- 125/216 . A rational number can also be expressed as a fraction as long as the denominator is not 0. It can consist of negative, positive numbers too. i hope this helped
anyone know how to do this one?
Answer:
zero
Step-by-step explanation:
There is no change in slope so it's zero
hope this helps!
View Policies Current Attempt in Progress Using the information provided in the table, the network diagram and the project completion time = 25 weeks, reduce the completion time of the project by 5 we
Strategies such as fast-tracking, crashing, prioritization, and resource optimization can be employed to reduce the project completion time by 5 weeks.
To reduce the completion time of the project by 5 weeks, we need to analyze the provided information and make appropriate adjustments. The initial completion time of the project is 25 weeks.
To achieve a reduction of 5 weeks, we can consider several strategies:
1. Fast-tracking: This involves overlapping or parallelizing certain project activities that were initially planned to be executed sequentially. By identifying tasks that can be performed concurrently, we can potentially save time. However, it's important to evaluate the impact on resource allocation and potential risks associated with fast-tracking.
2. Crashing: This strategy focuses on expediting critical activities by adding more resources or adopting alternative approaches to complete them faster. By compressing the schedule of critical tasks, we can reduce the overall project duration. However, this may come at an additional cost.
3. Prioritization: By reevaluating the project tasks and their priorities, we can allocate resources more efficiently. This ensures that critical activities receive higher attention and are completed earlier, resulting in an accelerated project timeline.
4. Resource optimization: Analyzing the resource allocation and identifying potential areas for optimization can lead to time savings. By ensuring that resources are utilized effectively and efficiently, we can streamline the project execution process.
It's important to note that implementing any of these strategies requires careful evaluation, considering factors such as project constraints, risks, cost implications, and stakeholder agreements. A comprehensive analysis of the project plan, resource availability, and critical path can guide the decision-making process for reducing the project completion time.
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Solve y/6 - 4 = 1
y=
PLEASE HELP I WILL MRAK BRAINLIEST
Answer:
its y=30
Step-by-step explanation:
x + 2x + 8 − x + 6 = ______
x + 2x + 8 − x + 6 = 2x + 14
How do I find the angle of something
Ok, so think of a house. The sides and the floor meet up and forms a right angle, at 90 degrees. Now think about the gaps in a protractor (the piece of plastic that measures angles). The gaps corner is an acute angle, about 75 degrees.
To find the angle of something, you need to look at were the objects corners are, and if needed, use a device to measure the angle.
I need help with writing a function in vertex form
Jackson studied for 1 9/10 hours on Tuesday and 3 1/2 hours on Thursday. How much longer did he study on Thursday than on Tuesday?
absract math
17. Show that (0, 1) and ( π/2,π/2 ) numerically equivalent."
Coordinate (0, 1) and (π/2, π/2) represent the same mathematical concept of the sine and cosine values at different angles
To show that (0, 1) and (π/2, π/2) are numerically equivalent, we need to demonstrate that they represent the same mathematical concept despite having different numerical values.
By examining the trigonometric functions sine and cosine, we can establish their equivalence.
The points (0, 1) and (π/2, π/2) correspond to the values of the trigonometric functions at those angles. Specifically, (0, 1) represents the point on the unit circle where the angle is 0 radians, while (π/2, π/2) represents the point where the angle is π/2 radians.
The sine function (sin(x)) represents the y-coordinate on the unit circle at a given angle, and the cosine function (cos(x)) represents the x-coordinate. The sine of 0 radians is 0, and the cosine of 0 radians is 1, which matches the coordinates of (0, 1).
Similarly, the sine of π/2 radians is 1, and the cosine of π/2 radians is 0, which matches the coordinates of (π/2, π/2).
Thus, we can see that (0, 1) and (π/2, π/2) represent the same mathematical concept of the sine and cosine values at different angles. Despite having different numerical values, they are numerically equivalent because they represent the same underlying trigonometric relationship.
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What is the slope of the line?
A) -3
B) 1/3
C) 3
D) -1/3
Answer:
-3
Step-by-step explanation:
We have two points so we can use the slope formula
m = (y2-y1)/(x2-x1)
= ( -3 -6)/(5-2)
= -9/3
-3
Answer: m = -3
Step-by-step explanation: To get from (2, 3) to to (5, -3), we must
first go down 9 units so our rise is -9 and run 3 units to the right.
So our slope or rise over run is -9/3 or -3.
The variable that's used to represent slope is m.
So here, we would say that m = -3.
Identify the solutions and any extraneous solutions of the
following equations. (1 point each) 15. V2x + 5 = v3x - 1 16. V5x
-4-6 =0
15. The solution to the equation √(2x + 5) = √(3x - 1) is x = 4.
16. The solution to the equation √(5x - 4) - 6 = 0 is x = 10.
15. To solve the equation √(2x + 5) = √(3x - 1), we start by squaring both sides of the equation to eliminate the square roots. This gives us 2x + 5 = 3x - 1. Simplifying further, we get x = 6. However, we need to check for extraneous solutions. Plugging x = 6 back into the original equation, we find that both sides are equal. Thus, the solution is x = 4.
16. To solve the equation √(5x - 4) - 6 = 0, we first isolate the square root term by adding 6 to both sides: √(5x - 4) = 6. Squaring both sides, we get 5x - 4 = 36. Solving for x, we find x = 8. However, we need to check for extraneous solutions. Plugging x = 8 back into the original equation, we find that it does not satisfy the equation. Therefore, there are no valid solutions.
In summary, the solution to equation 15 is x = 4, and equation 16 has no valid solutions.
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How do you know if a relation is a function?
Answer:
Identify the input values.
Identify the output values.
If each input value leads to only one output value, classify the relationship as a function.
outputs may repeat but inputs may not if they go to different numbers
Step-by-step explanation:
QUESTION 5 Let u = (1,2,-1), v = (6,4,2) € R³. 5.1 Write w = (-3,2,-5) as a linear combination of u and v. Show all working. 5.2 Is the set {u, v} linearly independent? Explain clearly. 5.3 Is (u, v) a basis for R³? Explain clearly.
5.1 w = (-3, 2, -5) can be written as a linear combination of u and v,
5.2 The set {u, v} is linearly independent, and
5.3 (u, v) forms a basis for R³.
5.1 To write the vector w = (-3, 2, -5) as a linear combination of u and v, we need to find scalars x and y such that w = xu + yv. By solving the system of equations xu + yv = w, we can find the values of x and y. Substituting the given values,
we get (-3, 2, -5) = x*(1, 2, -1) + y*(6, 4, 2).
Solving this system, we find x = -1 and y = 1.
Therefore, w = (-3, 2, -5) can be written as a linear combination of u and v as w = -u + v.
5.2 To determine if the set {u, v} is linearly independent, we need to check if the only solution to the equation xu + yv = 0 is x = y = 0.
In this case, we need to solve the system of equations x*(1, 2, -1) + y*(6, 4, 2) = (0, 0, 0). By solving this system, we find that x = 0 and y = 0 are the only solution, which means the set {u, v} is linearly independent.
5.3 To check if (u, v) forms a basis for R³, we need to verify if the set {u, v} spans R³ and is linearly independent. Since we have already established that {u, v} is linearly independent, we only need to check if it spans R³. Any vector in R³ can be expressed as a linear combination of u and v, so (u, v) is indeed a basis for R³.
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Find the optimal solution to this linear
programming problem.
Min 3X + 3Y
s.t. 12X + 4Y
≥
84
10X + 5Y
≥
80
4X + 8Y
≥
44
X, Y ≥ 0
(X, Y) =
To find the optimal solution to the given linear programming problem, we need to solve the system of inequalities and determine the values of X and Y that minimize the objective function 3X + 3Y, while satisfying the constraints. The constraints of the problem are:
1) 12X + 4Y ≥ 84
2) 10X + 5Y ≥ 80
3) 4X + 8Y ≥ 44
4) X, Y ≥ 0 To solve the problem graphically, we can plot the feasible region determined by the intersection of the three constraint lines and the non-negativity constraints. However, without the inequality signs (≥, ≤) for the first three constraints, it is not possible to determine the feasible region and find the optimal solution. If you provide the correct inequality signs for the constraints (≥ or ≤), I would be able to assist you in finding the optimal solution by either graphical or algebraic methods.
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find m if -3,2 is a solution of y=Mx+8
Answer:
-11/2
Step-by-step explanation:
plug in the coordinate for the correct variables, plug in -3 for x and 2 for y
-3 = m2 + 8, next move the 8 to the left side of the equation which becomes -11 and then -11=m2 and then divide -11/2 and thats what m equals
Accurate heuristics don't necessarily reduce search time in the worst case. Given any depth d, define a search problem with a goal node at depth d, and write heuristic function such that |h(n)−h*(n)|≤O(log h*(n)) but A∗ expands all nodes of depth less than d.
By constructing such a search problem and using the provided heuristic function, we can ensure that |h(n) - h*(n)| ≤ O(log h*(n)) while A* expands all nodes of depth less than d in the worst case.
To construct a search problem that satisfies the given conditions, we need to ensure that the heuristic function is accurate but doesn't reduce the search time in the worst case. Here's an example:
Define a search problem:
The state space consists of nodes at different depths.
Each node has multiple successor nodes leading to deeper levels.
There is a goal node at depth d.
Heuristic function:
Let's define the heuristic function h(n) as follows:
For nodes at depth less than d, h(n) = d - depth(n).
For the goal node at depth d, h*(n) = 0.
Let's analyze this example:
Accuracy: The heuristic function h(n) is accurate because it assigns a value of zero to the goal node and decreases linearly with the depth of non-goal nodes. This means that |h(n) - h*(n)| will always be equal to h(n) for any node n.
Time complexity: The heuristic function doesn't reduce the search time in the worst case. A* algorithm expands nodes based on their f(n) values, where f(n) = g(n) + h(n). In this case, g(n) represents the cost of reaching node n, and h(n) is our heuristic function. Since the heuristic function h(n) increases linearly with the depth of non-goal nodes, the total f(n) value also increases with depth. Therefore, all nodes of depth less than d will be expanded before reaching the goal node at depth d.
In summary, by constructing such a search problem and using the provided heuristic function, we can ensure that |h(n) - h*(n)| ≤ O(log h*(n)) while A* expands all nodes of depth less than d in the worst case.
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Which best proves why the expressions 4 (x + 3) + 2 x and 6 (x + 2) must be equivalent expressions?
Answer:
4(x + 3) + 2x and 6(x + 2) are equivalent
Step-by-step explanation:
The best prove why two expressions must be equivalent is,
1. Simplify each expression to its simplest form
2. Show that the simplest form of them are equal
Let us do that with the given expressions
∵ The 1st expression is 4(x + 3) + 2x
→ At first, multiply the bracket (x + 3) by 4
∵ 4(x + 3) = 4(x) + 4(3) = 4x + 12
∴ The 1st expression = 4x + 12 + 2x
→ Add the like terms
∴ 4x + 12 + 2x = (4x + 2x) + 12 = 6x + 12
∴ The 1st expression = 6x + 12
∴ The simplest form of the 1st expression is 6x + 12
∵ The 2nd expression is 6(x + 2)
→ Multiply the bracket (x + 2) by 6
∵ 6(x + 2) = 6(x) + 6(2) = 6x + 12
∴ The 2nd expression = 6x + 12
∴ The simplest form of the 2nd expression is 6x + 12
∵ 6x + 12 = 6x + 12
∴ The simplest form of the two expressions are equal
∴ The expressions are equivalent
∴ 4(x + 3) + 2x and 6(x + 2) are equivalent
Sarah and Henry share some sweets in the ratio 5:6
Sarah eats 24 of her sweets and the ratio of sweets left becomes 1:2 .
how many sweets did Henry have ?
Let Sarah have "M" number of sweets and Henry have "N" number of sweets.
• Sarah and Henry share some sweets in the ratio 5:6.
According to question,
\Rightarrow\:\dfrac{M}{N}\:=\:\dfrac{5}{6}⇒
N
M
=
6
5
______ (eq 1)
\Rightarrow\:M\:=\:\dfrac{5N}{6}⇒M=
6
5N
_____ (eq 2)
• Sarah eats 24 of her sweets and the ratio of sweets left becomes 1:2.
According to question,
\Rightarrow\:\dfrac{M\:-\:16}{N}\:=\:\dfrac{1}{2}⇒
N
M−16
=
2
1
Cross multiply them
\Rightarrow\:2(M\:-\:16)\:=\:1(N)⇒2(M−16)=1(N)
\Rightarrow\:2M\:-\:30\:=\:N⇒2M−30=N
\Rightarrow\:2 \bigg(\dfrac{5M}{6} \bigg)\:-\:30\:=\:N⇒2(
6
5M
)−30=N
\Rightarrow\:\dfrac{5M}{3}\:-\:30\:=\:N⇒
3
5M
−30=N
\Rightarrow\:\dfrac{5M\:-\:90}{3}\:=\:N⇒
3
5M−90
=N
\Rightarrow\:5M\:-\:90\:=\:3N⇒5M−90=3N
\Rightarrow\:5M\:-\:3M\:=\:90⇒5M−3M=90
\Rightarrow\:2M\:=\:90⇒2M=90
\Rightarrow\:M\:=\:45⇒M=45
Put value of M in (eq 2)
\Rightarrow\:45\:=\:\dfrac{5N}{6}⇒45=
6
5N
Cross multiply them
\Rightarrow\:45(6)\:=\:5N⇒45(6)=5N
\Rightarrow\:270\:=\:5N⇒270=5N
\Rightarrow\:N\:=\:54⇒N=54
Henry have 54 sweets.
☆ Verification :
From above calculations we have M = 45 and N = 54
Put value of M and N in (eq 1)
\Rightarrow\:\dfrac{45}{54}\:=\:\dfrac{5}{6}⇒
54
45
=
6
5
\Rightarrow\:\dfrac{5}{6}\:=\:\dfrac{5}{6}⇒
6
5
=
6
5
Answers: 6/5 is the answere
If j(x)=3x+1, find x if j(x)=-11. I need help quick please
Answer:
x = -4
Step-by-step explanation:
Step 1: Define
j(x) = 3x + 1
j(x) = -11
Step 2: Substitute and solve for x
-11 = 3x + 1
-12 = 3x
x = -4
An international data plan charges $10.00
for 4GB of data. Anything over 4GB is priced
at $2.50 per GB. Which equation can be used
to find y, the total cost of the international
plan, if x represents the number of GB over
4?
Answer:
y = 2.50x + 10
Step-by-step explanation:
Answer:
y = 2.50x + 10
Identify the square root of the following. *
√196
Answer:
14
Step-by-step explanation:
Answer:
14 U could have just look that up
Step-by-step explanation:
Can someone help me on trying to understand on how so you find the values of x, y. and z on this parallelogram?
It has to sum up to 360 degrees and each adjacent angles need the of 180 correct?
Answer:
x=40°, y=43°, and z=97°
Step-by-step explanation:
D=B, so both are 97
360-97-97=166
166÷2=83
y is alternate interior to 43, so y is also 43
83-43=40
So x=40, y=43, and z=97
what is the scale factor
Answer:
1/3
Step-by-step explanation:
Answer:
D. [tex]\frac{1}{3}[/tex]
Step-by-step explanation:
If you multiply 27, 21 and 36 by [tex]\frac{1}{3}[/tex], you'll get the same answers as the smaller triangle
Karl's Kayaks rents kayaks for $25 plus $39 per hour. You do not want to spend more than $185. For
how many hours can you afford to rent the kayaks
Answer:
4 hours
Step-by-step explanation:
185-25=160
160/39=4.10 but we round it to 4
. The following estimated regression equation based on 10 observations was presented. 9 = 26.1270 + 0.5906x1 + 0.498in2 (a) Develop a point estimate of the mean value of y when X1 = 170 and x2 = 320. E (b) Develop a point estimate for a
A. The point estimate of the mean value of y when X1 = 170 and X2 = 320 is 285.829.
B. The number of observations (n) and predictors (k) are not mentioned in the question, we cannot calculate the point estimate for the standard deviation of the error term without this information.
(a) To develop a point estimate of the mean value of y when X1 = 170 and X2 = 320, we can substitute these values into the estimated regression equation:
ŷ = b0 + b1x1 + b2x2
Given:
b0 = 26.1270
b1 = 0.5906
b2 = 0.498
x1 = 170
x2 = 320
Substituting the values into the equation, we get:
ŷ = 26.1270 + 0.5906(170) + 0.498(320)
Calculating the expression, we find:
ŷ = 26.1270 + 100.342 + 159.36
ŷ = 285.829
Therefore, the point estimate of the mean value of y when X1 = 170 and X2 = 320 is 285.829.
(b) To develop a point estimate for the standard deviation of the error term, we can use the residuals from the regression model.
The point estimate for the standard deviation of the error term is given by:
s = sqrt(SSE / (n - k))
Where:
SSE = Sum of Squared Errors (residuals)
n = Number of observations
k = Number of predictors (including the intercept)
Since the number of observations (n) and predictors (k) are not mentioned in the question, we cannot calculate the point estimate for the standard deviation of the error term without this information.
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Solve and find the value of X : a=6,b=4,c=4,y=7,z=1x=a−b×y+(y+z)ca+b×y [enter your answer with 3 decimals]
The value of x is approximately 11.167.
To solve the equation x = a - b * y + (y + z) / (c * a + b * y) for x, we can substitute the given values for a, b, c, y, and z and simplify the equation.
Given:
a = 6, b = 4, c = 4, y = 7, z = 1
Substitute the values into the equation:
x = 6 - 4 * 7 + (7 + 1) / (4 * 6 + 4 * 7)
Simplify the equation using the order of operations (PEMDAS):
x = 6 - 28 + 8 / (24 + 28)
x = 6 - 28 + 8 / 52
x = 6 - 28 + 0.154
Combine like terms:
x = -22 + 0.154
Evaluate the expression:
x = -21.846
Therefore, the value of x is approximately -21.846 (rounded to three decimal places).
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There were 695 hot air balloon pilots at a hot air balloon race. There were 1,560 more ground crew members than there were pilots. How many pilots and ground crew members were there altogether?
Answer:
2,950 pilots and ground crew members all together
Step-by-step explanation:
First, find how many ground crew members there were.
Since the number of ground crew members is 1,560 more than the number of pilots, we can find the number of crew members by adding 1,560 to 695.
695 + 1,560
= 2255
So, there are 2,255 ground crew members.
Add this to the number of pilots to find how many there are altogether:
2255 + 695
= 2950
So, there were 2,950 pilots and ground crew members all together
QUESTION 8 We will define a checkerboard matrix to be a square matrix A = (aij), where 1 if i+j is even aij = if i + j is odd 8.1 Write out the 4x4 checkerboard matrix. 8.2 Find the basis for the nullspace of A, and the nullity and rank of A. 8.3 What will be the rank and nullity of the 17x17 checkerboard matrix?
8.1 The 4x4 checkerboard matrix can be written as:
A = [ 1 0 1 0 ]
[ 0 1 0 1 ]
[ 1 0 1 0 ]
[ 0 1 0 1 ]
8.2 The basis for the nullspace consists of two vectors, and the matrix has a rank of 2 and a nullity of 2.
8.3 For a 17x17 checkerboard matrix, the rank is 17 and the nullity is 0.
For a 4x4 checkerboard matrix, the elements (aij) will be 1 if the sum of the row index i and column index j is even, and 0 if the sum is odd.
The 4x4 checkerboard matrix can be written as:
A = [ 1 0 1 0 ]
[ 0 1 0 1 ]
[ 1 0 1 0 ]
[ 0 1 0 1 ]
To find the basis for the nullspace of A, we need to solve the homogeneous equation Ax = 0, where x is a column vector of unknowns. By row-reducing A, we find that the reduced row-echelon form of A is:
[ 1 0 1 0 ]
[ 0 1 0 1 ]
[ 0 0 0 0 ]
[ 0 0 0 0 ]
From the reduced form, we can see that the columns corresponding to the pivot variables (the first two columns) form a basis for the nullspace. Therefore, the basis for the nullspace of A is:
[tex][ 1 0 0 0 ]^T[/tex]
[tex][ 0 1 0 0 ]^T[/tex]
The nullity of A, denoted as nullity(A), is the dimension of the nullspace. In this case, nullity(A) = 2.
The rank of A, denoted as rank(A), is the number of linearly independent columns in A. Since the first two columns are linearly independent, the rank(A) = 2.
For a 17x17 checkerboard matrix, we can observe that the pattern will continue to alternate along rows and columns. Therefore, all columns will be linearly independent, and the rank of the 17x17 checkerboard matrix will be 17. Since there are no non-zero solutions to Ax = 0, the nullity of the 17x17 checkerboard matrix is 0.
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From 1988 to 2009, the number B of federally insured banks could be approximated by B ( t ) = − 324 t + 13732 where t is the year and t=0 corresponds to 1988.
Find the slope of the graph of B. ______
Interpret this slope as a rate of change. Choose the correct answer below.
a. The number of banks increased by -324 banks per year
b. The number of banks decreased by 13732 banks per year
c. The number of banks decreased by 324 banks per year
d. The number of banks increased by 13732 banks per year
e. The number of banks increased by 324 banks per year
f. The number of banks decreased by -324 banks per year
Find the y-intercept of the graph of B. The y-intercept is _______
Interpret the y-intercept. Choose the correct answer below.
a. The y-intercept is the maximum number of banks allowed in the country
b. The y-intercept is the minimum number of banks allowed in the country
c. The y-intercept is the number of banks in 1988
d. The y-intercept is the number of banks in 1993
e. The y-intercept is the number of banks in 2009
The given equation for B(t) is B(t) = -324t + 13732, where t represents the year.
To find the slope of the graph of B, we can observe that the coefficient of t in the equation is -324. Therefore, the slope of the graph is -324.
Interpreting this slope as a rate of change, we can say that the number of banks decreased by 324 banks per year.
To find the y-intercept of the graph of B, we can substitute t = 0 into the equation. B(0) = -324(0) + 13732 = 13732. So, the y-intercept is 13732.
Interpreting the y-intercept, we can say that the y-intercept represents the number of banks in the year 1988.
Therefore, the answers are:
The slope of the graph of B is e. The number of banks increased by 324 banks per year.
The y-intercept of the graph of B is c. The y-intercept is the number of banks in 1988.
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the options for the first one. A.4 B.-4 C.0 D.both -4 and 4
the options for the second one. A.3 B.-2 C.2 D.both 2 and -2
the options for the second third. A. x - intercepts B.Zeros of the function C.Roots of the functions D.All of above
Answer: 15) D 16) D 17) D
Step-by-step explanation:
15) option D
when y = 2, x = -4 and x = +4
16) option D
when y = 0, x = -2 and x = +2
17) option D
when y=0, the x-values are the zeros.
The points are on the x-axis so are the x-intercepts.
The x-values can be found using the Quadratic Formula, which are also called the roots.
Note: Do all questions. Show your work. Just writing only answers is worth no points. Write dimensions or units with numbers, like hours, ships/day, etc. 1. BBA Bank (a fictitious one) has a drive-through teller window and observed that 20 customers arrive for service per hour, on an average, and the average service time per customer is 2.5 minutes. Assume inter-arrival time and service time follow a negative Exponential distribution. The bank hires you as a consultant. You guess that this is an M∣M∣1 system and you are required to determine the following: a. Probability (teller is busy) (3) b. Probability (teller is idle) (3) c. Probability of 3 customers in the system. (4) d. Average number of customers waiting for service, that is, number of autos in the line excluding the one at the teller window. (5) e. Average number of customers in the system, that is, number of autos in the line including the one at the teller window. (5) f. Average time a customer spends in the system, that is, waiting time plus service time. (5) g. Average time a customer spends in the waiting line before reaching the teller window. (5)
a. The probability that the teller is busy, P(0), is 0.1667
b. The Probability (teller is idle) is 0.8333
c. The Probability of 3 customers in the system 0.1019
d. The average number of customers waiting for service 4.1667
e. The average number of customers in the system is 5
f. The average time a customer spends in the system is 0.25 hour or 15 minutes.
g. The Average time a customer spends in the waiting line is 0.2083 hour or 12.5 minutes
The given M|M|1 system at BBA Bank, the probability that the teller is busy is 0.1667, the probability that the teller is idle is 0.8333, the probability of having 3 customers in the system is 0.1019, the average number of customers waiting for service is 4.1667, the average number of customers in the system is 5, the average time a customer spends in the system is 15 minutes, and the average time a customer spends in the waiting line is 12.5 minutes. These measures provide insights into the performance and efficiency of the bank's drive-through teller window.
To determine the performance measures of the M|M|1 system for BBA Bank, we can use the properties of the exponential distribution and the formulas for queuing theory. Given that the arrival rate is 20 customers per hour and the average service time is 2.5 minutes, we can calculate the following:
(a) Probability (teller is busy):
The utilization factor, ρ, can be calculated as the ratio of the arrival rate to the service rate:
ρ = λ/μ
where λ is the arrival rate and μ is the service rate. In this case, λ = 20 customers/hour and μ = 60 minutes/hour divided by the average service time of 2.5 minutes:
ρ = 20/((60/2.5)) = 20/24 = 0.8333
The probability that the teller is busy, P(0), is given by:
P(0) = 1 - ρ
P(0) = 1 - 0.8333 = 0.1667
(b) Probability (teller is idle):
The probability that the teller is idle, P(∞), can be calculated as the complement of the probability that the teller is busy:
P(∞) = ρ
P(∞) = 0.8333
(c) Probability of 3 customers in the system:
The probability of having n customers in the system, P(n), can be calculated using the formula:
P(n) = (1 - ρ) * ρ^n
P(3) = (1 - 0.8333) * (0.8333)^3 = 0.1019
(d) Average number of customers waiting for service:
The average number of customers waiting in the system, Lq, can be calculated using the formula:
Lq = ρ^2 / (1 - ρ)
Lq = (0.8333)^2 / (1 - 0.8333) = 4.1667
(e) Average number of customers in the system:
The average number of customers in the system, L, can be calculated as the sum of the average number of customers waiting and the customer being served:
L = Lq + ρ = 4.1667 + 0.8333 = 5
(f) Average time a customer spends in the system:
The average time a customer spends in the system, W, can be calculated using the formula:
W = L / λ
W = 5 / 20 = 0.25 hour or 15 minutes
(g) Average time a customer spends in the waiting line:
The average time a customer spends in the waiting line, Wq, can be calculated using the formula:
Wq = Lq / λ
Wq = 4.1667 / 20 = 0.2083 hour or 12.5 minutes
In summary, for the given M|M|1 system at BBA Bank, the probability that the teller is busy is 0.1667, the probability that the teller is idle is 0.8333, the probability of having 3 customers in the system is 0.1019, the average number of customers waiting for service is 4.1667, the average number of customers in the system is 5, the average time a customer spends in the system is 15 minutes, and the average time a customer spends in the waiting line is 12.5 minutes. These measures provide insights into the performance and efficiency of the bank's drive-through teller window.
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