If f(x) =
√x-10+3, which inequality
x-10+3, which inequality can be used to find the domain of f(x)?
O
0x20
Ox-1020
O √x-10
x-10+320
Answers
Inequality can be used to find the domain of f(x) is \($\quad x \geq 20$\)
Interval Notation: \($[20, \infty)$\)
What is Domain definition?
The domain of a function is the set of input or argument values for which the function is real and defined.
Find non - negative values for radicals: \($x \geq 20$\)
The function domain
\($$x \geq 20$$\)
Range of\($\sqrt{\frac{1}{2} x-10}+3:\left[\begin{array}{cc}\text { Solution: } & f(x) \geq 3 \\ \text { Interval Notation: } & {[3, \infty)}\end{array}\right]$\)
Function range definition
The set of values of the dependent variable for which a function is defined
The range of an radical function of the form \($c \sqrt{a x+b}+k$\) is \($f(x) \geq k$\)\($k=3$\)
\($f(x) \geq 3$\)
Axis interception points of \($\sqrt{\frac{1}{2} x-10}+3$\) : None
x - axis interception points of \($\sqrt{\frac{1}{2} x-10}+3$\) : None
y - axis interception point of \($\sqrt{\frac{1}{2} x-10}+3$\) : None
None
Extreme Points of \($\sqrt{\frac{1}{2} x-10}+3: \quad$\)Minimum (20,3)
First Derivative Test definition
Suppose that \($x=c$\) is a critical point of f(x) then, If \($f^{\prime}(x) > 0$\) to the left of \($x=c$\)and \($f^{\prime}(x) < 0$\) to the right of x=c then x=c is a local maximum. If \($f^{\prime}(x) < 0$\) to the left of \($x=c$\) and \($f^{\prime}(x) > 0$\) to the right of \($x=c$\) then \($x=c$\) is a local minimum. If \($f^{\prime}(x)$\) is the same sign on both sides of \($x=c$\) then \($x=c$\) is neither a local maximum nor a local minimum.
\($$f^{\prime}(x)=\frac{1}{2 \sqrt{2} \sqrt{x-20}}$$\)
Find intervals: Increasing:20 \($ < x < \infty$\)
Plug \($x=20$\) into \($\sqrt{\frac{1}{2} x-10}+3: \quad 3$\)
Minimum (20,3)
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Related Questions
Find the solution to the linear system of differential equations satisfying the initial conditions x(0)=5 and y(0)=3:
Answers
The solution of the system of differential equations satisfying the initial conditions x(0)=5 and y(0)=3 is given by:
\($x(t) = e^{at} (5 \cos(bt) + 3 \sin(bt))$\)
\($y(t) = e^{ct} (5 \sin(bt) - 3 \cos(bt))$\)
Let x(t) and y(t) be the solution of the linear system of differential equations:
\($\frac{dx}{dt} = ax + by$\)
\($\frac{dy}{dt} = cx + dy$\)
with initial conditions x(0)=5 and y(0)=3.
The general solution of the system of differential equations can be written as:
\($x(t) = e^{at} (X0 \cos(bt) + Y0 \sin(bt))$\)
\($y(t) = e^{ct} (X0 \sin(bt) - Y0 \cos(bt))$\)
where X0 = 5 and Y0 = 3.
Therefore, the solution of the system of differential equations satisfying the initial conditions x(0)=5 and y(0)=3 is given by:
\($x(t) = e^{at} (5 \cos(bt) + 3 \sin(bt))$\)
\($y(t) = e^{ct} (5 \sin(bt) - 3 \cos(bt))$\)
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Keisha has been given a List of 5 bands and asked to place a vote. Her vote must have the names of her favorite and second favorite bands from the list. How many different bands are possible ? Please help me
Answers
Answer:
2
Step-by-step explanation:
because she hast to pick her favorite and second favorite
Five equations are given. Each equation has infinitely many solutions, only 1 solution, or no solution at all. Select all of the equations that have only 1 solution. A. 5 a − a + 4 = 4 B. 9 b − 1 = 9 b C. x − 5 3 = − 4 D. n − 3 n − 2 = − 2 ( n + 1 ) E. 2 3 t = 3 2
Answers
Here are some equations. The equal symbol (=) connects the two sides of the expression to form the equation. The equation 2x+1 = 9 is composed of the left hand side (LHS) of 2x+1 and the right hand side (RHS) of 9. (RHS).
The equations that have only 1 solution are:
B. 9b - 1 = 9b
This equation simplifies to -1 = 0, which is a contradiction. Therefore, this equation has no solution, not infinitely many solutions or only 1 solution.
C. x - (5/3) = -4
Subtracting (5/3) from both sides gives:
x = -4 + (5/3)
x = -7/3
This equation has only 1 solution.
D. n - 3/n - 2 = -2(n + 1)
Multiplying both sides by (n - 2) gives:
n - 3 = -2(n^2 - n - 2)
n - 3 = -2n^2 + 4n + 4
Rearranging terms gives:
2n^2 - 3n - 7 = 0
Using the quadratic formula, we get:
n = (3 ± sqrt(73))/4
This equation has only 1 solution.
Therefore, the equations that have only 1 solution are C and D.
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Equation E has only one solution.
What is an equation?
An equation is a mathematical statement that shows that two expressions are equal. It consists of two sides, a left-hand side and a right-hand side, separated by an equal sign (=).
Equations that have only one solution are those that when solved result in a single unique value for the variable. The equations that have only 1 solution are:
B. 9 b − 1 = 9 b (When we simplify the equation, we get -1 = 0 which is not possible. Therefore, this equation has no solution.)
D. n − 3 n − 2 = − 2 ( n + 1 ) (This equation can be simplified as 1 = -1, which is not true. Therefore, this equation has no solution.)
E. 2 3 t = 3 2 (We can solve this equation for t to get t = 2.25. Therefore, this equation has only one solution.)
Hence, equation E has only one solution.
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Note: Figure not drawn to scale
If X - 11 units and h=9 units, then what is the area of the triangle shown above?
O A. 49.5 square units
B. 99
square
units
C. 40.5 square units
D. 198 square units
Answers
Answer:
A) 49.5
Step-by-step explanation:
11*9=99
99/2= 49.5
Mrs.bowlin Organise the markers and two sets of staff were nine markers in each is that how many sets of markers did Mrs. Bowlin create
Answers
Answer:
18 sets of markers
Owen receives $10 and puts it into his savings account. He adds $0.50 to the account each day for a number of days, d, after that. He writes the expression 10+0.5(d−1) to find the amount of money in his account after d days. Which statement about his expression is true?
Answers
Answer:
it is the sum of the initial amount and the additional amount after d days
Step-by-step explanation:
yw!!.
The number of small air bubbles per 3 feet by 3 feet plastic sheet has a Poisson distribution with a mean number of two per sheet. What percent of these sheets have no air bubbles
Answers
The percentage of the sheets with no air bubbles is given as follows:
13.53%.
What is the Poisson distribution?In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following mass probability function:
\(P(X = x) = \frac{e^{-\mu}\mu^{x}}{(x)!}\)
The parameters are listed and explained as follows:
x is the number of successes that we want to find the probability of.e = 2.71828 is the Euler number\(\mu\) is the mean in the given interval or range of values of the input parameter.The mean for this problem is given as follows:
\(\mu = 2\)
The proportion of these sheets with no air bubbles is P(X = 0), hence it is given as follows:
P(X = 0) = e^-2 = 0.1353 = 13.53%.
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I need help solving this
Answers
Answer:
I think it's C.
Step-by-step explanation:
The Correct and Right Answer Is C
The annual number of passengers going through Hartsfield-lackson Adanta International Aipport betwoen 2000 and 2008 can be modeled as p(t)=−0.102t3+1.39t2−3.29t+7925 where output is measured in mation passengers and c is the number of years since 2000.t (a) Numerically estimote rho(0) to the nearest thousand passengers: passengers per year therpret the result: At the end of 2005, the number of pabserigers going through Hartsfieitf tocksan intematsonal Airport was by approvinately per year. (b) Calcuiate the percentage rafe of change of p ot t=5.
Answers
(a) To estimate p(0), we substitute t = 0 into the given function:
p(0) = -0.102(0)^3 + 1.39(0)^2 - 3.29(0) + 7925
= 0 + 0 - 0 + 7925
= 7925
(b) To calculate the percentage rate of change of p at t = 5, we need to find the derivative of p(t) with respect to t:
p'(t) = -0.306t^2 + 2.78t - 3.29
(a) To estimate p(0), we substitute t = 0 into the given function:
p(0) = -0.102(0)^3 + 1.39(0)^2 - 3.29(0) + 7925
= 0 + 0 - 0 + 7925
= 7925
Therefore, p(0) is approximately 7925 passengers per year.
Interpretation: At the end of 2005, the number of passengers going through Hartsfield-Jackson Atlanta International Airport was approximately 7925 passengers per year.
(b) To calculate the percentage rate of change of p at t = 5, we need to find the derivative of p(t) with respect to t:
p'(t) = -0.306t^2 + 2.78t - 3.29
Now, we substitute t = 5 into the derivative:
p'(5) = -0.306(5)^2 + 2.78(5) - 3.29
= -7.65 + 13.9 - 3.29
= 3.96
The percentage rate of change can be calculated as (p'(5)/p(5)) * 100:
Percentage rate of change = (3.96/ p(5)) * 100
To determine the value of p(5), we substitute t = 5 into the given function:
p(5) = -0.102(5)^3 + 1.39(5)^2 - 3.29(5) + 7925
Now, we can calculate the percentage rate of change:
Percentage rate of change = (3.96/ p(5)) * 100
Please provide the value of p(5) in order to calculate the exact percentage rate of change.
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please help me it is due soon i will give brianleyest
Answers
In Luther v Borden, 1849 the Supreme Court abdicated its role in clarifying whether the people had the right to abolish their state governments. The great statesman Daniel Webster argued that people do indeed possess the right to overthrow their government. T or F?
Answers
In Luther v. Borden, 1849, it is true that the Supreme Court abdicated its role in clarifying whether the people had the right to abolish their state governments. The great statesman Daniel Webster argued that people do indeed possess the right to overthrow their government.
Luther v. Borden, 1849 was a United States Supreme Court case that decided whether the court should intervene in a political question, specifically the Rhode Island Dorr Rebellion of 1841 to 1842. In this case, the court declined to do so and ruled that the authority to decide whether a rebellion against a state government exists or not, rests solely with the United States government. The Court refused to support either side in the rebellion and thus upheld the lower court's decision.
The decision is well-known for declaring that the federal government must guarantee a "republican form of government" in the states. In this context, Webster argued that people possess the right to overthrow their government as they possess the ultimate sovereignty. True.
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The average polar bear weighs 1200 pounds. the average grizzly bear weighs 800 pounds. the average black bear weighs 400 pounds. The average polar bear weighs how many times more than the average black bear?
The average grizzly bear weighs how many times more than the average black bear ?
Answers
Answer:
It weighs 3 times more than the average black bear.
Step-by-step explanation:
If a polar bear weighs 1200 pounds and a black bear weighs 400 pounds, all you have to do it take the weight of the polar bear and divide it by the weight of the black bear. So 1200/400=3. The polar bear weighs 3 times more than the average black bear.
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a sample of adults were asked about their alcohol consumption. researchers were interested if there was an equal preference for beer, wine, and liquor. here is an incomplete computer output for the corresponding chi-square test: category beer wine liquor test contribution onserved proportion expected to chi-sq 264 0.333333 216.667 10.3405 237 0.333333 216.667 1.9082 149 0.333333 216.667 21.1328 n 650 de chi-sq p-value the chi-square for this test is?
Answers
Using a significance level of 5%, we conclude that there is significant evidence that American adults who drink alcohol do not favor beer, wine, and liquor equally as the alcoholic drink they consume most often. So, the correct answer is D).
Using a significance level of 5%, the chi-square test statistic is 33.3815 with 2 degrees of freedom and a corresponding p-value of less than 0.0001.
Since the p-value is less than 0.05, we reject the null hypothesis of equal preference for beer, wine, and liquor and conclude that there is significant evidence that American adults who drink alcohol do not favor beer, wine, and liquor equally as the alcoholic drink they consume most often.
Therefore, the answer is (d) There is significant evidence that American adults who drink alcohol do not favor beer, wine, and liquor equally as the alcoholic drink they consume most often.
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--The given question is incomplete, the complete question is given
" Gallup asked a nationally representative sample of adults about their alcohol consumption. Those in the sample who drank alcohol were asked which they drank most often-beer, wine, or liquor. Do the data provide evidence of an equal preference for beer, wine, and liquor as the favored alcoholic drink of American adults who drink alcohol? Here is an incomplete Minitab output for the corresponding chi-square test: Test Contribution Category Observed Proportion Expected to Chi-Sq beer 264 0.333333 216.667 10.3405 wine 237 0.333333 216.667 1.9082 liquor 149 0.333333 216.667 21.1328 N DF Chi-Sq P-Value 650 Using a significance level of 5%, what should you conclude?
a. There are significantly more American adult drinkers who favor beer over wine or liquor.
b. The data are consistent with an equal distribution of beer, wine, and liquor as the alcoholic drink consumed most often by American adults who drink alcohol.
c. We are unable to conclude anything, because the test assumptions are not met.
d. There is significant evidence that American adults who drink alcohol do not favor beer, wine, and liquor equally as the alcoholic drink they consume most often."--
in general, ______________ are more problematic than ________________ because they can produce spurious relationships.
Answers
In general, observational studies are more problematic than randomized controlled trials because they can produce spurious relationships.
Observational studies rely on the natural variation in exposures and outcomes that occur in the population, without any intervention or manipulation by the researcher.
This can lead to confounding variables, which are factors that are associated with both the exposure and the outcome and can produce a false association between them.
Randomized controlled trials, on the other hand, assign participants to different exposure groups at random, which reduces the risk of confounding and allows for a more accurate assessment of causality.
Therefore, researchers must be cautious when interpreting observational studies and consider the potential for spurious relationships.
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A car travels a distance of 540 km in 6 hours what speed did he travel at
Answers
Speed = Distance / time
Speed = 540 km / 6 hours
Speed = 90 km/hr.
25% of the tickets sold at a school carnival were early-admission tickets. If the school sold 272 tickets in all, how
many early-admission tickets did it sell?
Answers
Answer:
HEY! it is me I am going to put my questions here
Step-by-step explanation:
RELATED QUESTION..........
Answers
BCGF is a quadrilateral, angles sum to 360 degrees.
B+C+G+F=360
B and G are right angles, formed by a radius and a tangent.
90+C+90+F=360
C+F=180
C and F are supplementary.
2x + 146 + 4x + 238 = 180
6x + 384 = 180
6x = -204
x = -34
Answer: -34, second choice
The water usage at a car wash is modeled by the equation w(x) = 5x3 9x2 − 14x 9, where w is the amount of water in cubic feet and x is the number of hours the car wash is open. the owners of the car wash want to cut back their water usage during a drought and decide to close the car wash early two days a week. the amount of decrease in water used is modeled by d(x) = x3 2x2 15, where d is the amount of water in cubic feet and x is time in hours. write a function, c(x), to model the water used by the car wash on a shorter day. c(x) = 5x3 7x2 − 14x − 6 c(x) = 4x3 7x2 − 14x 6 c(x) = 4x3 7x2 − 14x − 6 c(x) = 5x3 7x2 − 14x 6
Answers
The function to model the water used by the car wash on a shorter day is (C) \(4x^{3} +7x^{2} -14x-6\).
What is a function?A function is an expression, rule, or law in mathematics that describes a relationship between one variable (the independent variable) and another variable (the dependent variable).To find the function to model the water used by the car wash on a shorter day:
Given that the amount of water used on normal days is given by the equation:
\(W(x) =5x^{3} +9x^{2} -14x+9\) ......(1)The amount of decrease in water used is modeled by the equation:
\(D(x)=x^{3} +2x^{2} +15\) ......(2)To get the function \(C(x)\) that models the water used by the car wash on a shorter day you subtract equation (2) from equation (1).
\(5x^{3} +9x^{2} -14x+9-(x^{3} +2x^{2} +15)\\5x^{3} +9x^{2} -14x+9-x^{3} -2x^{2} -15\\4x^{3} +7x^{2} -14x-6\)Therefore, the function to model the water used by the car wash on a shorter day is (C) \(4x^{3} +7x^{2} -14x-6\).
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The correct question is shown below:
The water usage at a car wash is modeled by the equation w(x) = 5x3 9x2 − 14x 9, where w is the amount of water in cubic feet and x is the number of hours the car wash is open. the owners of the car wash want to cut back their water usage during a drought and decide to close the car wash early two days a week. the amount of decrease in water used is modeled by d(x) = x3 2x2 15, where d is the amount of water in cubic feet and x is time in hours. write a function, c(x), to model the water used by the car wash on a shorter day.
(A) c(x) = 5x3 7x2 − 14x − 6
(B) c(x) = 4x3 7x2 − 14x 6
(C) c(x) = 4x3 7x2 − 14x − 6
(D) c(x) = 5x3 7x2 − 14x 6
help me on the following, in the picture IK U WANT THE POINTS JUST HELLPPP , ONLY IF UKNOW THE ANSWER/ NO LINKS and swrong answers
Answers
Answer:
any coordinate outside the parabola is a reasonable answer.
Step-by-step explanation:
test the solution (0, 0) and if if that number is greater than 0 then the rest of the points outside on the inside are.
x = 0
y = 0
(0)^2 - 2(0) - 3 = -3
-3 is less than 0 so (0, 0) is not a possibility so any coordinate outside the parabola is a reasonable answer.
I need help or my math teacher will call my parents up about it and I will die
Answers
Answer:
113mm^2
Step-by-step explanation:
Firstly work out the area of the pink circle using πr^2, which is the area of a process.
So we do πx12^2, which gives us 144π mm we have to divide this by two to get half of a circle hence giving us the area of the semi-circle. 72π mm is the area of the semicircle.
Then we must find the area of the white circle to subtract it from our semicircle. Since we know the radius is 12 from the center to anywhere, we know that the white circles diameter is 12, using our formula for the area all we need to do is πx6^2. This gives us 36π.
72π - 36π = 36π
36π is equal to 113.0973355 and since we need it too 3sf it will be 113mm^2
(78000/ 19110) x100=
Answers
Answer:
= 408.16 or =20000/49
Step-by-step explanation:
= (78000/ 19110) x100
= (200/49)*100
=20000/49
78000/19110=4.081
4.081*100= 408.1
The answer is 408 to whole number
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Members of the drama club spend $795 on t-shirts and sweatshirts for a play.They purchase 20 more tshirts than sweatshirts.They buy tshirts for $12 each sweatshirt for $25 each. How many tshirts do the drama club buy
Answers
Answer:
The drama club bought 35 t-shirt
Step-by-step explanation:
Let
t-shirt = x
Sweatshirt = y
Total cost = $795
They purchase 20 more tshirts than sweatshirts
x = y + 20
The equation is:
12x + 25y = 795
12(y + 20) + 25y = 795
12y + 240 + 25y = 795
37y = 795 - 240
37y = 555
y = 15
Substitute y = 15 into
x = y + 20
x = 15 + 20
x = 35
t-shirt = 35
Sweatshirt = 15
The drama club bought 35 t-shirt
Knowledge check 01 all of the following are one of the building blocks of financial statement analysis except: multiple choice solvency profitability marketing prospects liquidity efficiency
Answers
e) external analyst services
These are the analytical tools that are used by analysts for further analysis of the the financial statements. These ratios give a insight and better understanding of the financial statements. Without this analysis financial statements are mere figures.
The building blocks of financial statement analysis includes the ratios that are used to compute the liquidity, efficiency, solvency, profitability, and market prospects.
Below are some ratios related to different types of analysis:
Analysis and their Ratios -
Liquidity and Efficiency : Acid-test ratio, Current ratio.
Solvency : Debt-to-equity ratio, Debt-to-capital ratio.
Profitability : Return on assets, Return on equity.
Market Prospects : Dividend yield ratio, Price earning ratio.
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Disclaimer : The complete question is given below.
Question:
All of the following are one of the building blocks of financial statement analysis except
a) solvency
b) profitability
c) marketing prospects
d) liquidity and efficiency
e) external analyst services
ASAP! I WILL GIVE BRAINLEST
The satellite dish is shaped like a paraboloid of revolution. This means that it can be formed by rotating a parabola around its axis of symmetry. The receiver is to be located at the focus. If the dish is 72 feet across at its opening and 9 feet deep at its center, where should the receiver be placed?
Find the equation of the parabola.
How far above the vertex should the receiver be placed?
Answers
Answer:
45 feet
Step-by-step explanation:
The parabola that forms the shape of the satellite dish has its axis of symmetry along the vertical direction, and the vertex at the bottom center of the dish. Let's assume that the vertex of the parabola is at the origin (0,0) of a coordinate system, and the opening of the dish is along the x-axis. Then the equation of the parabola is:
y = a x^2
where "a" is a constant that determines the shape of the parabola. We can find the value of "a" using the given dimensions of the dish.
At the opening of the dish, which has a diameter of 72 feet, the y-coordinate is zero. Therefore, we have:
0 = a (36)^2
Solving for "a", we get:
a = 0
This means that the equation of the parabola is simply y = 0, which is a horizontal line passing through the origin. Clearly, this is not the correct equation for the parabola.
To find the correct equation, we need another point on the parabola. We are given that the depth of the dish at its center, which corresponds to the focus of the parabola, is 9 feet. Using the definition of a parabola, we know that the distance from any point on the parabola to the focus is equal to the distance from that point to the directrix. Since the axis of symmetry is vertical, the directrix is a horizontal line located 9 feet below the vertex.
The equation of the directrix is therefore:
y = -9
We can use this to find another point on the parabola. Consider a point (x,y) on the parabola that is equidistant from the focus (0,9) and the directrix y = -9. The distance from (x,y) to the focus is:
d = sqrt(x^2 + (y-9)^2)
The distance from (x,y) to the directrix is simply y + 9. Therefore, we have:
sqrt(x^2 + (y-9)^2) = y + 9
Squaring both sides and simplifying, we get:
x^2 = 4y(18-y)
This is the equation of the parabola. We can now find the x-coordinate of the receiver, which is located at the focus (0,9). Setting y = 9 in the equation of the parabola, we get:
x^2 = 4(9)(9) = 324
Therefore, the x-coordinate of the receiver is:
x = ±18
Since the dish is 72 feet across at its opening, the receiver should be located at a distance of 36 feet from the center of the dish. Therefore, the receiver should be placed at a height of:
y = 9 + 36 = 45 feet
above the vertex of the parabola.
The equation �=112�y=1\frac{1}{2}xy=1
2
1
x represents the number of cups of dried fruit, y, needed to make x pounds of granola. Determine whether each point would be on the graph of this proportional relationship.
Choose Yes or No for each point.
Answers
The coordinates (2,1) will be on graph but (1,3) is not on graph.
What is a coordinate?
A coordinate is a set of two or more numbers or variables that identify the position of a point, line, or plane in a space of a given dimension. Coordinates are used to pinpoint a particular location, such as a specific point on a map or a specific point in a mathematical equation.
This means that for every 1.5 cups of dried fruit, there is 1 pound of granola. The graph of this proportional relationship would be a line that goes through the origin and has a slope of 1.5. For the point (2,1), the x-coordinate (2) is exactly 1.5 times the y-coordinate (1). This means that if you used 2 cups of dried fruit, you would get 1 pound of granola. Therefore, this point would be on the graph of the proportional relationship, so the answer is Yes. However, for the point (1,3), the x-coordinate (1) is not 1.5 times the y-coordinate (3). This means that if you used 1 cup of dried fruit, you would not get 3 pounds of granola.
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A cut in an undirected graph is a separation of the vertices V into two disjoint subsets S and T. The size of a cut is the number of edges that have one endpoint in S and the other in T. Let MAX-CUT = {(G, k)| G has a cut of size k or more}. Show that MAX-CUT is NP-complete. You may assume the result of Prob- lem 7.26. (Hint: Show that #SAT
Answers
The cut separates the variables from their negations, each clause will have at least one true literal, satisfying the 3SAT instance.
To show that MAX-CUT is NP-complete, we need to demonstrate two things: First, that MAX-CUT is in the NP complexity class, meaning that a proposed solution can be verified in polynomial time. Second, we need to reduce a known NP-complete problem to MAX-CUT, showing that MAX-CUT is at least as hard as the known NP-complete problem.
MAX-CUT is in NP:
To verify a proposed solution for MAX-CUT, we can simply check if the cut separates the vertices into two disjoint subsets S and T, and count the number of edges that cross the cut. If the number of crossing edges is equal to or larger than k, we can accept the solution. This verification process can be done in polynomial time, making MAX-CUT a member of the NP complexity class.
Reduction from a known NP-complete problem:
We will reduce the known NP-complete problem, 3SAT, to MAX-CUT. The 3SAT problem involves determining if a given Boolean formula in conjunctive normal form (CNF) is satisfiable, where each clause contains exactly three literals.
Given an instance of 3SAT with n variables and m clauses, we construct a graph G for MAX-CUT as follows:
Create a vertex for each variable and its negation, resulting in 2n vertices.
For each clause (a ∨ b ∨ c), introduce three additional vertices and connect them in a triangle. Label one vertex as a, another as b, and the third as c.
Connect the variable vertices with the corresponding clause vertices. For example, if the variable is x and it appears in the clause (a ∨ b ∨ c), create edges between x and a, x (negation of x) and b, and x and c.
Now, we claim that there exists a cut in G of size k or more if and only if the 3SAT instance is satisfiable.
If the 3SAT instance is satisfiable, we can assign truth values to the variables such that each clause evaluates to true. We can then define the cut by placing all true variables and their negations in one subset S, and the remaining variables and their negations in the other subset T. The number of crossing edges in the cut will be at least k, as each clause triangle will have at least one edge crossing the cut.
If there exists a cut in G of size k or more, we can use it to derive a satisfying assignment for the 3SAT instance. Assign true to all variables in subset S and false to those in subset T.
Therefore, we have successfully reduced 3SAT to MAX-CUT, showing that MAX-CUT is NP-complete. This conclusion is based on the assumption that 3SAT is already a known NP-complete problem, as stated in Problem 7.26.
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A presidential candidate plans to begin her campaign by visiting the capitals in 3 of 47 states. What is the probability that she selects the route of three specific capitals? P(she selects the route of three specific capitals) 97290 (Type an integer or a simplified fraction:)
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The probability that she selects the route of three specific capitals is 97290/1.
The probability that she selects the route of three specific capitals can be calculated by taking the total number of possible routes and dividing it by the total number of routes that involve the three specific capitals. To calculate the total number of possible routes, multiply the number of states (47) by the number of routes that could be selected from each state (2). This gives 94 total possible routes.
47 x 46 x 45 ways = 97290 ways
= 1/97290
Now, calculate the number of routes that involve the three specific capitals. Since the president is selecting three states, the total number of routes will be 3. Therefore, the probability that she selects the route of three specific capitals is 3/94, which can be simplified to 97290/1.
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Which name does not apply to the angle pair?
Answers
Answer:
your answer is D) Right Angles
Step-by-step explanation:
hope it helps!!!
Help ASAP!!!!!!! It’s to confusing
Answers
Answer:
every one equals 32
Step-by-step explanation:
Every hour equals 32 tickets sold
Answer:
32
Step-by-step explanation:
all equal 32
1. Use the geometric series formula 1 / 1−y=[infinity]∑n=0 y^n1 to express the function as a series:
1 /1−sin^5 x =[infinity]∑n=0
2. To test the series [infinity]∑n=1e^−5n for convergence, you can use the Integral Test. (This is also a geometric series, so we could also investigate convergence using other methods.)
Find the value of ∫1 to [infinity] e^−5x dx=
Answers
1 / (1 - sin^5 x) = [infinity]∑n=0 [sin^5 x]^n * [cos^2 x * (2 - cos^2 x)]^(-1)
the integral ∫1 to [infinity] e^−5x dx converges to a finite value, the series [infinity]∑n=1 e^−5n also converges.
1. To express the function 1 / (1−sin^5 x) as a series using the geometric series formula, we need to first identify the value of y. In this case, y is sin^5 x. Therefore, we have:
1 / (1−sin^5 x) = 1 / [1 - (sin x)^5]
= 1 / [1 - (sin x)^2 * (sin x)^3]
= 1 / [cos^2 x * (1 - sin^2 x) * (1 + sin^2 x + (sin^2 x)^2)]
= 1 / [cos^2 x * cos^2 x * (1 + sin^2 x + (sin^2 x)^2)]
= 1 / [cos^4 x * (1 + sin^2 x + (sin^4 x)/(sin^2 x))]
= 1 / [cos^4 x * (1 + sin^2 x + sin^2 x * (1 - cos^2 x)/(sin^2 x))]
= 1 / [cos^4 x * (1 + sin^2 x + (1 - cos^2 x)/sin^2 x)]
= 1 / [cos^4 x * (2 - cos^2 x)/sin^2 x]
= sin^2 x / [cos^2 x * (2 - cos^2 x)]
Now, we can substitute y = (sin x)^5 into the geometric series formula:
1 / (1 - y) = [infinity]∑n=0 y^n
to get:
1 / (1 - sin^5 x) = sin^2 x / [cos^2 x * (2 - cos^2 x)]
= [infinity]∑n=0 [sin^5 x]^n * [cos^2 x * (2 - cos^2 x)]^(-1)
Therefore,
1 / (1 - sin^5 x) = [infinity]∑n=0 [sin^5 x]^n * [cos^2 x * (2 - cos^2 x)]^(-1)
2. To find the value of ∫1 to [infinity] e^−5x dx using the integral test, we need to first verify that the function f(x) = e^−5x is positive, continuous, and decreasing for x ≥ 1. Since f(x) is a continuous, positive, and decreasing function, we can apply the integral test to determine the convergence of the series.
The integral test states that if a series [infinity]∑n=1 a_n converges and a_n is a positive, decreasing function of n, then the corresponding improper integral ∫1 to [infinity] a(x) dx also converges. Similarly, if the improper integral diverges, then the series also diverges.
In this case, we have:
a_n = e^(-5n)
a_n is a decreasing function of n, since e^(-5n) is a decreasing exponential function.
Therefore, we want to find the value of the improper integral:
∫1 to [infinity] e^−5x dx
Using the substitution u = -5x, du = -5dx, and limits of integration u(1) = -5 and u([infinity]) = -∞, we can rewrite the integral as follows:
∫1 to [infinity] e^−5x dx = ∫-5 to -∞ (1/-5) * e^u du
= (1/-5) * [e^u]_-5^-∞
= (1/-5) * [e^(-∞) - e^(-5)]
= (1/-5) * [0 - 1/e^5]
= 1/(5e^5)
Since the integral ∫1 to [infinity] e^−5x dx converges to a finite value, the series [infinity]∑n=1 e^−5n also converges.
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