Hamiltonian due to interaction of spin magnetic field:
H = -u.B
H = α/√2[tex]\left[\begin{array}{ccc}1&1\\1&-1\\\end{array}\right][/tex]
What is a matrix?
In mathematics, a matrix (multiple matrices) is a rectangular array or table of numbers, symbols, or expressions, arranged in rows and columns, and used to represent mathematical objects or properties of such objects. The
matrix represents a linear mapping without additional information and allows explicit computation in linear algebra. The study of matrices is therefore a large part of linear algebra, and most properties and operations in abstract linear algebra can be expressed in matrices. For example, matrix multiplication represents the construction of linear maps.
Not all matrices are related to linear algebra. This is especially true for graph theory, occurrence matrices, and adjacency matrices. This article focuses on matrices in the context of linear algebra. Unless otherwise stated, all matrices either represent linear maps or can be considered linear maps.
Square matrices (matrices with the same number of rows and columns) play an important role in matrix theory. Square matrices of a given dimension form non-commutative rings, one of the most common examples of non-commutative rings. The determinant of a square matrix is the numerical value associated with the matrix that underlies the study of square matrices. For example, a square matrix has nonzero determinant and is invertible only if the eigenvalues of the square matrix are the roots of the polynomial determinant.
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!!50 Points!! Pls help ASAP!!!
Simplify
[tex](3\sqrt{-5} + 2)(4\sqrt{-12} - 1)[/tex]
The simplification of the expression (3√-5 + 2)(4√-5 -1) gives 5√-5 - 62
How to simplify numbers and expressions?Simplification is a way of determining an answer for a complex calculation that may involve numbers on division, multiplication, square roots, cube roots, plus and minus, etc.
Given: the expression (3√-5 + 2)(4√-5 -1)
(3√-5 + 2)(4√-5 -1)
Clear the parenthesis:
(3√-5 + 2)(4√-5 -1) = 3√-5(4√-5) + 3√-5 (-1) + 2(4√-5) + 2(-1)
= -60 - 3√-5 + 8√-5 -2
= 5√-5 - 62
Therefore, the expression (3√-5 + 2)(4√-5 -1) simplifies to 5√-5 - 62
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Thomas took a 140-mile bus trip to visit his grandparents. His trip is outlined on the graph below.
1) Explain what might have happened in the interval between D and E.
2) State the interval in which the bus traveled the fastest.
3) State how many miles per hour the bus was traveling during this interval.
4) What was the average rate of speed, in miles per hour, for Thomas's entire bus trip?
1) The bus took a 30-minute stop in the interval between D and E
2) The bus traveled the fastest in the interval between C and D
3) The bus was traveling at 60 miles per hour during this interval.
4) The average rate of speed of the bus over the entire journey is 35 miles /hr
What is the speed of an object?Speed is the rate at which an object's position changes, measured in meters per second. For example, if an object starts at the origin, and then moves three meters in three seconds, its speed is one meter per second. The equation for speed is simple: distance divided by time
Given here for the first question we have there was no change in the distance in the interval between D and E thus bus must have stopped for 30 minutes.
2) The interval in which the bus travelled the fastest is between C and D
3) The Speed of the bus in this interval is S = 90/1.5
= 60 miles/hr
4) The average speed of the bus over the entire trip is Sₐ = 140/4
= 35 miles/hr
Hence the above answers are correct ones.
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What i the value of x in the equation 0. 2(x 1) 0. 5x = –0. 3(x – 4)?
Negative even-halve
–1
1
Seven-halve
The value of x for the given equation is 1.
What is solving an equation?
A General Rule for Equation Solving
Remove parentheses from each side of the equation and combine similar phrases to make it simpler.
To separate the variable term on one side of the equation, use addition or subtraction.
To find the variable, use division or multiplication.
Consider the given expression
0.2(x + 1) + 0.5x = -0.3(x - 4)
Simplifying,
0.2x + 0.2 + 0.5x = -0.3x + 1.2
0.7x + 0.2 = -0.3x + 1.2
Add 0.3x on both sides,
0.7x + 0.2 + 0.3x = -0.3x + 1.2 + 0.3x
x + 0.2 = 1.2
Subtract 0.2 from both sides,
x + 0.2 - 0.2 = 1.2 - 0.2
x = 1
Hence, the value of x for the given expression is, 1.
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ANSWER FOR BRAINLY!!
(x+1)(2x-4)=0
rewrite the equation as a compound statement and find 2 solutions to the equation.
Answer: [tex]x=-1, 2[/tex]
Step-by-step explanation:
If [tex]ab=0[/tex], then [tex]a=0[/tex] and/or [tex]b=0[/tex].
So, [tex]x+1=0[/tex] or [tex]2x-4=0[/tex].
[tex]x+1=0 \implies x=-1\\\\2x-4=0 \implies x=2\\\\\therefore x \in \{-1, 2\}[/tex]
The surface area of the figure shown is 192 cm².
What is the value of x?
8 cm
6 cm
10 cm
X cm
The value of x = 6cm.
What is surface area?
The area is the area occupied by a two-dimensional flat surface. It has a square unit of measurement. The surface area of a three-dimensional object refers to the space occupied by its outer surface.
From the given figure,
Surface area(A) = [tex]192cm^{2}[/tex]
Height = 8cm
a base side = 6 cm
b base side = 10 cm
c base side = x cm
Therefore, TSA= 2B +Px = [tex]2 \times\frac{1}{2} \times8\times6 +(6+10+8)x[/tex]
=48+24x
Thus, 60+24x=192
[tex]x=\frac{192-48}{24}[/tex]
x= 6cm
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Calculate how far each of the following travels. A helicopter flies at a speed of 72miles per hour for 10 minutes
Answer:
12 miles
Step-by-step explanation:
72 x [tex]\frac{10}{60}[/tex] = 12
The sum of three numbers is 23.
Three times the smallest is 4 less than the largest, while the sum of the largest
and smallest is 16. Use a linear system in three variables to find the three numbers.
The three numbers are __ , __
and __
Answer:
The three numbers are 3, 7 and 13
Step-by-step explanation:
Considering the 3 numbers: x, y and z
x + z = 16
z=16-x
3x=z-4
3x=16-x-4
4x=12
x=3
3+z=16
z=16-3
z=13
3+13+y=23
y=23-13-3
y=7
Hope this helps :)
He creates paths through the garden along line segment a b, line segment x y, and line segment z y. Which correctly compares the lengths of the paths?the path along line segment x y is longer than the path along line segment z y, and the path along line segment z y is longer than the path along line segment a b. The path along line segment a b is longer than the path along line segment z y, and the path along line segment z y is longer than the path along line segment x y. The path along line segment z y is longer than the path along line segment x y, and the path along line segment x y is longer than the path along line segment a b. The path along line segment z y is longer than the path along line segment a b, and the path along line segment a b is longer than the path along line segment x y.
The journey along Line segment X Y is longer than the path along Line segment Z Y, and the path along Line segment Z Y is longer than the path along Line segment A B, hence Option-A is the correct answer.
Given that,
He makes paths that follow Line segments A and B, X and Y, and Z and Y through the garden.
We have to find which accurately compares the pathways' lengths.
We know that,
The length of the third side in a triangle is proportional to the angle Ф if the lengths of the two sides that make up the angle Ф are fixed.
XY > YZ because XY's opposing angle (79°) is greater than YZ's (65°).
As before, YZ > AB because the opposing angle of YZ (65°) is greater than that of AB (36°).
Since both sides of the angle are the same length (8 feet),
As a result, assertion A. is accurate. (Answer)
Therefore, The journey along Line segment X Y is longer than the path along Line segment Z Y, and the path along Line segment Z Y is longer than the path along Line segment A B, hence Option-A is the correct answer.
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Answer:
It's A
Step-by-step explanation:
I got 100 on the quiz
There are 6 circles and 10 squares. What is the simplest ratio of circles to total shapes?.
The simplest expression of the ratio of circles to squares is 5:3.
Describe ratio.
A relationship between two quantities is called a ratio, and it is expressed as one quantity divided by the other.
There are 6 circles and 10 squares.
These steps can be used to determine the ratio of two quantities:
Finding the quantities of both scenarios for which we are calculating the ratio is the first step. There are 10 squares and 6 circles in this instance.
It should be expressed as a fraction, a/b. Thus, we represent it as 10/6.
If you can, make the fraction even simpler. The ultimate ratio will be shown by the simple fraction.
In this case, 10/6 can be reduced to 5/3.
As a result, the simplest expression of the ratio of circles to squares is 5:3.
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how many baskets did Elijad make during his seoncd game
Answer:
[tex]\huge\boxed{\sf 3 \ baskets}[/tex]
Step-by-step explanation:
Let the baskets missed in game 2 be x.
Says that, the shooting of both the games were accurate.
So,
Using proportionality:[tex]\displaystyle \frac{Baskets \ made \ in \ game \ 1}{Baskets \ missed \ in \ game \ 1 } =\frac{Baskets \ made \ in \ game \ 2}{Baskets \ missed \ in \ game 2} \\\\\frac{15}{5} =\frac{x}{1} \\\\Cross \ Multiply\\\\15 \times 1 = 5 \times x\\\\15 = 5x\\\\Divide \ 5 \ to \ both \ sides\\\\3 = x\\\\x = 3\\\\\rule[225]{225}{2}[/tex]
(pythagorean theorem mc) a right triangle has a leg length of square root of 11 and a hypotenuse length of 9. determine the length of the other leg of the right triangle. square root of 18 square root of 81 square root of 70 22
The length of the other leg of the given right triangle is [tex]\sqrt{70}[/tex].
RIGHT TRIANGLE
A triangle is classified as a right triangle when it presents one of your angles equal to 90º. The greatest side of a right triangle is called the hypotenuse. And, the other two sides are called legs.
The math tools applied for finding angles or sides in a right triangle are the trigonometric ratios or the Pythagorean Theorem.
The Pythagorean Theorem says (hypotenuse)²= (leg1)²+(leg2)². And the main trigonometric ratios are: sin (x) , cos (x) and tan (x) .
The question gives:
leg1=[tex]\sqrt{11}[/tex]hypotenuse = 9Therefore, from the Pythagorean Theorem, you will find the value of the other leg. See below.
[tex]9^2={\sqrt{11}^2 +{leg_2}^2[/tex]
[tex]81=11 +{leg_2}^2[/tex]
[tex]{leg_2}^2=81-11\\ \\ {leg_2}^2=70\\ \\ leg_2=\sqrt{70}[/tex]
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Consider all right circular cylinders for which the sum of the height and circumference is 60 cm. What is the radius of the one with maximum volume?.
By using the concept of maxima, it can be calculated that
Volume is maximum if the radius of the cone is 40 cm
What is maxima of a function?
Maxima of a function gives the maximum value of the function on a certain interval or on the whole domain.
Let the height of the cylinder be h and radius be r
Sum of height and circumference = h + 2πr
By the problem,
h + 2πr = 60
h = 60 -2πr
Volume(V) = π[tex]r^2[/tex] h
= π[tex]r^2[/tex](60 - r)
= 60π[tex]r^2[/tex] - π[tex]r^3[/tex]
[tex]\frac{dV}{dr} = 120\pi r - 3 \pi r^2[/tex]
For maximum volume
[tex]\frac{dV}{dr} = 0[/tex]
[tex]120\pi r - 3 \pi r^2 = 0\\[/tex]
[tex]3 r^2 = 120 r\\3r = 120 [r \neq 0]\\r = \frac{120}{3}\\r = 40 \ cm[/tex]
[tex]\frac{d^2V}{dr^2} = 120 \pi - 6 \pi r\\[/tex]
Putting r = 40
[tex]\frac{d^2V}{dr^2} = 120 \pi - 6 \pi \times 40\\\frac{d^2V}{dr^2} =120\pi - 240 \pi\\\frac{d^2V}{dr^2} = -120\pi < 0[/tex]
Hence Volume is maximum
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a riverboat travels at an average of 14km per hour in still water. the riverboat travels 110km east up the Ohio river and 110km west down the same river in a total of 17.5 hours. To the nearest tenth of a kilometer per hour, what was the speed of the current of the river?
The speed of the current of the river will be 4.47 km per hour.
What is speed?Speed is defined as the length traveled by a particle or entity in an hour. It is a scale parameter. It is the ratio of length to duration.
We know that the speed formula
Speed = Distance/Time
A riverboat goes at a normal of 14km each hour in still water. the riverboat voyages 110km east up the Ohio waterway and 110km west down a similar stream in a sum of 17.5 hours.
Let 'y' be the speed of the current of the river. Then the equation is given as,
T = T₁ + T₂
17.5 = [110 / (14 + y)] + [110 / (14 - y)]
17.5 / 110 = (14 - y + 14 + y) / (196 - y²)
196 - y² = 176
y² = 20
y = 4.47 km per hour
Then the speed of the current of the river will be 4.47 km per hour.
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A partial payment is made on the date indicated. Use the United States rule to determine the balance due on the note at the date of maturity. (The Effective Date is the date the note was written.) Assume the year is not a leap year.Principal Rate Effective Date Partial Payment Maturity Date Amount Date $2000 4% April 1 $1000 May 1 June 1 The balance due on the note at the date of maturity is $nothing.(Round to the nearest cent as needed.)
The balance due on the note at the date of maturity is $1010.13.
What is meant by partial payment?The offering of a payment by cheque for less than the whole amount sought by the creditor is known as partial payment. It is standard practice to grant debt forgiveness in exchange for the tender of a "payment-in-full" check.
Given permission to apply the US rule
In the United States, a year is deemed to have 360 days for the purposes of calculating interest.
Given that the principal is $ 2000,
Interest rate = 4% = 0.04
The effective date is April 1.
The whole amount is $1,000.
May 1 = partial payment
Duration = 30 days (May 1 - April 1)
Therefore, interest = PTR = 2000 * 0.04 * 30/360 = $6.67
So value = $ 2000 + $6.67 = $6.67
Partial payment received = $1,000
As a result, value = $ 1006.67 = $2006.67 - $ 1000
Time T= 31
Interest = PTR * 31/360 * 0.04 = 3.46
So the total value is $1006.67 + $3.47 = $1010.13.
The note's outstanding balance at maturity is $1010.13.
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Question 5 of 5
A linear function contains the following points.
y
What are the slope and y-intercept of this function?
OA. The slope is 5.
The y-intercept is (0, -3).
B. The slope is 5/6
The y-intercept is (0, -3).
OC. The slope is 5/6
The y-intercept is (-3,0).
OD. The slope is. 6/5
The y-intercept is (0, -3).
PLEASE HELP ME QUICK, HURRY!!!!!
The slope and y-intercept of this function is
B. The slope is 5/6, The y-intercept is (0, -3
How to find the slope and the y-interceptLinear function is a function of the form y = mx + c
m = slope
c = intercept
The slope is represented as m, The slope by definition is the ratio change of the output values to the input values
the slope, m calculated using the points (0, -3) and (6, 2)
m = (y₀ - y₁) / (x₀ - x₁)
m = (2 - -3) / (6 - 0)
m = (5) / (6)
m = 5/6
c = y intercept, from the table
c = -3
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Rewrite the equation shown in slope-intercept form. Identify the slope of the equation, the x intercept and the y-intercept. Then graph the equation. Make sure to show all your work in the
spaces provided.
Answer:
Slope-intercept form: y = 3x + 5
Slope: 3
y-intercept: 5
Step-by-step explanation:
What is slope-intercept form?When the equation of a line is written in the slope-intercept form, we can easily identify the slope from the coefficient of x and the y-intercept from the constant.
The slope-intercept form is expressed as y = mx + c, where m is the slope and c is the y-intercept. Note that other textbooks might use y = mx + b instead, but both have the same meaning as both m and b or c will be replaced by a certain value.
Rewriting in slope-intercept formTo rewrite an equation in the slope-intercept form, isolate the y term on the right-hand side of the equation and divide the whole equation by the coefficient of y.
Working
[tex]y - ( - 7) = 3(x - ( - 4))[/tex]
Expand:
[tex]y + 7 = 3(x + 4)[/tex]
[tex]y + 7 = 3x + 12[/tex]
Subtract both sides by 7:
[tex]y = 3x + 12 - 7[/tex]
[tex]\bf{y = 3x + 5}[/tex]
Since the coefficient of y is already 1, we have arrived at our final answer.
Identifying slopeWhen the equation is in the slope-intercept form, the slope can be identified from the coefficient of x. This refers to the number that comes before x which is multiplied to x.
Thus, the slope is 3.
Identifying the y-interceptThe y-intercept is the value of the constant in the slope-intercept form. This refers to the number that is not attached to any variables.
Hence, the y-intercept is 5.
Note that the y-intercept can also be found by substituting x = 0 into the equation.
Identifying the x-interceptThe x-intercept occurs at y = 0. Since we have the equation of the line, we can substitute y = 0 into the equation to find the value of x when y = 0.
Working
y = 3x + 5
When y = 0,
0 = 3x + 5
3x = -5
Divide both sides by 3:
[tex]x = - \frac{5}{3} [/tex]
Graphing the equationI have attached the graph as an image. If only a sketch is required, there are 3 pieces of information that needs to be labelled on the graph:
AxesInterceptsEquation of the lineAdditional resources:
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In a multiple regression problem involving two independent variables, if b1 is computed to be +2.0, it means thatthe estimated mean of Y increases by 2 units for each increase of 1 unit of X1, without regard to X2.the estimated mean of Y increases by 2 units for each increase of 1 unit of X1, holding X2 constantthe estimated mean of Y is 2 when X1 equals zero.the relationship between X1 and Y is significant.
The solution is y will increase by 2 units with every unit increase in x1
What do you mean by multiple regression?A single dependent variable and several independent variables can be analyzed using the statistical technique known as multiple regression. In order to predict the value of the single dependent value, multiple regression analysis uses independent variables whose values are known.
What is the difference between multiple and single regression?X and Y variables are the sole ones in simple linear regression. Two or more x variables and one y variable make up multiple linear regression. For instance, basic linear regression is used when we forecast rent using only square footage.
In the above question:-
Interpretation of b1=2
it means that y will increase every 2 units for every unit increase in x1
holding x2 constant.
Hence option B is the correct answer.
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In the diagram, A and B are points
on the circle AC and BC are
tangents to the circle and meet
at C. Work out the angle ACB.
The angle ACB would be 80 degrees.
What is tangent to a circle?
The tangent always makes a single-point contact with the circle. It never intersects the circle twice. Tangents from an external point to a circle have the same length.
In the given figure,
angle AOB = 40.
So angle OBA = 40.
Then the angle AOB = 180 - 40 -40
AOB = 100.
By the theorem of circles,
AOB + ACB = 180
100 + ACB = 180
ACB = 80.
Hence, the angle ACB would be 80 degrees.
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The letters that spell MISSISSIPPI are each written on separate tiles lying facedown on a table. A tile
is selected at random, the letter is recorded, and then the tile is placed facedown on the table. Then
the process repeats.
What is the theoretical probability of choosing a tile with the letter P?
0 11
01/11
Answer: 2/11
Step-by-step explanation: there are 11 letters and 2 of them are p so the chances of getting p is 2/11
Answer:
2/11
Step-by-step explanation:
The total amount of tiles is 11. There are only 2 tiles that have the letter P on them.
Theoretical probability is the probability that should occur over many, many trials.
Theoretical probability = favorable outcomes / total outcomes.
In this scenario, the favorable outcome is the letter P. which there are two of. The total outcomes would be all the letters that make up the word MISSISSIPPI, or 11.
So your theoretical probability would be 2/11.
two essential features of all statistically designed experiments are
A comparison of multiple treatments and the use of the double-blind method are two fundamental components of all statistically designed experiments. (6) Compare various treatments; assign patients to treatments at random.
What is double-blind method?
A kind of clinical experiment in which neither the participants nor the researcher is aware of the treatment or intervention that each participant is receiving until the trial is complete.
This reduces the likelihood of skewed study outcomes.
What is a sample of a double-blind study?
Imagine, for instance, that researchers are looking into the effects of a novel medication.
The participants in a double-blind study would not be aware of who was receiving the real medication and who was receiving a placebo, and neither would the researchers who interact with them.
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The complete question is -
Two essential features of all statistically designed experiments are (2 Points) O use enough subjects; always have a control groups O always have a placebo group; use the double - blind method O use a block design; use chance to assign subjects to treatments O compare several treatments; use the double - blind method O compare several treatments; use chance to assign subjects to treatments.
how does weather existe? I need help
Answer: How Weather Exists
Step-by-step explanation:
~I think you mean, how weather exists.
You see, "Weather" exists because of the atmosphere. Evidently, weather is the movement of the atmospheric gases around the planet that creates weather. Places like the Moon or Mercury do not have weather - only temperature changes.
~Hope this helped :)
Solve it please I need this asap!
The angles ∠BCD and ∠BAD are opposite interior angles of a parallelogram. in which the alternate interior angles ∠BAC and ∠ACD and ∠BCA and ∠DAC are congruent, from which by angle addition property, ∠BCD ≅ ∠BAD
What are alternate interior angles?Alternate interior angles are angles formed by the transversal to two parallel lines on the inside of the parallel lines, and on opposite sides of the transversal.
The two column-method used to prove that ∠BCA is congruent to ∠DAC can be presented as follows;
Step [tex]{}[/tex] Statement Reason
1. [tex]\overline{BC}[/tex]║ [tex]\overline{AD}[/tex], [tex]\overline{AB}[/tex]║[tex]\overline{CD}[/tex] [tex]{}[/tex] Given
2. [tex]{}[/tex] ∠BCA ≅ ∠DAC ║ lines cut by a trans. form ≅ int. ∠s
3. [tex]{}[/tex] [tex]\overline{AC}[/tex]║[tex]\overline{CA}[/tex] [tex]{}[/tex] Reflexive Property
4. ∠BAC ≅ ∠ACD [tex]{}[/tex] Alternate interion angles
5. ∠BCA = ∠DAC[tex]{}[/tex] Definition of congruency
[tex]{}[/tex] ∠BAC = ∠ACD
6. ∠BCD = ∠BCA + ∠ACD[tex]{}[/tex] Angle addition property
∠BAD = ∠BAC + ∠DAC
7. ∠BCD = ∠BCA + ∠ACD [tex]{}[/tex] Substitution property
8. [tex]{}[/tex] ∠BCD = ∠BAD [tex]{}[/tex] Substitution property
9. ∠BCD ≅ ∠BAD [tex]{}[/tex] Definition of congruency
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State the domain and range of the graph below
Answer:
For the first graph:
Domain: -∞ [tex]\leq[/tex] x [tex]\leq[/tex] ∞
Range: y [tex]\geq[/tex] 3
For the second graph:
Domain: 0 [tex]\leq[/tex] x [tex]\leq[/tex] 4
Range: y [tex]\leq[/tex] 64
Step-by-step explanation:
For the second graph, I'm assuming that that point are stopping at (0,0) and (4,0) that's why I wrote that for the domain. If it doesn't stop, then the domain would be the same and the first graph.
name the property used in the equations
A) 3x+9y-1 = 3(x+3y) -1, here distributive property is used the 3 is multiplied with each term inside the bracket.
What is algebraic operations?
Any of the common arithmetic operations, such as addi - tion, subtraction, multiplication, division, raising to a complete number power, and taking roots, are considered basic algebraic operations in mathematics (fractional power). These operations could be applied to numerical data, in which case people are frequently referred to as arithmetic operations. Similar operations can be carried out on factors, algebraic expressions, and, more generally, on components of algebraic structures like groups and fields. Another way to describe an algebraic operation is as a function from the Cartesian power of the a set to the same set. The dot product is an example of an operation that can be described by compounding simple algebraic operations, and is referred to as an algebraic operation.
b) 7x+5y-5y = 7x
Here the Additive inverse of 5y is used which is -5y .
c) (x-4)(x+3) = 0
here distributive property of addition over multiplication is used.
d) 4x+5x = 5x+4x
here commutative property is used.
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Determine which integers in the set S: {−2, −3, −4, −5} will make the inequality 2p − 8 < 5p + 4 true.
S:{−2, −3}
S:{−3, −4}
S:{−4, −5}
S:{−2, −5}
Answer:
[tex]S=\{-2,-3\}[/tex]
Step-by-step explanation:
Required integers are: -2 and -3
A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point.
The area of the triangle as the function of x is [tex]x^2[/tex] / 2x - 4 and the domain of the function is x∈(2, ∞)
The point on the line = (2, 1)
The give three points are (0, y), (2, 1) and (x, 0)
The slope of the line is the change in y coordinates with respect to the change in x coordinates of the line
Consider the point (0, y) and (2, 1)
Slope = (1-y) / (2-0)
= (1-y) /2
Consider the point (2, 1) and (x, 0)
Slope = (0-1) / (x-2)
= -1/x-2
Both slopes are equal
(1-y) /2 = -1/x-2
(1-y) = -2 / (x-2)
y = 1 + (2/(x-2))
y = x / (x-2)
The area of the triangle = (1/2)bh
Where b is the base
h is the height
The area of the triangle = (1/2) × x × (x / (x-2))
= [tex]x^2[/tex] / 2x - 4
The domain is x∈(2, ∞)
Hence, the area of the triangle is [tex]x^2[/tex] / 2x - 4 and the domain of the function is x∈(2, ∞)
The complete question is
A right triangle is formed in the first quadrant by the x- and y-axes and a line through the point (2,1). Write the area of the triangle as a function of x, and determine the domain of the function.
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Calculate the average rate of change from x = −2 to x = 1 of the function below. f(x) = x2 + 5x − 12 First, find the change in y by evaluating the function at −2 and 1:
The average rate of change over the interval x = -2 and x = 1 is 4
How to determine the average rate of changeThe equation is given as
f(x) = x² + 5x - 12
The interval is given as:
x = -2 to x = 1
Start by calculating the functions f(-2) and f(1)
So, we have
f(-2) = (-2)² + 5(-2) - 12 = -18
f(1) = (1)² + 5(1) - 12 = -6
So, the average rate over these intervals is calculated as:
Average rate = [f(1) - f(-2)]/[1 - -2]
This gives
Average rate = [-6 - -18]/[1 - -2]
Evaluate
Average rate = 4
Hence, the average rate of change is 4
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Uan earn $5 per item he ell. How many item hould he ell to make a profit of $60 or more?
By resolving an equation, we may determine that Uan needs to sell 12 or more goods in order to generate a profit of $60 or more.
What do we mean by equations?A formula that describes how two expressions on each side of a sign are connected.
It typically has an equal sign and one variable.
Similar to this: 2x - 4 = 2.
Graphing, substitution, or elimination are the three methods that can be used to solve systems of equations.
To answer a problem by graphing, just plot the given equations on a graph and identify the point(s) where they all intersect.
So, a number of items Uan should sell to earn a profit of $60 or above:
We know that Uan earns $5 for each item he sells.
So, let x be the number of items Uan should sell.
Then, the equation will be:
5x = 60
Now, solve as follows:
5x = 60
x = 60/5
x = 12
Profit of $60: 12 items to sell
Profit more than $60: More than 12 items to sell
So, by resolving an equation, we may determine that Uan needs to sell 12 or more goods in order to generate a profit of $60 or more.
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the string of a kite is 100 meters long and it makes an angle of 60° with the horizontal line.find the height of the kite assuming that there is no slack in the string (â3=1.73)
The height of the kite assuming that there is no slack in the string is 86.6 m.
We have given that,
the string of a kite is 100 meters long and it makes an angle of 60°.
and we have to find the height of the kite.
So therefore we know the formula for right angled triangle,
i.e. sinθ = Opposite side/ Hypotenuse
Here θ = 60°
Opposite side = height of the kite = h (say)
Hypotenuse = length of string = 100 m
⇒ sin60° = [tex]\frac{h}{100}[/tex]
h = 100sin60°
= 100 × [tex]\frac{\sqrt{3} }{2}[/tex]
= 50 × 1.732
h = 86.6 m
Hence the height of the kite is 86.6 m.
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a farmer has a rectangular garden plot surrounded by 200 ft of fence. find the length and width of the garden if its area is 1275 ft2. width ft (smaller value) length ft (larger value)
By solving a quadratic equation, it can be calculated that
Length of the plot = 85 ft.
Width of the plot = 100 - 85 = 15 ft.
What is a quadratic equation?
Equation shows the equality between two algebraic expressions by connecting the two algebraic expressions by an equal to sign.
A two degree equation is known as quadratic equation.
Here a quadratic equation needs to be solved.
Let the length be l ft. and the width be b ft.
Perimeter = 2(l + b) ft.
By the problem,
2(l + b) = 200
l + b = [tex]\frac{200}{2}[/tex]
l + b = 100
b = (100 - l) ft.
Now,
Area = l [tex]\times[/tex] b sq. ft.
= [tex]l \times (100 - l)[/tex]
By the problem,
[tex]l \times (100 - l) = 1275\\100l - l^2=1275\\l^2 -100l+1275 = 0\\l^2 - 85l -15l + 1275 = 0\\l(l -85) - 15(l - 85) = 0\\(l -85)(l -15) =0\\l -85 = 0 \ or \ l -15 = 0\\l =85 \ or \ l =15[/tex]
If l = 15 ft.
b = 100 - 15 = 85 ft.
Which is not possible as length is always greater than the width
So, length of the plot = 85 ft.
Width of the plot = 100 - 85 = 15 ft.
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