To find h(k(x)), substitute k(x) into h(x):
h(k(x)) = 2x
= 2(2x - 4)
= 4x - 8
Thus, h(k(x)) = 4x - 8 would be your final answer.
maddy had a piece of ribbon that was 3 1/2 yards long each bow was made from a piece of the ribbon that was 3/4 yard long how many bows can Maddy make
Step-by-step explanation:
this is answered by "how often does 3/4 fit into 3 1/2 ?".
and that is what division is for :
3 1/2 / 3/4
(3×2 + 1)/2 / 3/4
7/2 / 3/4 = (7×4)/(2×3) = 28/6 = 14/3 = 4 2/3
Maddy can make 4 bows.
and she has some tape left, but not enough to make a 5th.
the average daily rainfall for the past week in the town of hope cove is normally distributed, with a mean rainfall of 2.1 inches and a standard deviation of 0.2 inches. if the distribution is normal, what percent of data lies between 1.9 inches and 2.3 inches of rainfall?
The percentage of data that falls between 1.9 inches and 2.3 inches of rainfall is roughly (D) 68% if the distribution is normal.
What is the percentage?In essence, percentages are fractions with 100 as the denominator.
We place the percent symbol (%) next to the number to indicate that the number is a percentage.
For instance, you would have received a 75% grade if you answered 75 out of 100 questions correctly on a test (75/100).
So, we determine each's z-score:
z-score = (x-mean)/SD
z1 = (1.9-2.1)/0.2 = -1
z2 = (2.3-2.1)/0.2 = 1
Then, the percentage we want to determine is:
P(-1<x<1)
For this, we utilize the normal scoring table:
P(-1<x<1) = P(x<1) -P(x<-1) = 0.68269
= 68%
Therefore, the percentage of data that falls between 1.9 inches and 2.3 inches of rainfall is roughly (D) 68% if the distribution is normal.
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Complete question:
The average daily rainfall for the past week in the town of Hope Cove is normally distributed, with a mean rainfall of 2.1 inches and a standard deviation of 0.2 inches. If the distribution is normal, what percent of data lies between 1.9 inches and 2.3 inches of rainfall? a) 95% b) 99.7% c) 34% d) 68%
f(x) = x² + 6x + 7; Find f(0)
Answer: f(0) = 7
Step-by-step explanation:
f(x) = x² + 6x + 7
f(0), we just need to put 0 in for all the x
f(0) = 0^2 + 6(0) +7
f(0) = 0 + 0 + 7
f(0) = 7
For each graph below, state whether it represents a function. Graph 1 Graph 2 Graph 3 24 . 864 7 21 6 & -2 4- -6- Function? Yes O No Graph 4 Yes ONo Graph 5 Yes O No Graph 6 60 Function? Yes O No Yes ONo Yes O No
Using the vertical line test, there are only two graphs that represent functions (graph function): graph, 1 and graph 6.
A graph function is a graph that represents a function on a coordinate plane, which means that every point on the graph fulfills the function equation. In a graph function, an x corresponds to only one y.
If we want to know whether a graph represents a function or not, we can perform a vertical line test. In this test, we can drag a vertical line from left to right on the graph. If the vertical line intersects with the graph at one point at a time, then it's a function.
Otherwise, if the vertical line intersects with the graph at two or more points at a time, then it's not a function. It could happen when an x value gives two or more y values (1 input results in multiple outputs), which means the graph is not a function.
Now we take a look at the given picture.
Graph 1: when we drag a vertical line from left to right on the graph, it intersects with the graph at one point at a time. So yes, the graph represents a function.
Graph 2: when we drag a vertical line from left to right on the graph, it intersects with the graph at two points. So no, the graph doesn't represent a function.
Graph 3: when we drag a vertical line from left to right on the graph, it intersects with the graph at one point at first. But later on, it crosses two points in the graph. So no, the graph doesn't represent a function.
Graph 4: when we drag a vertical line from left to right on the graph, it intersects with the graph at two points. So no, the graph doesn't represent a function.
Graph 5: when we drag a vertical line from left to right on the graph, it also intersects with the graph at two points. So no, the graph doesn't represent a function.
Graph 6: when we drag a vertical line from left to right on the graph, it intersects with the graph at one point at a time. So yes, the graph represents a function.
So there are only two graphs that represent functions: graph, 1 and graph 6 since the other graphs fail the vertical line test.
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Enter the equation of the following using line in slope-intercept form.
slope = 0; y-intercept is 4
Answer:
y=4
Step-by-step explanation:
slope doesn't exist
so y=0x+4 >>>>> y=4
What theorems apply to equilateral triangles?
Theorems apply for equilateral triangles are theorem 1 and 2, In theorem 1 each angle of an equilateral triangle is the same and measures 60 degrees each while in theorem 2 a triangle is said to be equilateral if and only if it is equiangular.
What is theorems?Theorem can be defined as a statement which has been proved true by a special kind of logical argument called a rigorous proof.
There are different types of theorem such as:
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#8: Donovan publishes a magazine of short stories. He keeps
track of the number of copies (y) of the magazine he has sold
after each of the first x days. After 1 day,
Donovan had sold 20 copies of his magazine. After 2 days, he
had sold 32 copies of his magazine. He connects these two
points with a straight line to estimate his
future sales. What is the equation of the line Donovan drew?
A.Y-20= 12(x - 2)
B.Y-32= 12(x - 20)
C.Y-2012(x - 1)
D.Y-2 = 12(x - 1)
The equation of the line would be y - 20 = 12(x - 1).
Option (C) is correct.
What is the slope-point form of the line?
For the line having slope "m" and the point (x1, y1) the equation of the line passing through the point (x1, y1) having slope 'm' would be
y - y1 = m(x - x1)
Let's suppose on the y-axis there will be a number of copies sold and on the x-axis, there will be a number of days.
On day 1, he sold 20 copies so point would be (1, 20)
On day 2, he sold 32 copies so point would be (2, 32)
Slope of the line = (y2 - y1)/(x2 - x1)
= (32 - 20)/(2 - 1)
m = 12
By using the slope-point form of the line,
y - y1 = m(x - x1)
y - 20 = 12(x - 1)
Hence, the equation of the line would be y - 20 = 12(x - 1).
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What will be the location of point ′ F ′ if the triangle wa dilated by a cale factor of 3
If the coordinates of point F are (x,y), then the coordinates of point F' after the dilation will be (3x, 3y).
To find the location of point F' after the triangle is dilated by a scale factor of 3, you need to multiply the coordinates of point F by the scale factor.
For example, if the coordinates of point F are (x,y), then the coordinates of point F' after the dilation will be (3x, 3y).
It's important to note that the dilation will change the size of the triangle, but not its shape. The angles of the triangle will remain the same, and the ratios of the side lengths will also remain the same.
If you have a specific triangle and the coordinates of point F, I can provide more detailed guidance on finding the coordinates of point F' after the dilation.
∴ If the coordinates of point F are (x,y), then the coordinates of point F' after the dilation will be (3x, 3y).
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The mapping diagram shows a functional relationship.
A mapping diagram shows a relation, using arrows, between domain and range for the following ordered pairs: (negative 1, 0), (0, 3), (3, negative 1), (9, 9).
Complete the statement.
f(3) is
The value of function f(3) is -1
What is a Mapping Diagram?
A mapping diagram is an image of two circles which shows two values corresponding to each other, such that one is the input value to the function and the other is the output value.
Given ordered pairs :
(- 1, 0), (0, 3), (3, - 1), (9, 9)
Domain and range from given ordered pairs.
Domain - means x - axis
Range - means y-axis
Now, take out domain and range like that,
Domain of the function : (-1, 0, 3, 9) [ input values]
Range of the function : (0, 3, -1, 9) [ output values]
Now, we need to determine the value of function f(3).
It means 3 is input value in f(3) so the output value at 3 is,
f(3) = -1
Hence, the value of function f(3) is -1.
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Throughout the reflection, make sure you have a copy of the student guide. in your experiment, you tested this prediction:
prediction: rna determines the sequence of amino acids in proteins and polypeptides by a two-step process: transcription of dna produces mrna in the nucleus, then translation of the mrna to trna takes place in the ribosome in the cytoplasm. you transcribed dna to make and built a sequence of amino acids through:____.
The two-step process of how RNA determines the sequence of amino acids in proteins and polypeptides begins with transcription.
During transcription, DNA is copied into messenger RNA (mRNA) in the nucleus. mRNA consists of codons that are read in triplets to code for specific amino acids. The codons are then translated into transfer RNA (tRNA) in the ribosome located in the cytoplasm. tRNA is made up of anticodons, which are complementary to the codons of mRNA. The anticodons of tRNA bind to the codons of mRNA, forming a codon-anticodon pair. This codon-anticodon pair carries the amino acid that corresponds to the codon of mRNA. This carried amino acid is then added to a growing polypeptide chain.
The process is summarized by the following formula: DNA → mRNA → tRNA → Polypeptide. DNA is first transcribed into mRNA. The codons of mRNA are then translated into the anticodons of tRNA, which carry the corresponding amino acid. Finally, the amino acid is added to the polypeptide chain, forming a protein. This two-step process of transcription and translation is how RNA determines the sequence of amino acids in proteins and polypeptides.
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Answer: mRNA, Translation
Step-by-step explanation: trust me bro
A construction company can remove 1/2 metric tons of dirt from a construction site in 1/4 hours. What is the unit rate in metric tons per hour?
Answer: 2 metric tons per hour
Step-by-step explanation:
We know that to get from 1/4 hour to 1 hour we have time 4
So we take 1/2 times 4 = 4/2 = 2 metric tons per hour!
What are the 3 ways to prove similarity?
The 3 ways to prove similarity are Angle - Angle (AA), Side - Angle - Side (SAS), and Side - Side - Side (SSS).
The three theorems are as follows :
Two triangles are said to be compared according to the Angle-Angle (AA) principle if they have two pairs of corresponding angles that are congruent.The Angle-Side-Side (SAS) According to the theorem, two triangles are similar if two of their sides are proportionate to two of their corresponding sides in another triangle and their corresponding included angles are congruent.The Side-Side-Side (SSS) theorem states that two triangles are comparable if their sides are proportionate.To learn more about similarity here:
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suppose that for a certain coastal city, in any given year the probability of a major hurricane hitting is 0.5, the probability of flooding is 0.3, and the probability of both a major hurricane and flooding is 0.2. what is the probability of flooding given that a major hurricane hits?
Using the concept of conditional probability and using the values provided, The answer is 0.4.
What do you mean by probability?Probability is the likelihood that something will occur. When we don't know how an event will turn out, we might discuss the likelihood of various outcomes.
What do you mean by conditional probability?The possibility of an event or outcome happening contingent on the occurrence of a prior event or outcome is known as conditional probability. The probability of the prior event is multiplied by the current likelihood of the subsequent, or conditional, occurrence to determine the conditional probability.
p(h)=0.5
p(f)=0.3
p(h ∩ f) = 0.2
p(f/h) = p(h ∩ f) / p(h)
=0.2/0.5
=0.4
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Which system is equivalent to startlayout enlarged left-brace 1st row y = negative 2 x squared 2nd row y = x minus 2 endlayout?.
The required inverse of the given function is expressed as [tex]y=\sqrt{\frac{x+10}{7} }[/tex]
A relation between a collection of inputs and outputs is known as a function. A function is, to put it simply, a connection between inputs in which each input is connected to precisely one output. Each function has a range, codomain, and domain. The usual way to refer to a function is as f(x), where x is the input.
Replacing y with x
x = 7y² – 10
Making y the subject of the formula:
7y² = x+ 10
Dividing both sides by 7
7y²/7 = (x+10)/ 7
y² = (x+10)/ 7
Taking the square root of both sides
[tex]\sqrt{y^{2} } =\sqrt{\frac{x+10}{7} }[/tex]
[tex]y=\sqrt{\frac{x+10}{7} }[/tex]
Hence the required inverse of the given function is expressed as [tex]y=\sqrt{\frac{x+10}{7} }[/tex]
Complete question:
Which equation is the inverse of y = 7x² – 10
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How do you know if the y-intercept is positive or negative?
A positive y-intercept means that the line crosses the y-axis above the origin, and a negative y-intercept means that the line crosses below the origin. Arbitrary lines can be defined simply by changing the values of m and b.
Slope Intercept:
The slope intercept of the straight line is used to find the equation of the straight line. The slope intercept formula requires knowing the slope of the line and the intersection of the line with the y-axis. Consider a line with slope 'm' and y-intercept 'b'. The slope-intercept shape formula for a straight line with slopes 'm' and 'b' as y-intercept is given as y = mx + b.
Positive Intercept:
A positive slope refers to a line that is slanted and slopes upward when viewed from left to right. The positive slope of a straight line can be calculated using the formula m = (y2 - y1)/(x2 - x1) = Tanθ = f'(x) = dy/dx. A positive slope means that the two quantities represented along the two axes of the coordinate system increase or decrease simultaneously. A line with a positive slope is a positive value for the ratio of climb to run. A straight line with a positive slope makes an acute angle with the positive x-axis.
Positive slope (m) = +slope/run = +Δy/Δx
Negative Intercept:
Negative slope means that the coordinates of the two given points are negatively related. A line with a negative slope has m < 0, and the angle θ it makes with the positive x-axis is obtuse θ, such that 90° < θ < 180°. The slope-to-run ratio of the negative slope line is negative. where slope is the change in y value, denoted by Δy, and run is the change in x value, denoted by Δx.
Negative slope = -rise/run = -Δy/Δx
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approximately 12,000 years ago, people began to domesticate grain in what we now call the middle east. this involved a process that slowly spread around the world over the next 6,000 to 7,000 years. in 1984, 8 percent of u.s. households had a computer, but that number was above 70 percent in 2008. how do these two things relate to social change?
From the examples given in the question, we can say that Both are illustrations of social change that is taking place at various rates.
What is the Percentage?The Latin phrase "per centum," which means "by the hundred," is where the English word "percentage" comes from. Percentage segments are those with a numerator of 100. In other words, it is a connection where the whole is always deemed to be valued 100.
As per the given data,
Approximately 12,000 years ago, people began to domesticate grain in what we now call the Middle East. And it spread around the world over 6,000 to 7,000 years.
In this, it is taking a lot of time, which means that the rate is slower.
But on the other hand,
In 1984, 8 percent of u.s. households had a computer, but that number was above 70 percent in 2008.
But in this case, in around 24 years there are 70% of households have computers, which means that the rate is higher.
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Answer question pls, ixls
man runs for an hour and a half at 8 miles an hour due south. he then continues south on a bus traveling at 55 miles an hour for 24 minutes. how far d
A man runs for an hour and a half at 8 miles an hour due south. he then continues south on a bus traveling at 55 miles an hour for 24 minutes. He will traveled 34 miles.
He covered 8 miles the first hour, and then half of that in his next half hour,
So, 8+4 = 12 miles.
The bus took 24 minutes, which is [tex]\frac{2}{5}[/tex] of an hour.
[tex]\frac{60}{5} = 12[/tex]
So, after 12 minutes on the bus, he travelled [tex]\frac{55}{5} = 11[/tex] miles
And after 24 minutes, 22 miles.
In total 12+22=34 miles.
Speed(v) is related to distance(d) and time(t) by
v= [tex]\frac{d}{t}[/tex].
Equivalently, d= v×t
So, d= 8×1.5+55×24×[tex]\frac{1}{60}[/tex]
= 34 miles.
Therefore, he traveled 34 miles.
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What is the value of the discriminant in the quadratic equation 2x² 5x 3 0?
The value of the discriminant in the quadratic equation 2x² 5x + 3 = 0 is 1.
The discriminant of a quadratic equation is the expression under the square root sign in the quadratic formula. It is used to determine the number and type of solutions for the equation. In the case of the equation 2x² 5x + 3 = 0, the discriminant can be found by using the quadratic formula and substituting the values for a, b, and c. The formula is given by b² - 4ac, where a = 2, b = 5 and c = 3. So the discriminant is then 5² - 4(2)(3), which is equal to 1. This means that the equation has a single real solution, which is -1.
Discriminant = b² - 4ac
= 5² - 4(2)(3)
= 1
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In Lucy's coin collection, 15% of the coins are quarters.
If there are 12 quarters, find the total number of coins in Lucy's coin collection
The required total number of Lucy's coins,when we have 15% of quarter coins are 80.
What is percent?percentage, a general worth demonstrating 100th pieces of any amount. One percent (represented 1%) is a hundredth part; in this way, 100% addresses the total and 200 percent determines two times the given amount. rate.
According to question:Quarters coins = 12
Let x be total number of coins Lucy has,then
15% of x = 12
15x/100 = 12
x = 1200/15
x = 80 coins
Thus, there are 80 coins in total.
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Explain how to find the value of a line's y-intercept using its point-slope equation.
Answer:
Given the point-slope form of a line, y = mx + b, to get the y-intercept of a line, you first have to solve for b by setting x = 0. This gives you the b-value of y = 0 + b = b. We see that b is the y-intercept.
Step-by-step explanation:
Answer:
See below
Step-by-step explanation:
The point-slope form of a line looks like
y - [tex]y_{1}[/tex] = m( x - [tex]x_{1}[/tex])
The number that is the place of the 'm' is the slope.
For example:
If we had the equation
y - 3 = 12(x +4)
The slope would be 12.
What is nature of roots formula?
The nature of the roots for the equation ax² + bx + c is
α + β = -b/a and αβ = c/a
where α and β are the two roots of the equation.
What is an expression?An expression is a way of writing a statement with more than two variables or numbers with operations such as addition, subtraction, multiplication, and division.
Example: 2 + 3x + 4y = 7 is an expression.
We have,
A quadratic equation ax² + bx + c.
Now,
There will be two roots of the equation α and β,
Now,
The nature of the roots are
α + β = -b/a
αβ = c/a
Thus,
The roots are α and β and α + β = -b/a and αβ = c/a.
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The nature of roots of quadratic depends on its discriminant, formula is D=√(b²-4ac).
What is root of quadratic equation?It is the value of the unknown variable for which the quadratic equation holds true. Its nature can be real, imaginary, rational, irrational, equal and unequal as well that depends on the value of the discriminant.
For equation ax²+ bx + c =0 , expression b²- 4ac is called discriminant and denoted by D.
Given equation is , ax²+ bx + c =0
then, according to quadratic formula for the roots of quadratic equation
x = {-b±√(b²-4ac)}/4ac
Let us take a real number
k > 0
√k can be defined and is a positive quantity but √-k can not be defined.
The quantity which is under square root in the expression for roots is
b²-4ac, which is called discriminant of the quadratic equation. It is represented by D.
If D>0 and is perfect square then the equation will have two real, rational and distinct roots.
If D>0 and is not a perfect square then the equation will have two real, irrational and distinct roots.
If D=0 the equation will have two real and equal roots.
If D<0, the equation will have no real roots.
Hence, nature of roots formula for quadratic equation depends on the value of √b²-4ac.
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How do you use slope to determine if a line is parallel or perpendicular and solve problems?
Lines that have the same slope are said to be parallel, and if the slope of one line is equal to the negative reciprocal of the slope of the other, the two lines are said to be perpendicular; otherwise, they are merely intersecting lines.
A line's slope in mathematics is defined as the ratio of the change in the y coordinate to the change in the x coordinate.
Both the net change in the y-coordinate and the net change in the x-coordinate are denoted by y and x, respectively.
m = Δy/Δx = dy/dx = change in y/change in x
where "m" represents a line's slope.
Additionally, the slope of the line may be shown as
tan θ = Δy/Δx
A line's slope often indicates the steepness and direction of the line.
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m = Δy/Δx = dy/dx = change in y/change in x
where "m" represents a line's slope.
Additionally, the slope of the line may be shown as
tan θ = Δy/Δx
What will be the nature of roots of quadratic equation 5x² 4x 5 0?
The nature of roots of quadratic equation 5x²- 4x +5= 0
are unreal.
An algebraic equation of the second degree in x is a quadratic equation. The quadratic equation is written as ax2 + bx + c = 0, where x is the variable, a and b are the coefficients, and c is the constant term.
Nature of Roots of the Quadratic Equation
The symbols alpha () and beta () are typically used to denote a quadratic equation's roots. Here, we'll learn more about how to identify a quadratic equation's nature without locating the equation's roots. Additionally, look at the formulas for calculating the product and sum of the equation's roots.
Without actually locating the roots (, ) of the equation, the nature of the roots of a quadratic equation can be determined. By using the discriminant value, which is a component of the quadratic equation solution formula, this is made achievable. The discriminant of a quadratic equation is denoted as "D" and has the value b2 - 4ac. Using the discriminatory value as a basis, the nature of the roots of the quadratic equation can be predicted.
Discriminant: D = b2 - 4ac
D > 0, the roots are real and distinct
D = 0, the roots are real and equal.
D < 0, the roots do not exist or the roots are imaginary.
From the given quadratic equation, we can observe that:
a = 5
b = -4
c = 5
On finding discriminant, we get:
D = b² - 4ac
D = (-4)² - (4 × 5 × 5)
D = 16 - 100
D = -84
We can notice that, D(-84) < 0, which means the roots are unreal. Therefore, no roots exist!
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The nature of roots of quadratic equation 5x²- 4x +5= 0
are unreal.
An algebraic equation of the second degree in x is a quadratic equation. The quadratic equation is written as ax2 + bx + c = 0, where x is the variable, a and b are the coefficients, and c is the constant term.
Nature of Roots of the Quadratic Equation
Discriminant: D = b2 - 4ac
D > 0, the roots are real and distinct
D = 0, the roots are real and equal.
D < 0, the roots do not exist or the roots are imaginary.
From the given quadratic equation, we can observe that:
a = 5
b = -4
c = 5
D = b² - 4ac
D = (-4)² - (4 × 5 × 5)
D = 16 - 100
D = -84
We can notice that, D(-84) < 0, which means the roots are unreal. Therefore, no roots exist!
Solve the systems of equations graphed on the coordinate axes below. y=-4x-1 y=2x-1
The plane intersects the paraboloid in an ellipse. Find the points on this ellipse that are nearest to and farthest from the origin. Point farthest away occurs at.
The point on the ellipse that is farthest from the origin will be the point that is closest to one of the foci.
To find the points on the ellipse that are nearest to and farthest from the origin, we need to consider the equation of the paraboloid and the equation of the ellipse.
The equation of a paraboloid is given by:
[tex]z = ax^2 + by^2 + c[/tex]
where a, b, and c are constants.
The equation of an ellipse is given by:
[tex](\frac{x}{a})^2 + (\frac{y}{b})^2 = 1[/tex]
where a and b are the semi-major and semi-minor axes of the ellipse, respectively.
If the plane intersects the paraboloid in an ellipse, then the equations of the paraboloid and the ellipse must be equal at the points of intersection. Therefore, we can set the equation of the paraboloid equal to the equation of the ellipse and solve for the x and y values.
[tex]z = ax^2 + by^2 + c[/tex]
[tex](\frac{x}{a})^2 + (\frac{y}{b})^2 = 1[/tex]
Substituting the equation of the paraboloid into the equation of the ellipse, we get:
[tex](\frac{x}{a})^2 + (\frac{y}{b})^2+c = 1[/tex]
We can then solve this equation for the values of x and y that correspond to the points on the ellipse that are nearest to and farthest from the origin.
The point farthest away from the origin occurs at the point on the ellipse that is farthest from the center, which is called the "foci" of the ellipse. The distance from the center of the ellipse to the foci is called the "eccentricity" of the ellipse. The eccentricity is denoted by the letter "e" and is calculated using the following formula:
[tex]e = \frac{\sqrt{(a^2 - b^2)}}{a}[/tex]
where a and b are the semi-major and semi-minor axes of the ellipse, respectively.
Therefore, to find the point on the ellipse that is farthest from the origin, we need to calculate the eccentricity of the ellipse and then use this value to determine the location of the foci. The point on the ellipse that is farthest from the origin will be the point that is closest to one of the foci.
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Is the 3x 1 conjecture?
The gradient of 3x+ 1 is conjecture states that this function can be iterated over and over until it yields the number 1, starting from any positive integer n.
Describe conjecture.A conjecture turns into a theorem when it is rigorously proven. A conjecture is not merely a tool for professional mathematicians; it is an essential step in problem-solving. It is extremely uncommon for a problem's solution to be immediately obvious while tackling problems on a daily basis. Instead, establishing a conjecture about the answer and then proving it through evidence are steps in the problem-solving process that include reviewing cases and the problem's structure. When one observes a pattern that holds true in numerous instances, hypotheses start to form. A pattern may be true in many instances, but that does not guarantee that it will be true in all instances. For a mathematical observation to be fully accepted, conjectures must be proven.
Given
The 3x + 1
Conjecture states that this function can be iterated over and over until it yields the number 1, starting from any positive integer n.
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What is the square root of 2 as a power of 2?
Answer:
[tex]2^{\frac{1}{2}}[/tex]
Step-by-step explanation:
Taking the square root of a number is equivalent to putting that number to the power of 1/2.
Step-by-step explanation:
x^(a/b)
means
[tex] \sqrt[b]{ {x}^{a} } [/tex]
the numerator of the exponent is the actual "to the power of" factor.
the denominator defines the degree of the root to be taken.
as any integer number is also a fraction like
6 = 6/1
this applies to any form of rational numbers like
x^0.25 = x^(25/100) = x^(1/4) =
[tex] \sqrt[4]{ {x}^{1} } = \sqrt[4]{x} [/tex]
with irrational numbers as exponents this gets trickier, as with infinite numbers without a pattern after the decimal point we have a problem to precisely specify the degree of the root, as we also have a problem to specify the irrational number itself.
we try to find the origin of the irrational number (like a square root)
[tex] {x}^{ \sqrt{3} } [/tex]
or
[tex] \sqrt[ \sqrt{3} ]{ {x}^{2} } = {x}^{2 \div \sqrt{3} } [/tex]
so,
[tex] \sqrt{2} = \sqrt[2]{2} = {2}^{1 \div 2} [/tex]
14. In the figure (not drawn to scale), MO bisects LMN, mZNMO = (6x + 19), and m
14). Solve for x and find ZLMN.
L
M
A. x= 11, m LMN = 198°
B. x= 11, m ZLMN = 170°
N
C. x=8, m ZLMN = 25°
D. x=8, m ZLMN = 54°
The value of x is 11 and ∠LMN = 190°
What is Linear Equation in One Variable?
A linear equation is a one-variable equation of a straight line. The variable's only power is 1. Linear equations in one variable of the form ax + b = 0 are solved using simple algebraic techniques.
Solution:
Since, MO bisects the angle
we can say that, ∠LMO = ∠OMN
6x + 19 = 9x - 14
3x = 33
x = 11
so, ∠LMO = 66 + 19 = 85°
Therefor the ∠LMN = 190°
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Solve by completing the square : x^2+4x-7=0 and 2x^2+8x-1=0
By completing the squares we will see that the solutions are:
a) For 2x^2 + 8x - 1 = 0:
x = -2 ±√4.5
b) For x^2+4x-7 = 0
x = ±√11 - 2
How to complete squares?Remember that the perfect square trinomial is:
(a + b)^2 = a^2 + 2ab + b^2
The first quadratic equation is:
2x^2 + 8x - 1 = 0
Dividing by 2 we get:
x^2 + 4x - 1/2 = 0
We can rewrite this as:
x^2 + 2*2*x - 1/2 = 0
Adding and subtracting 2^2 = 4, we get:
x^2 + 2*2*x + 4 - 4 - 1/2 = 0
(x + 2)^2 - 4 - 1/2 = 0
(x + 2)^2 = 4.5
Now we can solve this:
x + 2 = ±√4.5
x = -2 ±√4.5
These are the two solutions.
The other quadratic equation is:
x^2+4x-7 = 0
We can rewrite this as:
x^2 + 2*2*x - 7 = 0
Again, add and subtract 4 to get:
x^2 + 2*2*x + 4 - 4 - 7 = 0
(x + 2)^2 = 11
Solving this we get:
x = ±√11 - 2
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