Answer:
I dont like sqaures.
Step-by-step explanation: Their pointy.
Estimate 58,088-39,625 by first rounding each number to the nearest thousand.
Answer:
Here is the answer, I took it as simple I could.
Lily mixed a batch of punch for the class party. Her recipe called for 2 1/2 cups of orange juice, 1 1/2 cups lemonade, and 3 quarts of ginger ale. How many 1-cup servings does the recipe make?
Answer:
16 1-cups servings
Step-by-step explanation:
First, you'll need to convert the 3 quarts to cups. One quart is 4 US cups, so quarts is 12 US cups. Next, add the 2 1/2 cups of OJ and 1 1/2 cups of lemonade together, which gives you 4 US cups. Add your 12 cups and 4 cups together and you get 16. You'll get 16 1-cup servings.
A testtaker who scores at the 5th stanine is scoring A. above average. B. below average. C. within the average range. D. in an unspecifiable range
Based on the information provided, a test-taker who scores at the 5th stanine is scoring:
C. within the average range.
To better understand this, let's break down the terms and concepts involved:
Stanine: Stanine (short for "standard nine") is a method used to scale test scores on a nine-point scale.
The stanine scores range from 1 (lowest) to 9 (highest).
This method is typically used in educational assessments to categorize students' performance.
Average: In the context of test scores, the average score refers to the middle range of scores that is typical for the majority of test-takers.
In the case of stanine scores, the average range falls between 4 and 6.
This means that students scoring in stanines 4, 5, and 6 are considered to be within the average range.
Now, to answer your question, a test-taker who scores at the 5th stanine is considered to be within the average range of performance.
This is because their score falls between the 4th and 6th stanines, which, as mentioned earlier, encompass the average range.
It's important to note that this doesn't mean the student is performing poorly; rather, their performance is on par with the majority of other test-takers.
Option C. within the average range is correct.
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Which of the following is not a solution to sin(x)=1
90 Degrees
450 Degrees
270 Degrees
-270 Degrees
Answer:
450 degrees is the answer
A manufacturer has a steady annual demand for 15,000 cases of sugar. It costs $10 to store 1 case for 1 year, $30 in set up cost to produce each batch, and $16 to produce each case. Find the number of cases per batch that should be produced to minimize cost.
The number of cases per batch that should be produced to minimize cost is: 300 units
How to find the economic order quantity?The number of cases per batch that should be produced to minimize cost can be found by using the Economic Order Quantity.
The Economic Order Quantity (EOQ) is a calculation performed by a business that represents the ideal order size that allows the business to meet demand without overspending. The inventory manager calculates her EOQ to minimize storage costs and excess inventory.
Thus:
Number of cases per batch = √((2 * Setup costs * annual demand)/ holding costs for the year)
Solving gives:
√((2 * 30 * 15000)/10)
= √90000
= 300 units
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The image below showcases a right triangle .
My questions:
What is a c, what does that represent
What is this problem asking for me
How do I solve this problem? Are there any formulas in place?
The perimeter of triangle is 66.24 units.
What is triangle?
In Euclidean geometry, any 3 points, once non-collinear, verify a unique triangle and at the same time, a unique plane
Main body:
according to question :
c = 28
let the vertices be A,B,C
∠A= 30°
by using trigonometric ratios,
BC/ AB = sin30°
AB = C = 28
BC/28 = sin30°
BC = 28*sin30°
BC= 28*(1/2)
BC = 14
similarly
AB /CA = cos 30°
28/CA = √3/2
CA = 28*√3/2
CA = 14/√3
CA = 24.24
Hence , perimeter = AB +BC +CA = 28+14+24.24
=66.24 units
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Part A Since triangle 2 is a right triangle, write an equation applying the Pythagorean Theorem to the triangle. Part B Since the equations for both triangles have a2 + b2, you can think of the two equations for c2 and n2 as a system of equations. Substitute what a2 + b2 equals in the first equation for a2 + b2 in the second equation. After you substitute, what equation do you get? Part C Now, take the square root of both sides of the equation from part B and write the resulting equation. Part D Is there any way for this equation to be true? How? Part E What does this show about the relationship between the two triangles? Part F Does this mean that triangle 1 is a right triangle? Why or why not?
Part A: Since triangle 2 is a right triangle, we can apply the Pythagorean Theorem to it. The theorem states that the sum of the squares of the legs of a right triangle is equal to the square of the hypotenuse. So, the equation for triangle 2 would be c2 = a2 + b2.
Part B: Both equations for triangles 1 and 2 have a2 + b2. We can think of them as a system of equations. So, we can substitute what a2 + b2 equals in the first equation for a2 + b2 in the second equation. After substituting, we get the equation: n2 = c2 - 2ac.
Part C: To get the resulting equation, we take the square root of both sides of the equation from part B. So, we get: n = sqrt(c2 - 2ac).
Part D: Yes, there is a way for this equation to be true. It would be true if triangle 1 is a right triangle with the hypotenuse being c and one leg being a.
Part E: This shows that there is a relationship between the two triangles. Specifically, it shows that the length of the third side of triangle 1 is related to the lengths of the other two sides of triangle 1 and the hypotenuse of triangle 2.
Part F: No, this does not necessarily mean that triangle 1 is a right triangle. It only means that if triangle 1 is a right triangle, then the equation from part C would be true.
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the angle associated with the total impedance is the angle by which the applied voltage leads the source current. true or false
False. The angle associated with the total impedance is the angle by which the source current leads the applied voltage.
What is angle?Angle is defined as the space or area between two lines or surfaces that meet at a certain point. It is measured in degrees, and can be categorized by its size (acute, right, obtuse, and reflex) or by its type (straight, reflex, complementary, supplementary, and vertical). Angle is an important concept in mathematics, geometry, physics, and engineering, and is used to describe the size of a turn, the shape of an object, and the directions of movements.
This is because the total impedance is defined as the ratio of the applied voltage to the source current. Since the applied voltage is divided by the source current, the source current will lead the applied voltage. This is due to the fact that the total impedance is a complex quantity, with both a real and imaginary component. If the total impedance only had a real component, then the angle associated with the total impedance would be zero, meaning the applied voltage and source current would be in phase. However, since the total impedance is complex, the angle associated with the total impedance is the angle by which the source current leads the applied voltage.
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6 mi Find the volume for the shape above. Use 3.14 for pi. Round to the nearest whole number.
Answer:
Step-by-step explanation: does anyone know? I really need this
The triangles are congruent, which sequence of motions will take triangle XYZ onto triangle BCA?
1: Translate XYZ along vector YC. Rotate X'Y'Z' around C by angle BCX'. Reflect X"Y"Z" over CB.
2: Translate XYZ along vector YC. Rotate X'Y'Z'around C by angle BCX Reflect X"Y"Z" over AC.
3:Translate XYZ along vector YC. Rotate X'YZ' around C by angle X'YA. Reflect X"Y"Z" over CB.
4: Translate XYZ along vector YC. Rotate X'Y'Z' around C by angle X'Y'A. Reflect X"Y"Z" over AC.
Answer:
3: အားနည်းချက်ကို YC တလျှောက်တွင် XYZ Translate ။ ထောင့် X'YA ဖြင့် C ပတ်လည် X'YZ ကိုလှည့်ပါ။ CB ရှိ“ X” Y“ Z” ကိုရောင်ပြန်ဟပ်ပါ။
Answer:
Translate XYZ using directed line segment YC. Rotate X'Y'Z' using C as the center so that X' coincides with B. Reflect X"Y"Z" across line CB.
Step-by-step explanation:
Its correct because I just got the answer right on the assignment
Hope this helps! :)
Overview
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Question Progress
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Here is a list of numbers:
20 29 30 33 37 40
41
46
50
Use four of the numbers to make a sequence and describe the rule
for continuing your sequence.
I
equences - Term-to-Term Rule
Overview
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Answer:
20 292 02 202 202
Step-by-step explanation:
A maintenance worker needs to wax a restaurant floor shaped like the image shown.
A six-sided figure. There is a horizontal base of 35 feet then another horizontal base below it of 20 feet. The longest side is to the right and is 40 feet. The top is 30 feet. There is a perpendicular from the vertex of the top to the first base of 35 that is labeled 15 feet.
If the wax cost $1.38 a square foot, how much will the wax cost to cover the floor?
$1,224.75
$1,569.75
$2,104.50
$2,449.50
Can you hurry pls?
Answer:
To calculate the area of the restaurant floor, we need to break it down into two parts: a trapezoid and a triangle.
The area of the trapezoid is given by the formula:
A = (b1 + b2) x h / 2
where b1 and b2 are the lengths of the two bases of the trapezoid, and h is the height.
In this case, the trapezoid has bases of 35 ft and 20 ft, and a height of 15 ft (the perpendicular from the vertex of the top to the first base of 35 ft). Substituting these values into the formula, we get:
A = (35 ft + 20 ft) x 15 ft / 2 = 525 square feet
The area of the triangle is given by the formula:
A = b x h / 2
where b is the base of the triangle, and h is the height.
In this case, the triangle has a base of 40 ft and a height of 15 ft (the same as the height of the trapezoid). Substituting these values into the formula, we get:
A = 40 ft x 15 ft / 2 = 300 square feet
Adding the area of the trapezoid and the area of the triangle, we get the total area of the restaurant floor:
525 square feet + 300 square feet = 825 square feet
Finally, to calculate the cost of waxing the floor, we can multiply the total area by the cost per square foot:
825 square feet x $1.38/square foot = $1,138.50
Therefore, the wax will cost $1,138.50 to cover the floor. The closest answer choice is $1,224.75, but that is not the correct answer.
The highest weekly wage of a group
of employees is $$$105.40. If the
range of the wages is $27.50, how
much does the lowest paid employee
receive?
(A.) $105.40
(B.) $77.90
(C.) $66.40
(D.) $27.50
what does f mean in f(x)=3/2x+2?
Also do the two points 0,2 and 2,5 work with the equation?
Answer:
f(x) means y
Points 0,2 and 2,5 work with the equation
Step-by-step explanation:
f(x) means y
For 0, 2
y = 3/2(0) + 2
y = 2
For 2,5
y =3/2(2) + 2
y = 5
Answer:
(0, 2) and (2, 5) are solutions to the function.
Step-by-step explanation:
Given the following linear function:
\(\displaystyle\mathsf{f(x)\:=\:\frac{3}{2}x\:+\:2 }\)
"f(x)" is a function notation.
"f" represents the function name. In function notations, you can use any letter to represent a function name. However, "f" is commonly used to express function notations.
"x" is your input variable. This is the x-values that you substitute into the function to get an output, or y-value.
f(x) is your output value.
We are also given the following points: (0, 2) and (2, 5).
In order to determine whether these points work with the given function, simply substitute their values into the function.
(0, 2) where x = 0, y = 2:\(\displaystyle\mathsf{f(x)\:=\:\frac{3}{2}x\:+\:2 }\)
\(\displaystyle\mathsf{f(0)\:=\:\frac{3}{2}(0)\:+\:2 }\) ⇒ The function notation, "f(0)" simply means the function of zero, which corresponds to the input value of zero (0).
Next, substitute the value of y = 2 into the function as an output value:
\(\displaystyle\mathsf{2\:=\:\frac{3}{2}(0)\:+\:2 }\)
2 = 0 + 2
2 = 2 (True statement). This means that the point, (0, 2), is a solution to the given function. The point (0, 2) represent the given function's y-intercept, which is the point on the graph where it crosses the y-axis. The y-coordinate of 2 is the value of b in the slope-intercept form, y = mx + b.
(2, 5) where x = 2, y = 5:We will do the same process for (2, 5):
\(\displaystyle\mathsf{f(x)\:=\:\frac{3}{2}x\:+\:2 }\)
\(\displaystyle\mathsf{f(2)\:=\:\frac{3}{2}(2)\:+\:2 }\)
Substitute the value of y = 5 into the function as an output value:
\(\displaystyle\mathsf{5\:=\:\frac{3}{2}(2)\:+\:2 }\)
5 = 3 + 2
5 = 5 (True statement). This means that the point, (2, 5), is also a solution to the given function.
The first three terms in a pattern are shown in the
image below. How many circles would be in the
5th term?
A. 10
B. 15
C. 18
D.20
Problem Description: An example of arithmetic progression would be a series of integers (which we will call terms) like: 3, 7, 11, 15, 19, 23, 27, 31, ... Note that 3 is the first term, 7 is the second term, 11 is the 3rd term, etc. 4 is the common difference between any two consecutive terms. Now, if we know that the progression has 100 terms, we would be interested in calculating the 100th term as well as the sum and the float average of all 100 terms. The following formulas can be used to calculate these items: LastTerm = FirstTerm + (NumberOfTerms - 1) x CommonDifference Sum of all terms = NumberOfTerms x (FirstTerm + LastTerm) / 2 Average of all terms = (Sum of all terms) / NumberOf Terms The program should adhere to the following pseudocode: 1. Prompt for and read the first term 2. 3. Prompt for and read the common difference Prompt for and read the number of terms Calculate the last term (see formula above) 4. 5. Calculate the sum of all the terms (see formula above) Calculate the average of all the terms (see formula above) 7. Display the results 6. Your program must match the following sample run (between the lines of dashes). Note that the 3, 3, and 100 on the first three lines were entered by the user. You should also check results for other set of inputs as well. Enter first term: 3 Enter common difference: 3 Enter number of terms: 100 The last term is 300 The sum of all the terms is 15150 The average of all the terms is 151.5
The last term is 300
The sum of all the terms is 15150.0
The average of all the terms is 151.5
Here is an example solution in Python that follows the given pseudocode:
# Prompt for and read the first term
first_term = int(input("Enter first term: "))
# Prompt for and read the common difference
common_difference = int(input("Enter common difference: "))
# Prompt for and read the number of terms
number_of_terms = int(input("Enter number of terms: "))
# Calculate the last term
last_term = first_term + (number_of_terms - 1) * common_difference
# Calculate the sum of all the terms
sum_of_terms = number_of_terms * (first_term + last_term) / 2
# Calculate the average of all the terms
average_of_terms = sum_of_terms / number_of_terms
# Display the results
print("The last term is", last_term)
print("The sum of all the terms is", sum_of_terms)
print("The average of all the terms is", average_of_terms)
If you run this code and enter the values from the sample run (first term: 3, common difference: 3, number of terms: 100), it will produce the following output:
The last term is 300
The sum of all the terms is 15150.0
The average of all the terms is 151.5
The program prompts the user for the first term, common difference, and number of terms. Then it calculates the last term using the given formula. Next, it calculates the sum of all the terms and the average of all the terms using the provided formulas. Finally, it displays the calculated results.
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The perimeter of a rectangle is 110, and the width is 4 times
the length. What is the width?
O 33
O 44
O 66
O 55
Answer:
44
Step-by-step explanation:
come on now.
the perimeter is
2×length + 2×width = 110
width = 4×length
so, we use the second equation in the first :
2×length + 2×(4×length) = 110
2×length + 8×length = 110
10×length = 110
length = 110/10 = 11
width = 4×length = 4×11 = 44
Verbal
4. When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?
Step-by-step explanation:
A parenthesis is used when the number next to it is NOT part of the solution set
like : all numbers up to but not including 3 .
Parens are always next to infinity when it is part of the solution set .
A bracket is used when the number next to it is included in the solution set.
Z is..? The problem is attached as a png!
Answer:
Step-by-step explanation:
What i would do is multiply both sides by 2z + 1
so you get z-6=20z+10
-19z=16
z=-16/19
Answer:
z is every integer and the study of z and its properties is a general understanding save the satisfying of your rate of change
A sample statistic addresses the parameter it is designed to estimate when
A the sample is large
B the target population and the sampled population are equivalent
C the sample is smaller than the target population
D the parameter is valid
A sample statistic handles the parameter it is intended to estimate so when target population and indeed the sampled population are equal.
What are statistics?Data gathering, organization, analysis, interpretation, as well as presentation are all topics covered in the field of statistics. It is customary to start with a statistical population or perhaps a statistical model to be investigated when using statistics to solve a scientific, industrial, or social problem.
To what end do we study statistics?To sum up, the five reasons to studying statistics are to be able to do research efficiently, to read and assess journal articles, to even further develop critical thinking and analytical skills, to behave as an informed consumer, and to be aware of when you need to seek outside statistical support.
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Completa el cuadro,escribiendo en la casilla de que corresponde los nombres de los elementos presentes en cada obra.
Answer:
Cada unidad integra los cinco bloques, según el ajuste curricular de Lengua y ... Escribo la letra de la palabra del recuadro, que corresponde al ... Completo el cuadro con las características antes señaladas. ... literarios con nuevas versiones de escenas, personajes u otros elementos. ... oración que presente al animal.
Step-by-step explanation:
Let A be a matrix with linearly independent columns.
Which one of these statements must be true?
A. The equation Ax=b never has a solution for all b.
B. The equation Ax=b always has a solution for all b.
C. The equation Ax=b has a solution for all b precisely when it has more columns than rows.
D. There is no easy way to tell if Ax=b has a solution for all b.
E. The equation Ax=b has a solution for all b precisely when it has more rows than columns.
F. The equation Ax=b has a solution for all b precisely when it is a square matrix.
G. none of the above
Let A be a matrix with linearly independent columns. None of the above statements is not true. So the option G is correct.
In mathematics, a matrix can be thought of as a rectangular array of numbers, symbols, or expressions. Matrices are commonly used in linear algebra, which is the study of mathematical operations involving vectors and matrices.
Matrices are used to represent linear equations, and they can be used to describe transformations such as rotations and reflections. Matrices can also be used to represent graph theory data, and to represent systems of equations.
A matrix with linearly independent columns is one in which each column is not a linear combination of the other columns. In other words, the columns are all independent from each other and cannot be expressed as a linear combination of the other columns. This means that the columns are all different and no two columns are the same.
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5. explain the observed change in beak characteristics using the following concepts in your answer: competition, survival of the fittest, inheritance.
The observed change in beak characteristics can be explained through the concepts of competition, survival of the fittest, and inheritance.
In an environment with limited resources, birds with different beak shapes compete for food. This competition leads to the survival of the fittest, where only individuals with beak shapes best suited for accessing available food sources will survive and reproduce.
This advantageous trait (the suitable beak shape) is then inherited by their offspring, ensuring the continuation of the successful trait in future generations. Over time, this process results in the prevalence of the advantageous beak shape within the population.
The observed change in beak characteristics can be explained by the concepts of competition, survival of the fittest, and inheritance.
As a result of competition for limited resources such as food, birds with beaks that were better suited to their environment were more likely to survive and reproduce. This is known as survival of the fittest. Over time, these advantageous beak characteristics were inherited by their offspring, leading to a shift in the overall beak characteristics of the population.
Therefore, the observed change in beak characteristics is a result of natural selection acting on inherited traits that allowed for better survival and reproduction in a competitive environment.
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If f(x) = 6x+7 find f(q+1)
Answer:
6q + 13Step-by-step explanation:
Given
f(x) = 6x+7To find f(q+1),
substitute x with q+1 in the given equation:
f(q + 1) = 6(q + 1) + 7 = 6q + 6 + 7 = 6q + 13Answer:
Given :-f(x) = 6x + 7
Solution :-Let's put c as q + 1 and f as 6
6(q + 1) + 7
(6 × q) + (6 × 1) + 7
6q + 6 + 7
6q + 13
\( \\ \)
what is 3⁵÷3⁷ using indicies?
Answer:
9
Step-by-step explanation
First you multiply 3 five times, then you multiply 3 seven times and you divide the results of each multiplication
Answer:
0.111111111111111
Step-by-step explanation:
=> 3⁵÷3⁷
Note 3⁵ means multiplying 3 five times
And 3⁷ means multiplying 3 7 times
=> 3×3×3×3×3÷3×3×3×3×3×3×3
multiply the values and then divide
=> 243÷2,187
divide 243 by 2187
=> 0.111111111111111
What is the primary difference between an experiment and an observational study?
The key difference between observational studies and experimental designs is that a well-done observational study does not influence the responses of participants, while experiments do have some sort of treatment condition applied to at least some participants by random assignment.
What is an observational study?An observational study is used in fields such as epidemiology, social sciences, psychology, and statistics to draw conclusions from a sample to a population where the independent variable is not under the researcher's control due to ethical concerns or logistical constraints.
Observational studies involve the study of participants who have had no forced change in their circumstances, i.e., no intervention.
Although participants' behavior may change while being observed, the goal of observational studies is to look into the 'natural' state of risk factors, diseases, or outcomes.
Observational studies are those in which researchers examine the impact of a risk factor, diagnostic test, treatment, or other intervention without attempting to change who is or is not exposed to it.
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Find the standard deviation if the mean of a sample is 72, a z-score of 3 Raw Score: 42 Raw Score: 81
Answer:
a) when x = Raw Score: 42
Standard deviation = σ = -10.3
b) Raw Score: 81
Standard deviation = σ = 2.67
Step-by-step explanation:
Find the standard deviation if the mean of a sample is 72, a z-score of 3
Z score formula is given :
z = (x-μ)/σ,
where
x is the raw score
μ is the population mean = 73
σ is the population standard deviation
z = 3
a) when x = Raw Score: 42
z = (x-μ)/σ
3 = 42 - 73/σ
Cross Multiply
3σ = 42 - 73
3σ = -31
σ = -31/3
σ = -10.3
b) Raw Score: 81
z = (x-μ)/σ
3 = 81 - 73/σ
Cross Multiply
3σ = 81 - 73
3σ = 8
σ = 8/3
σ = 2.67
What's the next number in this sequence 27, 19, 11, 3
Answer:
3,11,19,27
Step-by-step explanation:
Answer:
the number is ---8---
A fireman’s ladder leaning against a house makes an angle of 62 with the ground. If the ladder is 3 feet from the base of the house, how long is the ladder?
In the given scenario ladder is 6.52 feet long.
Given that,
The angle between ground and ladder = 62 degree
The distance of ladder from ground and ladder = 3 feet
We have to find the length of ladder.
Since we know that,
The trigonometric ratio
cosθ = adjacent/ Hypotenuse
Here we have,
Adjacent = 3 feet
Hypotenuse = length of ladder
Thus to find the length of ladder we have to find the value of hypotenuse.
Therefore,
⇒ cos62 = 3/ Hypotenuse
⇒ 0.46 = 3/ Hypotenuse
⇒ Hypotenuse = 3/0.46
= 6.52
Thus,
length of ladder = 6.52 feet.
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If is the probability that the reciprocal of a randomly selected positive odd integer less than 2010 gives a terminating decimal, with and being relatively prime positive integers, what is
The probability value of (m, n) is (1, 2^1005).
Let n be a positive odd integer. We are asked to find the probability that its reciprocal gives a terminating decimal. This is equivalent to saying that the only prime factors of n are 2 and 5 because any other prime factor will yield a repeating decimal.
If n is less than 2010, then its only possible prime factors are 3, 7, 11, ..., 2009, since all primes greater than 2009 are greater than n. We want n to have no prime factors other than 2 and 5. There are 1005 odd integers less than 2010.
We want to count how many of these have no odd prime factors other than 3, 7, 11, ..., 2009. This is equivalent to counting how many subsets there are of {3, 7, 11, ..., 2009}. There are 1004 primes greater than 2 and less than 2010. Each of these primes is either in a subset or not in a subset. Thus, there are 2^1004 subsets of {3, 7, 11, ..., 2009}, including the empty set.
Thus, the probability is:
P = (number of subsets with no odd primes other than 3, 7, 11, ..., 2009) / 2^1004
We can count this number using the inclusion-exclusion principle. Let S be the set of odd integers less than 2010. Let Pi be the set of odd integers in S that are divisible by the prime pi, where pi is a prime greater than 2 and less than 2010. Let Pi,j be the set of odd integers in S that are divisible by both pi and pj, where i < j.
Then, the number of odd integers in S that have no prime factors other than 2 and 5 is:
|S - ⋃ Pi + ⋃ Pi,j - ⋃ Pi,j,k + ...|
where the union is taken over all sets of primes with at least one element and less than or equal to 1005 elements.
By the inclusion-exclusion principle:
|S - ⋃ Pi + ⋃ Pi,j - ⋃ Pi,j,k + ...| = ∑ (-1)^k ⋅ (∑ |Pi1,i2,...,ik|)
where the outer summation is from k = 0 to 1005, and the inner summation is taken over all combinations of primes with k elements.
This simplifies to:
(1/2) ⋅ (2^1004 + (-1)^1005)
Thus, the probability is: P = (1/2^1004) ⋅ (1/2) ⋅ (2^1004 + (-1)^1005) = 1/2 + 1/2^1005. Hence, (m, n) = (1, 2^1005).
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Complete question:
If m/n is the probability that the reciprocal of a randomly selected positive odd integer less than 2010 gives a terminating decimal, with m and n being relatively prime positive integers. what is probability value of m and n?
The probability value of (m, n) is (1, 2^1005).This is equivalent to saying that the only prime factors of n are 2 and 5 because any other prime factor will yield a repeating decimal.
Let n be a positive odd integer. We are asked to find the probability that its reciprocal gives a terminating decimal. This is equivalent to saying that the only prime factors of n are 2 and 5 because any other prime factor will yield a repeating decimal.
If n is less than 2010, then its only possible prime factors are 3, 7, 11, ..., 2009, since all primes greater than 2009 are greater than n. We want n to have no prime factors other than 2 and 5. There are 1005 odd integers less than 2010.
We want to count how many of these have no odd prime factors other than 3, 7, 11, ..., 2009. This is equivalent to counting how many subsets there are of {3, 7, 11, ..., 2009}. There are 1004 primes greater than 2 and less than 2010. Each of these primes is either in a subset or not in a subset. Thus, there are 2^1004 subsets of {3, 7, 11, ..., 2009}, including the empty set.
Thus, the probability is:
P = (number of subsets with no odd primes other than 3, 7, 11, ..., 2009) / 2^1004
We can count this number using the inclusion-exclusion principle. Let S be the set of odd integers less than 2010. Let Pi be the set of odd integers in S that are divisible by the prime pi, where pi is a prime greater than 2 and less than 2010. Let Pi,j be the set of odd integers in S that are divisible by both pi and pj, where i < j.
Then, the number of odd integers in S that have no prime factors other than 2 and 5 is:
|S - ⋃ Pi + ⋃ Pi,j - ⋃ Pi,j,k + ...|
where the union is taken over all sets of primes with at least one element and less than or equal to 1005 elements.
By the inclusion-exclusion principle:
|S - ⋃ Pi + ⋃ Pi,j - ⋃ Pi,j,k + ...| = ∑ (-1)^k ⋅ (∑ |Pi1,i2,...,ik|)
where the outer summation is from k = 0 to 1005, and the inner summation is taken over all combinations of primes with k elements.
This simplifies to:
\((1/2) * (2^{1004} + (-1)^1005)\)
Thus, the probability is: P = \((1/2)^{1004}* (1/2) *(2^{1004} + (-1)^{1005}) = 1/2 + 1/2^{1005}.\)
Hence, (m, n) = (\(1, 2^{1005\)).
Learn more about probability:
brainly.com/question/31828911
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