Answers
The given limit is 0.
To solve the given limit, we can recognize the sum as a Riemann sum and convert it into an integral.
The given sum can be rewritten as:
[tex]\lim_{n \to \infty} \sum_{i=1}^{n} \frac{3}{n} \sqrt{1+\frac{3i}{n}}[/tex]
Let's rewrite it in terms of integration:
[tex]\lim_{n \to \infty} \frac{3}{n} \sum_{i=1}^{n} \sqrt{1+\frac{3i}{n}}[/tex]
Since we are taking the limit as n approaches infinity, we can approximate the sum as an integral.
The integral that corresponds to the given sum is:
[tex]\lim_{n \to \infty} \frac{3}{n} \sum_{i=1}^{n} \sqrt{1+\frac{3i}{n}} \approx \lim_{n \to \infty} \frac{3}{n} \int_{0}^{1} \sqrt{1+3x} ,dx[/tex]
To solve this integral, we can use a change of variables.
Let u = 1 + 3x, then du = 3dx.
The integral becomes:
[tex]\lim_{n \to \infty} \frac{3}{n} \int_{0}^{1} \sqrt{1+3x} ,dx = \lim_{n \to \infty} \frac{1}{n} \int_{1}^{4} \sqrt{u} ,du[/tex]
Integrating [tex]\sqrt{u}[/tex], we get,
[tex]\lim_{n \to \infty} \frac{1}{n} \int_{1}^{4} \sqrt{u} ,du = \lim_{n \to \infty} \frac{1}{n} \left[\frac{2}{3} u^{3/2}\right]_{1}^{4}[/tex]
Substituting the limits, we have:
[tex]\lim_{n \to \infty} \frac{1}{n} \left[\frac{2}{3} (4)^{3/2} - \frac{2}{3} (1)^{3/2}\right][/tex]
Simplifying further:
[tex]\lim_{n \to \infty} \frac{1}{n} \left[\frac{2}{3} (8 - 2)\right] = \lim_{n \to \infty} \frac{1}{n} \left[\frac{12}{3}\right] = \lim_{n \to \infty} \frac{4}{n} = 0[/tex]
Therefore, the given limit is 0.
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Related Questions
a number cube is tossed 60 times Determine the experimental probability of landing on a number less than two
Answers
Answer:
1.67% approx
Step-by-step explanation:
To determine the experimental probability of landing on a number less than two when a number cube is tossed 60 times, we need to count the number of times the number on the cube is less than two and divide it by the total number of tosses.
Let's denote the event of landing on a number less than two as "A." We'll count the number of successful outcomes, where the number on the cube is less than two.
Assuming the number cube is fair and unbiased, it has six sides numbered from 1 to 6. Out of these, only one side has a number less than two, which is one.
Now, we can calculate the experimental probability using the formula:
Experimental Probability (P(A)) = Number of successful outcomes / Total number of tosses
In this case, the number of successful outcomes is the number of times the cube lands on a number less than two, which is one. The total number of tosses is given as 60.
Therefore, the experimental probability of landing on a number less than two is:
P(A) = 1 (successful outcomes) / 60 (total number of tosses)
= 1/60
≈ 0.0167 or 1.67%
So, the experimental probability of landing on a number less than two is approximately 0.0167 or 1.67%.
obove answer is correct i think
a new social media sit is increasing its user base by approximately 4% per month. If the site currently has 35.930 users, what will the approximate user base be 10 months from now?
Answers
Answer:
The approximate user base of the social media site 10 months from now would be approximately 52,374.
Step-by-step explanation:
To calculate the approximate user base of the social media site 10 months from now, considering a 4% increase per month, we can use the following steps:
1. Calculate the monthly growth factor: 1 + (4% / 100) = 1 + 0.04 = 1.04
2. Apply the growth factor to the current user base for each month:
Month 1: 35,930 * 1.04 = 37,387.2 (approx.)
Month 2: 37,387.2 * 1.04 = 38,868.49 (approx.)
...
Month 10: Previous Month * 1.04
By repeating this calculation for each month, we can determine the approximate user base 10 months from now.
Month 1: 37,387.2
Month 2: 38,868.49
Month 3: 40,391.33
Month 4: 41,957.61
Month 5: 43,569.34
Month 6: 45,228.24
Month 7: 46,936.32
Month 8: 48,695.44
Month 9: 50,507.45
Month 10: 52,374.40 (approx.)
Therefore, the approximate user base of the social media site 10 months from now would be approximately 52,374.
what is an example of "an extension of a function f"?
Answers
An example of an extension of a function f is adding additional input-output pairs beyond the original domain and range of f.
What is an example of "an extension of a function f"?The extension of a function f means new function that preserves the properties and behavior of the original function on its domain while adding new values or expanding its domain.
For example, we will consider the function f(x) = x^2 defined on the interval [0, 1]. An extension of this function could be g(x) = x^2 for x in [0, 1] and g(x) = 0 for x outside [0, 1].
This extension maintains the quadratic behavior of f on [0, 1] while assigning a constant value of 0 to points outside that interval.
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Linear equations linear equations
Answers
A graph of the linear equation y = 2x - 1 in slope-intercept form is shown in the image attached below.
What is the slope-intercept form?In Mathematics and Geometry, the slope-intercept form of the equation of a straight line is given by this mathematical equation;
y = mx + b
Where:
m represent the slope or rate of change.x and y are the points.b represent the y-intercept or initial value.Based on the information provided above, we have the linear equation:
y = 2x - 1
Lastly, we would use an online graphing calculator to plot the given linear equation as shown in the graph attached below.
In conclusion, the slope of this linear equation is equal to 2 and it does not represent a proportional relationship.
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Complete Question:
Plot the linear equation on the coordinate axis shown on the right.
What is the sum of 104, 25, -11 to come of it
Answers
Answer:
118
Step-by-step explanation:
104+25+(-11)
129-11
118
Therefore, our answer is 118
Graph the image of this quadrilateral after a dilation with a scale factor of 2 centered at the origin. Use the polygon tool to graph the quadrilateral.
Answers
The new coordinates of the quadrilateral ABCD after the dilation is:
A' = (-6, 6)
B' = (-8, -4)
C' = (4, -8)
D' = (0, 0)
We have,
To find the coordinates of the new quadrilateral after a dilation with a scale factor of 2 centered at the origin, we multiply the x and y coordinates of each point by the scale factor.
Given the original coordinates of the quadrilateral ABCD:
A = (-3, 3)
B = (-4, -2)
C = (2, -4)
D = (0, 0)
To dilate the coordinates with a scale factor of 2, we multiply each coordinate by 2:
A' = 2 x (-3, 3) = (-6, 6)
B' = 2 x (-4, -2) = (-8, -4)
C' = 2 x (2, -4) = (4, -8)
D' = 2 x (0, 0) = (0, 0)
Thus,
The new coordinates of the quadrilateral ABCD after the dilation is:
A' = (-6, 6)
B' = (-8, -4)
C' = (4, -8)
D' = (0, 0)
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From least to greatest, What are the x–coordinates of the three points where the graphs of the equations intersect? If approximate, enter values to the hundredths.
,
,
Answers
The x-coordinates where the graphs of the equations intersect are x = -1 and x = 3
How to determine the x-coordinates where the graphs of the equations intersect?From the question, we have the following parameters that can be used in our computation:
y = 2x
y = x² - 3
The x-coordinates where the graphs of the equations intersect is when both equations are equal
So, we have
x² - 3 = 2x
Rewrite the equation as
x² - 2x - 3 = 0
When the equation is factored, we have
(x + 1)(x - 3) = 0
So, we have
x = -1 and x = 3
Hence, the x-coordinates are x = -1 and x = 3
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Question
From least to greatest, What are the x–coordinates of the three points where the graphs of the equations intersect? If approximate, enter values to the hundredths.
y = 2x
y = x² - 3
Find the equation of a parabola with focus (3, 4) and directrix y = 1.
Answers
The equation of the parabola with focus (3, 4) and directrix y = 1 is [tex]x^2 - 6x - 12y + 57 = 0.[/tex]
To find the equation of a parabola with a given focus and directrix, we can use the standard form of the equation of a parabola:
[tex]4p(y - k) = (x - h)^2[/tex]
where (h, k) represents the coordinates of the vertex, and p is the distance between the vertex and the focus or directrix.
In this case, the focus is given as (3, 4), which means the vertex of the parabola will also be located at (3, 4).
The directrix is given as y = 1.
First, let's find the value of p, which is the distance between the vertex and the focus (or the vertex and the directrix). In this case, p will be the distance between the vertex (3, 4) and the directrix y = 1.
Since the directrix is a horizontal line, the distance between the vertex and the directrix is the vertical distance, which is |4 - 1| = 3.
Now that we have the value of p, we can substitute it into the equation:
[tex]4p(y - k) = (x - h)^2[/tex]
Plugging in the values (h, k) = (3, 4) and p = 3, we get:
[tex]4(3)(y - 4) = (x - 3)^2[/tex]
Simplifying further:
[tex]12(y - 4) = (x - 3)^2[/tex]
Expanding the equation:
[tex]12y - 48 = x^2 - 6x + 9[/tex]
Bringing all terms to one side:
[tex]x^2 - 6x - 12y + 57 = 0[/tex]
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The graph of � = ∣ � ∣ y=∣x∣y, equals, vertical bar, x, vertical bar is shifted down by 9 99 units and to the right by 4 44 units. What is the equation of the new graph? Choose 1 answer: Choose 1 answer: (Choice A) � = ∣ � − 9 ∣ − 4 y=∣x−9∣−4y, equals, vertical bar, x, minus, 9, vertical bar, minus, 4 A � = ∣ � − 9 ∣ − 4 y=∣x−9∣−4y, equals, vertical bar, x, minus, 9, vertical bar, minus, 4 (Choice B) � = ∣ � − 4 ∣ − 9 y=∣x−4∣−9y, equals, vertical bar, x, minus, 4, vertical bar, minus, 9 B � = ∣ � − 4 ∣ − 9 y=∣x−4∣−9y, equals, vertical bar, x, minus, 4, vertical bar, minus, 9 (Choice C) � = ∣ � − 4 ∣ + 9 y=∣x−4∣+9y, equals, vertical bar, x, minus, 4, vertical bar, plus, 9 C � = ∣ � − 4 ∣ + 9 y=∣x−4∣+9y, equals, vertical bar, x, minus, 4, vertical bar, plus, 9 (Choice D) � = ∣ � − 9 ∣ + 4 y=∣x−9∣+4y, equals, vertical bar, x, minus, 9, vertical bar, plus, 4 D � = ∣ � − 9 ∣ + 4 y=∣x−9∣+4
Answers
An equation of the new graph is: A. y = ∣x - 4∣ - 9.
What is a translation?In Mathematics and Geometry, the translation of a graph to the right simply means a digit would be added to the numerical value on the x-coordinate of the pre-image:
g(x) = f(x - N)
Conversely, the translation of a graph downward simply means a digit would be subtracted from the numerical value on the y-coordinate (y-axis) of the pre-image:
g(x) = f(x) + N
Since the parent function y = ∣x∣ was translated 4 units to the right and 9 units down in order to produce the graph of the image, we have:
y = ∣x - 4∣ - 9
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Missing information:
The question is incomplete and the complete question is shown in the attached picture.
in long division what is the working and answer for 348 divided by 4?
Answers
4 will divide the 348 is 8 and 2 left, so you will write the two and move the 8 down, which is now 28, then divide the 28 by 4. It will be 7.
Find the number that belongs
in the green box.
[?]
37°
64°
21.1
30
Round your answer to the nearest tenth.
Answers
The number that belongs in the green box is 32.8
How to find the number that belongs in the green boxFrom the question, we have the following parameters that can be used in our computation:
The triangle
The third angle in the triangle is calculated as
Third angle = 180 - 37 - 64
Evaluate
Third angle = 79
If the number that belongs in the green box is x, then we have
x/sin(79) = 30/sin(64)
This gives
x = sin(79) * 30/sin(64)
Evaluate
x = 32.8
Hence, the number that belongs in the green box is 32.8
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what is a Pythagoras theorem
Answers
The Pythagoras theorem states that in a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides.
What is mathematical representation of Pythagoras theorem?From the given Pythagoras theorem above, the mathematical representation of the theorem would be written below as follows;
C² = a²+b²
Where:
'C' represents the hypotenuse which is the longest part of the triangle and is always diagonal connecting the two other points.'a' represents the perpendicular length that forms the angle 90° of the right angle triangle.'b' represents the base of the triangle.Learn more about Pythagorean formula here:
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Multiply
19(x + 1 + 9z)
Answers
The product of the expressions 19(x + 1 + 9z) is 19x + 19 + 171z
How to evaluate the product of the expressionsFrom the question, we have the following parameters that can be used in our computation:
19(x + 1 + 9z)
When the brackets are opened, we have
19(x + 1 + 9z) = 19 * x + 19 * 1 + 19 * 9z
Evaluate the products of the expression
So, we have the following representation
19(x + 1 + 9z) = 19x + 19 + 171z
Hence, the product of the expressions 19(x + 1 + 9z) is 19x + 19 + 171z
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Find the number that belongs
in the green box.
[?]
37°
64°
21.1
30
Round your answer to the nearest tenth.
Answers
The value of x is 33.4
How to determine the valueTo determine the value in the green box, we need to know that there are six different trigonometric identities.
These identities are enumerated as;
sinecosinetangentcotangentsecantcosecantUsing the sine identity, we have;
sin θ = opposite/hypotenuse
We have that;
θ = 64 degrees
Opposite = 30
Hypotenuse = x
Substitute the values, we have;
sin 64 = 30/x
cross multiply
x = 30/0. 8987 = 33. 4
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log base 4 of 9
HELPPPPPPPP
Answers
Answer:
1.58 ish
Step-by-step explanation:
For a logx y = z, x^z = y
So here, log4 9= 1.58 or so
1.5849625007
Please Help
8x + 1
115⁰
Answers
180-115=65
8x+1=65
(remove the +1 by -1 from both sides)
8x=64
(remove the 8• by dividing by 8 on each side)
64/8
x=8
ANSWER: x=8
De los 12 jugadores del equipo 2/8 son delanteros
Answers
There are a total of 3 forwards on the team.
How many forwards are there on the team?A fraction represents a part of the whole or group of objects.In a fraction, numerator and denominator are separated by a horizontal bar known as the fractional bar
Given:
2/8 of the players are forwards.
Total number of players on the team = 12
We will determine the total number of forwards on the team by multiplying the total number of players by the fraction representing the forwards.
Fraction representing the forwards:
= 2/8
= 1/4
Total forwards on the team:
= 12 * (1/4)
= 3.
Translated question:
Of the 12 players on the team, 2/8 are forwards. What are the total forward in the team?
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Which of the following shows a 90° counterclockwise rotation of the blue figure
Answers
En uno de los corrales de la granja de mi abuela hay gallinas y conejos. Aún cuando todos se mueven, logré contar 40 cabezas y 106 patas.
a) Explica cuáles son las incógnitas del problema.
b) Escribe una ecuación que represente el número de animales que hay en el corral.
c) Escribe una ecuación que represente el número total de patas.
d) ¿cuántos conejos y cuántas gallinas hay en el corral?
Answers
There are 27 chickens and 13 Rabbits in the pen on your grandmother's farm, based on the given information of 40 heads and 106 legs.
a) The unknowns in the problem are the number of chickens and rabbits in the pen. We don't know the exact quantities of each animal, which is what we need to determine.
b) Let's denote the number of chickens as "C" and the number of rabbits as "R." To represent the total number of animals in the pen, we can write the equation: C + R = total number of animals.
c) The total number of legs can be represented by the equation: 2C + 4R = total number of legs. This equation accounts for the fact that chickens have two legs, while rabbits have four.
d) To find the solution, we can use the given information: 40 heads and 106 legs. Since each animal has one head, the total number of animals is equal to the total number of heads, which is 40.
Using the equation C + R = 40, we can solve for one variable in terms of the other. For example, if we solve for C, we get C = 40 - R.
Substituting this value of C into the equation 2C + 4R = 106, we can solve for R:
2(40 - R) + 4R = 106
80 - 2R + 4R = 106
2R = 26
R = 13
Therefore, there are 13 rabbits in the pen.
To find the number of chickens, we can substitute the value of R into the equation C = 40 - R:
C = 40 - 13
C = 27
Hence, there are 27 chickens in the pen.
In conclusion, there are 27 chickens and 13 rabbits in the pen on your grandmother's farm, based on the given information of 40 heads and 106 legs.
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the bus comes at 8:05. It takes me 31 minutes to get to the bus stop. What time should I leave to catch the bus?
Answers
You should leave at 7:34 to catch the bus that comes at 8:05.
We have,
To catch the bus that arrives at 8:05, you should leave with enough time to reach the bus stop 31 minutes before the bus's arrival time.
To catch the bus that arrives at 8:05, you need to be at the bus stop before the bus arrives. Since it takes you 31 minutes to get to the bus stop, you should leave your starting point early enough to allow for that travel time.
By subtracting 31 minutes from the bus's arrival time of 8:05, you determine the time at which you should depart.
In this case, subtracting 31 minutes gives you 7:34, meaning you should leave at 7:34.
To calculate the departure time, subtract 31 minutes from the bus's arrival time:
= 8:05 - 31 minutes
= 7:34
Thus,
You should leave at 7:34 to catch the bus that comes at 8:05.
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These shapes are similar.
Find X.
5
X
5
30
24
30
Answers
The value of x is 4.
To determine the value of x, we can use the concept of similarity between shapes.
Similar shapes have corresponding sides that are proportional to each other.
Given the dimensions of the first shape as 5, x, and 5, and the dimensions of the second shape as 30, 24, and 30, we can set up the following proportion:
5/x = 30/24
To solve for x, we can cross-multiply:
30 · x = 5 · 24
30x = 120
Dividing both sides of the equation by 30:
x = 120 / 30
x = 4
Therefore, the value of x is 4.
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URGENT!! ILL GIVE BRAINLIEST! AND 100 POINTS
PLEASE ANSWER!!
Answers
Answer:
Step-by-step explanation:
a. 2r + 5 = b because you have you have 5 more blue marbles than double the red *r= red and b=blue
b. to do the substitution you know that the red marbles plus the blue marbles add up to 77 so
b + r=77
now we can isolate for b and then substitute that value into the first equation
b=77-r and then this goes to 2r+5=77-r
c. now you just have to get like terms to their side of the equal sign
add r to both sides: 2r+r+5=77-r+r= 3r+5=77
and then you subtract 5 from both sides which makes 3r=72
to find r, divide both sides by 3
r=24
there are 24 red marbles
Best way to solve this type of equation Ax+by=c
Answers
The best way to solve an equation of the form Ax+by=c could be either substitution, completing the squares or graphical method depending on the type of equation given.
Quadratic equation
The equation in the form Ax+by=c is a quadratic problem which could be approached in different ways depending on the specific equation given and the information provided.
The methods of approach are substitution, completing the squares or graphical method which all have proven to be suitable ways to arrive at the solution.
Hence, either of the three methods are good ways to solve such equation.
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Find w=a+bi=, where a and b are real numbers
√-8+6i
Answers
Given statement solution is :- The given expression √(-8 + 6i) does not have a real solution in the form w = a + bi, where a and b are real numbers.
To find the complex number in the form w = a + bi, where a and b are real numbers, we can solve the given expression.
Let's solve √(-8 + 6i) step by step:
First, we can express -8 + 6i in the form a + bi:
-8 + 6i = -8 + 6i + 0i = -8 + 6i + [tex]0i^2[/tex]
Now, we can take the square root of -8 + 6i:
√(-8 + 6i) = √((-8) + 6i + [tex]0i^2[/tex])
Since [tex]i^2[/tex] = -1, we can simplify the expression further:
√(-8 + 6i) = √(-8 - 6 + 6i)
= √(-14 + 6i)
Now, we want to express -14 + 6i in the form a + bi:
-14 + 6i = -14 + 6i + 0i = -14 + 6i +[tex]0i^2[/tex]
Taking the square root, we have:
√(-14 + 6i) = √((-14) + 6i + [tex]0i^2[/tex])
Since [tex]i^2[/tex] = -1, we can simplify the expression further:
√(-14 + 6i) = √(-14 - 6 + 6i)
= √(-20 + 6i)
Now, we want to express -20 + 6i in the form a + bi:
-20 + 6i = -20 + 6i + 0i = -20 + 6i + [tex]0i^2[/tex]
Taking the square root, we have:
√(-20 + 6i) = √((-20) + 6i + [tex]0i^2[/tex])
Since [tex]i^2[/tex] = -1, we can simplify the expression further:
√(-20 + 6i) = √(-20 - 6 + 6i)
= √(-26 + 6i)
Now, we want to express -26 + 6i in the form a + bi:
-26 + 6i = -26 + 6i + 0i = -26 + 6i +[tex]0i^2[/tex]
Taking the square root, we have:
√(-26 + 6i) = √((-26) + 6i + [tex]0i^2[/tex])
Since [tex]i^2[/tex] = -1, we can simplify the expression further:
√(-26 + 6i) = √(-26 - 6 + 6i)
= √(-32 + 6i)
We continue this process until we reach a point where the expression simplifies. However, in this case, we encounter an expression that cannot be simplified further, as the square root of a negative number is not defined in the set of real numbers.
Therefore, the given expression √(-8 + 6i) does not have a real solution in the form w = a + bi, where a and b are real numbers.
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Answer Question 11 please
Answers
Answer:
increasing exponential function.
Step-by-step explanation:
We can see from the table that the function is not increasing by the same amount as it goes up by 1 in x, so it is not linear.
It must be exponential.
It is increasing at it increases in x, so it is an increasing exponential function.
What is (fg)(x)?
f(x)=2x
g(x)=2x2+3x
Write your answer as a polynomial or a rational function in simplest form
Answers
Given statement Function Multiplication Simplified solution is :- (fg)(x) = [tex]4x^2[/tex] + 6x.
To find (fg)(x), we need to first find the product of f(x) and g(x) and simplify the expression.
f(x) = 2x
g(x) = [tex]2x^2[/tex] + 3x
To find (fg)(x), we substitute g(x) into f(x):
(fg)(x) = f(g(x)) = f([tex]2x^2[/tex] + 3x)
Now we substitute f(x) = 2x into the expression above:
(fg)(x) = 2([tex]2x^2[/tex] + 3x)
Distribute the 2:
(fg)(x) = [tex]4x^2[/tex] + 6x
So, (fg)(x) = [tex]4x^2[/tex] + 6x.
Function Multiplication Simplified solution is :- (fg)(x) = [tex]4x^2[/tex] + 6x.
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Decide whether pairs of angles <1 and <2, <6 and<1 , and <5 and <6 are alternate interior angles, same-side interior angles, corresponding angles, or alternate exterior angles. Question content area bottom Part 1 Choose the statement that correctly describes angles 1 and 2.
*PLEASE ANSWER EACH QUESTION. IN ALL 3 PICS. THANKS
Answers
<1 and <2 are same-side interior angles. Option D
<1 and <5 are same-side interior angles. Option B
<6 and <8 are corresponding angles. Option C
How to determine the anglesTo determine the angles, we need to know the following;
Corresponding angles are angles formed in matching corners with the transversal when two parallel lines are intersected by any other lineSame side interior angles are two angles that are found on the interior of two lines and exactly on the same side of the transversal. Adjacent angles are equalNow, from the information given, we have that;
1. <1 and <2 are same-side interior angles because they are found on the same side of the transversal
2. We can see that < 1 and <5 are also same-side interior angles because they are two angles found on the exact same side of the transversal
3. Since corresponding angles are angles on matching corners of a transversal, < 6 and < 8 are corresponding angles
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if m∠2=3x+14° and m∠4=2x+6°, what is
Answers
The angle m∠HAT in the line segment is 54 degrees.
How to find the angle in a line?When line segment intersect, angle relationships are formed such as
vertical angles, linear angles etc.
Therefore, the angle ∠HAT can be found using the relationship of the angles as follows:
m∠MAH = 3x + 14
m∠HAT = 2x + 6
Therefore,
m∠MAH + m∠HAT = 90 degrees
3x + 14 + 2x + 6 = 90
5x + 20 = 90
5x = 90 - 20
5x = 70
x = 24
Therefore,
m∠HAT = 2x + 6 = 2(24) + 6 = 48 + 6 = 54 degrees
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The sum of 18 and a number, x, equals negative 40. Part A Which equation represents this situation? x-18=-40. x-18-40 18+x=-40. 18 + x = 40
Answers
Answer:
18+x = -40
Step-by-step explanation:
The sum of 18 and a number x is translated to "18+x"
Equals means "="
Negative forty is "-40"
Put them all in order and you get 18+x = -40
Mrs. Brown wants to assign eight homework
problems to her Algebra 1 class tonight. If
there are eleven problems she could choose
from, how many different homework sets
could she assign?
A
88
B
165
C
495
D
990
Answers
Mrs. Brown can assign 165 different homework sets to her Algebra 1 class.
We have,
The number of different homework sets Mrs. Brown could assign can be calculated using the combination formula.
In this case, she wants to choose 8 problems out of 11 available problems. The formula for combinations is given by:
C(n, r) = n! / (r! * (n - r)!)
Where n is the total number of items to choose from and r is the number of items to be chosen.
Using this formula:
C(11, 8) = 11! / (8! x (11 - 8)!)
C(11, 8) = 11! / (8! x 3!)
Simplifying the factorial expressions:
C(11, 8) = (11 x 10 x 9 x 8!) / (8! x 3 x 2 x 1)
The factorial terms cancel out:
C(11, 8) = (11 x 10 x 9) / (3 x 2 x 1)
C(11, 8) = 165
Therefore,
Mrs. Brown can assign 165 different homework sets to her Algebra 1 class.
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