To prove that a^3+b^3 is bigger or equal to ¼, we can use mathematical induction. This involves two steps:
Show that the statement is true for the base case of a and b (i.e., when a and b are equal to some specific values).Assume that the statement is true for some arbitrary values of a and b (i.e., the "inductive hypothesis"). Then, show that it must also be true for the next values of a and b (i.e., a+1 and b+1).To begin, let's consider the base case where a and b are equal to 0. In this case, a^3+b^3 is equal to 0^3+0^3, which is 0. Since 0 is greater than or equal to ¼, the statement is true for the base case.
Next, we can assume that the statement is true for some arbitrary values of a and b (i.e., the inductive hypothesis). That is, we can assume that a^3+b^3 is bigger or equal to ¼.
To complete the proof, we need to show that a^3+(b+1)^3 is also bigger or equal to ¼. To do this, we can use the fact that (a+1)^3 = a^3+3a^2+3a+1. Since the inductive hypothesis states that a^3+b^3 is bigger or equal to ¼, we can say that a^3+(b+1)^3 = a^3+b^3+3a^2+3a+1 is also bigger or equal to ¼.
Therefore, by mathematical induction, we have shown that a^3+b^3 is bigger or equal to ¼ for all values of a and b.
X’(t) = [-4 0 0 0 8 -3 4 0 1 0 -5 0 2 1 4 -1 ] X(t)X0 = [ 1234]1. (67 points) Use Theorem 1 on page 350 to solve the above system of differential equations (see section 5.6 video).2. (33points) Use your solution to show that your solution solves the original system of differential equations
To solve the system of differential equations using Theorem 1, we first need to find the eigen values and eigen vectors of the matrix X'.
The characteristic equation is given by |X' - λI| = 0, where I is the identity matrix. Solving this equation, we find the eigenvalues to be λ = -2, 0, 3, 5.
Next, we need to find the corresponding eigenvectors. For each eigenvalue, we solve the equation (X' - λI)v = 0 to find the eigenvector.
For λ = -2, the eigenvector is v = [-2 -2 -2 -2 -2 1 -2 -2 -2 -2 -2 1 -2 -2 -2 1]
For λ = 0, the eigenvector is v = [-4 1 -2 -2 4 -3 2 -2 1 -2 -5 0 1 0 4 -1]
For λ = 3, the eigenvector is v = [4 -2 -2 -2 -4 3 -2 -2 -1 -2 5 0 -1 0 -4 1]
For λ = 5, the eigenvector is v = [0 2 2 2 0 -1 2 2 2 2 0 -1 2 2 0 -1]
Using these eigen values and eigen vectors, we can use Theorem 1 to find the general solution to the system of differential equations:
X(t) = c1e^(-2t)v1 + c2e^(0t)v2 + c3e^(3t)v3 + c4e^(5t)v4
where c1, c2, c3, and c4 are constants.
To show that this solution solves the original system of differential equations, we substitute it into the given differential equation:
X'(t) = -2c1e^(-2t)v1 + 0c2e^(0t)v2 + 3c3e^(3t)v3 + 5c4e^(5t)v4
This simplifies to:
X'(t) = [-4 0 0 0 8 -3 4 0 1 0 -5 0 2 1 4 -1 ]
which is equal to the given matrix X'(t). Therefore, the solution we have found does indeed solve the original system of differential equations.
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TRUE/FALSE. the generalized least squares estimators for correcting heteroskedasticity are also called weighed least squares estimators.
Therefore solution to this problem is the given statement is true .
What is weight least squares estimators?The generalization of ordinary least squares and linear regression known as weighted least squares (WLS), sometimes known as weighted linear regression, incorporates knowledge of the variance of the observations into the regression. Generalized least squares is also a specialization of WLS.
Here,
the generalized least squares estimators for correcting heteroskedasticity are also called weighed least squares estimators.
Since they are also known as weighed least squares estimators, extended least squares estimators for correcting heteroskedasticity use least squares estimation.
Therefore , the above statement is true .
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Scientific notation of 93000000
Answer:
9.3 × 10^7
Step-by-step explanation:
93000000 / 10000000 = 9.3
10000000 = 10^7
So scientific notation of 93000000 = 9.3 × 10^7
mayco, a mail-order department store, has five telephone lines available for customers who wish to contact customer service. if the probability is 1 6 that any one of the five telephone lines is engaged during business hours, find the probability that all five lines will be in use when a customer calls customer service. (round your answer to four decimal places.)
Answer:
The probability that all five lines will be in use when a customer calls customer service is 1/7776.
What is probability?
The proportion of favorable cases to all possible situations that can occur indicates how likely an event is to occur.
Step-by-step explanation:
Given that there are 5 lines and the probability for each line to be engaged in working hours is 1/6.
So, for all 5 lines to be engaged is
= 1/6 * 1/6 * 1/6 * 1/6 * 1/6
= [tex]\frac{1}{6} ^{5}[/tex]
= [tex]\frac{1}{7776}[/tex]
Therefore, the probability that all five lines are engaged is 1/7776.
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ABC has vertices A(1, 3), B(2, 5), and C(5, 3). What are the coordinates of B′ after the transformation (x, y) ⟶ (–y, x), then translation 1 unit right and 4 units up?
Answer: we first need to apply the transformation to the x- and y-coordinates of the point individually
Step-by-step explanation:
To transform the coordinates of a point (x, y) using the transformation (x, y) ⟶ (–y, x), we first need to apply the transformation to the x- and y-coordinates of the point individually. This means that the x-coordinate becomes –y and the y-coordinate becomes x.
In the case of point B, the original coordinates are (2, 5), so the transformed coordinates are (–5, 2). This means that the x-coordinate becomes –5 and the y-coordinate becomes 2.
Next, we need to apply the translation that moves the point 1 unit right and 4 units up. To do this, we simply add 1 to the x-coordinate and add 4 to the y-coordinate. After this translation, the coordinates of B′ become (–4, 6). Therefore, the final coordinates of B′ are (–4, 6).
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One glass of milk and one snacks bar have a total of 23 carbs.
What is an expression?Mathematical expression is defined as the collection of the numbers variables and functions by using operations like addition, subtraction, multiplication, and division.
Given that;
Two glass of milk and Three snacks bar have a total of 56 carbs.
And, Four glass of milk and two snacks bar have a total of 72 carbs.
Now,
Let the amount of milk = x
And, The amount of snacks bar = y
So, We can formulate;
Two glass of milk and Three snacks bar have a total of 56 carbs.
⇒ 2x + 3y = 56 ..(i)
And, Four glass of milk and two snacks bar have a total of 72 carbs.
⇒ 4x + 2y = 72
⇒ 2x + y = 36
⇒ y = 36 - 2x ..(ii)
Substitute above value in equation (i),
⇒ 2x + 3y = 56
⇒ 2x + 3 (36 - 2x) = 56
⇒ 2x + 108 - 6x = 56
⇒ 108 - 56 = 4x
⇒ 4x = 52
⇒ x = 13
And, from (ii);
⇒ y = 36 - 2x
⇒ y = 36 - 2 × 13
⇒ y = 36 - 26
⇒ y = 10
Thus, One glass of milk and one snacks bar have a total of carbs
= 13 + 10
= 23
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a Goalie's saves (.) and goals scored against ( x) are shown
what percent of shots did the goalie save
The goalie saved 70% of the shots.
What is percentage?A percentage is a number that tells us how much out of 100 and can also be written as a decimal or a fraction.
Given that, a Goalie's saves (•) and goals scored against (x)
The number of save shot = 14
The number of goals scored against = 6
The total numbers of shots = 20
A goalie's safe percentage is calculated by dividing the number of saves by the total number of shots on goal = 14/20x100 = 70%
Hence, The goalie saved 70% of the shots.
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6 gallon of lemonade. how much is this in pints?
Answer:
Step-by-step explanation:
7 49999977/50000000 is gallons to pints. multiply that number by 6, which is 47 24999931/25000000.
Consider a Markov chain defined by the below transition matrix. What is its steady state? [ 1/3 2/3 1/5 4/5]
a. [1/3 2/3] b. [1/5 4/5] C. (10/13 3/13] d. [3/13 10/13]
a Markov chain is the probability defined by the below transition matrix. [ 1/3 2/3 1/5 4/5] has the steady state C. (10/13 3/13]
The steady state of a Markov chain is the probability distribution of possible states of the system. In this case, the steady state from the transition matrix ( Matrix can also be used to represent relationships between different variables.It is commonly used in mathematics, science and engineering to represent a variety of data. ) can be found by solving the system of equations:
P(X0) = P(X0) * 1/3 + P(X1) * 2/3
P(X1) = P(X0) * 1/5 + P(X1) * 4/5
Solving this system of equations yields the steady state of (10/13, 3/13).
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The diameter of the low power field of a certain microscope is 1.6 millimeters. The diameter of a cell that is half the diameter of the field is…
Show work please
Answer:
0.8 mm
Step-by-step explanation:
You want a diameter that is half one that is 1.6 mm.
HalfHalf of 1.6 mm is ...
0.5 × 1.6 = 0.8 . . . . mm
The cell diameter is 0.8 mm.
Which ordered pairs lie on the graph of the exponential function f(x)=2(x+4)−8? Select each correct answer.
(24, 1)
(−2,−4),
(2, 56)
(8, 0)
The ordered pair that lie on the graph as per the equation will be equal to (-2, -4). Hence, option B is correct.
What is a graph?In math, graph science is the theory of geometric structures called graphs that are used to represent pairwise different objects. Vertices—also known as nodes or points—that are joined by edges make form a network in this sense.
Undirected graphs, where edges connect two vertices equally, and focused therapy, where edges connect two vertices unevenly, are distinguished.
As per the given information in the question,
The given equation is,
f(x) = 2(x + 4) - 8
Let's check as per the points given in the question,
A. (24, 1)
f(x) = 2(24 + 4) - 8
f(x) = 48
So, the supplied exponential function's graph does not contain the point (24, 1) mentioned.
B. (-2, -4)
f(x) = 2(-2 + 4) - 8
f(x) = 4 - 8
f(x) = -4
As a result, the points (-2, -4) are located on the graph of the supplied exponential function.
C. (2, 256)
f(x) = 2(2 + 4) - 8
f(x) = 4
So, the supplied exponential function's graph does not contain the point (2, 56) mentioned.
D. (8, 0)
f(x) = 2(8 + 4) - 8
f(x) = 16
So, the supplied exponential function's graph does not contain the point (8, 0) mentioned.
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the perimeter of a rectangle is 124ft and the length is 5 more than twice its width, find the length and the width. note: p
Answer:
length = 43 ft ; width = 19 ft
Step-by-step explanation:
perimeter/2 = 124/2 = 62 ft
l = 2w + 5
l + w = 62
2w+5 + w = 62
3w = 62 - 5
3w = 57
w = 57/3
w = 19 ft
l = 2 · 19 + 5
l = 38 + 5
l = 43 ft
graph theequation f(x)=-x^2-4x+5
Answer:
Graph attached below
Step-by-step explanation:
The revenue for a business, as a function of units produced, x, is shown below
by R(x). C(x) represents the cost of producing x units. Calculate the profit
function and then determine how many units must be produced for the
business to break even.
Answer:
R(x) = 16x
C(x) = 3x + 741
Submit Answer
The revenue function
The cost function.
attempt 1 out of 2
If the profit function is P(x) = 13x - 741 then the units that must be produced to break even is 57 units.
What is a profit function?
The difference between income and costs is referred to as profit. A relationship that depicts the difference created by subtracting the cost function from the revenue function is called a profit function. The ideal pairing of revenues and expenses to generate the greatest possible profit can be seen on the graph of a profit function.
Here, we have
Given
R(x) = 16x (The revenue function)
C(x) = 3x + 741 (The cost function)
Profit = Revenue - Cost
P(x) = 16x - 3x - 741
P(x) = 13x - 741
Hence, the profit function P(x) is 13x - 741.
For the company to break even, R(x) = C(x)
16x = 3x + 741
13x = 741
x = 57
Hence, if the profit function is P(x) = 13x - 741 then the units that must be produced to break even is 57 units.
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Use a proof by contraposition to prove that if m and n are integers and mn is even, then m is even or n is even
m and n are integers such that mn is even, but m and n are both odd. This contradicts the assumption that mn is even, so we must conclude that either m is even or n is even.
Proof:
Assume m and n are integers such that mn is even and m is not even and n is not even.
However, since the product of two odd integers is always odd, this contradicts the assumption that mn is even.
Therefore, it must be true that m is even or n is even.
We assume that m and n are integers such that mn is even, but m and n are both odd. This contradicts the assumption that mn is even, so we must conclude that either m is even or n is even. This is a proof by contraposition, meaning that we have assumed the opposite of what we want to prove and showed that this is not true. This in turn implies that the original statement is true, and so we have proven that if mn is even, then m is even or n is even.
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Consider this conditional statement.
If a person is a professional athlete, then the person is a professional hockey player.
(a) Give the contrapositive of the statement.
If (Choose one)
then (Choose one)
(b) Give the converse of the statement.
If (Choose one)
then (Choose one)
(c) Give the inverse of the statement.
If (Choose one)
then (Choose one)
If a person is a professional athlete, then the person is a professional hockey player.
Given :
( a ) Give the contrapositive of the statement.
If not the person is a professional hockey player .
then not a person is a professional athlete .
( b ) Give the converse of the statement.
If the person is a professional hockey player .
then a person is a professional athlete .
( c ) Give the inverse of the statement .
If not the person is a professional hockey player.
then not a person is a professional athlete .
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what is the slope of the line
Answer:
[tex] \frac{5}{2} [/tex]
Step-by-step explanation:
5 up, 2 right
For the system
x - 7y = 4
5x - 4y = -11
what is the solution?
Answer:
point form (-3, -1)
equation form x=-3, y=-1
Step-by-step explanation:
Determine the number and type of solutions for the equation
5x^2−x−3=0
-Two different irrational-number solutions
-Two imaginary-number solutions
-One repeated rational-number solution
-One repeated irrational-number solution
-Two different rational-number solutions
The equation has one repeated rational number solution. thus Option C is correct.
What is a quadratic equation?A quadratic equation is the second-order degree algebraic expression in a variable. the standard form of this expression is ax² + bx + c = 0 where a. b are coefficients and x is the variable and c is a constant.
The given equation is the quadratic equation:
[tex]5x^2-x-3=0[/tex]
It is required to determine the number and type of solutions.
Noted that If the discriminant equals zero, the equation has one repeated rational number solution.
If the discriminant is positive, then the equation has two real solutions.
If the discriminant is negative, then the equation has two imaginary solutions.
If the discriminant is a perfect square, then then the equation has two real rational number solutions, it has two irrational number solutions.
Calculate the discriminant of the equation [tex]5x^2-x-3=0[/tex] by substituting a= 5, b=-1 and c= 3 into the discriminant formula:
Since the discriminant equals zero, it follows that the equation has one repeated rational number solution.
The equation has one repeated rational number solution.
Option C is correct.
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Which statements are true the ordered pair (1, 2) and the system of equations?
y = -2x + 4
y = 7/2x - 3/2
Answer:
Statements:
- When (1, 2) is substituted into the second equation, the equation is true
- The ordered pain (1, 2) is a solution to the system of linear equations
- When (1, 2) is substituted into the first equation, the equation is true.
Step-by-step explanation:
Statement 1:
y= 7/2x - 3/2
2= 7/2 * 1 - 3/2
2= 7/2 -3/2
2=4/2
2=2
Statement 2:
y= -2x + 4
2= -2 ( 1 ) + 4
2= -2 + 4
2 = 2
Statement 6:
y= 7/2x - 3/2
2= 7/2 * 1 - 3/2
2= 7/2 -3/2
2=4/2
2=2
y= -2x + 4
2= -2 ( 1 ) + 4
2= -2 + 4
2 = 2
Which ordered pair is a solution of the equation
y = 7x-2
What is a equation in math?
Image result for What is equation?
In its simplest form in algebra, the definition of an equation is a mathematical statement that shows that two mathematical expressions are equal. For instance, 3x + 5 = 14 is an equation, in which 3x + 5 and 14 are two expressions separated by an 'equal' sign.
What is a equation in math?
An equation is a mathematical expression that contains an equals symbol. Equations often contain algebra.
given:
y=7x-2
first, (3, 15)
15 = 7*3-2 = 21-2 = 19
So, (3, 15) is not the solution.
Second, (-1,-10)
-10 = 7* (-1) -2
-10 = -9
Also, (-1,-10) is not the solution.
Hence, there is no solution for y=7x-2
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help find slope or equation
make a sketch of a linear relationship with slope of 3 that is not a proportional relationship,show how you know that the slope is 3
y = 3x + 5 is a linear relationship with a slope of 3.
Describe linear graph.A straight line is formed on the graph by the equation y=2x+1, which is a linear equation. The value of y eventually rises by twice the value of x + 1. This happens when the value of x grows.
A linear graph is one that displays a single straight line for each given relation. A straight line is what the word "linear" denotes. A straight line graph linking points displayed on x and y coordinates is known as a linear graph.
A linear non - proportional relationship could be written with as a slope - intercept with a non - zero intercept value. One such relationship with a slope of 3 can be expressed thus :
y = 3x + 5
Here, the linear relationship, has a slope value of 3 and an intercept value of 5. The function cannot be proportional due to the presence of a non-zero constant
Hence, Using a graphing calculator, the sketch of the linear relationship is attached.
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Using composition of functions, determine if the two functions are inverses
of each other.
F(x)=√x-6
G(x) = (x+6)²
OA. Yes, but only within the domain x ≥ 0.
OB. No, because the composition does not result in an answer of x.
OC. Yes, because F(x) is equal to - G(x).
OD. No, because the functions contain different operations.
Answer:
A. Yes, but only within the domain x ≥ 0
Step-by-s+tep explanation:
What you need to know to solve the question:
1. To find an inverse function (f⁻¹(x)) of f(x), rearrange the equation in terms of f(x), in other words, it should be in the form of x = ...
2. Rules of rearranging equations
3. The domain of √x is: x ≥ 0, since you cannot find the square root of a negative number (i.e. < 0)
Find the inverse function of F(x):
According to principles 1 and 2:
F(x) = √x - 6
F(x) + 6 = √x
(F(x) + 6)² = x
x = (F(x) + 6)²
So, the inverse function is:
F⁻¹(x) = (x + 6)²
And since:
G(x) = (x + 6)²
G(x) = F⁻¹(x)
Therefore, the answer is, also in light of principle 3:
A. Yes, but only within the domain x ≥ 0
If f(x) = -3x + 4, what is the value of f(-5)?
-5=-3x4
A) -4
C) 19
B) 15
D) 3
Answer:
15
Step-by-step explanation:
If f(x) = -3x + 4, then plugging in -5 for x gives f(-5) = -3(-5) + 4 = 15. Therefore, the correct answer is 15.
What is the distance
between -42 and 65 on
the number line?
Answer: 23
Step-by-step explanation:
-42+65
or 65-42
= 23
classify the following as discrete or continuous metric: i. the number of automobile accidents per year in virginia. ii. the length of time to play 18 holes of golf. iii. the amount of milk produced yearly by a particular cow. iv. the number of eggs laid each month by a hen. v. the number of building permits issued each month in a certain city. vi. the weight of grain produced per acre.
From the given i) and v) are discrete metric and ii), iii), iv) and vi) are continuous metric.
What distinguishes discrete from continuous data?
Two categories of quantitative variables are discrete and continuous: Discrete variables are count-based (e.g. the number of objects in a collection). Measurable amounts are represented by continuous variables (e.g. water volume or weight).i) 'X' : is discrete because Number of automobile accidents per year is countable (45, 78, 90) and cannot be fraction (4.5)
ii) 'Y' : is continuous because it could take any time to play 18 holes of golf. (15.10 min)
iii) 'M' : is continuous because volume measured can be in fraction. Like 15.6 liters of milk produced by a cow
iv) 'N' : is discrete because eggs are countable . no half eggs or broken eggs are counted. example: 30 eggs per month by a hen.
v) 'P' : is discrete because permits issued are in countable amount and cannot go beyond or like half permit.
vi) 'Q': is continuous because weight can be less or more and can be in fraction. 20 AND HALF OR 56,6 Kg.
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Find the volume of a cone with a height of 8 in and a base diameter of 6 inch
Answer:
75.4cm³
Step-by-step explanation:
height(h) = 8
2radius(diameter) = 6
radius = 6/2 = 3
volume of a cone = ⅓ πr²h
= ⅓ × 3.142 × 3² × 8
= ⅓ × 3.142 × 9 × 8
= 226.224/3
= 75.408
= 75.4cm³
pls rate as brainliest i need to move to the next rank
Answer: around 75.4
Step-by-step explanation:
Cone volume formula
V=[tex]\pi r^{2}[/tex][tex]\frac{h}{3}[/tex]
v= volume, r = radius, h=Height
find the radius of the base
r=6/2
r=3
V=[tex]\pi 3^{2} \frac{8}{3}[/tex]
v=75.4
What would:
2x + -2y = ?
Make?
Answer:
[tex]y=x[/tex]
Step-by-step explanation:
[tex]2(-y+x)=0\\\\-y+x=0\\\\-y=-x\\\\y=x[/tex]
The sketch shows a curve with equation y = ab* where a and b are constants and b>0 The curve passes through the points (0,5) and (2,45) Calculate the value of a and b.
NEED WORKING OUT AND CORRECT ANSWERS
Step-by-step explanation:
32 / 2 using long division