show that, given one nth root of z, the others are obtained by multiplying it by the nth roots of unity. (z is complex number)
It is proven that the other nth roots of z can be found by multiplying the initial one by the nth roots of unity.
A complex number satisfying the equation [tex]z^{n-1} = 0[/tex] is known as the nth root of unity. There are about n nth roots of a unit, according to the fundamental theorem of algebra. We know that,[tex]1=e^{2\pi i}[/tex]. Then, the nth root is written as [tex]\omega=e^{2\pi i/n}[/tex].
The integer of powers of this root between 0 and n-1 provides the other nth roots. Then, the roots are [tex]1, \omega, \omega^2 ,\cdots \cdots, \omega^{(n-1)}[/tex]. Now, factorize the equation [tex]z^{(n-1)} = 0[/tex] we get the required identity as, [tex]z^{(n-1)} = (z- 1)(z^{(n-1)}+ z^{(n-2)} + \cdots \cdots+ z + 1) = 0[/tex].
This indicates that the equation [tex]1+\omega+ \omega^2+\cdots \cdots+ \omega^{(n-1)}[/tex] is true for all nth roots of unity other than 1.
Roots of unity also have the great quality of being periodic, which results from the fact that n = 1. It is equal to a power of that power mod n if the power is bigger than n.
When ω is a third root of unity, [tex]\omega^5 + \omega^4 + 1 = \omega^2 + \omega + 1 = 0[/tex].
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Identify whether or not each pair of terms can be combined to simplify the expression (−4t − 7) + (3t+8).
-4r and -7
3r and 8
-41 and 3r
Answer: (-4t - 7) + (3t + 8) cannot be simplified further as the terms have different variables.
(-4r and -7) and (3r and 8) cannot be combined as these terms have different variables.
(-41 and 3r) cannot be combined as these terms have different variables.
Step-by-step explanation:
show that the improper integral g(x0dx is divergent but the average value of g on the interval 4,infinoty is finite
To show that the improper integral g(x)dx is divergent, but the average value of g on the interval 4, infinity is finite, we can consider a function g(x) = 1/x, which is mostly used as an example for a divergent integral.
The improper integral of g(x) from 4 to infinity is given by:
[tex]\int\limits {g(x)} \, dx = \int\limits {(1/x)} \, dx[/tex] from 4 to infinity
= lim (upper limit -> infinity) [ln|x|] from 4 to x
= [ln|x|] from 4 to infinity
= ln(infinity) - ln(4) = infinity - ln(4) = infinity
As the limit of the integral is infinity, the integral is divergent.
However, the average value of g on the interval [4, ∞) is given by:
average value = (1/∞ - 4) * ∫(1/x)dx from 4 to infinity
= (1/∞ - 4) * (infinity - ln(4))
= -4 + ln(4)
= ln(4) - 4
which is a finite value when the integral is divergent
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Solve for x: 2 over 3 equals 8 over quantity x plus The swimming team has competed in 45 races this season. They have won 30 races so far. How many races will the team need to win today for the team to have a 75% success rate?
Answer:
x = 12
4 races
Step-by-step explanation:
2/3 = 8/x
Cross multiply.
2x = 8 × 3
x = 4 × 3
x = 12
Answer: x = 12
0.75 × 45 = 33.75
Since the number of races must be an integer, they must win a total of 34 races to have at least a 75% success rate.
They already won 30 races.
34 - 30 = 4
Answer: 4 races
Annual Tuition and Fees $6,125.00
Annual Room and Board $2,112.00
Annual Cost of Books and Supplies $1,250.00
Other One-Time Fee $750.00
Annual Scholarship and Grants $5,000.00
Using the information from the table, identify the equation in slope-intercept form that models the total cost of attendance. (4 points)
y = 5,237
y = 10,237x
y = 9,487x + 750
y = 4,487x + 750
The equation in slope-intercept form that models the total cost of attendance is y = 4487x + 750.
What are lines and their slopes?We know lines have various types of equations, the general type is
Ax + By + c = 0, and the equation of a line in slope-intercept form is
y = mx + b.
Where slope = m and b = y-intercept.
the slope is the rate of change of the y-axis with respect to the x-axis and the y-intercept is the (0,b) where the line intersects
the y-axis at x = 0.
To find the equation in the slope-intercept form,
We'll sum all the annual fees and subtract the scholarship amount and 'x' will represent the number of years while the one-time fee will be our constant term.
So, (6125 + 2112 + 1250 - 5000).
= 4487.
Therefore, The equation in slope-intercept form is,
y = 4487x + 750.
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Set B is the set of positive two-digit even numbers less than 36 that do not contain the digit 2 .
The required set of positive two-digit even numbers less than 36 that do not contain the digit 2 is 11,13,14,15,16,17,18,19,30,31,33,34,35.
What is an integer?An integer is a whole number (not a fractional number) that can be positive, negative, or zero. Zero is not a fraction or decimal of any number. It is neither positive nor negative.
Given:
Set D is the set of positive two-digit even numbers less than 36 that do not contain the digit 2.
According to the given question, we have
2-digit numbers are numbers that have two digits and they start with the number 10 and end with the number 99.
Starting at 11 and going up the two-digit numbers even numbers less than 36 that do not contain the digit 2.
11,13,14,15,16,17,18,19,30,31,33,34,35.
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Matt has a piggy bank with 250 coins all dimes and quarter.The total dollar amount in the piggy bank is $39.25. How of each type of coin are in the piggy bank?
The piggy bank has 200 dimes and 50 quarters.
To find this, we can set up the following equation:
(number of dimes x $0.10) + (number of quarters x $0.25) = $39.25
We know that the piggy bank contains 250 coins, so we can substitute that in:
(x x $0.10) + (250-x x $0.25) = $39.25
Now we can solve for x, the number of dimes:
x0.10 + (250-x)0.25 = 39.25
x0.10 + 62.5 - x0.25 = 39.25
0.1x + 62.5 - 0.25x = 39.25
-0.15x = -23
x= 153
So there are 153 dimes in the piggy bank, and 250-153 = 97 quarters.
But we know that the piggy bank contains 250 coins, so we can check that 153 dimes + 97 quarters = 250 coins.
Determine another point on the line given two points on the line.
(0, 5), (4, 1)
Another point on the line is (2, 3)
How to find another point on the lineTo find another point on the line the equation of the line will be solved as used to get other points
The slope, m of the lines is calculated the formula
m = (y₀ - y₁) / (x₀ - x₁)
where
m = slope
x₂ and x₁ = points in x coordinates
y₂ and y₁ = points in y coordinates
The slope, m of the linear function is calculated using the points (0, 5), (4, 1)
m = (y₀ - y₁) / (x₀ - x₁)
m = (5 - 1) / (0 - 4)
m = (4) / (-4)
m = -1
equation passing through point (0, 5)
(y - y₁) = m (x - x₁)
y - 5 = -1 (x - 0)
y - 5 = -x
y = -x + 5 equation of the line
for x = 2
y = -x + 5 = -2 + 5 = 3
the point is (2, 3)
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A school staff meeting had 5 teachers in attendance, 60% of whom were first-year teachers.
How many first-year teachers were in the meeting?
Answer: 3
Step-by-step explanation: 60 x 5/100 = 3
There were three first-year teachers in the meeting.
complete the table by using your knowledge of the formula for simple interest. principle amount saved (dollars) 425, interest rate 7% time (years) 5 how many interest Earned (dollars)
Answer:
$ 148.75
Step-by-step explanation:
Simple interest:Principle = $ 425
Rate = 7%
Time = 5 yeats
[tex]\sf \boxed{Simple \ Interest = \dfrac{Principle*rate*time}{100}}[/tex]
[tex]= \dfrac{425*7*5}{100}\\\\=\dfrac{17*7*5}{4}\\\\\\=\dfrac{595}{4}\\\\=148.75[/tex]
Lisa earns $215.20 a week. She is married and claims 3 allowances. Starting next week she will receive a $40 per week raise. After the raise, how much will she pay in federal income taxes Annually?
After the raise, Lisa much will pay $9 in federal income taxes Annually.
Explain the term federal income taxes?The federal tax is a tax that is levied on the yearly profits of people, companies, and some other legal entities. Federal income taxes are due on all compensation, including salaries, monetary gifts from employers, business profits, tips, winnings from gaming, bonuses, and unemployment benefits.For the stated question:
Lisa's salary = $215.20 a week.
After increment of $40 per week.
$215.20 + $40 = $255.2
She gets 3 allowances and is married.
After raise,
Federal income taxes = $9. (taken from the table)
Thus, After the raise, Lisa much will pay $9 in federal income taxes Annually.
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a population of bacteria growing exponentially can be modeled as P(t)= P0e kt , where is the time in hours and P0 is the initial population. if the population has a doubling time of 3 hours, calculate the growth constant .
A population of bacteria growing exponentially can be modeled as P(t)= P0e kt, where is the time in hours and P0 is the initial population. if the population has a doubling time of 3 hours, the growth constant is k = ln(2) / 3 = 0.231049.
The growth constant (k) can be calculated by taking the natural log of 2 (ln(2)) and dividing it by the doubling time (3 hours). Therefore, the growth constant is 0.231049. This equation models exponential growth, which is a type of growth where the rate of increase is proportional to the population size. In this case, the population doubles every 3 hours.
This exponential growth is modeled using the equation P(t)= P0e^kt, where P(t) is the population size at a given time, P0 is the initial population size, and k is the growth constant. By finding the growth constant, we can predict the population size at any given time. The growth constant is found by taking the natural log of 2 (ln(2)) and dividing it by the doubling time (3 hours).
This equation is used to find the amount of growth per unit time and is the same regardless of the population size. This equation can be used to predict the population size at any given time, allowing us to estimate the population size in the future.
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99
2
12
. At the library, Herb spent 1/6 hour looking for a book,1/4 hour of
reading, and 1/2 hour doing research on the computer. How man
hours did Herb spend at thelibrary?
2
3
Herb spent 55 hours at the library.
What is a fraction?Fractions defines a part of a whole and are represented as a numerical value. A fraction can be a portion or section of any quantity out of a whole, where the whole can be any number, a specific value, or a thing.
Parts of a FractionThe denominator indicates the number of parts in which the whole has been divided into. It is placed in the lower part of the fraction below the fractional bar.The numerator indicates how many sections of the fraction are represented or selected. It is placed in the upper part of the fraction above the fractional bar.Total time spent in the Library = 1/6 + 1/4 + 1/2
Total time spent = 11/12 x 60 mins
Total time spent = 55 hours
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(1 point) the following question is from the four steps of statistical inference. if you respond with a summary of all four steps, you will receive no credit on this problem: what is assumed when considering the likelihood of the occurrence of an outcome? hint: this is a question about statistics, not probability.
When evaluating the likelihood of an outcome, the claim or hypothesis statement is assumed. The required answer is a hypothesis claim or statement.
It is possible to test one's assumptions regarding a population parameter through the statistical analysis known as hypothesis testing. The association between two statistical variables can then be estimated by this hypothesis.
We start by writing the claim mathematically and determining if it is the null or alternative hypothesis. The null hypothesis is the assertion that there is no change or no relationship between the two variables; otherwise, the alternative hypothesis is the assertion that tells that there is a change or relationship.
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What is the value of x?
Answer:
X = 11
Step-by-step explanation:
The total degrees of a line is 180
They gave us 39 and 13x-2
We can get rid of the 39 by subtracting it from 180
180-39=141
That means 13x-2 = 141
Now to find X. Adding 2 on both sides, gets rid of the 2 on the left side
13x = 143
Dividing 13 on both sides, we can get X alone
X = 11
Which of the following is equivalent to the expression j ^ 41 ?
The equivalent expression of i⁴¹ is i.
What are the equivalent expressions?Expressions that are equivalent to the same thing, even when they have distinct appearances. When we enter the same value(s) for the variable, two algebraic expressions that are equivalent have the same value (s).
Given:
An expression i⁴¹.
Simplifying,
i⁴¹ = {(i⁴)¹⁰} i
i⁴¹ = {(1)¹⁰} i {i⁴ = 1}
i⁴¹ = i
Therefore, the equivalent expression is i.
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For a normal distribution with mean 0 and standard deviation 1, which of the following Python lines outputs the probability P(x<0.325)?
Question 1 options:
a)
import scipy.stats as st
print(st.norm.cdf(0.325, 0, 1))
b)
import pandas as pandas
print(st.norm.pdf(0.325, 0, 1))
c)
import scipy.stats as st
print(st.norm.sf(0.325, 0, 1))
d)
print(normal(0.325, 0, 1))
Python lines to get the output for the probability P(x<0.325) are -
a) import scipy.stats as
st print(st.norm.cdf(0.325, 0, 1))
Hence, option a is the correct answer.
The Python code -
import scipy.stats as
st print(st.norm.cdf(0.325, 0, 1))
This line of code imports the scipy.stats module as "st", and then uses the norm.cdf function to calculate the cumulative distribution function (CDF) of a normal distribution with mean 0 and standard deviation 1, evaluated at x=0.325.
The CDF gives the probability that a random variable from the normal distribution is less than or equal to a certain value, so this line of code is outputting the probability P(x<0.325).
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What is the distance between the points (-7.5, -11) and (2, -11)?
Answer:
9.5
Step-by-step explanation:
Distance (d) = [tex]\sqrt{(2-(-7.5)^2)+(-11-(-11)^2)}\\ \sqrt{(9.5)^2+(0)^2}\\ \sqrt{90.25} \\9.5[/tex]
Answer the following
One can infer some information about a line's slope just by looking at its graph, especially in comparison to other lines plotted on the same coordinate plane.
What is Identify slope from a graph?Slope has a meaning in mathematics that is remarkably similar to what it means in everyday language. When describing the steepness and direction of lines in mathematics, slope is utilized. The three lines' graphs are displayed below; take a look:Lines A and B will be examined first. Line B is steeper than line A, if these lines were hills, in your mental model. Line A and line B have different slopes.Then, as you walk from left to right, observe how lines A and B incline up. Assuming a positive slope for these two lines. Between left and right, Line C slopes downward. It slopes down on line C. The rise and run of the line may be determined using two of its points, and the slope can be calculated. The rise and run are terms for the vertical and horizontal changes between two places, respectively. Divide the rise by the run to get the slope.
Rise / run = a slope.
A slope of 1.2 will be present on this line.regardless of the line's two points that you choose. Try calculating the slope between the point ( 6, 3 ) and the origin, ( 0, 0). The ascent will equal 3 and the run will equal 6. Rise run equals 3 / 6 plus 1 2 is the slope. It remains the same.The Complete Question is line's slope.
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Following is a statement of a theorem which can be proven using calculus or precalculus mathematics. For this theorem, a, b, and c are real numbers. Theorem If f is a quadratic function of the form f (x) = ax^2 + bx + c and a < 0, then the function f has a maximum value when x = -b/2a. Using only this theorem, what can be concluded about the functions given by the following formulas? ? (a) g (x) = -8x^2 + 5x - 2 (b) h (x) = -(1/3)x^2 + 3x (c) k (x) = 8x^2 - 5x - 7 (d) j (x) = -(71/99)x^2 + 210 (e) f (x) = 4x^2 - 3x + 7 (f) F (x) = -x^4 + x^3 + 9
(a) The maximum value of g(x) = -8[tex]x^2[/tex] + 5x - 2 is 5/16.
(b) The maximum value of h(x) = -(1/3)[tex]x^2[/tex] + 3x is 9/2.
(c) The maximum value of k(x) = 8[tex]x^2[/tex] - 5x - 7 is 5/16.
(d) The maximum value of j(x) = -(71/99)[tex]x^2[/tex] + 210 is 0.
(e) The maximum value of f(x) = 4[tex]x^2[/tex] - 3x + 7 is 3/8.
(f) The maximum value of F(x) = -[tex]x^4[/tex] + [tex]x^3[/tex] + 9 is 0/0. We cannot say anything about that function.
The general form of a quadratic function;
f(x) = a[tex]x^2[/tex] + bx + c and a < 0,
Then the function f has a maximum value when x = -b/2a
(a) The function is g(x) = -8[tex]x^2[/tex] + 5x - 2
We can find the maximum value at x = -b/2a.
On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = -8 and b = 5
Now putting the value at x = -b/2a
x = -5/2(-8)
x = (-5)/(-16)
x = 5/16
(b) The function is h (x) = -(1/3)[tex]x^2[/tex] + 3x
We can find the maximum value at x = -b/2a.
On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = -(1/3) and b = 3
Now putting the value at x = -b/2a
x = (-3)/2(-1/3)
x = (3×3)/2
x = 9/2
(c) The function is k(x) = 8[tex]x^2[/tex] - 5x - 7
We can find the maximum value at x = -b/2a.
On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = 8 and b = -5
Now putting the value at x = -b/2a
x = -(-5)/(2×8)
x = 5/16
(d) The function is j(x) = -(71/99)[tex]x^2[/tex] + 210
We can find the maximum value at x = -b/2a.
On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = -(71/99) and b = 0
Now putting the value at x = -b/2a
x = 0/2(-(71/99))
x = 0
(e) The function is f(x) = 4[tex]x^2[/tex] - 3x + 7
We can find the maximum value at x = -b/2a.
On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = 4 and b = -3
Now putting the value at x = -b/2a
x = -(-3)/(2×4)
x = 3/8
(f) The function is F(x) = -[tex]x^4[/tex] + [tex]x^3[/tex] + 9
We can find the maximum value at x = -b/2a.
On comparing with f(x) = a[tex]x^2[/tex] + bx + c, we get a = 0 and b = 0
Now putting the value at x = -b/2a
x = 0/0
We cannot say anything about that function.
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The complete question is given below:
Andrew and his brothers want to put their money together to buy a new game system . Costs $168.69 , how much money will each boy need to contribute so they put in equal amount.
The amount each boy will have to contribute is $56.23
How to calculate the amount that each boy will contribute?
Andrew and his two brother want to contribute money together to buy a new game system
The cost of the game system is $168.69
Each boy want to contribute the same amount
Therefore the amount that each boy will contribute can be calculated as follows
There are 3 boys in total, Andrew and his two brothers
= 168.69/3
= 56.23
Hence each boy will contribute an amount of $56.23
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Complete the square to write the equation of the sphere in standard form. x2 + y2 +Z2 + 9x-2y + 12z + 20 = 0 2 V149 Find the center and radius. (x,y,z)=(1-2 center V149 radius
Center of the sphere would be (-4.5, 1, -6) and radius be 6.086.
Write the equation of the sphere?
A sphere's general equation is (x - a)2 + (y - b)2 + (z - c) 2 = r2, where (a, b, c) represents the sphere's center, r represents the radius, and x, y, and z are the coordinates of points on the sphere's surface.
To complete the square, we need to add and subtract the square of half of the x coefficient and y coefficient, and the square of half of the z coefficient.
The equation will be : [tex](x + 9/2)^2 + (y - 1)^2 + (z + 6)^2 = (9/2)^2 + 1^2 + 6^2[/tex]
So the standard form of the equation of the sphere is,
[tex](x + 4.5)^2 + (y - 1)^2 + (z + 6)^2 = 37[/tex]
The center of the sphere is (x, y, z) = (-4.5, 1, -6) and the radius is √37 ≈ 6.086
Therefore, Center of the sphere would be (-4.5, 1, -6) and radius be 6.086.
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John had a yard full of dogs and chickens. He counts 25 heads and 78 legs. How many dogs does John have?
In the problem, it is solved that the number of dogs John has is 14
How to solve for the number of dogs John hasThe problem is solved by expression the word problem into a simultaneous equation as follows
let d be the number of dogs and c d number of chickens
d + c = 25 (since all must have a head)
4d + 2c = 78 (2 legs for chicken and 4 legs for dog)
using calculator to solve gives
d = 14 and c = 11
hence the number of dogs is 14
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Assume A,B,C are invertible matrices of appropriate dimension. Solve for X:
AX + B = CX
The solution of the given equation is X = B(C - A)⁻¹.
To solve for X in the equation AX + B = CX, we can rearrange the equation as follows:
AX - CX = -B
Factor out X on the left side:
(X(A - C)) = -B
To isolate X, we can multiply both sides of the equation by the inverse of (A - C), assuming it exists:
X = -(A - C)⁻¹ × B
X = (C - A)⁻¹ × B
Therefore, the solution for X is:
X = B(C - A)⁻¹
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The sum of four consecutive integers; n = first integer of the four
The sum of four consecutive integers is,
⇒ 4n + 6
What is Addition?The process of combining two or more numbers is called the Addition. The 4 main properties of addition are commutative, associative, distributive, and additive identity.
We have to given that;
⇒ n = first integer of the four.
Now, Let four consecutive integers with n is first integer of the four are,
⇒ n, n + 1, n + 2, n + 3
Hence, The sum of four consecutive integers is,
⇒ S = n + (n + 1) + (n + 2) + (n + 3)
= 4n + 6
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11) The dimensions, in feet, of a rectangle are represented by expressions, as shown in the
diagram below. Explain your work.
(2.5m) ft.
(4n+ 5) ft.
The area of the rectangle is and using the length and breadth
What is the area of the rectangle?In a two-dimensional plane, a rectangle area is a space it encloses. Known for having 4 sides and 4 vertices, a rectangleis a closed shape. The sum of the spaces occupied by a rectangle three sides is known as its area. The product of a rectangle breadth and length is the usual formula for calculating its area.The area that is included within the perimeter of a flat object or figure is generally considered to be the definition of the word "area." The square units used for measuring are square metres as the standard unit.area of rectangle = length × breadth
area of rectangle = 2.5m × (4n +5)
area of rectangle = 10mn + 12.5m
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Question:
The dimensions, in feet, of a rectangle, are represented by expressions, as shown in the diagram below. The area of the rectangle is 10mn + 12.5m. Explain your work.
Sudoku is a puzzle consisting of squares arranged in 9 rows and 9 columns. The 81 squares are further divided into nine 3 ✕ 3 square boxes. The object is to fill in the squares with numerals 1 through 9 so that each column, row, and box contains all nine numbers. However, there is a requirement that each number appear only once in any row, column, or box. Each puzzle already has numbers in some of the squares.
Alisha wants to do a statistical study to determine how long it takes people to complete a Sudoku puzzle. Her plan is as follows.
Download 10 different puzzles from the Internet.
Find 10 friends willing to participate.
Ask each friend to complete one of the puzzles and time him- or herself.
Gather the completion times from each friend.
Describe some of the problems with Alisha's plan for the study. (Note: Puzzles differ in difficulty, ranging from beginner to very difficult. SELECT ALL THAT APPLY.)
A. There are actually only 9 different Sudoku puzzles.
B. Some Sudoku puzzles don't have solutions.
C. It is not clear that all puzzles have the same difficulty.
D. Self-timing may lead to some inaccurate measurements.
E. Friends having different levels of experience.
The problems with Alisha's statistical study are that the difficulty levels of the sudoku puzzles might be different, self timing can lead to inaccurate measures of time and also the experience and skill level of her friends will be different.
In her statistical study, Alisha asked 10 of her friends to try 10 different sudoku puzzles from the internet and time themselves while doing it. In this study there are a few errors. The first one being that the difficulty level of the puzzles that her friends will be solving are going to be different.
Another issue is that since her friends are self-timing, there can be inaccurate measures of time and lastly, the experience and skill of her friends for sudoku is going to create a bias in her study.
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Find the average rate of change of each function on the interval specified for the real numbers b or h in the simplest form.
1) the rate change of function f(x) in given interval =4(b+1)
2) the rate change of function g(x) in given interval =1
3) the rate change of function j(x) in given interval =[tex]3h^2+9h+9[/tex]
To find out the average rate change of function on given interval is -
Use the graph of f to determine f (a) and f (b) for the provided range [a,b]. Remember that the graph of f's point with the y-coordinate of f (a) and the x-coordinate of an is called f. Similar to that, f ( b ) is the y-coordinate of the point on the f graph whose x-coordinate is b.
Step 2: Calculate the average rate of change of f over the range [a, b] using the average rate of change formula.
[tex]\frac{f(b)-f(a)}{b-a}[/tex]
1)
[tex]f(x)=4x^2-7 \ on\ [1,b]\\\\f(1)=4-7=-3\\\\f(b)=4b^2-7\\\\the \ rate \ change =\frac{4b^2-7+3}{b-1}\\\\=4(b+1)[/tex]
2)
[tex]p(x)=3x+4 \ on\ [2,2+h]\\\\p(2)=10\\\\p(2+h)=10+h\\\\The\ average\ rate \ change= \frac{10+h-10}{2+h-2}\\\\=1[/tex]
3)
[tex]j(x)=3x^3 \ on \ [1,1+h]\\\\j(1)=3\\\\j(1+h)= 3(1+h)^3\\\\\\so, the\ rate\ change = 3h^2+9h+9[/tex]
The complete Question is: -
For the following exercises, find the average rate of change of each function on the interval specified for real numbers b or h in simplest form.
1. f(x)= 4x^2 - 7 on [1, b]
2. p(x)= 3x+4 on [2,2+h]
3. j(x)= 3x3 on [1, 1+h]
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Quadrilateral DEFG is similar to quadrilateral HIJK. Find the measure of side JK.
Round your answer to the nearest tenth if necessary.
Quadrilateral DEFG is similar to quadrilateral HIJK The measure of side JK is 8.
Since quadrilateral DEFG is similar to quadrilateral HIJK, we know that the ratio of corresponding side lengths is equal. This means that if we know the length of one side of one of the quadrilaterals, we can use that ratio to find the length of the corresponding side in the other quadrilateral.
For example, if we know that side DE is 8 units long, we can use the ratio of side JK to side DE (x/8) to find the length of side JK. We can also use the same ratio of side IJ to side DF which is x/4.
Let's assume that the length of side DE is 8 units. We can use the ratio of side JK to side DE (x/8) to find the length of side JK.
[tex]x/8 = x/4[/tex]
Cross-multiplying and simplifying the equation we get
[tex]4x = 8x[/tex]
[tex]x = 8[/tex]
Therefore, the measure of side JK is 8 units.
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Example 1. 15+6x3 This problem is modeled after example 1. Find the requested pattern. (Enter the first three numbers of the pattern as a comma-separated list.) three pattern
Therefore, the 3 patterns are
3,6,9,3,6,9
Explanation:
Given that,
we will carry out the multiplications of succession counting number by 3 and if there is more than a single digit answer .we add the digits.
we are looking for is a pattern for there single digit numerals.
carryout the plan
1x3=3→3
2x3=6→6
3x3=9→9
4x3=12→1+2=3
5x3=15→1+5=6
and so on
continue with some additional at terms.
6x3=18→1+8=9
7x3=21→2+1=3
∵ Therefore the 3 patterns are
3,6,9,3,6,9
A pattern is defined in mathematics as a sequence of repeating objects, shapes, or numbers. A pattern can be associated with any type of event or object. A pattern has a rule that tells us which objects belong to the pattern and which objects do not belong to the pattern.
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