2 (a) Given a table with n numbers, where n is at least 2, design an algorithm for finding the minimum and maximum of these numbers, that uses at most 3n/2 comparisons. Provide an argument that your algorithm indeed uses at most 3n/2 comparisons. You need to analyse the number of comparisons that your algorithm uses and prove that it is at most 3n/2. [10 marks] (Note: You should not use sorting here, because it uses (nlog n) comparisons. An algo- rithm that uses more, but still linear number, say cn, of comparisons, for some small constant c, can still attract some but appropriately fewer marks
Answers
The algorithm uses at most 3n/2 comparisons.
To design an algorithm that finds the minimum and maximum of n numbers using at most 3n/2 comparisons, we can employ a technique known as "tournament method" or "pairwise comparison."
Here's the algorithm:
Initialize two variables, min and max, with the first number from the table.
Set the index i = 2.
While i ≤ n, do the following:
a. Compare the (i-1)th and ith numbers from the table.
b. If the (i-1)th number is smaller than the ith number:
Compare the (i-1)th number with min.
Compare the ith number with max.
c. If the (i-1)th number is greater than the ith number:
Compare the ith number with min.
Compare the (i-1)th number with max.
d. Increment i by 2.
If n is odd, compare the last number with both min and max.
Return min and max as the minimum and maximum of the given table.
To analyze the number of comparisons, let's consider the worst-case scenario. In the worst case, the numbers in the table are sorted in descending order.
In each iteration of the while loop, we compare two numbers, which makes 1 comparison. Since the loop iterates n/2 times, the total number of comparisons within the loop is n/2.
If n is odd, we perform two additional comparisons to compare the last number with both min and max.
Therefore, the total number of comparisons in the worst case is (n/2) + 2.
Using mathematical inequality, we can show that (n/2) + 2 ≤ 3n/2.
(n/2) + 2 ≤ 3n/2
(n + 4) ≤ 3n
4 ≤ 2n
2 ≤ n
Since the given condition states that n is at least 2, the inequality holds true for all valid values of n.
Hence, the algorithm uses at most 3n/2 comparisons.
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Related Questions
Wyatt' Ga Mileage
\text{Gallon (\(x\))}Gallon (x) \text{Mile (\(y\))}Mile (y)
55 153153
77 214. 2214. 2
1010 306306
1414 428. 4428. 4
How much farther can Camden' car travel on one gallon of ga than Wyatt' car?
Answers
Both automobiles' lines have the equation "y = 3x," and their respective gas mileage is 3 miles per gallon. Wyatt's automobile gets more miles per gallon of gas than Camden's vehicle does.
We must compute the mileage (the distance driven per gallon of fuel) for each automobile using the provided information to determine the difference between the ranges of travel for Camden's and Wyatt's vehicles.
We can determine the miles for Wyatt's automobile using the data points (5, 15) and (7, 21). The equation for the slope of the line connecting these two locations is
(y2 - y1) / (x2 - x1) = (21 - 15) / (7 - 5) = 6/2 = 3
"Y = mx + b" is the equation for the line connecting these two points, where m denotes the line's slope and b its y-intercept. We may determine the equation of the line by adding the values of m as well as the dimensions of one of the endpoints into this equation:
y = 3x + b
15 = 3(5) + b
15 = 15 + b
0 = b
"Y = 3x" is the equation for the line passing thru the points (5, 15) and (7, 21).
We can determine the miles for Camden's automobile using the data points (10, 30) and (14, 42). The equation for the slope of the line connecting these two locations is
(y2 - y1) / (x2 - x1) = (42 - 30) / (14 - 10) = 12/4 = 3
"Y = mx + b" is the equation for the line connecting these two points, where m denotes the line's slope and b its y-intercept. We may determine the equation of the line by incorporating the values of m as well as the dimensions of one of the endpoints into this equation:
y = 3x + b
30 = 3(10) + b
30 = 30 + b
0 = b
"Y = 3x" is the equation for the line passing through the points (10, 30) & (14, 42).
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The question is -
Computes the miles per gallon of your car via the miles traveled and the number of gallons used. Also, if you enter the cost per gallon and how many miles you drive a day, it will estimate your monthly and yearly gas expenses with data points (5, 15) and (7, 21).
Which answer correctly describes the transformation from the preimage to the
image?
a translation 10 units left and 8 units down
a reflection over the x-axis
a translation 10 units right and 8 units up
a rotation of 180°
a reflection over the y-axis
Answers
Translation of 10 units left and 8 units down
================================================
Explanation:
Always start with the preimage. Let's say we focus on point A located at (3,6). Its corresponding point is A' located at (-7,-2). To go from (3,6) to (-7,-2), we move 10 units to the left and 8 units down.
The motion "10 units to the left" means we subtract 10 from the x coordinate. The "8 units down" part means we subtract 8 from the y coordinate.
Overall, the translation rule is \((x,y) \to (x-10,y-8)\)
If we applied that rule to A(3,6), then...
\((x,y) \to (x-10,y-8)\\\\(3,6) \to (3-10,6-8)\\\\(3,6) \to (-7,-2)\\\\\)
which is what the diagram shows going from A to A'
Let's do the same for point B(3,1)
\((x,y) \to (x-10,y-8)\\\\(3,1) \to (3-10,1-8)\\\\(3,1) \to (-7,-7)\\\\\)
This indicates B(3,1) moves to B ' (-7,-7) which is what the diagram shows. I'll let you confirm that C(6,6) moves to C ' (-4,-2) using that translation rule.
Will give branlist Which situation could be represented by the graph?
Answers
there are 20 socks in a drawer.
• 2/5 of them are white
• 1/4 of them are black
• The rest are gray.
How many socks are gray?
help ‼️‼️
Answers
Answer:
7
Step-by-step explanation:
Make the fractions have a denominator of 20.
2/5 times the top and bottom by 4 (because 5 times 4 = 20) = 8/20
So 8 socks are white
1/4 times 5 = 5/20
So 5 socks are black
Add them and subtract from 20 to get the number of gray socks
20-13=7
Still struggling with this.
Answers
Answer:
22
Step-by-step explanation:
You add up the value of each X and you get a final answer of 22. Have a great day!
what is one trillion to the tenth power?
Answers
Answer:
1e+120
Step-by-step explanation:
What is the part of line having 1 endpoint and extending in one direction?
Answers
A part of a line that has 1 endpoint and extends indefinitely in only one direction is called a ray.
A ray is named using its endpoint first, and then any other point on the ray
Properties of ray:
A line is a series of points placed together that continue infinitely.When this line is restricted from one direction and is extended in the other direction indefinitely, it forms a ray.It has just one starting point and does not have an opposite end and goes through and cuts many points and lines and is often used to draw angles, and we cannot measure the length of a ray.To know more about ray:
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Using a graph, table of values, and/or equations solve the following problem.
A sofa and a love seat together costs S351. The sofa costs double the love seat. How much does the love seat cost?
Round and write your answer to the tenths place value. Do not include units
Answers
Answer:
\(L= 117.0\)
Step-by-step explanation:
Represent the sofa with S and the love seat with L.
So, we have:
\(S + L = 351\) --- Total Cost
\(S = 2L\)
Required
Determine the value of L
Substitute 2L for S in the first equation.
\(2L + L = 351\)
\(3L= 351\)
Divide both sides by 3
\(\frac{3L}{3}= \frac{351}{3}\)
\(L= \frac{351}{3}\)
\(L= 117.0\)
Hence, a love seat costs 117.0
Using Graph
See attachment for graph
S is plotted on the y-axis and L is on the x-axis.
The green line represents \(S + L = 351\) while the orange line represents \(S = 2L\)
When the line of the graph is traced to the x-axis, we have:
\(L= 117.0\)
What will be the value stored in the variable x after the execution of the following code snippet?
int a = 10;
int b = 20;
int c = 2;
int x = b / a /*c*/;
a) 1
b) 2
c) 4
d) The code has a syntax error
Answers
The value stored in the variable x after the execution of the following code snippet will be b) 2.
The right-click context menu option or a combination of hotkeys can be used to add code snippets, which are compact chunks of reusable code, to a code file. Although try-finally and if-else blocks, for example, are frequently used code blocks found in code snippets, you can also use them to add whole classes or methods.
Learn more about using the templates tool to create reusable emails. Applications and web pages use snippets. Snippets are made to be reusable and to add functionality, like connecting various parts of a program.
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what is the exponent of the greatest power of 40 that will divide into 80! with no remainder ?
Answers
Answer: 19
Step-by-step explanation: For each power of 40, you need 3 prime factors of 2 and 1 prime factor of 5.
In 80, the number of factors of 5 is 16+3 = 19.
In 80, the number of factors of 2 is 40+20+10+5+2+1 = 78.
The number of prime factors of 2 in 80! is more than 3 times the number of prime factors of 5; that means the number of prime factors of 5 is what limits the power of 40 that divided into 80! gives a whole number answer. So
Which expression is equal to x−9x−4+x2−x+5x−4 ?
Responses
x2−42x−8
x2−4x−4
x2−x−42x−8
x2−x−4x−4
Answers
Starting with the x terms:
X - 9x - x = -9x
Next, the constants:
-4 + 5(-4) = -4 - 20 = -24
Putting it all together:
X-9x-4+×2-x+5×-4 = -8x - 24
Therefore, none of the expressions listed is equal to X-9x-4+×2-x+5×-4.
The answer and what is the value of a
Answers
Answer:a=40
Step-by-step explanation:
angles on a straight line add to 180. This means that the missing angle that isn't a is 40. Angles in a triangle add to 180 so a=40
The number of bacteria in a refrigerated food product is given by N(T)=21T^2 −103T+73,5
Answers
(a) The composite function N(T(t)) is given by N(T(t)) = 575t^2 + 65t − 31.25. (b)The bacteria count reaches 6752 at approximately 1.88 hours (rounded to two decimal places)
a. To find the composite function N(T(t)), we substitute the expression for T(t) into N(T). Let's calculate N(T(t)) step by step.
Given: N(T) = 23T^2 − 56T + 1 and T(t) = 5t + 1.5.
Substituting T(t) into N(T), we have:
N(T(t)) = 23(T(t))^2 − 56(T(t)) + 1.
Replacing T(t) with its expression:
N(T(t)) = 23(5t + 1.5)^2 − 56(5t + 1.5) + 1.
Expanding and simplifying:
N(T(t)) = 23(25t^2 + 15t + 2.25) − 280t − 84 + 1.
N(T(t)) = 575t^2 + 345t + 51.75 − 280t − 83.
N(T(t)) = 575t^2 + 65t − 31.25.
Therefore, the composite function N(T(t)) is given by N(T(t)) = 575t^2 + 65t − 31.25.
b. To find the time when the bacteria count reaches 6752, we need to solve the equation N(T(t)) = 6752. Let's set up the equation and solve it.
Given: N(T(t)) = 575t^2 + 65t − 31.25 and we want to find t.
Setting N(T(t)) equal to 6752:
575t^2 + 65t − 31.25 = 6752.
Rearranging the equation to make it quadratic:
575t^2 + 65t − 31.25 - 6752 = 0.
Combining like terms:
575t^2 + 65t - 6783.25 = 0.
This is a quadratic equation in the form of At^2 + Bt + C = 0, where A = 575, B = 65, and C = -6783.25. We can solve this quadratic equation using various methods, such as factoring, completing the square, or using the quadratic formula. In this case, we will use the quadratic formula:
t = (-B ± √(B^2 - 4AC)) / (2A).
Substituting the values:
t = (-(65) ± √((65)^2 - 4(575)(-6783.25))) / (2(575)).
Calculating inside the square root:
t = (-65 ± √(4225 + 4675300)) / 1150.
t = (-65 ± √(4679525)) / 1150.
t = (-65 ± 2162.24) / 1150.
We have two solutions:
t₁ = (-65 + 2162.24) / 1150 ≈ 1.8819 (rounded to two decimal places).
t₂ = (-65 - 2162.24) / 1150 ≈ -1.9250 (rounded to two decimal places).
Since time cannot be negative in this context, the bacteria count reaches 6752 at approximately 1.88 hours (rounded to two decimal places).
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please help me on this
Answers
Answer:
reflection across the y- axis
Step-by-step explanation:
Under a reflection in the y- axis
a point (x, y ) → (- x, y )
Thus
The given transformation rule (x, y ) → ( - x, y )
Represents a reflection in the y - axis
Ross had 85 papers to grade from his students. He has graded 60 of them. Approximately what percentage of papers does he have left to grade?
Answers
well, our origin amount will be 80, so 80 is the 100% or all papers, now, she has graded 60, so what's leftover is 20, what's 20 off of it in percentage?
\(\begin{array}{ccll} amount&\%\\ \cline{1-2} 80 & 100\\ 20& x \end{array} \implies \cfrac{80}{20}~~=~~\cfrac{100}{x} \\\\\\ 4 ~~=~~ \cfrac{ 100 }{ x }\implies x=\cfrac{100}{4}\implies x=25\)
How do you rearrange an equation so that the X is alone ?
Example: (g=6x),
(Z=m-x)
Or (u=2x-2)
Please help !
Answers
The water level in a plastic pool changed
by −8 gallons each hour due to a small
hole in the bottom. After 6 hours, the pool
contained 132 gallons. How much water
was in the pool originally?
Answers
Answer: 180 gallons
Step-by-step explanation:
If you take 8 times 6 you get 48, now take 132 plus 48 and you get 180.(hope this helped):)
Four integers have a mean of 12, a median of 12, a mode of 12 and a range of 6.
Find the four integers.
Answers
Write add, divide, multiply, and subtract in the correct order to complete the following sentence.
When using the order of operations to evaluate an expression,
Always:____and____before you____and____.
Answers
Answer:
what grade are you??.......
Glen Davis is shooting free throws. Making or missing free throws doesn't change the probability that he will make his next one, and he makes his free throws 74%
Answers
Answer:
0.00456
Step-by-step explanation:
Given: Percentage of free throws = 74%
Required: What is the probability of Glen Davis making none of his next 4 free throw attempts?
Since, there is a free throw 74% of the time, thus, the time he misses = 26% of the time.
To miss four in a row, it is requires to miss the first throw, the second throw, the third throw and also the fourth throw.
Thus, the probabilities of Gien missing will be multiplied.
By using the probability formula -
P ( MMMM) = 26/100 × 26/100 × 26/100 × 26/100
= 456976/ 100000000
= 0.00456
Therefore, the probability that he will have next 4 free attempts is 0.00456.
what is the probability that you reach into the jar and randomly grab a quarter and then, without replacement, another quarter? express your answer as a fraction or a decimal number rounded to four decimal places.
Answers
P = 0.0688
The first thing is to calculate the total number of coins:
17 + 10 + 19 + 23 = 69
Probability
Probability is simply how likely something is to happen. Whenever we're unsure about the outcome of an event, we can talk about the probabilities of certain outcomes—how likely they are. The analysis of events governed by probability is called statistics.
Fraction
A fraction is a part of a whole. In arithmetic, the number is expressed as a quotient, in which the numerator is divided by the denominator. In a simple fraction, both are integers. A complex fraction has a fraction in the numerator or denominator. In a proper fraction, the numerator is less than the denominator.
The probability in this case will be the product between the probability of each event, the probability of each event would be the number of favorable cases divided by the total number of coins, therefore:
P(nickel) = 19/69
P(penny) = 17/68
Therefore
P = 19/69 * 17/68
P = 323/4692
P = 0.0688
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how to copy and paste
Answers
Answer:
control + C then Control + V
Step-by-step explanation:
Answer:
If you have a PC, press Windows, Print Screen.
Step-by-step explanation:
Hope this helps..
Simplify: 1/3(15y-6)
I NEED HELP PLZ!!!
Answers
1/3(15y-6)=17
Answer:
5y - 2
Step-by-step explanation:
1/3 ( 15y - 6 ) = (15y - 6) / 3 = 5y - 2
Done! Hope you learned how to do this/understood this and have a great day! Please mark me as brainliest, vote 5.0 on my answer and thank me to show some support! Bye!
Solve -mk-110>75 for m
Show ur work
Answers
Answer:
\(m<-\frac{185}{k};\quad \:k>0\)
Step-by-step explanation:
\(-mk-110>75\\\\\mathrm{Add\:}110\mathrm{\:to\:both\:sides}\\\\-mk-110+110>75+110\\\\Simplify\\\\-mk>185\\\\\mathrm{Multiply\:both\:sides\:by\:-1\:\left(reverse\:the\:inequality\right)}\\\\\left(-mk\right)\left(-1\right)<185\left(-1\right)\\\\Simplify\\\\mk<-185\\\\\mathrm{Divide\:both\:sides\:by\:}k;\quad \:k>0\\\\\frac{mk}{k}<\frac{-185}{k};\quad \:k>0\\\\Simplify\\\\m<-\frac{185}{k};\quad \:k>0\)
Find the distance between the following points (-3, -2) and (1, 2)
Answers
Answer: 8 graph units.
Step-by-step explanation:
The difference between -3 and 1 is 4
The difference between -2 and 2 is 4
4 on the x axis + 4 on the y axis = 8 graph units total.
Add the two expressions.
-6.56 +11 and 3.30 - 2
Enter your answer in the box
Pls help
Answers
Answer:
5.74
Step-by-step explanation:
solve the equation for all values of x in simplest form (x+1)^2=15
Answers
Answer: \(x=-1 \pm \sqrt{15}\)
Step-by-step explanation:
\((x+1)^2 =15\\\\x+1 =\pm \sqrt{15}\\\\x=-1 \pm \sqrt{15}\)
Use the formulas to find the volume of each figure. Show your work.
Answers
The volume of the cone is 7225.66 m³ and the volume of the hemisphere is 209.44 cubic feet.
From the general formula of the volume of the cone,
Volume = πr²h
in the given case,
Radius (r) = 10 m
Height (h) = 23 m
Thus, Volume = π*10²*23
Volume = 7225.66 m³
The formula for the volume of a hemisphere is:
V = (2/3) * π * r³
where π is pi (approximately 3.14159), and r is the radius of the hemisphere.
Substituting the given value of r = 10 ft, we get:
V = (2/3) * π * (10 ft)³
= (2/3) * π * 1000 ft³
= (2/3) * 314.16 ft³
≈ 209.44 ft³
Therefore, the volume of the hemisphere is approximately 209.44 cubic feet.
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How to find the point of intersections on these equations? (x+6)^2 =-12 (y+2) x^2/81 + y^2/100 =1
Answers
Answer:
Graphically.
Step-by-step explanation:
We have the equations:
(x + 6)^2 = -12*(y + 2)
(x^2/81) + (y^2/100) = 1
First, we can see that the second one is an ellipse, and we can write it as:
(x/9)^2 + (y/10)^2 = 1
This adds some complexity to our problem.
The first equation is just a quadratic equation, that we can write in standard form as:
y = (x^2 + 12*x + 36)/(-12) - 2
y = (-1/12)*x^2 - x - 5
Then our two equations are:
(x/9)^2 + (y/10)^2 = 1
y = (-1/12)*x^2 - x - 5
This is actually really hard to solve analytically, we may end with a quartic equation or something like that (this happens for the nature of the ellipse equation)
So now that we have two rather simple equations that we know how to graph (or that we can just input the equations in a program and graph them) we can find the intersections visually.
You can see the graph of this below:
Of course, this only works to find estimations for the intersections, where one intersection is (3.35, -9.28) and the other is at (-8.69, -2.6)
hey, can someone please double check these questions for me / or help me solve them if they are incorrect? thank you U-U
Answers
Question 11
The directrix is a horizontal line, which means the parabola opens either upward or downward. In this case, it opens downward. This is because all answer choices have a negative leading coefficient. Also, it's because the focus is below the directrix.
For vertically opening parabolas, we use this form
4p(y-k) = (x-h)^2
where (h,k) is the vertex and p is the focal distance, aka the distance from the vertex the focus. To find (h,k), we start at the focus (0,-4) and move directly up until we reach the directrix y = 4. We'll arrive at (0,4). The midpoint of (0,-4) and (0,4) is (0,0) which is the vertex's location. So (h,k) = (0,0).
Note that in moving from (0,-4) to (0,4) is a span of 4 units. So this is the value of p.
Plug h = 0, k = 0, p = 4 into the equation mentioned and solve for y
4p(y-k) = (x-h)^2
4*4(y-0) = (x-0)^2
16y = x^2
y = (1/16)x^2
The only adjustment we need to make is to change the 1/16 to -1/16 so that the parabola opens downward.
Answer: Choice D. y = -(1/16)x^2===============================================
Question 3
The given equation is in the form y = ax^2+bx+c
In this case,
a = 2b = 4c = 3Let's compute the x coordinate of the vertex h
h = -b/(2a)
h = -4/(2*2)
h = -1
This h value is plugged into the original function to find k
f(x) = 2x^2+4x+3
f(-1) = 2(-1)^2+4(-1)+3
f(-1) = 1
We find that h = -1 and k = 1 pair up together. In short, (h,k) = (-1,1) is the vertex.
Answer: Choice B. (-1,1)