How do you prove that between any two irrationals there is a rational number and vice versa?
To prove that there exists a rational number between any two irrational numbers, we can use the fact that there exists a rational number between any two real numbers.
Let a and b be two irrational numbers with the property that a b. Then a and b are not rational by definition, which means they cannot be written in the form p/q, where p and q are integers and q is not equal to zero.
We can define a new number c = (a + b) / 2 because a b. This number is a rational number because it can be expressed as a fraction in lowest terms as (a + b) / 2.
As a result, we have demonstrated that a rational number c exists between any two irrational numbers a and b.
A similar argument can be used to demonstrate that there exists an irrational number between any two rational numbers.
Consider a and b to be two rational numbers such that a b. We can then define a new number c = (a + b) / 2.
If c is also rational, we're done because we can just use c as our desired number between a and b. However, if c is irrational, we have found our desired number between a and b, which is also irrational.
Assume that a and b are both even integers to see why c must be irrational in some cases. Then their average c is also even, implying that c can be written as p/q, where p and q are integers and q is not equal to zero, with both p and q being even. However, because p and q are both even, this contradicts the fact that they must have no common factors.
As a result, we have demonstrated that there exists an irrational number c between any two rational numbers a and b.
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pls help me its urgent ty
Each of the quadratic function should be factorized as follows;
x² - 6x + 8 ⇒ (x - 4)(x - 2).
x² - 4x - 32 ⇒ (x - 8)(x + 4).
x² + x - 42 ⇒ (x - 6)(x + 7).
What is a quadratic function?In Mathematics, the standard form of a quadratic function is given by this mathematical expression;
ax² + bx + c = 0
Where:
a and b represents the coefficients of the first and second term in the quadratic function.c represents the constant.Next, we would solve each of the quadratic function by using the factorization method as follows;
x² - 6x + 8 = 0
x² - 4x - 2x + 8 = 0
x(x - 4) - 2(x - 4) = 0
(x - 4)(x - 2) = 0
x² - 4x - 32 = 0
x² - 8x + 4x - 32 = 0
x(x - 8) + 4(x - 8) = 0
(x - 8)(x + 4) = 0
x² + x - 42 = 0
x² - 6x + 7x - 42 = 0
x(x - 6) + 7(x - 6) = 0
(x - 6)(x + 7) = 0
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which of the following shows 87 and 1/2% as a fraction? 7/8, 5/8, -8/75, none of these
Answer:
Step-by-step explanation:
87.5% = 87.5/100 = 875/1000= 7/8
Ben says this system of equations has one solution. Is he correct? Explain
Answer:
what system of equations? and ben is a bad name, try and come up with another one
Step-by-step explanation:
Resuelve los siguientes ejercicios sobre raíces cuadradas.
15. Observa la siguiente suma de radicales. Luego, de-
termina sí las proposiciones son verdaderas (V) o
falsas (F).
-10√3+√12+8√3
Turea
Solo es posible sumar el primer y el tercer
sumando, ya que son los únicos semejantes.
al
La suma de los tres radicales es igual a cero.
La suma de los tres radicales es
1-2√3+√2.
La diferencia entre el tercer y el primer l
sumando es igual a 18√3.
Answer:
La proposición "La suma de los tres radicales es igual a cero" es falsa. Si sumamos los tres radicales, tenemos:
-10√3 + √12 + 8√3 = (-10 + 8)√3 + √12 = -2√3 + √12.
La proposición "La suma de los tres radicales es 1-2√3 + √2" es falsa, ya que la suma real de los tres radicales es -2√3 + √12.
La proposición "La diferencia entre el tercer y el primer sumando es igual a 18√3" es falsa, ya que la diferencia real entre el tercer y el primer sumando es 8√3 - (-10√3) = 8√3 + 10√3 = 18√3.
freight train cars in a train yard there are tank cars, boxcars, and flatcars. how many ways can a train be made up consisting of tank cars, boxcars, and flatcars? (in this case, order is not important.)
The number of ways can a train made up consisting of tank cars, box cas and flat cars is 166320
Total number of tank cars = 4
Total number of Box cars = 112
Total number of flat cars = 5
Here we have to apply the concept of combinations
Number of ways of selecting 2 tank car from 5 = [tex]4C_2[/tex]
= 4! / 2!( 4-2) !
= 6
Number of ways of selecting 5 Box cars from 12 = [tex]12C_5[/tex]
= 12! / 5!(12-5)!
= 792
Number of ways of selecting 3 flat cars from 7 = [tex]7C_3[/tex]
= 7! / 3!(7-3)!
= 35
Number of possible ways can a train be made up = 6 × 792 × 35
Multiply the numbers
= 166320
Therefore, the number of possible ways is 166320
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The given question is incomplete, the complete question is :
Freight Train CarsIn a train yard there are 4 tank cars. 12 boxcars, and 7 flatcars. How many ways can a train be made up consisting of 2 tank cars, 5 boxcars, and 3 flatcars?
Graph the line with Slope 7 and y-intercept -8
Answer:
y=7x-8
Step-by-step explanation:
You start off with the equation y= mx+b, where m = the slope (7/1) and b, which is (0,-8). Then you can graph. You first put a dot at (0,-8). Then you count up to 7 and then go to the right 1. You continue the pattern until the line is over the graph. The equation would be y= 7x-8
Help please Hurry
Based on the data, which of the following best describes the experimental probability of the arrow landing on a black or white section of the spinner? (1 point)
• a
O b
• c
• d
The spinner is equally likely to land on black or white.
The spinner is 3 times more likely to land on white.
The information cannot be concluded from the data.
The spinner is 3 times more likely to land on black.
Answer:
it's d. answer d. 57/76 is the probability of landing on black. that is roughly 3 times as much as the probability of landing in white 19/76. 19 X 3 = 60 which is roughly around the ballpark of 57.
Someone help me out on this, please!
Find the equation of the ellipse if the major and minor axes have lengths 2 and 5, the major axis is vertical, and the center of the ellipse is (-1,-2)
Answer: The equation of an ellipse with major axis of length 2, minor axis of length 5, center at (-1, -2), and major axis aligned with the vertical axis, can be given by:
((x + 1)/2)^2 + ((y + 2)/5)^2 = 1
The standard form of an ellipse is:
(x^2)/a^2 + (y^2)/b^2 = 1,
where a and b are the lengths of the major and minor axes, respectively. To convert the equation to this form, we can divide both the x and y terms by the corresponding axis lengths, and shift the center of the ellipse to the origin.
Step-by-step explanation:
Problem #1 - Luca went
grocery shopping for
some last-minute picnic
supplies. He knows that
the sliced turkey was
$3.50 per 1 1/4-pounds
and coleslaw was $1.35
per 1/2 pound. How
much would it cost him if
he bought 3 pounds of
turkey and 1 1/2 -pounds
of coleslaw?
Problem #2 - Fill in the
Why?
The required it would cost $12.45 to buy 3 pounds of turkey and 1 1/2 pounds of coleslaw.
What are expression models?The expression model is defined as the model of the given situation in the form of an equation using variables and constants.
Here,
To calculate the cost of the turkey, we need to find the price per pound and then multiply it by the number of pounds. The price per pound is,
= $3.50 ÷ (1 1/4 pounds)
= $2.80 per pound
To find the cost of 3 pounds of turkey, we can multiply the price per pound by the number of pounds:
= 3 pounds × $2.80 per pound = $8.40
To find the cost of the coleslaw, we can simply multiply the price per half pound by the number of half pounds,
1 1/2 pounds = 3 half pounds
= 3 half pounds × $1.35 per half pound
= $4.05
Finally, we can add the cost of the turkey and the coleslaw to find the total cost:
$8.40 + $4.05 = $12.45
Therefore, it would cost $12.45 to buy 3 pounds of turkey and 1 1/2 pounds of coleslaw.
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who will be in your sample? when, where, and how will you survey those people? describe the procedure you will use to target respondents.
My sample will include people who are older. I will survey those people through an online survey that I will send out email or post
I will use targeted ads on social media platforms ,I will also use search engine optimization (SEO)to ensure that my survey is seen by the right people. The survey will be conducted over a two weeks. In order to ensure that I get a representative sample of people, I will use demographic and psychographic targeting. This will include targeting people based on their age, gender, location, interests, and other relevant characteristics. My sample will include people who are older. I will survey those people through an online survey that I will send out email or post
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Park rangers released 4 fish into a pond in year 0. Each year, there were three
times as many fish as the year before. How many fish were there after x
years? Write a function to represent this scenario.
A. f(x) = 3(x)4
B. f(x) = 4(3)x
C. f(x) = 3(4)x
D. f(x) = 4(x)3
The complete question:
Park rangers released 4 fish into a pond in year 0. Each year, there were three times as many fish as the year before. How many fish were there after x years? Write a function to represent this scenario.
An function which best represent the given scenario is, 4 x 3ˣ. So option B is correct.
What is geometric progression?In algebra, in sequence we study various progressions, one of the progression is geometric, in this progression for every two consecutive terms, the common ratio is the same.
Formula for nth term of G.P.,
Tₙ = a×rⁿ
where a is first term and r is common ratio
Given that,
Park rangers released 4 fish into a pond in year 0.
Each year, there were three times as many fish as the year before.
The number of fish after x years = ?
After 1 year,
the number of fish = 4 x 3
After 2 year,
the number of fish = 4 x 3 x 3
= 4 x 3²
After 3 year,
the number of fish = 4 x 3² x3
= 4 x 3³
Similarly, after x years
the number of fish = 4 x 3ˣ
Hence, the expression is 4 x 3ˣ
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jyothi is the human resource manager in a company. For the post of 2 data analyst, she has the resume of 3 men and 2 women
1) What will be the probability that no man will be selected
2)What is the probability that both men will get selected
3)If she gets one more resume of a woman, what is the probability that only one man gets selected
Answer: The probability that no man will be selected is the number of ways that only women can be selected divided by the total number of combinations of candidates for the 2 data analyst positions. There are 2 ways to choose 2 women from the 2 available women, and there are C(4,2) = 6 ways to choose 2 candidates from the 3 men and 2 women, so the probability that no man will be selected is 2/6 = 1/3.
The probability that both men will get selected is the number of ways to choose 2 men from the 3 available men divided by the total number of combinations of candidates for the 2 data analyst positions. There are C(3,2) = 3 ways to choose 2 men from the 3 available men, and there are C(5,2) = 10 ways to choose 2 candidates from the 3 men and 2 women, so the probability that both men will get selected is 3/10 = 3/5.
If Jyothi gets one more resume of a woman, there will be a total of 3 men and 3 women. The probability that only one man gets selected is the number of ways to choose 1 man and 1 woman from the 3 men and 3 women divided by the total number of combinations of candidates for the 2 data analyst positions. There are C(3,1) * C(3,1) = 9 ways to choose 1 man and 1 woman, and there are C(6,2) = 15 ways to choose 2 candidates from the 3 men and 3 women, so the probability that only one man gets selected is 9/15 = 3/5.
Step-by-step explanation:
A pole that is 2.8 m tall casts a shadow that is 1.01 m long. At the same time, a nearby tower casts a shadow that is 45.5 m long. How tall is the tower? Round
your answer to the nearest meter.
Answer: 147m
Step-by-step explanation: the tower's height can be calculated using the following equation: height = (shadow length/pole shadow length) * pole height Therefore, the tower's height is: height = (45.5/1.01) * 2.8 height = 146.5 m The tower's height is approximately 147 m, rounded to the nearest meter
Please help me by finding the degree measure of angle x!
On solving the provided question, we cans ay that in this triangle by Pythagorean theorem [tex]A^2 = B^2 + C^2[/tex] => AC = [tex]\sqrt{115}[/tex]
What is triangle?A triangle is a polygon since it has three sides and three vertices. It is one of the basic geometric shapes. The name given to a triangle containing the vertices A, B, and C is Triangle ABC. A unique plane and triangle in Euclidean geometry are discovered when the three points are not collinear. Three sides and three corners define a triangle as a polygon. The triangle's corners are defined as the locations where the three sides converge. 180 degrees is the result of multiplying three triangle angles.
here, in this triangle by Pythagorean theorem
[tex]A^2 = B^2 + C^2[/tex]
AC = [tex]\sqrt{14^2 - 9^2}[/tex]
AC = [tex]\sqrt{196 - 81}[/tex]
AC = [tex]\sqrt{115}[/tex]
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As Bin is driving her car, she notices that after 1 hour her gas tank has 7.25 gallons left and after 4 hours of
driving, it has 3,5 gallons of gas left in it.
(3) Represent this information as two coordinate
pairs in the form (1, g), Where A is the
number of hours driven and g is the gallons of Gas left
Bin had 7.25 gallons of gas to start with using the unitary method.
Unitary method is a problem-solving technique that involves finding the value of one unit and using it to calculate the value of other units. In this case, we will use unitary method to find the number of gallons Bin had to start with.
Firstly, we need to find the rate of gas consumption. The slope of the line, which is the change in the gas level over time, gives us this information. Since the slope is -1.25, this means Bin is losing 1.25 gallons of gas per hour.
Next, we need to find the total time Bin has been driving. From the problem, we know that after 1 hour, she had 7.25 gallons left, and after 4 hours, she had 3.5 gallons left. This means that she drove for 3 hours (4-1).
Using the unitary method, we can calculate the total amount of gas Bin had to start with. We know that she lost 1.25 gallons of gas per hour, and she drove for 3 hours. Therefore, she lost a total of 3.75 gallons of gas during the trip.
To find the initial gas level, we need to subtract the amount of gas she lost from the gas level she had at the end of the trip. This gives us:
Initial gas level = Gas level at end of trip + Gas lost during the trip
Initial gas level = 3.5 gallons + 3.75 gallons
Initial gas level = 7.25 gallons
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Complete question:
As Bin is driving her car, she notices that after 1 hour her gas tank has 7.25 gallons left and after 4 hours of driving, it has 3.5 gallons of gas left in it. The rate at which she is LOSING gas is 1.25 gallons....So, the slope is -1.25. How many gallons did she have to START?
An expression is given.
Which expression is equivalent to the given expression?
A
−13x+12-13x+12−13x+12
B
10x2−7x−1210x^2-7x-1210x
2
−7x−12
C
10x2−23x+1210x^2-23x+1210x
2
−23x+12
D
7x2−23x−77x^2-23x-77x
2
−23x−7
The equation -13x+12-(-13x+12)-13x+12 = -13x+12 is equivalent to the given expression that is -13x+12.
10x^2 - 23x + 12 = (5x - 3)(2x - 4) = 0
So the solutions to the equation are x = 3/5 or x = 2.
Therefore, this expression is not equivalent to -23x + 12 for all values of x.
10x^2 - 7x - 12 = 0 can be factored as (2x - 3)(5x + 4) = 0.
So the solutions to the equation are x = 3/2 or x = -4/5.
Therefore, this expression is not equivalent to -7x - 12 for all values of x.
-13x+12-(-13x+12)-13x+12 = -13x+12
-13x+12 = -13x+12
This expression is equivalent to -13x+12 .
7x^2 - 23x - 77 = (7x + 11)(x - 7) = 0
So the solutions to the equation are x = -11/7 or x = 7.
Therefore, this expression is not equivalent to -23x - 7 for all value x.
The given expressions of each case are not equivalent to the given equations for all values of x, except −13x+12-(-13x+12)-13x+12 = -13x+12 which is equivalent to -13x+12 .
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______The given question is incorrect, the correct question is given below:
An expression is given on equation
Which expression is equivalent to the given expression of each case
?
A
10x2−23x+1210x^2-23x+1210x = −23x+12
B
10x2−7x−1210x^2-7x-1210x = −7x−12
C
−13x+12-(-13x+12)-13x+12 = -13x+12
D
7x2−23x−77x^2-23x-77x = −23x−7
Which investment results in the greatest total amount? Investment A: 3000$ invested for 7 years compounded semiannually at 7% Investment B: $5,000 invested for 4 years compounded quarterly at 3.2%.
With the given compound interests, Investment B results in greatest total amount.
What exactly is compound interest?
Compound interest is interest charged on a loan or deposit. It is the most widely utilised idea in our everyday lives. The compound interest for an amount is determined by both the principal and the interest earned over time. This is the primary distinction between compound and simple interest.
Compound interest calculation formula:
A=P(1+r/n)ⁿˣ
Where,
A Equals Amount
P stands for principal.
r = interest rate
n is the number of times interest is compounded each year.
x = time (in years)
Alternatively, the formula may be written as follows:
CI = A – P
Where CI stands for Compound Interest.
Now,
For A
principal = $3000, Rate = 7% compounded semiannualy and time = 7 years
amount=3000(1+7/200)¹⁴
A=3000*1.61
=$4856
For B
principal = $5000, Rate = 3.2% compounded quaterly and time = 4 years
amount = 5000(1+3.2/400)¹⁶
A=5000*1.13
=$5680
hence,
Investment B results in greatest total amount.
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5. People Population The country of Elbonia was established in 2012. Its population at the time was 5
million and is growing at an annual rate of 3.2%. The neighboring country of Neecapia was established
at the same time. Its population was 4.1 million and is growing at an annual rate of 6.5%.
Answer:
Step-by-step explanation:
yes because the montly rate is 6.5%
i need to reask this:
I need help with this since my school switched teachers and we didnt take it and the new teacher is refusing to teach us it
let's pry open the triangles a bit and plug their values for the pythagorean theorem to get the hypotenuse, Check the picture below, those are only the first 4 triangles, notice the hypotenuse.
now let's move like the crab, backwards, let's do D first and then C
D)
well, since for the 1st triangle the root has a "2", that means "1+1", and it's pretty much the same for any subsequent triangle, so the rule must be
[tex]{\Large \begin{array}{llll} P_{n}=\sqrt{n+1} \end{array}}\hspace{5em}\textit{rule for the }n^{th}~triangle[/tex]
C)
what does that make the 9th one?
[tex]{\Large \begin{array}{llll} P_{9}=\sqrt{9+1}\implies P_9=\sqrt{10} \end{array}}[/tex]
Luke sells 386 dollars worth of candy bars as part of a school club fundraiser. If the candy bars cost 297 dollars, what equation can we make to find out how much money Luke raised as the variable x ?
Given that axb2x c3x = 2, where a, b, and c are] positive real numbers, then x =
Pls help thank you will mark the Brainliest
The exponential function that represents the given scenario is
y = (30.5)(0.7)ˣ.
The exponential function is a type of mathematical equation that can be used to determine if anything is increasing or decreasing exponentially, such as population, money, or price.
The general form of an exponential function is given by
f(x) = abˣ or y = abˣ
Where 'x' is a variable and a and b are constants
Here we have
The graph shows the number of games system sold since 2005.
As we know an exponential function is represented as:
=> y = abˣ
From the given graph x = 0 and y = 30.5
=> 30.5 = ab⁰
=> 30.5 = a(1)
=> a = 30.5
When x = 1, y = 21.35
=> 21.35 = (30.5)(b¹)
=> b = 0.7
So the required exponential function can be written as
=> y = (30.5)(0.7)ˣ
Therefore,
The exponential function that represents the given scenario is
y = (30.5)(0.7)ˣ.
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tyrone drew a logo and a dilation of the same logo on the coordinate plane. describe the dilation. then write a coordinate rule to represent it.
Dilation in coordinate geometry refers to the process of increasing or stretching the size of the originally drawn geometrical figure to a new figure by multiplying the co-ordinates of the original figure with a scale factor and obtain new co-ordinates to plot and form on the graph.
In the given figure, we observe that the original co-ordinates of the logo drew by Tyrone are:
A (-2,0) ; B (-2,3) ; C(2,2) ; D(2,-1)
From the graph, we see that the co-ordinates are multiplied by a scale factor of 5/2 and hence turn to new co-ordinates:
A'(-5,0) ; B'(-5, 15/2); C'(5,5) ; D'(5, -5/2)
Thus, we see that the rule followed to form the dilation of the logo on the coordinate plane was (5/2x, 5/2y).
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In one trip to the hardware store you buy 15 boards, 3 boxes of nails, and a saw. One a second trip you buy 9 more boards and a box of nails. Define and use variables to represent the total cost.
The variables representing the total cost of the items bought are x and y.
Where X is the total cost of the items bought during the first trip and Y is the total cost of the items bought during the second trip.
Items bought on the first trip:
Boards = 15
Boxes of nails = 3
Total cost = x
Items bought on the second trip:
Boards = 9
Boxes of nails = 1
Total cost = y
Variables:Let a piece of board = b
Let a box of nails = n
Let the total cost for the first trip = x
Let the total cost for the second trip = y
Equations:x = 15b + 3n
y = 9b + n
20. Mr. Lewinger has to choose 2 students to
lead the class field trip. If he draws from a
list of 6 boys and 6 girls, what is the
probability that the names of 2 girls will be
drawn?
A.5/24
B.5/22
C.1/7
D.5/6
Answer:
B. 5/22
----------------------
Total number of students is:
6 + 6 = 12Probability of a first girl is 6 of 12:
girls / total = 6/12 = 1/2Probability of a second girl is 5 of 11 since both numbers reduced by one:
girls / total = 5/11Probability of the two events is:
1/2*5/11 = 5/22Correct choice is B.
what is -2x+4=-6x-17
A bucket contains 1 1/2 gallons of water. Is the amount of water in the bucket greater than less than or equal to 6 quarts
One gallon is equivalent to 4 quarts, so 1 1/2 gallons is equal to 1 1/2 x 4 = 6 quarts. To determine if the amount of water in the bucket is greater than, less than, or equal to 6 quarts, we need to convert the amount from gallons to quarts.
A gallon is a unit of volume commonly used in the United States, while a quart is a unit of volume commonly used in both the US and the UK.
One gallon is equivalent to 4 quarts, so 1 1/2 gallons is equal to 1 1/2 x 4 = 6 quarts.
Therefore, the amount of water in the bucket is equal to 6 quarts.
It's important to be able to convert between different units of measurement to effectively communicate and understand various quantities. In the case of volume, it's crucial to use consistent units when comparing the amount of liquids in containers or when measuring the volume of substances.
Conversion between units of measurement also plays a significant role in many fields, such as cooking, baking, construction, and engineering, where precise measurements are critical.
In conclusion, the amount of water in the bucket is equal to 6 quarts, and the ability to convert between different units of measurement is an important skill in various areas of life and work.
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3. Zaidi says the table does not represent a function because all of the inputs have the same output 20. Is Zaidi correct?
From the table, the conclusion can be made that for any value of the input, the value of the output remains the same. Then the correct option is C.
What is a function?A function is an argument, concept, or regulation that demonstrates an association between two variables. Functions may be located throughout mathematics and are necessary for the expansion of significant linkages.
If the number of values of the variable 'y' is more than one for a given value of 'x', then it is not a function.
The table is given below.
x y
- 5 20
0 20
5 20
10 20
From the table, the conclusion can be made that for any value of the input, the value of the output remains the same. Then the correct option is C.
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The complete question is attached below.
12x (3+2’2) divided by 2-10
Answer:
-5.5
Step-by-step explanation:
[12*(3/(2^2))]/2-10 [Rewritten to add brackets.
Use PEDMAS to determine order of operation.
P =- parentheses first:
[12*(3/(2^2))]/2-10
[12*(3/(4))]/2-10 [squared the 2]
[12*(3/4)]/2-10 [Removed excess parentheses]
[12*(3/4)]/2-10 [Next parenthses: 12*(3/4) Divide and Multioply]
[9]/2 - 10 [Divide and Multiply]
4.5 - 10 [Add/Subtract]
-5.5 Result
