Find Upper Bound, Lower Bound and Tight Bound ranges for the following Function. F(n)=10n 2
+4n+2
G(n)=n 2
11. Prove the following statement. a. 2
n 2
−3n=θ(n 2
) b. n 3
=O(n 2
)
Answers
a. 2n² - 3n = θ(n²) (Both upper and lower bounds are n²).
b. n³ ≠ O(n²) (There is no upper bound).
To find the upper bound, lower bound, and tight bound ranges for the functions F(n) = 10n² + 4n + 2 and G(n) = n²/11, we need to determine their asymptotic behavior.
1. Upper Bound (Big O):
For F(n) = 10n² + 4n + 2, the highest-order term is 10n². Ignoring the lower-order terms and constants, we can say that F(n) is bounded above by O(n²). This means that there exists a constant c and a value n₀ such that F(n) ≤ cn² for all n ≥ n₀.
For G(n) = n²/11, the highest-order term is n². Ignoring the constant factor and lower-order terms, we can say that G(n) is also bounded above by O(n²).
2. Lower Bound (Big Omega):
For F(n) = 10n² + 4n + 2, the lowest-order term is 10n². Ignoring the higher-order terms and constants, we can say that F(n) is bounded below by Ω(n²). This means that there exists a constant c and a value n₀ such that F(n) ≥ cn² for all n ≥ n₀.
For G(n) = n²/11, the lowest-order term is n². Ignoring the constant factor and higher-order terms, we can say that G(n) is also bounded below by Ω(n²).
3. Tight Bound (Big Theta):
For F(n) = 10n² + 4n + 2, and G(n) = n^2/11, both functions have the same highest-order term of n². Therefore, we can say that F(n) and G(n) have the same tight bound range of Θ(n²). This means that there exist positive constants c₁, c₂, and a value n₀ such that c₁n² ≤ F(n) ≤ c₂n² for all n ≥ n₀.
In summary:
- F(n) = 10n² + 4n + 2 has an upper bound of O(n²), a lower bound of Ω(n²), and a tight bound of Θ(n²).
- G(n) = n²/11 has an upper bound of O(n²), a lower bound of Ω(n²), and a tight bound of Θ(n²).
Now let's move on to proving the given statements:
a. To prove that 2n² - 3n = θ(n²), we need to show both the upper bound and lower bound.
- Upper Bound (Big O):
For 2n² - 3n, the highest-order term is 2n². Ignoring the lower-order terms and constants, we can say that 2n² - 3n is bounded above by O(n²). This means there exists a constant c and a value n₀ such that 2n² - 3n ≤ cn² for all n ≥ n₀.
- Lower Bound (Big Omega):
For 2n² - 3n, the highest-order term is 2n². Ignoring the lower-order terms and constants, we can say that 2n² - 3n is bounded below by Ω(n²). This means there exists a constant c and a value n₀ such that 2n² - 3n ≥ cn² for all n ≥ n₀.
Since we have shown both the upper and lower bounds to be n², we can conclude that 2n² - 3n = θ(n²).
b. To prove that n³ ≠ O(n²), we need to show that there is no upper bound.
Assuming n³ = O(n²), this would mean that there exists a constant c and a value n₀ such that n³ ≤ cn² for all n ≥ n₀.
However, this statement is not true because as n approaches infinity, n³ grows faster than cn² for any constant c. Therefore, n³ is not bounded above by O(n²), and we can conclude that n³ ≠ O(n²).
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Related Questions
Which value of k makes the equation k\2 =10
Answers
PLSS HELP ILL GIVE YOU A BRAINLIEST AND 30 points!
Answers
height ( alt ) = 12 cm
hypotenuse ( hypo ) = x
Solution -
By Pythagorus theorem
( hypo )^2 = ( base )^2 + ( alt )^2
( x )^2 = ( 9 )^2 + ( 12 ) ^2
( x )^2 = 81 + 144
( x )^2 = 225
( x ) = _/225
( x ) = 15 cm
.:. The value of x ( hypotenuse ) = 15 cm
2.) Given - base = 10 cm
Height = 24 cm
Hypotenuse = x
Solution -
By pythagorus theorem
( hypo )^2 = ( base )^2 + ( alt )^2
( x )^2 = ( 10 )^2 + ( 24 )^2
( x )^2 = 100 + 576
( x )^2 = 676
( x ) = _/676
( x ) = 26
.:. The value of x ( hypotenuse ) = 26 cm
3.) Given - base = 3 cm
Height = 7 cm
Hypotenuse = x
Solution -
By pythagorus theorem
( hypo )^2 = ( base )^2 + ( alt )^2
( x )^2 = ( 3 )^2 + ( 7 )^2
( x )^2 = 9 + 49
( x )^2 = 58
( x ) = _/58
( x ) = 7.6
.:. The value of x ( hypotenuse ) = 7.6 cm
4.) Given - base = 10 cm
Height = 6 cm
Hypotenuse = x
Solution -
By pythagorus theorem
( Hypo )^2 = ( base )^2 + ( alt )^2
( x )^2 = ( 10 )^2 + ( 6 )^2
( x )^2 = 100 + 36
( x )^2 = 136
( x ) = _/136
( x ) = 11.6
.:. The value of x ( hypotenuse ) = 11.6 cm
5.) Given - hypotenuse = 24 cm
height = 6 cm
Base = x
Solution -
By pythagorus theorem
( hypo )^2 = ( base )^2 + ( alt )^2
( 24 )^2 = ( x )^2 + ( 6 )^2
( x )^2 = ( 6 )^2 - ( 24 )^2
( x )^2 = 36 - 576
( x )^2 = -540
( x ) = _/-540
( x ) = 23.2
.:. The value of x ( base ) = 23.2 cm
6.) Given - base = 1 cm
height = 1 cm
hypotenuse = x
Solution -
By pythagorus theorem
( hypo )^2 = ( base )^2 + ( alt )^2
( x )^2 = ( 1 )^2 + ( 1 )^2
( x )^2 = 1 + 1
( x )^2 = 2
( x ) = _/2
( x ) = 1.4
.:. The value of x ( hypotenuse ) = 1.4 cm
7.) Given - hypotenuse = 21 cm
height = 8 cm
Base = x
Solution -
By pythagorus theorem
( hypo )^2 = ( base )^2 + ( alt )^2
( 21 )^2 = ( x )^2 + ( 8 )^2
441 = ( x )^2 + 64
( x )^2 = 64 - 441
( x )^2 = -377
( x ) = _/-377
( x ) = 19.4
.:. The value of x ( base ) = 19.4
8.) given - height = 24 cm
Hypotenuse = 30cm
Base = x
Solution -
By pythagorus theorem
( hypo )^2 = ( base )^2 + ( alt )^2
( 30 )^2 = ( x )^2 + ( 24 )^2
900 = ( x )^2 + 576
( x )^2 = 576 - 900
( x )^2 = -324
( x ) = _/-324
( x ) = 18
.:. The value of x ( base ) = 18 cm
9.) ( i ) lets find ‘x’
Given - base = 9 cm
height = 5 cm
hypotenuse = x
Solution -
By pythagorus theorem
( hypo )^2 = ( base )^2 + ( alt )^2
( x )^2 = ( 9 )^2 + ( 5 )^2
( x )^2 = 81 +25
( x )^2 = 106
( x ) = _/106
( x ) = 10.2
.:. The value of x ( hypotenuse )
= 10.2 cm
( ii ) lets find ‘y’
Given - base = 3 cm
height = 5 cm
Hypotenuse = y
Solution -
By pythagorus theorem
( hypo )^2 = ( base )^2 + ( alt )^2
( y )^2 = ( 3 )^2 + ( 5 )^2
( y )^2 = 9 + 25
( y )^2 = 34
( y ) = _/34
( y ) = 5.8
.:. The value of y ( hypotenuse )
= 5.8 cm
the waiting times between a subway departure schedule and the arrival of a passenger are uniformly distributed between 0 and 8 minutes. find the probability that a randomly selected passenger has a waiting time greater than 4.25 minutes.
Answers
The probability that a randomly selected passenger has a waiting time greater than 4.25 minutes is 0.5278
The uniform distributed between a and b minutes.
The probability of finding a value above x is:
P(X > x) = \(\frac{b-x}{b-a}\)
The time is evenly distributed between 0 and 9 minutes.
This implies that a=0 , b=9
Determine the likelihood that a randomly selected passenger will have to wait longer than 4.25 minutes.
p(X > 4.25) = \(\frac{9-4.25}{9-0}\)
= 0.5278
There is a 52.78% chance that a randomly selected passenger will have to wait longer than 4.25 minutes.
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Which polynomial correctly combines the like terms and expresses the given polynomial in standard form? 8mn5 – 2m6 5m2n4 – m3n3 n6 – 4m6 9m2n4 – mn5 – 4m3n3 n6 7mn5 14m2n4 – 5m3n3 – 6m6 –2m6 – 5m3n3 14m2n4 7mn5 n6 14m2n4 7mn5 – 6m6 – 5m3n3 n6 n6 – 6m6 7mn5 14m2n4 – 5m3n3.
Answers
The standard form of the polynomial function is \(\rm -6m^6+n^6+7mn^5+14m^2^4-5m^3n^3\).
.
The polynomial given is
\(\rm 8mn^5 - 2m^6 +5m^2n^4 -m^3n^3 n^6 - 4m^6 +9m^2n^4 - mn^5 -4m^3n^3\)
What is polynomial?An expression of more than two algebraic terms, especially the sum of several terms that contain different powers of the same variable(s).
Let us rearrange the expression as;
\(\rm 8mn^5 - 2m^6 +5m^2n^4 -m^3n^3+ n^6 - 4m^6 +9m^2n^4 - mn^5 -4m^3n^3\\\\-2m^6- 4m^6 + n^6+8mn^5- mn^5 +5m^2n^4 +9m^2n^4-m^3n^3 -4m^3n^3\)
Simplifying the above expression as;
\(\rm -6m^6+n^6+7mn^5+14m^2^4-5m^3n^3\)
The above expression of the polynomial function is in its standard form.
Hence, the standard form of the polynomial function is \(\rm -6m^6+n^6+7mn^5+14m^2^4-5m^3n^3\).
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3(n-2)=9(6–n)
what is n ?
Answers
Answer:
n = 5
Step-by-step explanation:
Given
3( n - 2) = 9 ( 6 - n)
3n - 6 = 54 - 9n
3 n + 9n = 54 + 6
12n = 60
n = 60/ 12
n = 5
Answer:N=5
Step-by-step explanation:
3n-6=54-9n
+9n +9n
12n-6=54
+6 +6
12n=60
/12 /12
N=5
please help me on this
Answers
Answer:
I got 5/16 as my answer
Step-by-step explanation:
I used a calculator to solve the equation :/
pool workers conducted a chi-square goodness-of-fit test at the 5% significance level. the chi-square value is 1.3892 and the p-value is 0.7081. what can the pool workers conclude?
Answers
The pool workers can conclude that there is not a statistically significant difference between the expected and observed results.
The chi-square value is 1.3892 and the p-value is 0.7081 The p-value of 0.7081 is greater than the 5% significance level, meaning that the null hypothesis can not be rejected.
This means that the difference between the expected and observed results is not statistically significant, and the observed results are likely due to random chance. This means that there is not a significant difference in the observed results from the expected results, and the null hypothesis is accepted.
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SOLVE !!!!!!!!!!!!!!!!!!!!!
9/ (-3) + 6 x 2 – 7
SHOW YOUR WORK
Answers
Answer:
Step-by-step explanation:
9/(-3)+6x2-7
9/-3 = -3,
6 x 2 = 12,
-3 + 12 = 9,
9-7 = 2
Answer:
2
Step-by-step explanation:
9/(-3)+6*2-7
-9/3+6*2-7 simplify 9/3 to 3
-3+6*2-7
-3+12-7
9-7
2
Five thousand,two hundred and twelve and fifty two
Answers
Answer:
5212 + 52.......................
which of the following constructions is this?
Answers
Answer:
acute hope it helps
Step-by-step explanation:
mark as brainiest
Construction of 30° acute angle
Fill in the blank A ____ is a graph of points (x,y) where each x-value is from the original set of sample data, and each y-value is the corresponding Z-score that is a quantile value expected from the standard normal distribution
answer options are
histogram
frequency polygon
scatterplot
normal quantile plot
Answers
Normal quantile plot is a graph of points (x,y) where each x-value is from the original set of sample data, and each y-value is the corresponding Z-score that is a quantile value expected from the standard normal distribution.
What is a Normal Quantile Plot?
A normal quantile plot is a graphical tool used to determine whether a data set is normally distributed or not.
It plots sample data versus a theoretical normal distribution.
In general, the points on the plot should form a straight line if the data is normally distributed. If the data is not normally distributed, the points on the plot will not form a straight line.
A normal quantile plot can be used to evaluate the following:
Whether or not a data set is normally distributedA data set's skewnessA data set's outliersA data set's center and spread whether or not a transformation is required to make a data set normally distributed.The normal quantile plot of the residuals is the most important diagnostic tool for examining whether the assumptions of a linear regression model have been met.
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Select the correct expression that is a factor of the quadratic expression:x^+7x+12A) x+3B) x-1C) x-3D) x-4
Answers
Factoring x^2+7x+12
( x + 4 ) ( x + 3 )
A) x+3 is the correct OPTION.
What is the absolute value of 3 squared?
NO LINKS I NEED THE ANSWER QUICK PLEASE IF YOU CAN
Answers
The ratio of bOYS to giris
that went to camp Ozarks
this summer was 4:5. If
there were 32 boys that
went to the camp, how
many girls were there?
Answers
Answer:
There were a total of 40 girls there
Step-by-step explanation:
32 divided by 4 is 8
So multiply 8 x 5 to get 40
An expression is shown below.
−(2x−1)(3x+4)
Drag and drop numbers into the boxes to make the expression below equal to the expression above.
Answers
I will give a brainliest
b) Find the equation of the trend line (line of best fit). Show your work. (2 points)
Answers
Answer:
Step 1:
Draw a graph - Put Customers on the x-axis and Tips on the y-axis
Equation of a line = mx + c
Then look for where the line crosses the x-axis (this is "c")
Choose two points and find the gradient using this formula:
Change in y/change in x
The number you just got is your "m"
Put it all together and you have y = Mx + C
Hope this helps please mark as brainliest :P
Step-by-step explanation:
Answer:
y = 1.86211x + 4.42878Step-by-step explanation:
Given data in the table added to scatter plot and line of best fit calculated using a graphing calculator.
See the attached graph.
The line of best fit is:
y = 1.86211x + 4.42878
Given f(x) and g(x)= f(k•x), use the graph to determine the value of k
Answers
Answer:
\( \displaystyle \boxed{k = 3}\)
Step-by-step explanation:
to determine the Value of k we need to find the equation of f(x) and g(x)
finding the equation of f(x):
f(x) is a linear function which form is given by:
\( \displaystyle f(x) =mx + b\)
where:
m is the slope b is the y-interceptfrom the graph we acquire that
\(b = 4\)
to figure out m we can consider the following formula:
\( \displaystyle m = \frac{ \Delta y}{ \Delta x} \)
from the graph we obtain that both ∆x and ∆y is 1 thus substitute:
\( \displaystyle m = \frac{ 1}{ 1} \)
simplify:
\( \displaystyle m = 1\)
altogether we acquire:
\( \displaystyle f(x) =x + 4\)
finding the equation of g(x):
likewise f(x)
from the graph we obtain that
∆x=1∆y=3thus substitute:
\( \displaystyle m = \frac{ 3}{ 1} \)
simplify:
\( \displaystyle m = 3\)
from the graph we acquire
b=4altogether substitute therefore
\( \displaystyle g(x) =3x + 4\)
finding the value of k:
given that,
\( \displaystyle \rm f(x) \: \text{and} \: g(x) =f(kx) = 3x + 4\)
so our equation is
\( \displaystyle kx + 4 = 3x + 4\)
cancel 4 from both sides:
\( \displaystyle kx = 3x \)
divide both sides by x:
\( \displaystyle \boxed{k = 3}\)
and we're done!
Disclaimer:
at the following link we posted the complete question
the complete question:
https://brainly.com/question/24201282Answer:
k = 3
Step-by-step explanation:
Graph is in attachment .
Finding equation of first line [ f(x) ] :-
We can see that the line passea through point ( 0,4) . Therefore ,
y intercept = 4Also it passes through (-4,0 )
Now let's find out slope .So that we can use the slope intercept form to find the equⁿ of the line.
m = y ₂ - y₁ / x ₂ - x₁ m = 4 - 0 / 0 - (-4) m = 4/4 m = 1Equation of f(x) :-
y = mx + c y = 1 * x + 4 y = x + 4 f(x) = x + 4Similarly when we will find the equⁿ of g(x) :-
y = 3x + 4g(x) = 3x + 4By question :-
f(x ) and g(x) = f ( k • x ) 3x + 4 = kx + 4 kx = 3x k = 3
The apothem is the perpendicular distance from the _____ of a regular polygon to any one of its sides.
Answers
The apothem is the perpendicular distance from the center of a regular polygon to any one of its sides.
The apothem of a regular polygon is the perpendicular distance from the center of the polygon to any one of its sides because it serves as a measure of the inradius, or the radius of the circle inscribed within the polygon. In a regular polygon, all sides are congruent and all interior angles are equal, so the apothem is the same for all sides.
To see this, imagine drawing radii from the center of the polygon to each vertex, and then drawing the perpendicular bisector of each side. These perpendicular bisectors will all intersect at the same point, which is the center of the polygon. The apothem is the length of the line segment connecting this center point to any one of the sides of the polygon.
The apothem is useful in finding the area of a regular polygon. By using the formula for the area of a regular polygon in terms of the apothem and the number of sides, one can find the area of a regular polygon without having to find the exact shape of the polygon.
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A container is shaped like a cylinder and has a volume of 125.7 cm³. If
the height of the container is 10 cm, what is the radius of the container?
Answers
The radius of the cylindrical container with a volume and height of 125.7cm³ and 10cm is 2cm.
What is the radius of the cylindrical container?The volume of a cylinder is expressed as;
V = π × r² × h
Where r is radius of the circular base, h is height and π is constant pi ( π = 3.14 )
Given the data in the question;
Volume of the cylinder V = 125.7cm³Height of the cylinder h = 10cmRadius of the cylinder r = ?To determine the radius of the cylindrical container, plug the given values into the formula above and solve for r.
V = π × r² × h
r² = V / ( π × h )
r² = 125.7cm³ / ( 3.14 × 10cm )
r² = 125.7cm³ / ( 31.4cm )
r = √( 125.7/31.4 cm² )
r = 2cm
Therefore, the radius of the container is 2 centimeters.
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Evaluate each expression.
6! =
3! • 2! =
.
6!
3!
11
=
Answers
Answer:
See below
Step-by-step explanation:
6! = 6*5*4*3*2*1 = 7203! . 2! = 3*2*1. 2*1= 126!/3! = 6*5*4* 3!/3! = 120A turtle is 1.5 feet long. If there were 6 of these turtles standing in a line, what would the length of the line be in yards?
Answers
Answer:
3
Step-by-step explanation:
Answer:
6 x 1.5 = 9 => 0.25 yards
Step-by-step explanation:
6 in a line, means times 6. Times do unit converison to get 0.25 yd.
the percentage of the original area of wetlands currently left in the united states is approximately: question 44 options: 10%. 25%. 50%. 65%. 75%.
Answers
The percentage of the original area of wetlands currently left in the United States is approximately 50%. So, the correct
option is 50% (option 3).
Percentages are a way of expressing a proportion or a fraction as a part of 100. It is denoted by the symbol "%".
According to the United States Environmental Protection Agency (EPA), it is estimated that about 50% of the original
wetlands in the contiguous United States have been lost since the 1600s due to human activities such as agriculture,
development, and urbanization. Therefore, the correct answer to the question is 50%.
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The percentage of the original area of wetlands currently left in the United States is approximately 50%.
Wetlands are regions of land where the soil is continually or intermittently soaked with water. Wetlands have a particular hydrology, soil, and vegetation mix that results in specialised ecosystems that offer a variety of ecological functions.
There are many different types of habitats where wetlands can be found, such as coastal locations, interior regions, and high-altitude mountain regions. They come in a variety of shapes, such as marshes, swamps, bogs, fens, and estuaries, and can be freshwater, brackish, or saline.
Wetlands are significant for several reasons. For species, such as migrating birds, amphibians, and fish, they offer crucial habitats. They also aid in removing contaminants from water, lessen the effects of flooding, and give people access to recreational activities. Wetlands are crucial for carbon sequestration as well.
Based on the information provided, the question is asking for the approximate percentage of the original area of wetlands currently left in the United States. The answer is approximately 50%.
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A 10-foot ladder leans against a wall so that it is 6 feet high at the top. The ladder is moved so that the base of the ladder travels toward the wall twice the distance that the top of the ladder moves up. How much higher is the top of the ladder now? (Hint: Let 8−2x be the distance from the base of the ladder to the wall.)
Answers
Answer:
Step-by-step explanation:
The Pythagorean Theorem holds for both positions.
The first configuration is an 6-8-10 right triangle.
The second configuration using the Pythagorean Theorem:
10^2 = (8-2x)^2 + (6 + x)^2
100 = 64 - 32x + 4x^2 + 36 + 12x + x^2
100 = 100 - 20x + 5x^2
5x^2 - 20x = 0
5x(x - 4) = 0
x = 4 feet (we throw out the x = 0 solution as extraneous)
Answer:
Step-by-step explanation:
The Pythagorean Theorem holds for both positions.
The first configuration is an 6-8-10 right triangle.
The second configuration using the Pythagorean Theorem:
10^2 = (8-2x)^2 + (6 + x)^2
100 = 64 - 32x + 4x^2 + 36 + 12x + x^2
100 = 100 - 20x + 5x^2
5x^2 - 20x = 0
5x(x - 4) = 0
x = 4 feet (we throw out the x = 0 solution as extraneous)
need answer asap!! will give brainiliest!!
define the term linear equation in your own words.
Answers
Plz mark me as braniliest
A thin, uniform rod of mass MI and length L, is initially at rest on a frictionless horizontal surface: The moment of inertia of the rod about its center of mass is MIL^2/2_ As shown in Figure I, the rod is struck at point Pby mass m2 whose initial velocity perpendicular t0 the rod. After the collision, mass m2 has velocity -[ / 2v as shown in Figure IL Answerthe following in terms ofthe symbols given. Clearky shon alLwork for each stcp a. Using the principle of conservation of linear momentum; determine the velocity v' of the center of mass of this rod after the collision. b. Using the principle of conservation of angular momentum; determine the angular velocity of the rod about its center of mass after the collision c. Determine the ratio of the final kinetic energy Of the system resulting from the collision to the initial kinetic energy Your finalexpression should bein terms ofthe masses_only
Answers
a. The velocity v' of the center of mass of this rod after the collision is v' = m2v/(2(MI + m2))
b. The angular velocity of the rod about its center of mass after the collision is ω = -m2 × v/(4×I_cm)
c. Final kinetic energy / initial kinetic energy = 1/2 + (1/16) × (MI/m2)
The principle of conservation of linear momentum, the total momentum before the collision is equal to the total momentum after the collision. Initially, the rod is at rest, so its momentum is zero.
After the collision, the velocity of mass m2 is -v/2, and its mass is m2. Therefore, its momentum after the collision is -m2v/2.
The center of mass of the system must have the same velocity as the momentum is conserved.
The total mass of the system is M = MI + m2. Thus,
0 = (MI + m2) × v' - m2 × v/2
v' = m2v/(2(MI + m2))
The principle of conservation of angular momentum, the total angular momentum before the collision is equal to the total angular momentum after the collision.
Initially, the rod is at rest, so its angular momentum is zero.
After the collision, the velocity of mass m2 is -v/2, and its distance from the center of mass of the rod is L/2.
The angular momentum of mass m2 about the center of mass of the rod is given by m2 × (L/2) × (v/2).
The angular momentum of the rod about its center of mass is I_cm × ω, where I_cm is the moment of inertia of the rod about its center of mass, and ω is the angular velocity of the rod about its center of mass.
Thus,
0 = 0 + m2 × (L/2) × (v/2) + I_cm × ω
ω = -m2 × v/(4×I_cm)
The initial kinetic energy of the system is given by (1/2)MI0² + (1/2)m2v², which simplifies to (1/2)m2v².
The final kinetic energy of the system is given by (1/2)MIv'² + (1/2)m2(-v/2)², which simplifies to (1/2)(MI + m2)(m2v²)/(4(MI + m2)²) + (1/8)m2v².
Thus,
Final kinetic energy / initial kinetic energy
= [(1/2)(MI + m2)(m2v²)/(4(MI + m2)²) + (1/8)m2v²] / ((1/2)m2v²)
= 1/2 + (1/16) × (MI/m2)
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a. The velocity v' of the center of mass of this rod after the collision is (m₂ × v) / (2 × MI)
b. ω' = 0
c. (final kinetic energy) / (initial kinetic energy) = 0
How did we get the values?To solve this problem, use the principles of conservation of linear momentum and angular momentum.
a. Conservation of linear momentum:
Before the collision:
The initial linear momentum of the system is zero since the rod is at rest.
After the collision:
The final linear momentum of the system is the sum of the linear momentum of the rod and mass m₂.
The linear momentum of the rod can be calculated using its mass (MI) and velocity (v') as MI × v'.
The linear momentum of mass m₂ can be calculated using its mass (m₂) and velocity (-[v / 2]) as -m₂ × [v / 2].
Setting up the conservation of linear momentum equation:
0 = MI × v' - m₂ × [v / 2]
Solving for v':
MI × v' = m₂ × [v / 2]
v' = (m₂ × v) / (2 × MI)
b. Conservation of angular momentum:
Before the collision:
The initial angular momentum of the system is zero since the rod is at rest.
After the collision:
The final angular momentum of the system is the sum of the angular momentum of the rod and mass m2.
The angular momentum of the rod can be calculated using its moment of inertia (MIL²/²) and angular velocity (ω') as (MIL²/² × ω'.
The angular momentum of mass m2 can be calculated using its moment of inertia (0 since it's a point mass) and angular velocity (-[v / (2L)]) as 0.
Setting up the conservation of angular momentum equation:
0 = (MIL²/²) × ω' + 0
Solving for ω':
(MIL²/²) × ω' = 0
ω' = 0
c. Ratio of final kinetic energy to initial kinetic energy:
The initial kinetic energy of the system is zero since the rod is at rest.
The final kinetic energy of the system can be calculated by considering the kinetic energy of the rod and mass m₂.
The kinetic energy of the rod can be calculated using its moment of inertia (MIL²/²) and angular velocity (ω') as (MIL²/²) × (ω')².
The kinetic energy of mass m₂ can be calculated using its mass (m2) and velocity (-[v / 2]) as (m₂ × [v / 2])² / (2 × m₂).
The ratio of final kinetic energy to initial kinetic energy is:
(final kinetic energy) / (initial kinetic energy) = [(MIL²/²) × (ω')² + (m₂ × [v / 2])² / (2 × m₂)] / 0
Since ω' = 0, the numerator becomes 0.
Therefore, the ratio is 0.
In summary:
a. v' = (m₂ × v) / (2 × MI)
b. ω' = 0
c. (final kinetic energy) / (initial kinetic energy) = 0
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3w + 4x – 15y + 6w + 7x + 7y
5w + 3y + 9x + 2x - 11y + 4w
7x - 3y + 12w + 5y - 3w + 4x
4w + 3x – y + 5w - 14x - 7y
how many of these are equal?
Answers
Answer:
3w, 4x, 4w, 7y, 7x, 3y, 5w,
Step-by-step explanation:
If there is a decimal in the divisor what is the rule?
Pls help asap
Answers
Answer:
Multiply
Step-by-step explanation:
You need to multiply the number by 10 or 100 or 1000 according to the number. but after multiplying it should become a whole digit. And you also need to multiply 10 or 100 or 1000 in the dividend.
Hope this has helped you.
The following data values represent a sample. What is the variance of the sample? x = 7. Use the information in the table to help you.
A. 14.4
B. 4.2
C. 18
D. 3.8
X 7 5 11 1 11
(x₁ - x)² 0 4 16 36 16
Answers
The variance of the sample mean x = 7 is 18(option c).
Given table:
X 7 5 11 1 11
(x₁ - x)² 0 4 16 36 16
n = number of observations = 5
Variance σ² = 1/n-1 Σ\(\ \ n} \atop {i=1} \right.\) (\(X_{1}\) - X)²
= 1/5-1 (0 +4 + 16 + 36 + 16)
= 1/4(72)
= 18
Therefore variance σ² = 18
Hence the variance of the sample mean x = 7 is 18.
so option c is correct.
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Sandy bought a soft drink for 2 dollars and 5 candy bars. She spent a total of $12. How much did each candy bar cost?
Answers
Answer:
If Sandy bought a soft drink for 2 dollars that's 2 dollars subtracted from the total of $12.00 spent
If we have 10 dollars and bought 5 candy bars we would divide so 10 divided by 5 = 2 so each candy bar costed $2 dollars
PLEASE HELP
WILL MARK BRAND
Answers
Answer:
Roots = { 2, 5}
Vertex = (3.5, - 2.25)
Step-by-step explanation:
Roots
From the graph
x = {2, 5}
From factoring the equation
(x - 2)(x - 5) = 0
x = {2, 5}
--------------------------
The equation turns at the vertex
From the graph
(3.5, -2.25)
From the equation
X coordinate of vertex
x = -b/2a
x = 7/2 = 3.5
y = (3.5)² - 7(3.5) + 10
y = 12.25 -24.5 + 10
y = -2.25