The asymptotic upper and lower bounds for T(n) in the given recurrence [tex]\(T(n) = 2T\left(\frac{n}{2}\right) + n^4\)[/tex] are both [tex]\(\Theta(n^4)\)[/tex]
How To find asymptotic upper and lower bounds for T(n)To find asymptotic upper and lower bounds for [tex]\(T(n)\)[/tex] in the given recurrence [tex]\(T(n) = 2T(n/2) + n^4\),[/tex] we can use the Master Theorem. The Master Theorem is a useful tool for solving recurrence relations of the form [tex]\(T(n) = aT(n/b) + f(n)\)[/tex], where [tex]\(a \geq 1\), \(b > 1\),[/tex] and [tex]\(f(n)\)[/tex] is an asymptotically positive function.
In this case, we have[tex]\(a = 2\), \(b = 2\),[/tex] and [tex]\(f(n) = n^4\).[/tex] Comparing [tex]\(f(n) = n^4\)[/tex] with [tex]\(n^c\),[/tex] where [tex]\(c = \log_b(a)\),[/tex] we have [tex]\(f(n) = n^4\)[/tex] and[tex]\(n^c = n^{\log_2(2)} = n[/tex]). Since [tex]\(f(n) = n^4\)[/tex] grows faster than [tex]\(n\),[/tex]we fall into case 3 of the Master Theorem.
Case 3 states that if [tex]\(f(n) = \Omega(n^c)\)[/tex] for some constant [tex]\(c > \log_b(a)\),[/tex] and if [tex]\(a \cdot f(n/b) \leq kf(n)\)[/tex] for some constant [tex]\(k < 1\)[/tex] and sufficiently large [tex]\(n\)[/tex], then [tex]\(T(n)\)[/tex] has the same asymptotic upper and lower bounds as [tex]\(f(n)\).[/tex]
In our case,[tex]\(f(n) = n^4\)[/tex] and[tex]\(n^c = n\).[/tex] Since[tex]\(n^4 = \Omega(n)\)[/tex], we satisfy the condition for case 3. We also have[tex]\(a \cdot f(n/b) = 2 \cdot (n/2)^4 = (n/2)^4 = (1/16) \cdot n^4 \leq (1/16) \cdot n^4 = (1/16) \cdot f(n)\)[/tex] for sufficiently large [tex]\(n\).[/tex]
Therefore, by the Master Theorem, the asymptotic upper and lower bounds for [tex]\(T(n)\)[/tex] in the given recurrence [tex]\(T(n) = 2T(n/2) + n^4\)[/tex] are both[tex]\(\Theta(n^4)\).[/tex]
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Suppose that you began a one-year study of tuberculosis (TB) in a subsidized housing community in the Lower East Side of New York City on January 1st, 2020. You enrolled 500 residents in your study and checked on their TB status on a monthly basis. At the start of your study on January 1st, you screened all 500 residents. Upon screening, you found that 20 of the healthy residents were immigrants who were vaccinated for TB and so were not at risk. Another 30 residents already had existing cases of TB on January 1st. On February 1st, 5 residents developed TB. On April 1st, 5 more residents developed TB. On June 1st, 10 healthy residents moved away from New York City were lost to follow-up. On July 1st, 10 of the residents who had existing TB on January 1st died from their disease. The study ended on December 31, 2010. Assume that once a person gets TB, they have it for the duration of the study, and assume that all remaining residents stayed healthy and were not lost to follow-up.
Using the same TB scenario, what was the case-fatality rate among residents with TB over the course of the year?
25.0%
14.3%
20.0%
None of the above
The case-fatality rate among residents with TB over the course of the year is approximately 33.3%. None of the answer choices provided (25.0%, 14.3%, 20.0%) match this calculated value, so the correct answer is "None of the above."
The case-fatality rate among residents with TB over the course of the year can be calculated by dividing the number of residents who died from TB by the total number of residents with TB at the start of the study. Let's calculate it based on the given information.
On January 1st, there were 30 residents with existing cases of TB. On July 1st, 10 of these residents died from their disease.
Case-Fatality Rate = (Number of TB deaths / Number of residents with TB) * 100%
Case-Fatality Rate = (10 / 30) * 100% ≈ 33.3%
Therefore, based on the given information, the case-fatality rate among residents with TB over the course of the year is approximately 33.3%. None of the answer choices provided (25.0%, 14.3%, 20.0%) match this calculated value, so the correct answer is "None of the above."
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The number of hours Brittany has been hiking varies directly with the total distance she has traveled, in miles. She can hike 8 miles in 3 hours. How many miles can Brittany hike in 12 hours?
Answer:
24 miles
Step-by-step explanation:
for every 3 miles she can hike 8 miles so she can hike 24 miles in 12 hours
if the equation has infinitely many solutions for x, what is the value of b ?
If the equation has infinitely many solutions for x, the value of b that results in infinitely many solutions for x is b = 5/2.
To determine the value of b in the equation a(x + b) = 4x + 10, we can analyze the conditions that would result in infinitely many solutions for x.
First, let's simplify the equation:
a(x + b) = 4x + 10
Expanding the left side of the equation
ax + ab = 4x + 10
Now, let's rearrange the equation to isolate x:
ax - 4x = 10 - ab
Factor out x on the left side:
x(a - 4) = 10 - ab
For the equation to have infinitely many solutions, the coefficient of x (a - 4) must be zero, because any value of x would satisfy the equation.
Therefore, we have:
a - 4 = 0
Solving for a:
a = 4
Substituting the value of a into the equation:
4 - 4 = 0
0 = 0
The equation holds true, indicating that a = 4 satisfies the condition for infinitely many solutions. Now, we can find the value of b.
Substituting a = 4 into the original equation:
4(x + b) = 4x + 10
Expanding:
4x + 4b = 4x + 10
Since the x terms cancel out, we are left with:
4b = 10
Dividing both sides by 4:
b = 10/4
Simplifying:
b = 5/2
Therefore, the value of b that results in infinitely many solutions for x is b = 5/2.
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A large right triangle is going to be a part of a geometric sculpture, as shown below.
The hypotenuse will be 10 feet long. The length of one leg of the triangle is 4 feet less
than twice the other leg. Find the length of each leg, in feet, and separate them with a
comma.
H
ft
10
2x-4
Answer:
Step-by-step explanation:
Let's call one leg of the triangle "x" and the other leg "2x-4". We can use the Pythagorean theorem to solve for the lengths of the legs:
x^2 + (2x-4)^2 = 10^2
x^2 + 4x^2 - 16x + 16 = 100
5x^2 - 16x - 84 = 0
We can solve this quadratic equation by factoring or using the quadratic formula. Factoring gives us:
(5x + 14)(x - 6) = 0
So x = -14/5 or x = 6. We can discard the negative solution since we're dealing with lengths. Therefore:
x = 6
2x - 4 = 8
The lengths of the legs are 6 feet and 8 feet.
in a Cantilever beam of mass 5 kg carrying an end mass 3.85 kg .then the equivalent mass is 8 ib 5 kg 5 ft 7.5 kg in The piston travel diagrams the first order radius equal to R/L crank angle Crank radius Connected rod length The gas pressure force depend on * Piston acceleration Crank radius Piston velocity Piston travel 2 points 2 points 2 points
In a cantilever beam system, where a beam carries an end mass, the equivalent mass is the combined mass of the beam and the end mass. The equivalent mass is calculated by adding the mass of the beam and the end mass. In the context of piston travel diagrams, the gas pressure force depends on the piston acceleration, crank radius, and piston velocity, with the piston travel being influenced by the crank angle, crank radius, and connected rod length.
In a cantilever beam system, the equivalent mass is determined by adding the mass of the beam to the mass of the end mass. In the given example, the beam has a mass of 5 kg, and the end mass is 3.85 kg. Therefore, the equivalent mass of the system is 5 kg + 3.85 kg = 8.85 kg.
Regarding piston travel diagrams, the gas pressure force acting on the piston depends on several factors. The piston acceleration, which represents the rate of change of piston velocity with respect to time, influences the gas pressure force. A higher acceleration results in a greater gas pressure force acting on the piston.
The crank radius, which is the distance from the center of the crankshaft to the point where the connecting rod is attached, also affects the gas pressure force. The piston velocity, which is the rate at which the piston moves, is influenced by the crank radius. A larger crank radius leads to higher piston velocity and, consequently, greater gas pressure force.
The piston travel, representing the linear displacement of the piston, is influenced by the crank angle, crank radius, and connected rod length. These factors determine the motion of the piston and the overall behavior of the system.
In summary, the equivalent mass in a cantilever beam system is obtained by adding the mass of the beam and the end mass. In piston travel diagrams, the gas pressure force depends on the piston acceleration, crank radius, and piston velocity. The piston travel is influenced by the crank angle, crank radius, and connected rod length.
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8.
Felicity asked 100 students how they came to school one day.
Each student walked or came by bicycle or came by car.
49 of the 100 students are girls.
10 of the girls came by car.
16 boys walked
21 of the 41 students who came by bicycle are boys.
Work out the total number of students who walked to school.
Answer:
i think its 33
Step-by-step explanation:
10 Girls came by Car, 20 Girls came by bicycle.
19 Girls walked.
16 boys walked, 21 boys came with bicycle.
14 boys walked.
In total 33 students walked
Your cell phone plan costs $39.99 per month plus $0.11 for each text message you send or receive. You have at most $45 to spend on your cell phone bill. What is the maximum number of text messages that you can send or receive next month? The maximum number of text messages is nothing.
Answer:
The maximum number of text messages is 372.8Step-by-step explanation:
Given that the monthly cost is $39.99
and also $0.11 for each text (sent or recieved)
let y be the total cost
let x be the number of text
the model for the total cost is given as
y=3.99+0.11x
given that y=45
45=3.99+0.11x
45-3.99=0.11x
41.01=0.11x
divide both sides by 0.11
x=41.01/0.11
x=372.8
Write the prime factorization for the number 50.
Write the prime factorization for the number 120.
Someone please help me
Answer:
4: 2x2
7: 3+4
9: 27 divided by 3
11: 5+6
12: 6x2
14: 7x2
17: 9+8
21: 3x7
26: 11+15
27: 3x9
Step-by-step explanation:
We have to use equations to represent the following numbers: 4, 7, 9, 11, 12, 14, 17, 21, 26, 27 only using the numbers 1,2,3,4,5
12 Write the decimal equivalent of the following: answer in the space.
2 5/8=_____
Answer:
2.625
Step-by-step explanation:
2 + 5/8
2 + .625 = 2.625
5/8 = .625
Answer:
ill help
Step-by-step explanation:
mathstronauts has 3 bags of marbles. bag a has 345 marbles. bag c has 180 marbles.he says bag a has the most and bag c has the least marbles. how many marbles could be in bag b explain your reasoning
Step-by-step explanation:
i hope i have been useful buddy.
good luck ♥️ ♥️♥️.
Solve. 6x +12x = 72.
x=
PLEASE HELP I REALLY NEED HELP
Answer:
x=4
Step-by-step explanation:
6x+12x=72
18x=72
x=4
Can someone answer???my question answer and explanation I’ll give brainlist and points I asked a bunch of times
Answer:
A. see instructions below
B. w = 3
Step-by-step explanation:
A. Apparently, your (machine instructions, algebra) are ...
(multiply by 5, 5w)
(subtract 6, 5w-6)
(divide by 3, (5w-6)/3)
(add 2, (5w-6)/3 +2)
(output the number, 5)
__
B. Working backward, we want to undo each instruction.
(subtract 2, 5-2 = 3)
(multiply by 3, 3·3 = 9)
(add 6, 9+6 = 15)
(divide by 5, 15/5 = 3)
(input the number, 3)
The equation (5w-6)/3 +2 = 5 has the solution w = 3.
Sara makes $12 an hour working at the mall. Shae, her sister, just received
$100 for her birthday. Together they are saving to buy a $424 TV. How
many more hours does Sara need to work until that can purchase the TV?
And Show your work please
Answer:
x=27
Step-by-step explanation:
let x be the amount of hours she needs to work
she gets 12 dollars per hour, and they already have 100
12x+100=424
12x=324
x=324/12
x=27
Answer:
X=27
Step-by-step explanation:
I took the top number, 27 and I multiplied it by 12 and I got 324 and since Shae, her sister, has $100 and $324+$100=$400, she has to work 27 more hours.
If you need to like show equations then just take the 2 equations I did.
Could I please have BRAINLIEST?The demand for good X has been estimated by Qxd=12-3Px + 3Py. Suppose that good X sells at $2 per unit and good Y sells for $1 per unit. Calculate the own price elasticity. Multiple Choice -0.2 -0.67
Therefore, the own price elasticity of good X is approximately -0.67.(rounded to 2 decimal places)
To calculate the own price elasticity of good X, we need to use the formula:
Own Price Elasticity = (% change in quantity demanded) / (% change in price)
Given that the price of good X is $2 per unit, we can substitute the price (Px) into the demand equation:
Qxd = 12 - 3Px + 3Py
Substituting Px = $2 and Py = $1 into the equation, we get:
Qxd = 12 - 3(2) + 3(1)
= 12 - 6 + 3
= 9
Now, let's calculate the quantity demanded when the price changes by a small amount, let's say $0.10. We can substitute Px = $2.10 into the demand equation:
Qxd' = 12 - 3(2.10) + 3(1)
= 12 - 6.30 + 3
= 8.70
Using these values, we can calculate the percentage change in quantity demanded and the percentage change in price:
% change in quantity demanded = (Qxd' - Qxd) / Qxd * 100
= (8.70 - 9) / 9 * 100
= -0.30 / 9 * 100
= -3.33% (rounded to 2 decimal places)
% change in price = (Px' - Px) / Px * 100
= (2.10 - 2) / 2 * 100
= 0.10 / 2 * 100
= 5%
Now, let's calculate the own price elasticity using the formula:
Own Price Elasticity = (% change in quantity demanded) / (% change inprice)
= -3.33% / 5%
= -0.67 (rounded to 2 decimal places)
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Simplify. 3/5 · 3/9 help a guy out
Answer: 0.18 or 9/50
Step-by-step explanation:
Answer:
well I am not sure anymore and I don't know how to delete my answer so sorry
Step-by-step explanation
What are the five guidelines when solving equations?
Which equation does NOT represent a function?
A) y=12
B) 3x+4y=6
C) x=y^2-9
D) y=3x^2-5
Answer:
A
Step-by-step explanation:
Because it is the only one that is not asking you to solve. It is simply stating that y is equal to 12.
Hope this helps!
solve for x. 2x +5=33
Answer:
x = 14
Step-by-step explanation:
2x +5=33
Subtract 5 from each side
2x+5-5 = 33-5
2x = 28
Divide by 2
2x/2 =28/2
x = 14
find all missing angles. show your work.
The angles are as follows:
∠1 = 51°
∠2 = 34°
∠3 ≅ 95°
∠4 = 38°
∠5 ≅ 47°
∠6 = 74°
∠7 = 59°
The angles are as follows:
∠1 = 180 - 129 = 51° (angle on a straight line)
∠2 = 180 - 95 - 51 = 34° (sum of angles in a triangle is 180 degrees)
∠3 ≅ 95° (vertically opposite angles)
∠4 = 180 - 47 - 95 = 38° (sum of angles in a triangle)
∠5 ≅ 47°(vertically opposite angles)
∠6 = 180 - 47 - (180 - 121) = 74°(sum of angles in a triangles)
∠7 = 180 - 121 = 59°(angle on a straight line)
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Rae was asked to compute 243, 192 x 4. She wrote: 243, 192 x 4 - 8 | 16 12 | 4 | 36 | 8 1. What was Rae thinking about? Is her answer correct? 2. Translate Rae's answer into a number that the rest of the world can understand. 3. Use Rae's method to find the answers to these multiplication exercises. Be sure to translate your answers into familiar base 10 numbers. 156 x 3 2873 x 2 71, 181 x 5 3, 726, 510, 392 x 2
This number does not correspond to the correct product of 243,192 multiplied by 4.
Rae's method appears to involve breaking down the numbers into their components and performing the multiplication operations separately. Let's analyze her answer:
243, 192 x 4 - 8 | 16 12 | 4 | 36 | 8 1
It seems that she broke down the number 243,192 into its individual digits (2, 4, 3, 1, 9, and 2) and performed separate calculations for each digit. However, her approach is not mathematically accurate or coherent.
Now, let's translate Rae's answer into a number that can be understood in base 10:
243,192 x 4 - 8 | 16 12 | 4 | 36 | 8 1
If we reassemble the digits without the vertical separators, we get: 8161243681.
However, this number does not correspond to the correct product of 243,192 multiplied by 4.
To find the correct answers using Rae's method, let's apply it to the provided multiplication exercises:
156 x 3:
Applying Rae's method:
156 x 3 - 6 | 15 18 | 3
Reassembling the digits: 6153183
The correct answer: 156 x 3 = 468
2873 x 2:
Applying Rae's method:
2873 x 2 - 6 | 14 16 | 3
Reassembling the digits: 614163
The correct answer: 2873 x 2 = 5746
71,181 x 5:
Applying Rae's method:
71,181 x 5 - 5 | 35 25 | 5
Reassembling the digits: 535255
The correct answer: 71,181 x 5 = 355,905
3,726,510,392 x 2:
Applying Rae's method:
3,726,510,392 x 2 - 4 | 7 14 12 10 | 3
Reassembling the digits: 471412103
The correct answer: 3,726,510,392 x 2 = 7,453,020,784
So, while Rae's approach is not mathematically valid, we were able to compute the correct answers using traditional multiplication methods.
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Mineral hardness is rated from 1, the softest, to 10, the hardest. The table shows some minerals with their hardness ratings.
Which mineral listed in the table can be used to cut all of the other minerals?
A) Corundum
B) Diamond
C) Gypsum
D) Quartz
Frost wedging is a form of weathering caused by water seeping into the cracks of rocks. This type of weathering is common in regions that have very cold winters. How does frost wedging take place?
A) This is a physical weathering process — when water freezes, it expands, pushing the rock apart slowly over a period of time.
B) This is a chemical weathering process — carbonic acid formed by the reaction of water and carbon dioxide breaks the bonds that hold the rock together.
C) This is a physical weathering process — when water freezes, the temperature of the rock is decreased, causing the outer layers of the rock to crumble off.
D) This is a chemical weathering process — water reacts with the chemicals in the rock by a process call hydrolysis and breaks the bonds that hold the rock together.
-3.6,-5.4,-8.1,-12.15 arithmetic or geometric or neither.
Answer:
Geometric Sequence
Step-by-step explanation:
1. Check the difference.
The difference between the 1st and 2nd term
[tex]( - 5.4) - ( - 3.6) = - 1.8[/tex]
The difference between the 2nd and 3rd term
[tex]( - 8.1) - ( - 5.4) = - 2.7[/tex]
The difference is not the same. Therefore, it is not an arithmetic sequence.
2. Check the ratio
The ratio between the 1st and 2nd term
[tex]( - 5.4) \div ( - 3.6) = 1.5[/tex]
The ratio between the 2nd and 3rd term
[tex]( - 8.1) \div ( - 5.4) = 1.5[/tex]
The ratio is the same. Therefore, it is a geometric sequence.
Which method or methods listed below describe a solid system for keeping financial records?
1. Making a mental note of all purchases made during the week.
II. Recording money spent in a hand-held notebook.
III. Organizing receipts and transactions on a computer.
a. I only
b. I only
C. II and III
d. I, II, and III
Answer:
C :)
Step-by-step explanation:
Answer:
C
Step-by-step explanation:
Respond to the following required content in a minimum of 175
words:
Provide and discuss a specific example of an
algorithm.
In general, is there only one correct algorithm for a given
problem? Expla
Answer:
Its a perfect square root f ..
Step-by-step explanation:
The Sugar Sweet Company will choose from two companies to transport its sugar to market. The first company charges $6500 to rent trucks plus an additional fee of $125.25 for each ton of sugar. The second company charges $5897 to rent trucks plus an additional fee of $175.50 for each ton of sugar. 1)for what amount of sugar do the two companies charge the same ? []tons 2) what is the cost when the two companies charge the same? s[]
Answer:
The two companies charge the same for 12 tons of sugar.
Amount charged by each of the company [tex]=\$8003[/tex]
Step-by-step explanation:
Let x denotes amount of sugar for which the two companies charge the same.
Amount charged by the first company [tex]=6500+125.25x[/tex]
Amount charged by the second company [tex]=5897+175.50x[/tex]
To find amount of sugar for which the two companies charge the same,
solve [tex]6500+125.25x=5897+175.50x[/tex]
[tex]6500+125.25x=5897+175.50x\\6500-5897=175.50x-125.25x\\603=50.25x\\x=\frac{603}{50.25}\\ x=12[/tex]
So, the two companies charge the same for 12 tons of sugar.
Amount charged by the first company [tex]=6500+125.25(12)=8003[/tex]
Amount charged by the second company [tex]=5897+175.50(12)=8003[/tex]
Amount charged by each of the company [tex]=\$8003[/tex]
A map has a scale of 3 cm 18 km in Riverside Riverside in Smithville are 54 km apart then they are how far apart on the map
Divide actual km by scale km:
54/18 = 3
Multiply scale cm by that:
3 x 3 = 9
They are 9cm apart on the map
Morgan purchased a new car, which depreciates over time. Morgan’s car depreciates
according to the function y=62,150( . 9999)xx, where x represents the number of months since
the car was purchased. What is the range of the exponential function y based on its equation
and the context of the problem?
The range of the exponential function y based on its equation and the context of the problem is a set of positive real numbers less than the initial value of 62,150, representing the decreasing value of the car over time.
The given function represents the depreciation of Morgan's car over time, where x represents the number of months since the car was purchased. The function is y = 62,150 * (0.9999)^x.
To determine the range of the exponential function y, we need to consider the context of the problem. In this case, the range represents the possible values for the car's depreciation, or the value of y.
The base of the exponential function is 0.9999, which is less than 1. When a value between 0 and 1 is raised to a positive power, it decreases exponentially as the power increases. Therefore, as x (the number of months) increases, the value of y (the car's depreciation) decreases.
Since the base is less than 1, the range of the exponential function y is also less than the initial value of 62,150. As x approaches infinity, the value of y approaches 0, but it will never actually reach 0.
In practical terms, the range of the exponential function represents the decreasing value of the car over time. It indicates that the car's depreciation is continuous and will approach zero but will never completely reach zero.
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find out the value of x.Help me out pls!!!
Answer:
80
Step-by-step explanation:
the A and X are equal
I will give brainiest. need help asap!
f(6)=50(6)+30
Answer:
f=55
Step-by-step explanation:
f(6)=50(6)+30
then: 6F=330 using distributive, and communitive property
then: divide 6 on each side
and ur answer will be f=55
