the two varying quantities (x and y) and provide a starting point for the calculation.
Without the constant quantity, it would not be clear what the relationship is between x and y.
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3x + 2 > 2x - 3 > x - 12 solve the compound inequality
Answer:
We get that x > -5
Step-by-step explanation:
1. Split up the inequality into two parts
We think of it as a > b > c and know that it is the same as a > b and b > c.
Apply this to the compound inequality
3x + 2 > 2x-3 and 2x - 3 > x - 12
2. Start solving 3x + 2 > 2x-3
Subtract 2x from both sides
x + 2 >-3
Subtract 2 from both sides
x > -5
3. Start solving 2x - 3 > x - 12
Subtract x from both sides
x - 3 > -12
Add 3 to both sides
x > -9
4. Finish
Now we have to think how we started with 3x + 2 > 2x-3 and 2x - 3 > x - 12
Since it is an and we need to identify the interval in which it is true on both inequalities which is when the x overlap. In this case is from x > -5.
the base of a certain solid is the region r. cross sections through the solid taken perpendicular to the x-axis are squares. an integral that expresses the volume of the solid is:
The integral[tex]$\int_{-\infty}^{\infty} f(x)^2 dx$[/tex] expresses the volume of the solid, where f(x) is the side length of the square cross sections taken perpendicular to the x-axis in the region r.
The given integral [tex]$\int_{-\infty}^{\infty} f(x)^2 dx$[/tex] expresses the volume of the solid whose base is the region r. The integral is calculated by taking the cross sections of the solid perpendicular to the x-axis and consider them as squares. The side length of each square is denoted by f(x). The integral is calculated over the region r, which is the base of the solid. This integral is the sum of the areas of all the squares, which is equivalent to the volume of the solid. The integral is calculated by taking the lower limit of integration as[tex]$-\infty$[/tex] and the upper limit of integration as[tex]$\infty$[/tex]. Hence, the integral [tex]$\int_{-\infty}^{\infty} f(x)^2 dx$[/tex] expresses the volume of the solid with the base region r and square cross sections.
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Which expression is a factor of -2x2 - 11x - 20
No real Factors
(2x + 10)
(x - 10)
(2x - 5)
The given expression will not have real factors. The correct option is A.
What is an expression?The mathematical expression combines numerical variables and operations denoted by addition, subtraction, multiplication, and division signs.
Mathematical symbols can be used to represent numbers (constants), variables, operations, functions, brackets, punctuation, and grouping. They can also denote the logical syntax's operation order and other properties.
Given that the quadratic expression is -2x² - 11x - 20. The expression will be solved as below:-
E = -2x² - 11x - 20
Apply the quadratic factorization formula,
x = [ -b ± √( b² - 4ac ) ] / 2a
x = [ 11 ± √ ( 121 - ( 4 x - 2 x - 20 ) ] / ( 2 x - 2 )
x = [ 11 ± 39i } / -4
The roots of the quadratic equation are not real.
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Answer:
no real factors
Step-by-step explanation:
i took the test
Farrah borrowed $155 from her brother. She has already paid back $15. She plans to pay back $35 each month until the debt is paid off. Which describes the number of months it will take to pay off the debt? Select three options. x + 15 + 35 = 155 35 x + 15 = 155 35 x = 155 minus 15 It will take 8 months to pay off the debt. It will take 4 months to pay off the debt.
It will take 4 months to pay off the debt for Farah.
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
Numbers (constants), variables, operations, functions, brackets, punctuation, and grouping can all be represented by mathematical symbols, which can also be used to indicate the logical syntax's order of operations and other features.
Given that Farrah borrowed $155 from her brother. She has already paid back $15. She plans to pay back $35 each month until the debt is paid off.
The expression can be written as:-
35x + 15 = 155
35x = 155 - 15
35x = 140
x = 140 / 35
x = 4 months
Hence, the number of months will be 4.
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Angela plays soccer and golf for a
total of 125 minutes every day, She
plays soccer for 45 minutes longer
than she plays goif
Part A: Witte a pair of linear
equations to show the relationship
between the number of minutes
Angela plays soccer (x ) and the
number of minutes she plays golf (y)
every day.
Part B: How much does angela
spend playing golf everyday? Show
your work.
PartC: Is it possible for Angela to
have spent 80 minutes playing
soccer if she plays for a total of
exactly 125 minutes and plays
soccer for 45 minutes longer than
she plays golf?
It is possible for Angela to have spent 80 minutes playing soccer if she plays for a total of exactly 125 minutes and plays soccer for 45 minutes longer than she plays golf because 80+45 adds up to 125.
What is a linear equation?A linear equation is an equation that has the variable of the highest power of 1. The standard form of a linear equation is of the form Ax + B = 0.
Angela plays soccer and golf for a total of 125 minutes every day. She plays soccer for 45 minutes longer than she plays golf.
Part A: A pair of linear equations to show the relationship between the number of minutes Angela plays soccer (x) and the number of minutes she plays golf (y) every day.
x = y + 45
Part B: A pair of linear equations to show how much Angela spends playing golf every day.
125 - 45 = 80,
80/2 = 40
Part C: Yes, it is possible for Angela to have spent 80 minutes playing soccer if she plays for a total of exactly 125 minutes and plays soccer for 45 minutes longer than she plays golf because 80+45 adds up to 125.
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Kathleen has 21 marbles some of the marbles are green the other 12 marbles are blue How many green marbles does she have?
Answer: 9
Step-by-step explanation:
21 - 12 = 9
Business XYZ has the following projected budget
and actual budget for a given month:
Budget Actual
Payroll $24,000 25,275
Inventory 12,700 12,100
Rent
7,500 7,700
Services 5,000 5,100
Calculate the percent change that the total of all
categories is over budget.
[?] %
Give your answer as a percent rounded to the nearest tenth.
Enter
PH
Skip
Answer:
The total budget for all categories is $49,200 and the total actual is $50,175, for a percent change of 1.8%. Rounded to the nearest tenth, the percent change is 1.9%.
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to find the average amount of money that american college students spend on food per week, a researcher surveys 300 college students across the u.s.
The result of the calculation was $93.41 per week.
To find the average amount of money spent on food by American college students per week, a researcher conducted a survey of 300 college students across the U.S. The researcher collected data on the amount of money spent by each student on food per week and then calculated the average using the formula: Average = (sum of all the scores)/(number of scores). The researcher then added up the money spent by all 300 students and divided it by 300 to get the average amount of money spent on food per week by American college students. The result of the calculation was $93.41 per week.
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Find the exact circumference of each circle
The exact circumference for the circles are;
5. C = 94.26mi
6. C = 879.76in
7. C = 91.118in
8. C = 62.84m
9. C = 12.568m
10. C = 29.535cm
What is Circumference of a Circle?The circumference of a circle is its boundary or the length of the complete arc of a circle. The circumference of a circle is the perimeter of the circle. It is the total length of the boundary of the circle. The circumference of a circle is the product of the constant π and the diameter of the circle.
Circumference = πd
or
Circumference = 2πr
where π = 3.142 or 22/7
r = radius
d = diameter
5. C = 2πr
C = 2 x 3.142 x 15
C = 94.26mi
6. C = 2πr
C = 2 x 3.142 x 140
C = 879.76in
7. C = πd
C = 3.142 x 29
C = 91.118in
8. C = 2πr
C = 2 x 3.142 x 10
C = 62.84m
9. C = 2πr
C = 2 x 3.142 x 2.1
C = 12.568m
10. C = πd
C = 3.142 x 9.4
C = 29.535cm
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A 15 kg child is tossed up into the air by her parent. The child is 2.5 meters off the ground traveling 3 m/s.
The values of each type of energy the object has:
Kinetic energy is 67.5J and Gravitational Potential Energy is 435 J.
What is GPE? GPE stands for Gravitational Potential Energy. It is the energy stored in an object due to its position in a gravitational field. It is equal to the work done to move an object from a reference point to its current position in a gravitational field. GPE depends on an object’s mass, gravity, and height. The formula for calculating GPE is GPE = mgh, where m is the mass of the object, g is gravity, and h is the height of the object from the reference point. The unit of GPE is the Joule (J). An object with more mass, greater height, and a stronger gravitational field will have higher GPE.KE: Mass= 15 kg Velocity = 3 m/s Velocity2= 9 m²/s²
KE = ½ (15) (9) = 67.5 J
GPE: Mass= 15 kg Gravity (g)=9.8 m/s² Height = 2.5 m
GPE= (15) (9.8) (2.5) = 367.5 J
KE+GPE= 67.5 + 367.5 = 435 J
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Average monthly expenditure (£)
600
500
400
300
200
100
Clip 15
Key:
2010
2020
I
Transport Utility Food Housing Leisure
bills Category
a) Which category shows a decrease
in the average monthly expenditure
between 2010 and 2020?
leisure
b) How much more was the average
monthly expenditure on Food in
2020 than in 2010?
c) The total average monthly
expenditure in 2010 was £1200.
What was it in 2020?
Help on question c
Answer:
the total average monthly expenditure in 2020 was £2400.
Step-by-step explanation:
c) To find the total average monthly expenditure in 2020, we need to add up all the individual expenditures in each category. From the given data, we have:
Average monthly expenditure in 2020 = 600 + 500 + 400 + 300 + 200 + 100 = £2400
So, the total average monthly expenditure in 2020 was £2400.
Peyton had a balance of -$21.46 in his bank account according to his
last statement. He immediately deposited $35.50 into his account.
What was Peyton's account balance AFTER he made the deposit?
Answer: $14.04
Step-by-step explanation: -21.46+35.50 = 14.04
Construct a 3x3 matrix A, with nonzero entries, and a vector b in R3 such that b is not in the set spanned by the columns of A. Choose the correct answer below : A. A= | 1 1 1 | | 3 | B. | 1 1 1 | |4| . |2 2 2 | b = | 6 | A= | 2 2 2 | b = |5| . |3 3 3 | | 9 | | 3 3 3 | |6| . . C. A= | 1 1 1 | | 1 | D. | 1 2 3 | |3| . |2 2 2 | b = | 2 | A= | 2 1 1 | b = |2| . |3 4 5 | | 5 | | 3 3 2 | |1|
A 3x3 matrix A, with nonzero entries, and a vector b in R3 such that b is not in the set spanned by the of A= | 1 1 1 | | 1 | . |2 2 2 | b = | 2 |
In order for b to not be in the set spanned by the columns of A, we need to ensure that the system of equations Ax=b has no solution. This means that the system of equations needs to be inconsistent. We can achieve this by making sure that the determinant of A is not equal to zero, i.e. the matrix must be invertible.
The matrix given in option C satisfies this requirement. A = | 1 1 1 | | 2 2 2 | | 3 3 3 |
The corresponding vector b is | 1 | | 2 | | 3 |
Thus, the correct answer is C.
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all of the following are possible steps of scientific investigation except for . group of answer choices the development of one or more working hypotheses or models to explain facts development of observations and experiments to test the hypotheses assumption of conclusions without prior experimentation or observation the collection of scientific facts through observation and measurement
All of the following are possible steps of scientific investigation except for
C. assumption of conclusions without prior experimentation or observation
All of the following are possible steps of scientific investigation except for.
A. the development of one or more working hypotheses or models to explain facts
B. development of observations and experiments to test the hypotheses
C. assumption of conclusions without prior experimentation or observation
D. the collection of scientific facts through observation and measurement
All of the following are possible steps of scientific investigation except for
C. assumption of conclusions without prior experimentation or observation
At the core of biology and other sciences lies a problem-solving approach called the scientific method. The scientific method has five basic steps, plus one feedback step:
Make an observation.
Ask a question.
Form a hypothesis, or testable explanation.
Make a prediction based on the hypothesis.
Test the prediction.
Iterate: use the results to make new hypotheses or predictions.
The scientific method is used in all sciences—including chemistry, physics, geology, and psychology. The scientists in these fields ask different questions and perform different tests. However, they use the same core approach to find answers that are logical and supported by evidence.
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(linear algebra)
(a) Let U be the subspace of C5 defined by U = {(z1, z2, z3, z4, z5) ? C5 ; 6z1 = z2 and z3+2z4+3z5 = 0} Find a basis of U.
(b) Extend the basis in part (a) to a basis of C5.
(c) Find a subspace W of C5 such that C5 = U direct sum W
a) One possible basis for U is {(1/6,1,0,0,0), (0,0,1,-2,3)}.
b) A basis of C5 is {(1/6,1,0,0,0), (0,0,1,-2,3), (1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}.
c) The basis of W is {(1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}.
Linear Algebra Identificationa) To find a basis of U, we can use the two given equations to solve for two of the variables in terms of the others. From the first equation, we have z1 = z2/6. Substituting this into the second equation, we get z3 + 2z4 + 3z5 = 0. Therefore, one possible basis for U is {(1/6,1,0,0,0), (0,0,1,-2,3)}.
b) To extend this basis to a basis of C5, we need to add three more vectors that are not in the subspace U. These vectors can be any linearly independent vectors that are not in U. One possible set of additional vectors is {(1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}. Therefore, a basis of C5 is {(1/6,1,0,0,0), (0,0,1,-2,3), (1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}.
c) To find a subspace W of C5 such that C5 = U direct sum W, we can take the complement of U in C5. The complement of U is the set of all vectors in C5 that are not in U. Since we have already found a basis for C5, we can simply take the subset of basis vectors that are not in U to form a basis for W. In this case, the basis of W is {(1,0,0,0,0), (0,0,1,0,0), (0,0,0,1,0)}.
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let $s$ be a subset of $\mathbb{r}$ that is closed under multiplication (that is, if $a$ and $b$ are in $s$, then so is $ab$). let $t$ and $u$ be disjoint subsets of $s$ whose union is $s$. given that the product of any three (not necessarily distinct) elements of $t$ is in $t$ and that the product of any three elements of $u$ is in $u$, show that at least one of the two subsets $t$, $u$ is closed under multiplication.
As per the given subset, the set T and U are not closed under multiplication.
In math the term subset is defined through the example set A is a subset of another set B if all elements of the set A are elements of the set B.
Here let us consider that S be a subset of R that is closed under multiplication
Here we have also consider that T and U be disjoint subsets of S whose union is S.
Here we have given that the product of any three elements of T is in T and that the product of any three elements of U is in U, now we have to show that at least one of the two subsets T, U is closed under multiplication.
Here we have to start out with the assumption that neither T nor U is closed under multiplication and use proof by contradiction to prove that at least one of T or U must be closed under multiplication.
Here let us write this as,
=> t1,t2,t3∈T1,2,3 and
=> u1,u2,u3∈U1,2,3∈,
While it is given by the problem that, then T and U are disjoint subsets of S and therefore,
=> T∩U = ∅.
At this time we have take that
=> t3 = u1⋅u2
=> 3 = 1⋅2 and
=> u3=t1⋅t2
=> 3=1⋅2.
While we looking these statements are valid because T and U are not closed under multiplication, where as the product of u1⋅u21⋅2 must not be in the set U and the product of t1⋅t21⋅2 must not be in set T.
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Captain Ben has a ship, the H.M.S Crimson Lynx. The ship is five furlongs from the dread pirate Luis and his merciless band of thieves.
If his ship hasn't already been hit, Captain Ben has probability
2
3
3
2
start fraction, 2, divided by, 3, end fraction of hitting the pirate ship. If his ship has been hit, Captain Ben will always miss.
If his ship hasn't already been hit, dread pirate Luis has probability
1
3
3
1
start fraction, 1, divided by, 3, end fraction of hitting the Captain's ship. If his ship has been hit, dread pirate Luis will always miss.
If the Captain and the pirate each shoot once, and the pirate shoots first, what is the probability that the pirate hits the Captain's ship, but the Captain misses?
The probability that the pirate hits the Captain's ship, but the Captain misses is 1/6.
What is probability?It is the chance of an event to occur from a total number of outcomes.
The formula for probability is given as:
Probability = Number of required events / Total number of outcomes.
We have,
P(Captain Ben hit the ship) = 2/3
P(Captain Ben misses to hit the ship) = (1 - 2/3) = 1/3
P(Pirate hit the ship) = 1/3
P(Pirate misses to hit the ship) = (1 - 1/3) = 2/3
Now,
The probability that the pirate hits the Captain's ship, but the Captain misses.
= P(Pirate hit the ship) x P(Captain Ben misses to hit the ship)
= 1/3 x 1/3
= 1/6
Thus,
The probability is 1/6.
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let abcd be a regular tetrahedron and m,n distinct points in the planes abc and adc respectively. show that the segments mn,bn,md are the sides of a triangle.
The segments MN, BN, and MD are the sides of a triangle.
Let AM intersect BC at point P and let AN intersect CD at point Q. Spin A about PQ to obtain E where AE = AB. We want to prove that EMN is the required triangle. For that, it is adequate to show that EM = MD and EN = BN.
Let h be the distance of A from the BCD. Then the distance of A from the line PQ is minimum h, so point E exists provided that the side length maximum is 2h. It is √(3/2) h.
We have by construction AD = AE and EP = AP. Also, since P lies on the line BC, then PA = PD. Hence PA = PA, PD = PE, AD = AE, so PAD and PAE are congruent triangles. Hence E can also be acquired by spinning D about AP. Since M lies on AP, it gives EM = DM.
Similarly, with B and D interchanged and M and N interchanged, we get EN = BN.
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