Answer: 8/52
2/13=x/52
(2×52)/13=x
104/13=x
8=x
x=8
8 is the equivalent of the fraction using a denominator of 52. 2/13
Equivalent fractions can be defined as fractions that may have different numerators and denominators but they represent the same value. For example, 9/12 and 6/8 are equivalent fractions because both are equal to 3/4 when simplified.
All equivalent fractions get reduced to the same fraction in their simplest form as seen in the example given above. Explore the given lesson to get a better idea of how to find equivalent fractions and how to check if the given fractions are equivalent.
⇒[tex]\frac{2}{13} =\frac{x}{52}[/tex]
⇒[tex]\frac{2(52)}{13} = x[/tex]
⇒[tex]x = 8[/tex]
therefore, 8 is the equivalent of the fraction using a denominator of 52. 2/13
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.....................................................................
Answer:
b
Step-by-step explanation:
Answer:
The correct statement is: The temperature of the object is decreasing exponentially by 0.007% each minute.
Step-by-step explanation:
The exponential term in the function, [tex]e^{0.007 h}[/tex], is greater than 1. This means that the temperature is increasing over time.
However, the exponent is very small, so the increase in temperature is very slow. In fact, the temperature is increasing by only -0.007% each minute.
We can use the following formula to calculate the percent change in temperature each minute.
[tex]\boxed{\bold{Percent\: change = \frac{(new \:value - \:old \: value) }{ old\:value }* 100\%}}[/tex]
In this case, the new value is the temperature after one minute, and the old value is the temperature at the start of the minute.
So, the percent change in temperature is:
[tex]= \frac{10 + 160e^{-0.007 * 1} - 10) }{ 10} * 100\%[/tex]
≈ -0.007%
Therefore, the temperature of the object is decreasing exponentially by 0.007% each minute.
Urgent!!!
cos4α + 2 Sin² 2α
Answer:
Step-by-step explanation:
cos4α + 2 Sin² 2α is a trigonometric expression that involves the cosine and sine functions of the angle α. This expression can be simplified using trigonometric identities to get:
cos4α + 2 Sin² 2α = (cos² 2α)² + 2 (Sin 2α)²
This simplification involves using the double angle identity for sine and the identity for the square of cosine.
Brainliest Pls
Answer:1
Step-by-step explanation:
cos(4α) + 2sin²(2α)
First, let's expand cos(4α)
cos(4α) = cos²(2α) - sin²(2α)
Now, let's substitute this expression back into the original equation:
cos(4α) + 2sin²(2α) = (cos²(2α) - sin²(2α)) + 2sin²(2α)
= cos²(2α) + sin²(2α)=1 by
cos²(α) + sin²(α)=1
7. Given right triangle ABC below, determine sin(A).
The value of Sin A is 5/13.
Option A is the correct answer.
We have,
Sin A = Perpendicular / Hypotenuse
Sin A = BC / AB
And,
BC = 5
AB = 13
Substituting.
Sin A = 5/13
Thus,
The value of Sin A is 5/13.
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Can someone please answer and provide an explanation for these problems?
Answer:
details:
54)Write the midpoint formula::
Substitute
into
::
Swap the sides:
Swap the sides:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Calculate the product or quotient:
Rearrange variables to the left side of the equation:
Calculate the sum or difference:
Express solutions in ordered pairs::
(55)Write the midpoint formula::
Substitute
into
::
Swap the sides:
Swap the sides:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Calculate the product or quotient:
Rearrange variables to the left side of the equation:
Calculate the sum or difference:
Express solutions in ordered pairs::
(56)Write the midpoint formula::
Substitute
into
::
Swap the sides:
Swap the sides:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Calculate the product or quotient:
Rearrange variables to the left side of the equation:
Calculate the sum or difference:
Express solutions in ordered pairs::
(57)Write the midpoint formula::
Substitute
into
::
Swap the sides:
Swap the sides:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Multiply both sides of the equation by the common denominator:
Reduce the fractions:
Calculate the product or quotient:
Rearrange variables to the left side of the equation:
Calculate the sum or difference:
Express solutions in ordered pairs::
2/5 of Jim's savings is equal to 4/9 of Max's savings. Jim has $30 more than Max. 5 How much money does Max have?
As Jim has $30 more than Max, as per this, Max has $270.
Let's assume Max's savings as 'x' dollars.
We have:
2/5 of Jim's savings is equal to 4/9 of Max's savings.
(2/5) * Jim's savings = (4/9) * Max's savings
We also know that Jim has $30 more than Max.
Jim's savings = Max's savings + $30
So,
(2/5) * Jim's savings = (4/9) * Max's savings
(2/5) * (Max's savings + $30) = (4/9) * Max's savings
To solve for Max's savings, we can solve this equation:
(2/5) * (x + $30) = (4/9) * x
So, for x:
(2/5) * (x + $30) = (4/9) * x
Multiplying both sides by 45 (the least common denominator of 5 and 9) to eliminate the fractions:
18(x + $30) = 20x
18x + 540 = 20x
540 = 20x - 18x
540 = 2x
x = 270
Therefore, Max has $270.
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2/3x - 1/5x = x - 1
solve pls
are the side lengths of triangle Graph shows a triangle plotted on a coordinate plane. The triangle is at A(minus 7, 3), B(minus 3, 6), and C(5, 0). Type the correct answer in each box. If necessary, round any decimal to the nearest tenth. units units units
The side lengths for the triangle are given as follows:
AC = 12.4.AB = 5.BC = 10.How to calculate the distance between two points?Suppose that we have two points of the coordinate plane, and the ordered pairs have coordinates [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex].
The shortest distance between them is given by the equation presented as follows, derived from the Pythagorean Theorem:
[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
Then the length AC is given as follows:
[tex]AC = \sqrt{(5 - (-7))^2 + (0 - 3)^2}[/tex]
AC = 12.4.
The length AB is given as follows:
[tex]AB = \sqrt{(-3 - (-7))^2 + (6 - 3)^2}[/tex]
AB = 5.
The length BC is given as follows:
[tex]BC = \sqrt{(5 - (-3))^2 + (6 - 0)^2}[/tex]
BC = 10.
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50 Points! Multiple choice algebra question. Photo attached. Thank you!
The angle that is not a solution to the trigonometric equation sin(2θ) = 1 is given as follows:
a) 90º.
How to solve the trigonometric equation?The trigonometric equation in the context of this problem is defined as follows:
sin(2θ) = 1
The sine assumes a value of 1 at:
90º.
Hence the solution is obtained as follows:
2θ = 90º.
θ = 45º.
The equivalent angle to 45º on the third quadrant is given as follows:
180º + 45º = 225º.225 - 360 = -135º.Hence 225º and -135º are also solutions to the trigonometric equation.
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estimate the original volume of the pyramid in fig 15.28 given that it's frustum has a 4 m by 4 m squared top which is 12 m vertically above the square base which is 20 m by 20 m (assume that the original problem was raised to a point . Neglect the volume of the entrance of the right of the photograph)
Answer:
Step-by-step explanation:
To estimate the original volume of the pyramid, we can use the formula for the volume of a frustum of a pyramid:
V = (1/3)h(a^2 + ab + b^2)
Where V is the volume, h is the height of the frustum, a is the side length of the top square, and b is the side length of the bottom square.
In this case, the top square has a side length of 4 m, and the bottom square has a side length of 20 m. The height of the frustum is 12 m.
Plugging these values into the formula, we get:
V = (1/3) * 12 * (4^2 + 4*20 + 20^2)
Simplifying the equation:
V = (1/3) * 12 * (16 + 80 + 400)
V = (1/3) * 12 * 496
V = 12 * 496/3
V = 1984
Therefore, the estimated original volume of the pyramid is 1984 cubic meters.
PLEASE HELP!!!!!!
it’s a division equation
Answer:
x+1
Step-by-step explanation:
you have to rewrite the equestion, factor the expression, and than reduce it
The mean of three test is 74 what must be the the 4th score to get a mean of 78?
To achieve a mean of 78 with four scores, the fourth score must be 90.
We need to consider the current mean, the number of tests, and the desired mean in order to find the fourth score needed to achieve a mean of 78
Let's assume the three test scores are x1, x2, and x3.
We know that the mean of these three scores is 74. So we can assume that
The sum of these numbers is 74 or
(x1 + x2 + x3)/3 = 74
Let's denote it as x4 ,to find the fourth score we will use the formula to calculate the mean:
(x1 + x2 + x3 + x4)/4 = 78
To solve for x4, we can rewrite it as:
x1 + x2 + x3 + x4 = 78 * 4
Now, we can substitute the value of the mean of the first three scores into the equation:
(x1 + x2 + x3) + x4 = 312
we know that,
(x1 + x2 + x3) = 3 * 74
or,
3 * 74 + x4 = 312
Simplifying this, we get:
222 + x4 = 312
Subtracting 222 from both sides:
x4 = 312 - 222
x4 = 90
Therefore, to achieve a mean of 78 with four scores, the fourth score must be 90.
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An urn contains 11 red balls numbered 1 through 11, 11 yellow balls numbered 1 through 11, 11 green balls numbered 1 through 11, and 11 black balls numbered 1 through 11. If 4 balls are randomly selected, find the probability of getting
(a) a flush (i.e., all balls the same color).
(b) three of a kind. (Three of a kind is 3 balls of one denomination and a fourth ball of a different denomination. e.g., 5,5,5,2)
(c) two pairs. (A pair is two balls of the same denomination. e.g., 3,3. Two pairs is something like 3,3,5,5, i.e., a pair of one denomination and a second pair of a different denomination.)
please write the steps as well!! thank you so much
(a) Probability of getting a flush is 0.000546.
(b) Probability of getting three of a kind is 0.00451.
(c) Probability of getting two pairs is 0.0150.
We have,
(a)
To calculate the probability of getting a flush, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of balls = 11 (red) + 11 (yellow) + 11 (green) + 11 (black) = 44 balls
The number of ways to choose 4 balls (without replacement)
= 44C4 = 44! / (4! x (44 - 4)!) = 44! / (4! x 40!)
= 44 x 43 x 42 x 41 / (4 x 3 x 2 x 1)
= 7315
The number of favorable outcomes:
There are 4 different colors of balls, so we have 4 possible flushes (all red, all yellow, all green, all black).
Probability of getting a flush = Number of favorable outcomes / Total number of outcomes = 4 / 7315 ≈ 0.000546
(b)
To calculate the probability of getting three of a kind, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of ways to choose 4 balls = 44C4 = 7315 (as calculated in part (a)).
Number of ways to choose 3 balls of the same denomination: We have 11 denominations, so we can choose 3 balls of the same denomination in 11C1 = 11 ways.
Number of ways to choose the fourth ball of a different denomination: After choosing 3 balls of the same denomination, we have 3 denominations left from which we can choose the fourth ball. So, the number of ways to choose the fourth ball = 3.
Number of favorable outcomes = Number of ways to choose 3 balls of the same denomination x Number of ways to choose the fourth ball of a different denomination
= 11 x 3
= 33
Probability of getting three of a kind = Number of favorable outcomes / Total number of outcomes = 33 / 7315 ≈ 0.00451
(c)
To calculate the probability of getting two pairs, we need to determine the total number of possible outcomes and the number of favorable outcomes.
Total number of ways to choose 4 balls = 44C4 = 7315 (as calculated in part (a)).
Number of ways to choose the first pair: We have 11 denominations, so we can choose the first pair in 11C1 = 11 ways.
Number of ways to choose the second pair: After choosing the first pair, we have 10 denominations left from which we can choose the second pair. So, the number of ways to choose the second pair = 10.
Number of favorable outcomes
= Number of ways to choose the first pair x Number of ways to choose the second pair
= 11 x 10
= 110
Probability of getting two pairs = Number of favorable outcomes / Total number of outcomes = 110 / 7315 ≈ 0.0150
Thus,
(a) Probability of getting a flush = 4/7315 ≈ 0.000546
(b) Probability of getting three of a kind = 33/7315 ≈ 0.00451
(c) Probability of getting two pairs = 110/7315 ≈ 0.0150
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What is the meaning of "possibly some ui are not free, or even do not occur, in ϕ"?
The phrase "possibly some ui are not free, or even do not occur, in ϕ" refers to a logical formula or expression ϕ that contains variables ui.
Answer to the aforementioned questionWhen it says "possibly some ui are not free," it means that not all of the variables ui in ϕ are unrestricted or unconstrained. Some of them may have certain conditions or restrictions placed on them within the context of the formula ϕ.
On the other hand, when it says "or even do not occur, in ϕ," it means that some variables ui may not appear at all in the formula ϕ. This implies that those variables are not present or relevant in the expression ϕ and do not play a role in its interpretation or evaluation.
In summary, the phrase indicates that in the formula ϕ, some variables ui may have restrictions or conditions placed on them, while others may not be present or relevant within the context of the formula.
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a) The monthly basic salary of the married Chief of Army Staffs (COAS) General is Rs 79,200 with Rs 2,000 dearness allowance. He gets Dashain allowance which is equivalent to his basic salary of one month. He contributes 10% of his basic salary in Employee's Provident Fund (EPF) and he pays Rs 50,000 as the premium of his life insurance. Given that 1% social security tax is levied upon the income of Rs 6,00,000, 10% and 20% taxes are levied on the next incomes of Rs 2,00,000 and up to Rs 3,00,000 respectively. Answer the following questions. What is his monthly basic salary? (ii) Find his taxable income. (iii) Find the total income tax paid by him.
i) The monthly basic salary of the married COAS General is Rs 79,200.
ii. Taxable Income = Rs 1,60,400 - Rs 57,920 = Rs 1,02,480
iii. The total income tax paid by him is Rs 24.8.
How to calculate the valueHis taxable income can be calculated as follows:
Total Monthly Income = Rs 79,200 + Rs 2,000 + Rs 79,200 = Rs 1,60,400
EPF Contribution = 10% of Basic Salary = 10% of Rs 79,200 = Rs 7,920
Life Insurance Premium = Rs 50,000
Total Deductions = EPF Contribution + Life Insurance Premium = Rs 7,920 + Rs 50,000 = Rs 57,920
Taxable Income = Total Monthly Income - Total Deductions
Taxable Income = Rs 1,60,400 - Rs 57,920 = Rs 1,02,480
For income up to Rs 6,00,000: No income tax
For income between Rs 6,00,001 and Rs 8,00,000: 1% of the amount exceeding Rs 6,00,000
Taxable Income in this slab = Rs 2,480 (Rs 1,02,480 - Rs 6,00,000)
Income tax for this slab = 1% of Rs 2,480 = Rs 24.8
Therefore, the total income tax paid by him is Rs 24.8.
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Dana had to pay her friend $900, 3 months ago and she has to pay $580 in 3 months. If her friend was charging her an interest rate of 1.50% per month, what single payment would settle both payments today?
Dana had to pay her friend $900, 3 months ago, and she has to pay $580 in 3 months. If her friend was charging her an interest rate of 1.50% per month. $1499.45, single payment would settle both payments today.
Given:
The principal of a loan is the original sum borrowed.The interest rate per period is known as rate.The loan's duration is expressed in terms of time.Simple interest = Principal × Rate × Time
Simple interest = $900 × 0.015 × 3
Simple interest = $40.50
Present Value = Future Value / (1 + Rate x Time)
Present Value = $580 / (1 + 0.015 × 3)
Present Value = $580 / 1.045
Present Value = $558.95
Total amount $580 loan is $558.95.
Total amount owed = $940.50 + $558.95
Total amount owed = $1499.45
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Fractions such as
12 in.
1 ft
and
1 yd
3 ft
are called
fractions.
Fractions such as 12 in/ 1 ft and 1 yd/ 3ft are called unit conversion fractions
Unit conversion fractions are used to convert between different units of measurement within the same quantity.
The fraction 12 in/1 ft represents the conversion factor from inches to feet, where 12 inches is equivalent to 1 foot.
This fraction allows us to convert a certain length measured in inches to an equivalent length in feet.
Similarly, the fraction 1 yd/3 ft represents the conversion factor from yards to feet, where 1 yard is equivalent to 3 feet.
This fraction allows us to convert a certain length measured in yards to an equivalent length in feet.
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Fill in each blank so that the resulting statement is true
Fractions such as 12 in/ 1 ft and 1 yd/ 3ft are called _________fractions
10A34594-4103-466C-A0CC-22EBA
9EB47A6.jpeg
The total surface area of the cylinder is 414.7 cm².
What is the curved surface area of the cylinder?The curved surface area of the cylinder is calculated as follows;
C.S.A = 2πrh
where;
r is the radius of the cylinderh is the height of the cylinderThe curved surface area of the cylinder is calculated as;
C.S.A = 2π(6 cm) x (5 cm)
C.S.A = 188.5 cm²
The total surface area of the cylinder is calculated as follows;
T.S.A = 2πrh + 2πr²
circular area = 2πr² = 2π x (6cm)² = 226.2 cm²
The total surface area = 188.5 cm² + 226.2 cm² = 414.7 cm²
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Li Wei and Colleen have the same reading assignment. After one week Li Wei has read 90 pages and Colleen has read 126 pages. If Li Wei can you read 30 pages in an hour and Colleen henry 24 pages an hour, when will they be on the same page 
Equating the expressions, if Li Wei can read 30 pages in an hour and Colleen Henry 24 pages an hour, they will be on the same page in 6 hours.
What are equivalent expressions?Equivalent expressions are two or more algebraic expressions that have the same value when the variables are substituted with real numbers.
In this situation, we can determine the time that Li Wei and Colleen Henry can be on the same page by equating the expressions representing their reading rates.
The number of pages Li Wei can read in a week = 90
The number of pages Collen can read in a week = 126
Li Wei's reading rate per hour = 30 pages
Collen's reading rate per hour = 24 pages
Let the hours that Li Wei and Colleen can be on the same page= x
Expressions:The total number of pages Li Wei can read = 90 + 30x
The total number of pages Colleen can read = 126 + 24x
For Li Wei and Colleen to be on the same, the two expressions are equated.
90 + 30x = 126 + 24x
6x = 36
x = 6
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A Park is shaped like a trapezoid. It has an area of 7.2 mi.² one piece is 1.2 miles on the other bases 2.4 miles long. What is the height of the trapezoid
(Can't delete my answer)
A sales person starts working 40 hours per week at a job with 2 options for being paid . Option A is an hourly wage of $19. Option B is a commission rate of 8% on weekly sales.
How much does the sales person need to sell in a given week to earn the same amount with each option?
A. $9,500
B. $4,750
C. $760
D. $320
Given, Option A: Hourly wage is $19 and the salesperson works 40 hours per week. So, he will earn in a week [tex]\sf = 19 \times 40 = \$760[/tex]
Now, according to option b, he will get 8% commission on weekly sales.
Let. x = the amount of weekly sales.
To earn the same amount of option A, he will have to equal the 8% of x to $760
So, [tex]\sf \dfrac{8x}{100}=760[/tex]
Or, [tex]\sf 8x= 76000[/tex]
Or, [tex]\sf x= \dfrac{76000}{8}=9500[/tex]
the salesman needs to make a weekly sales of $9,500 to earn the same amount with two options.
#1 Write a positive or negative integer that represents the situation.
You run up 24 steps.
© 24
0-24
The integer number that represents the situation running up 24 steps is given as follows:
+24.
What are integer numbers?Integer number are numbers that can have either positive or negative signal, but are whole numbers, meaning that they have no decimal part.
For altitudes, the signs are given as follows:
Gain of altitude: positive integer.Loss of altitude: negative integer.In this problem, we have an increase of altitude of 24 steps, hence the integer number is given as follows:
+24.
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Pls help I’m stuck Tysm I really thankful
Answer:
23/41
Step-by-step explanation:
You want to know the fraction who failed, given they predicted they would fail.
StatsThe attached 2-way table shows the given numbers (black) and the numbers that make the totals correct (blue).
It shows 23 of the 41 who thought they would fail actually did. That fraction is ...
23/41 . . . . failed who thought they would
<95141404393>
MA.7.AR.4.1
Johnny and Eleanor went to their local gas station to collect information about the cost of
fuel for compact cars. They observed both regular and premium gas purchases that day and
recorded their data in the table below.
Gallons Purchased 11.5 7.2
10
14.3 6.8
9.7
Cost
$25.23 $15.80 $21.94 $40.63 $14.92 $27.56
Part A. Is there a proportional relationship between the number of gallons of gas sold and
the cost? Explain your answer.
Part B. If the relationship is not proportional, which data value or values should be
changed to make the relationship proportional? What could explain this
difference?
a) The fourth and sixth data points have significantly different ratios.
a) Let's calculate the ratios:
For the first data point:
= 11.5 gallons / $25.23 = 0.4555 gallons per dollar
For the second data point:
= 7.2 gallons / $15.80 = 0.4557 gallons per dollar
For the third data point:
= 10 gallons / $21.94 = 0.4556 gallons per dollar
For the fourth data point:
= 14.3 gallons / $40.63 = 0.3519 gallons per dollar
For the fifth data point:
= 6.8 gallons / $14.92 = 0.4555 gallons per dollar
For the sixth data point:
= 9.7 gallons / $27.56 = 0.3517 gallons per dollar
However, the fourth and sixth data points have significantly different ratios.
b) If the relationship is not proportional, the data values that should be changed to make the relationship proportional are the fourth and sixth data points.
The difference in ratios could be explained by factors such as fluctuations in gas prices or differences in gas grades.
To establish a proportional relationship, it would be necessary to collect data where the price per gallon remains constant for all data points or to separate the data based on gas grades and analyze each grade separately.
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Plssssss helllllp meeee 50.point and will mark Brainlyiest Solve: 4!
A.16
B.6
C.24
D.4
Answer:
C
Step-by-step explanation:
the expression ! ( factorial ) has the following meaning
n! = n(n - 1)(n - 2) ... × 3 × 2 × 1
Then
4! = 4 × 3 × 2 × 1 = 24
pls help it is due today
The property that should be used to simplify the expression 1/t⁻³ is [tex]\frac{1}{a^-m}[/tex] = [tex]a^m[/tex] (where a ≠ 0). Option A
How to identify the property to simplify the expression?In the given scenario or expression, this property implies that 1/t⁻³ = t³. The negative exponent in the denominator becomes a positive exponent when moved to the numerator.
Here is the step-by-step solution
1/t⁻³ = 1/(t⁻³)
1/(t⁻³) = (t⁻³)⁻¹
(t⁻³)⁻¹ = t⁽⁻³⁾ˣ⁽⁻¹⁾
t⁽⁻³⁾ˣ⁽⁻¹⁾ = t³
Therefore, the simplified expression is t³.
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Which expression is equivalent to 8.8x10^9/2.2x10^-3?
A 4 x 10^12
B 4 x 10^6
C 4 x 10^-3
D 4 x 10^-6
Exam the following data set.
{8,19,11,20,2,14,17,9,15}
Which answer shows the five number summary that could be used to create a box plot for this data?
The five number summary of the data set for the box plot would be :
Minimum value: 2Q1 : 9Median : 14Q3 : 19Maximum value: 20How to find the five number summary ?First, order the numbers in ascending order:
2, 8, 9, 11, 14, 15, 17, 19, 20
The minimum is the smallest number which is 2. The maximum is the largest number which is 20.
The median is the number in the middle which is 19. The Q1 is the number in the middle of the first half of the numbers which is 9. The Q3 is the number in the middle of the second half which is 19.
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Answer:
min=2
1st quartile= 8.5
median=14
3rd quartile=18
max=20
Step-by-step explanation:
Help !!
centimeters and millimeters <3
1 cm = 10 mm. Length of book : 28 cm. Length of book = 280 mm. This shows that measurements in mm are the same but look to have a higher number .
How to compare centimeters and millimeters ?The conversion factor provided states that 1 cm is equal to 10 mm, which means that 1 cm is made up of 10 individual millimeters.
The length of my book in when measured in millimeters is 280 mm.
In centimeters, this is therefore:
= 280 mm / 10 cm /mm
= 28 cm
Although the numerical value appears higher when expressed in millimeters, it is important to note that the actual length of the book remains the same.
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Suppose a website has about 1,500,000 users. When each member signs up,
he or she is assigned an account number.
A sample of users is to be chosen to take a survey. Using systematic
sampling of account numbers, which of these will result in the smallest
sample size?
OA. Choosing every 1500th account number
OB. Choosing every 15th account number
OC. Choosing every 150th account number
D. Choosing every 15,000th account number
Consider a tree T with n vertices, where n is an odd integer greater than or equal to 3. Let v be a vertex of T. Prove that there exists a vertex u in T such that the distance between u and v is at most (n-1)/2.
Our initial assumption that for every vertex u in T, the distance between u and v is greater than (n-1)/2 must be false. Thus, there must exist a vertex u in T such that the distance between u and v is at most (n-1)/2.
To prove that there exists a vertex u in tree T such that the distance between u and v is at most (n-1)/2, we can use a proof by contradiction.
Assume that for every vertex u in T, the distance between u and v is greater than (n-1)/2.
Since T is a tree with n vertices, it has n-1 edges. We can select a longest path P in T, which has a length of at least n-1. Let's denote the endpoints of this path as u and w, where u is the starting vertex and w is the ending vertex.
Now, consider the distance between u and v. Since we assumed that the distance between any vertex u and v is greater than (n-1)/2, the distance between u and v must be greater than (n-1)/2.
However, let's consider the distance between w and v. Since the path P is the longest path in the tree, the distance between w and v must be greater than or equal to the length of path P, which is at least n-1.
Now, let's consider the total number of vertices in the tree, which is n. Since n is an odd integer, (n-1)/2 is an integer as well.
If the distance between w and v is greater than or equal to n-1, and (n-1)/2 is an integer, then there must be a contradiction. This is because the distance between w and v is greater than (n-1)/2, violating our assumption.
Therefore, our initial assumption that for every vertex u in T, the distance between u and v is greater than (n-1)/2 must be false. Thus, there must exist a vertex u in T such that the distance between u and v is at most (n-1)/2.
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