Slack of a3 is 6 weeks.
What is slack?
Slack variables are variables that are introduced to an inequality constraint in an optimization problem to change it into an equality. An inequality constraint is replaced by an equality requirement and a non-negativity constraint on the slack variable when a slack variable is introduced. In particular, slack variables are employed in linear programming. The slack variable, like the other variables in the augmented constraints, can only have positive or zero values because of the simplex method. A constraint is binding at a certain candidate solution if its associated slack variable is zero because it limits the modifications that can occur after that.
There are two paths through the network: A1, A2 with completion time 5 + 8 = 13; A1, A3 with completion time 5 + 2 = 7. The project completion time is 13. LCT of A3 = 13. LST of A3 = 13 - 2 = 11. EST of A3 = ECT of A1 = 5. Slack of A3 = 11 - 5 = 6.
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(There are two paths through the network: A1, A2 with completion time 5 + 8 = 13; A1, A3 with completion time 5 + 2 = 7. The project completion time is 13. LCT of A3 = 13. LST of A3 = 13 - 2 = 11. EST of A3 = ECT of A1 = 5. Slack of A3 = 11 - 5 = 6.)
Which of the expressions below is equal to 6 (2 + 2)? Select all that apply. A) (6 x 2) + (6 x 2) B) 6 x 2 + 2 C) 12 x 2 D) 6 x 4
Question Progress
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Expand and simplify (x+3)(x + 5)
Answer:
x2 + 8x + 15
Step-by-step explanation:
First - take the first terms of the brackets (so, x and x) and multiply them together: for this example that gives x2.
Outside - take the two outermost terms of the brackets (x and 5) and multiply them together: for this example that gives 5x.
Inside - take the two innermost terms of the brackets (3 and x) and multiply them together: for this example that gives 3x.
Some people may draw lines between terms when doing this (as seen in the diagram above) leaving you with a “crab claw” around the brackets.
Write down everything you have worked out as one long expression:
x2 + 5x + 3x + 15
Then, we will simplify this expression by grouping any “like” terms (this just means adding together any bits with x where they have the same power).
This leaves us with our final answer, which is:
x2 + 8x + 15
a washer and a dryer cost combined. the cost of the washer is three times the cost of the dryer. what is the cost of the dryer?
$250 is the cost of the dryer.
What do we mean by cost?
The amount of money spent by a company on the creation or production of goods or services is referred to as the cost.It does not include the profit margin. From the standpoint of a seller, the cost is the amount of money spent to produce a good or product.To find the cost of the dryer:
The dryer is priced at x dollars.The washer costs 3 times as much.Then,
3x + x = 10004x = 1000x = 250Therefore, $250 is the cost of the dryer.
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The correct question is given below;
A washer and a dryer cost $1000 combined. The cost of the washer is three times the cost of the dryer. What is the cost of the dryer?
Which expression is equivalent to the given expression after using the distributive property?
6 · 13 + 6 · 7
6 · 7(13 + 6 )
6 · 7(13 + 6 )
6 · (13 + 6) · 7
6 · (13 + 6) · 7
6 (13 + 7)
6 (13 + 7)
6(13 + 6) + 7
6(13 + 6) + 7
The equivalent expression to the given expressions after using distributive property is [tex]6(13+7)[/tex]
As per the question statement, we are supposed to find the expression equivalent to the given expression [tex]6*13 + 6*7[/tex] after using the distributive property. Before solving this problem, we should know about the distributive property, which states that [tex]a(b+c) = a*b + a*c[/tex].
Therefore, [tex]6*13 + 6*7[/tex]
[tex]=6(13+7)[/tex], as per the definition of the distributive property.
Distributive property: The distributive Property says that it is essential to multiply each of the two numbers by the factor prior to performing the addition operation when a factor is multiplied by the sum or addition of two terms.To learn more about distributive property, click on the link given below:
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3. Two quantities, p and q, are in direct proportion. The sum of the values of q when p = 6 and when p = 11 is 51.
(a) Find an equation connecting p and q. (b) Find the value of p when q = 72.
Step-by-step explanation:
this just means
q = k×p
and we need to find k.
the sum of both q values when p = 6 and p = 11 is then
k×6 + k×11 = 51
17k = 51
k = 51/17 = 3
so, our equation is
q = 3p
and when q = 72
72 = 3p
p = 72/3 = 24
Assist me with this graph.
Answer:
3.1.1. A(-4, 0) B(0, -2) E(4, 0)
3.1.2. k = -16
3.1.3. (-∞, ∞)
3.1.4. [-16, ∞)
3.1.5. [-6, 4]
Step-by-step explanation:
Given functions:
[tex]\begin{cases}f(x)=x^2+k\\g(x)=-2x+8\\h(x)=\dfrac{6}{x-2}+1\end{cases}[/tex]
Question 3.1.1.Points A and B are the x-intercept and y-intercept of function h(x).
Point E is the x-intercept of function g(x).
The x-intercept of function h(x) is when h(x) = 0:
[tex]\begin{aligned}h(x) & = 0\\\implies \dfrac{6}{x-2}+1 & = 0\\\dfrac{6}{x-2} & = -1\\6 & = -1(x-2)\\6 & = -x+2\\4 & = -x\\x & = -4\end{aligned}[/tex]
Therefore, the coordinates of point A are (-4, 0).
The y-intercept of function h(x) is when x = 0:
[tex]\begin{aligned}h(x) & = \dfrac{6}{x-2}+1\\\implies h(0) & = \dfrac{6}{0-2}+1\\ & = -3+1\\& = -2\end{aligned}[/tex]
Therefore, the coordinates of point B are (0, -2).
The x-intercept of function g(x) is when g(x) = 0:
[tex]\begin{aligned}g(x) & = 0\\\implies -2x+8 & = 0\\-2x & = -8\\x & = 4\end{aligned}[/tex]
Therefore, the coordinates of point E are (4, 0).
Question 3.1.2.As function f(x) passes through point C (-6, 20), substitute the point into the equation of the function to find the value of k:
[tex]\begin{aligned}f(-6) & = 20\\\implies (-6)^2 + k & = 20\\36 + k & = 20\\k & = 20 - 36\\k & = -16\end{aligned}[/tex]
Hence, proving that the value of k is -16.
Question 3.1.3.The domain is the set of all possible input values (x-values).
The domain of function f(x) is unrestricted, therefore its domain is:
Solution: -∞ < x < ∞Interval notation: (-∞, ∞)Question 3.1.4.The range is the set of all possible output values (y-values).
The range of function f(x) is restricted, since the minimum point of the parabola is at (0, -16). Therefore its range is:
Solution: f(x) ≥ -16Interval notation: [-16, ∞)Question 3.1.5.[tex]\begin{aligned}g(x) - f(x) & \geq 0\\\implies g(x) &\geq 0+f(x)\\ \implies g(x) & \geq f(x)\end{aligned}[/tex]
Therefore, find the interval for which g(x) is greater than or equal to f(x).
From inspection of the given graph this is between points C and E:
Solution: -6 ≤ x ≤ 4Interval notation: [-6, 4]Each year a certain amount of money is deposited in an account that pays an annual
the interest rate so that at the end of each year the balance in the account is multiplied by a
the growth factor of z = 1+r. $500 is deposited at the start of the first year, and an additional $200
is deposited at the start of the next year, and $600 at the start of the following year.
b. What is the amount (to the nearest cent) in the account at the end of three years if the
interest rate is 2%?
Answer:
What is the amount (to the nearest cent) in the account at the end of three years if the interest rate is 2%? $1,350.68
Step-by-step explanation:
DATA:
Deposits (at beginning of year):
Year 1 = $500
Year 2 = $200
Year 3 = $600
growth factor: z = 1 + r
interest rate is 2% so replace r with 2%. Growth factor equation is now:
z = 1 + 2%
z = 1.02
Year 1 account balance end of year = account balance + growth factor
= $500 + 1.02
= $ 510 account balance at end year 1
Year 2 account balance end of year = (previous year ending balance + current year deposit) * growth factor
= ($510 + $200) * 1.02
= $724.20
Year 3 account balance end of year = (previous year ending balance + current year deposit) * growth factor
= ($724.20 + $600) * 1.02
=$1,350.684
rounded: = $1,350.68 (This answer assumes no money has been withdrawn at any time.)
Lesson 8 Practice Problems
1. A number line can represent positions that are north and south of a truck stop on a
highway. Decide whether you want positive positions to be north or south of the
truck stop. Then plot the following positions on a number line.
a. The truck stop
b. 5 miles north of the truck stop
c. 3.5 miles south of the truck stop
The graph can be viewed at the conclusion of the response. We want to graph a number line and certain points on it.
Let A denote the truck stop and then it is 0 miles away from the truck stop.
Let us take the north side of the truck stop as the negative side.
Then the south side of the truck stop is the positive side.
We are asked to represent the following positions on the number line:
a. The truck stop
b. 5 miles to the north of the truck stop
c. 3.5 miles to the south of the truck stop
Point A refers to the truck stop.
Let B denote the point 5 miles away from A towards the north side of the truck stop.
And let C denote the point 3.5 miles away from A towards the south side of the truck stop.
Refer to the attached image for the required number line.
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Ex) what is the ratio? Of 5:35
Answer:
1:
Step-by-step explanation:
5 divided by 5 is 1, and 35 divided by 5 is
write the slope-intercept form of the equations of the line described
1. through (1,3), perpendicular to y=x-4
2. through (3,1), parallel to y=-x-4
1. The equation of the perpendicular line is: y = -x + 4
2. The equation of the parallel line is: y = -x + 4
What are the Equations of Parallel and Perpendicular Lines?The equations of two lines that are parallel to each other will have the slope (m), while the equations of two lines that are perpendicular to each other will have slope values that are negative reciprocals of each other.
The slope-intercept form for the equation of a line is, y = mx + b, where b is the y-intercept and m is the slope of the line.
1. The slope (m) of y = x - 4 is 1. The line that is perpendicular to y = x - 4 would have a slope (m) of -1.
Substitute (a, b) = (1, 3) and m = -1 into y - b = m(x - a):
y - 3 = -1(x - 1)
Rewrite in slope-intercept form:
y - 3 = -x + 1
y = -x + 1 + 3
y = -x + 4
2. The slope (m) of y = -x - 4 is -1. The line that is parallel to y = -x - 4 would have a slope (m) of -1.
Substitute (a, b) = (3, 1) and m = -1 into y - b = m(x - a):
y - 1 = -1(x - 3)
Rewrite in slope-intercept form:
y - 1 = -x + 3
y = -x + 3 + 1
y = -x + 4
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These are fractions 4/5 x 3/8=?
Answer:
3/10
Step-by-step explanation:
4/5 * 3/8
We can rewrite the fractions as
4/8 * 3/5
4/8 simplifies to 1/2
1/2 * 3/5
Multiply the numerators
1*3 = 3
Multiply the denominators
2*5 = 10
Numerator over denominator
3/10
Determine whether the conjecture is true or false. Give a counterexample for any false conjecture.
If you have three points A, B , and C , then A, B, C are noncollinear.
The given conjecture "If you have three points A, B , and C , then A, B, C are noncollinear" is false.
How to Interpret Conditional Statements?Notice that the given conjecture does not necessarily be always true.
To be precise, there are no reasons as to why A, B and C must be noncollinear.
They could lie on the same line.
Therefore, the given conjecture "If you have three points A, B , and C , then A, B, C are noncollinear" is false.
A Counterexample of the given statement is; A, B, C are three points that lie on the same line and so they are collinear.
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Hexagonal chess is played on a regular hexagonal board comprised of 92 small hexagons in three colors. The chess pieces are arranged so that a player can move any piece at the start of a game.
a. What is the sum of the measures of the interior angles of the chess board?
The sum of the measures of the interior angles of the chess board is equal to 720°.
What is a polygon?A polygon can be defined as a two-dimensional geometric shape that comprises straight line segments and a finite number of sides. Also, some examples of a polygon include the following:
TriangleQuadrilateralPentagonHexagonHeptagonOctagonNonagonNote: The number of sides (n) of a hexagon is 6.
In Geometry, the sum of the interior angles of both a regular and irregular polygon is given by this formula:
Sum of interior angles = 180 × (n - 2)
Sum of interior angles = 180 × (6 - 2)
Sum of interior angles = 180 × 4
Sum of interior angles = 720°.
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If B is between A and C, and AB = 4x + 1, BC = 2x – 7, and AC = 24. What is the value of length of BC?
*
a. 3 units
b. 17 units
c. 7 units
d. 21 units
Answer:
a. 3 units
Step-by-step explanation:
The length of BC can be found in 3 main steps, namely the formation of equations, solving for x and substitution.
Let's rewrite the given lengths:
AB= 4x +1 -----(1)
BC= 2x -7 -----(2)
AC= 24 -----(3)
Form an equation which relates the 3 lengths:
Given that B is between A and C,
AC= AB +BC -----(4)
Substituting equations (1) to (3) to equation (4):
24= 4x +1 +2x -7
Simplify:
24= 6x -6
Add 6 to both sides:
6x= 24 +6
6x= 30
Divide both sides by 6:
x= 5
Substitute x= 5 into (2):
BC
= 2x -7
= 2(5) -7
= 10 -7
= 3 units
Thus, option a is correct.
Additional:
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Which inequality is represented by the graph?
Answer:
Step-by-step explanation:
x < 0
Answer : x is less than or equal to 0
First, the direction of the number line in the graph is the negative number side, so it has to be 0 or less.
If it is less than 0, the dot that starts at the point 0 must have a hollowed out part in the middle. If it is 0 or less, the dot that starts at the point 0 is completely filled with no hollowed middle part.
Since the dot is completely blacked out, the answer is the second option.
Nabhitha buys cookies and onions at the store.
She pays a total of $60.39.
She pays a total of $4.79 for the cookies.
• She buys 8 bags of onions that each cost the same amount.
.
.
Write and solve an equation which can be used to determine x, how much each bag
of onions costs.
The equation is 8x + 4.79= 60.39 and the value is x is $6.95
How to solve the equation ?Nabhitha pays a total of $60.39 for the cookies and onions
She pays a total of $4.79 for the cookies
She buys 8 bags of onions that each costs the same amount of money
The equation used to find the value of x is written as
8x + 4.79= 60.39
Collect the like terms
8x= 60.39 - 4.79
8x= 55.6
Divide both sides by the coefficient of x which is 8
8x/8 = 55.6/8
x= 6.95
Hence the price of each bag of onions is $6.95
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What is a number that is at most 30
Answer: 30
Because its thirty
The angle lies in which quadrant?
tan<0 and csc<0 are in Quadrant II
Una ciclopista para andar en bicicleta tiene 17/6 de km ¿cuantos km se recorren al dar dos vueltas a la pista
The number of kilometers when the person goes twice is 17km.
How to compute the value?From the information, it should be noted that the bicycle lane to ride a bicycle is 17/6 km.
Therefore, the number of kilometers when the person goes twice will be:
= 17/6 × 2
= 17 km
Therefore, the number of kilometers when the person goes twice is 17km.
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find parametric equations and symmetric equations for the line (use the parameter t.) the line through the point (−3,3,−1) and perpendicular to both ⟨1,1,0⟩ and ⟨
The second vector that's perpendicular to the line we want (call it L) is missing; I'll denote it by ⟨a, b, c⟩.
The cross product of ⟨1, 1, 0⟩ and ⟨a, b, c⟩ is perpendicular to both of these vectors. The direction vector of L will be parallel to this cross product.
Compute the cross product.
⟨1, 1, 0⟩ × ⟨a, b, c⟩ = ⟨1•c - 0•b, 0•a - 1•c, 1•b - 1•a⟩ = ⟨c, -c, b - a⟩
Then the vector equation of L is
L(t) = ⟨c, -c, b - a⟩ t + ⟨-3, 3, -1⟩
As parametric equations, we write
x(t) = ct - 3
y(t) = -ct + 3
z(t) = (b - a) t - 1
Eliminating the parameter above, we get
x + y = (ct - 3) + (-ct + 3) = 0 ⇒ x = -y
and substituting
x = ct - 3 ⇒ t = (x + 3)/c
into z(t), we get
z = (b - a) (x + 3)/c - 1 ⇒ x = c (z + 1)/(b - a) - 3
As symmetric equations, then, we write
x = -y = c (z + 1)/(b - a) - 3
The midpoint of KL is M(8, 3). One endpoint is L(8, 0). Find the coordinates of the other
endpoint K.
Answer: K(8,6)
Step-by-step explanation:
M(8,3) - the midpoint of KL L(8,0) K(x,y)=?
[tex]\displaystyle\\Mx=\frac{Kx+Lx}{2}[/tex]
Multiply both parts of the equation by 2:
[tex]2Mx=Kx+Lx\\2Mx-Lx=Kx+Lx-Lx\\2Mx-Lx=Kx\\Mx=8\ \ \ \ Lx=8\\Hence,\\2(8)-8=Kx\\16-8=Kx\\8=Kx\\\\2My-Ly=Ky\\My=3\ \ \ \ Ly=0\\Hence,\\2*(3)-0=Ky\\6-0=Ky\\6=Ky\\Thus,\\K(8,6)[/tex]
Which point is located at -3 1/2
Explanation:
The value -3 1/2, aka -3 & 1/2, is between -3 and -4. It's at the exact midpoint of those values.
Think of it like how 3 & 1/2 is between 3 and 4.
Or you can rewrite it in decimal form: -3 & 1/2 = -3.5
eric is randomly drawing cards from a deck of 52. he first draws a red card, places it back in the deck, shuffles the deck, and then draws another card. what is the probability of drawing a red card, placing it back in the deck, and drawing another red card? answer choices are in the form of a percentage, rounded to the nearest whole number.
The probability of drawing a red card, placing it back in the deck, and drawing another red card is 1/4.
According to the given question.
Total number of cards in a deck is 52.
As we know that total number of red cards in a well shuffled deck is 26.
Since, Eric first draws a red card, places it back in the deck, shuffles the deck, and then draws another card. Which means there is a replacement of cards.
So, the probability of drawing the first card is red = 26/52 = 1/2
And the probability of drawing the another card is red = 26/52 = 1/2
Therefore,
The probability of drawing a red card, placing it back in the deck, and drawing another red card
= 1/2 × 1/2
= 1/4
Hence, the probability of drawing a red card, placing it back in the deck, and drawing another red card is 1/4.
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john is playing a game with a standard deck of playing cards. he wants to draw a jack on the first try. which of the following statements is true?
John is playing a game with a standard deck of playing cards - this statement is true.
What is the purpose of playing cards?playing a card game, set of cards that are numbered or outlined (or both) and are utilized for messing around, for schooling, for divination, and for conjuring. Generally, Western playing a card game are made of rectangular layers of paper or slight cardboard stuck together to frame a level, semirigid material. Some have proposed that the playing a card game previously worked as "play cash" and addressed the stakes utilized for other betting games, and later turned out to be essential for the actual games. Others have proposed associations between playing a card game and chess or dice games, yet this is again theoretical. Different games, for example, a 25-card variation of Euchre which involves the Joker as the most elevated trump, make it one of the main in the game. Frequently, the Joker is a special case, and subsequently permitted to address other existing cards.
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Here are the city gas mileages for 13
different midsized cars in 2008.
16, 15, 22, 21, 24, 19, 20, 20, 21, 27,
18, 21, 48
What is the maximum?
Answer: 48
Step-by-step explanation: 48 is the highest mileages
A store buys 9 blue hats for every 6 red hats a. What’s is the ratio of blue hats to red hats b. What is the ratio of the blue hats all hats c. If the store purchased 45 hats, how many hats were blue
The ratio of blue hats to red hats IS 3:2 , the ratio of the blue hats all hats is 3:5 and If the store purchased 45 hats, 27 hats were blue
The ratio of blue hats to red hats is 9:6, which can be simplified to 3:2
Therefore the ratio of blue hats to red hats is 3: 2
The total number of hats is the summation of blue hats and red hats = 9+6= 15
So, The ratio of blue hats to all hats is 9:15, which can be simplified to 3:5
Therefore the ratio of blue hats to red hats is 3: 5
If the store purchased 45 hats, and we need to find out the number of blue hats
We have to assume the number of blue hats be x
Then the ratio becomes x: 45::3:5
We can also write this as
X= 3a and 45= 5a
The value of a comes out to be 9
So the number of blue hats purchased in the lot is 3a= 3 9 = 27
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Julie planted t fewer trees than Logan. Logan planted 6 trees. Write an expression that shows how many trees Julie planted.
Answer: 6 - t
Step-by-step explanation:
fewer trees than Logan is another word for subtraction.
Since Julie planted t fewer, and Logan planted 6 that means
6 - t
The expression that shows how many trees Julie planted would be; 6 - t
What is an expression?Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.
We are given that Julie planted t fewer trees than Logan. Logan planted 6 trees.
Here, fewer trees than Logan means subtraction.
Since Julie planted t fewer, and Logan planted 6 that means;
the difference between them; 6 - t
Therefore, the expression that shows how many trees Julie planted would be; 6 - t
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!HURRY PLSS!! 20 points!!
A transformation maps a preimage triangle to the image triangle shown in the coordinate plane below.
If the preimage triangle is reflected over the y-axis to get the image triangle, what are the coordinates of the vertices of
the preimage triangle?
The coordinates of the vertices of the preimage triangle = (4, 1), (6,1) and (6, -6).
Given the figure of the image triangle.
The coordinates of the image triangle are (-4, 1) , (-6,1) and (-6, -6).
We are asked to find the coordinates of the preimage triangle.
The preimage triangle was reflected over the y-axis .
The image triangle is lying on the 2nd and 3rd quadrants.
So the preimage triangle will lie on the 1st and 4th quadrants.
Then the coordinate (-4,1) will be the image of (4,1).
(-6, 1) will be the image of (6,1) and (-6,-6) will be the image of (6,-6).
Hence the coordinates of the vertices of the preimage triangle = (4, 1), (6,1) and (6, -6).
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solve the linear equation 4x-(2x-1)=x+5+x-6
The linear equation doesn't have a solution.
How to compute the value?The linear equation given is illustrated as: 4x-(2x-1) = x+5+x-6
This will be solved thus:
4x - 2x + 1 = x+5+x-6
4x - 2x + 1 = 2x - 1.
2x + 1 = 2x - 1
Collect like terms
2x - 2x = -1 - 1
0 = -2
This illustrates that the equation doesn't have a solution.
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Jessie surveyed some dentists and found that 42 out of 60 favored toothpaste with
fluoride. Can 42 out of 60 can be represented as a terminating or non-terminating
decimal?
Answer:non terminating
Step-by-step explanation: