We have proved that if the sequence (a_n) satisfies lim a_n = L = 0, then (a_n) is unbounded.
To prove that if the sequence (a_n) satisfies lim a_n = L = 0, then (a_n) is unbounded, we will use proof by contradiction.
Assume that (a_n) is bounded. Then, there exists a positive number M such that |a_n| ≤ M for all n in the natural numbers.
Since lim a_n = L = 0, we can choose an ε > 0 such that if n is sufficiently large, then |a_n - L| < ε. In other words, there exists a natural number N such that for all n ≥ N, |a_n - L| < ε.
Consider the case when n ≥ N and a_n > 0 (the case when a_n < 0 is similar). Then, we have:
a_n = L + (a_n - L) > L - ε
Since a_n ≤ M, we have:
0 ≤ a_n < M
Combining these inequalities, we get:
0 ≤ L - ε < a_n < M
This implies that a_n is bounded between two positive numbers, which contradicts the assumption that (a_n) is unbounded. Therefore, our initial assumption that (a_n) is bounded must be false, and hence (a_n) is unbounded when lim a_n = L = 0.
Therefore, we have proved that if the sequence (a_n) satisfies lim a_n = L = 0, then (a_n) is unbounded.
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An analysis of variance is used to evaluate the mean differences for a research study comparing four treatment conditions with a separate sample of n = 5 in each treatment. The analysis produces SSwithin treatments = 32, SSbetween treatments = 40, and SStotal = 72 For this analysis, what is MSwithin treatments?
The MSwithin treatments for this analysis is 2.
In analysis of variance (ANOVA), we partition the total variation in a set of data into two types of variation: variation within groups and variation between groups. This partitioning helps us to test whether the means of the groups are significantly different from each other or not.
The formula for calculating the mean square within treatments (MSW) is:
MSW = SSW / dfW
where SSW is the sum of squares within treatments and dfW is the degrees of freedom associated with the SSW.
To calculate MSW, we first need to calculate dfW, which is equal to the total number of observations minus the total number of groups. In this case, there are 4 groups, each with 5 observations, so there are a total of 20 observations:
dfW = 20 - 4 = 16
Next, we can use the given SSwithin treatments value of 32 to calculate SSW:
SSW = 32
Finally, we can plug in the values we have calculated to find MSW:
MSW = SSW / dfW
MSW = 32 / 16
MSW = 2
Therefore, the MSwithin treatments for this analysis is 2.
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"Evaluate the following continuous-time convolution integrals
(k) y(t)=e-yt u (t) x (u(t+2)-u(t))
This question is in the Signals and Sysytems 2nd edition."
The continuous-time convolution integral of y(t) is [tex]$y(t) = k e^{-yt} u(t) * (u(t+2)-u(t))$[/tex].
To evaluate this convolution integral, we first need to express the integrand as a piecewise function. Since u(t) is 1 for t >= 0 and 0 for t < 0, we can rewrite u(t+2)-u(t) as a piecewise function:
u(t+2)-u(t) =
1, 0 <= t < 2
0, t >= 2
0, t < 0
Now we can evaluate the convolution integral using the definition:
y(t) = ∫[tex]_0^t[/tex] x(τ)h(t-τ)dτ
Substituting the given functions for x(t) and h(t) and simplifying using the piecewise function for u(t+2)-u(t), we get:
y(t) = k ∫[tex]_0^t[/tex] [tex]e^}(-yt)}[/tex]dτ = [tex]k[-(1/y)e^{(-yt)}]_0^t = k(1 - e^{(-yt)})/y[/tex], t >= 0
Therefore, the continuous-time convolution integral of y(t) is [tex]$y(t) = k e^{-yt} u(t) * (u(t+2)-u(t))$[/tex] for t >= 0.
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A recent study found that the weight of a certain species of fish living in the Bahamas can be modeled by the function 0. 034213, where w is measured in grams and L, the length of the fish is measured in centimeters. Calculate the approximate length of a fish that weighs 250 grams. Round your answer to the nearest tenth of a centimeter.
L=____ centimeters
To solve this problem, we need to use the given function to find the length of a fish that weighs 250 grams. We can do this by setting the weight of the fish (w) to 250 grams and solving for the length of the fish (L):
w = 0.034213 L
250 = 0.034213 L
L = 250 / 0.034213
L ≈ 7304.4
Rounding to the nearest tenth of a centimeter, the approximate length of the fish is:
L ≈ 730.4 centimeters
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Two monomials are shown below. 28x²y 34x y What is the greatest common factor (GCF) of these monomials? A
7xy B
4x²y
C2xy
D2x²y
I need an answer asap
Answer: 2xy
Step-by-step explanation:
you need the biggest factor that goes into
6) Find the range of values for x using the
Triangle Inequality Theorem.
X
15.4
7.6
Jayclyn has 24 cupcakes. Out of the cupckaes, 1/3 of them have vanilla frosting. How many cupcakes have vanilla frosting
Answer:
8
Step-by-step explanation:
24 divided by 3 is 8
8 in each 1/3 so 8 vanilla
The current of a system is defined as the following function i(t) = Ae-1-1-1 Evaluate the rate of change of the current at time t = 0.01s and A = 20 0 28.79 O 18.79 O 10.79 O None 0 20.79
The rate of change of the current is -19.79.
You provided the function i(t) = Ae^(-t) and you'd like to evaluate the rate of change of the current at time t = 0.01s with A = 20.
Step 1: Find the derivative of the function i(t) with respect to time (t). This will give us the rate of change.
di/dt = -Ae^(-t)
Step 2: Substitute the given values for A and t into the derivative equation.
di/dt = -20e^(-0.01)
Step 3: Evaluate the expression.
di/dt ≈ -20 * 0.99004983 ≈ -19.79
So, the rate of change of the current at time t = 0.01s and A = 20 is approximately -19.79.
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A tin contains 5 cookies. Only 1 of them is chocolate flavoured. Noah picks a cookie at random from the tin. What is the probability that it is chocolate flavoured? Give your answer as a decimal.
Let A = {1,2,3,4}, give an example of a relation on A|| that is • reflexive and symmetric but not transitive. You may give your example in the form R = {(x,y),(y,w), (w,y), ... } or draw a directed graph reflexive and transitive but not symmetric. You may give your example in the form R = {(x,y),(y,w),(w,y),... } or draw a directed graph
For the first part of the question, let R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,1), (2,3), (3,2)}. This relation is reflexive because every element in A is related to itself, and it is symmetric because if (x,y) is in R, then (y,x) is also in R. However, it is not transitive because (1,2) and (2,3) are in R, but (1,3) is not in R.
For the second part of the question, let R = {(1,1), (2,2), (3,3), (4,4), (1,2), (2,3), (3,4), (1,3), (2,4)}. This relation is reflexive because every element in A is related to itself, and it is transitive because if (x,y) and (y,z) are in R, then (x,z) is also in R. However, it is not symmetric because (1,2) is in R, but (2,1) is not in R.
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Shannon’s Ice Cream Shop, Shenanigans, sells three types of ice cream: Peanut Butter Cup, Coffee Cream and Oreo. Last week Shenanigans sold a total of 489 ice cream cones that were either Coffee Cream or Oreo. In total, they sold 711 ice cream cones last week. What is the probability Shenanigan’s will sell a Peanut Butter Cup cone next? Express your answer as a percent.
The probability of Shenanigans selling a Peanut Butter Cup cone next is approximately 31.22%.
To calculate the probability of Shenanigans selling a Peanut Butter Cup cone next, we need to determine the number of cones sold that were not Peanut Butter Cup cones and divide that by the total number of cones sold.
Given that Shenanigans sold a total of 711 ice cream cones last week and 489 of those were either Coffee Cream or Oreo, we can subtract 489 from 711 to find the number of cones that were not Coffee Cream or Oreo;
Number of cones that were not Coffee Cream or Oreo = 711 - 489 = 222
So, out of the total 711 cones sold, 222 were not Coffee Cream or Oreo cones. Therefore, the remaining cones must be Peanut Butter Cup cones.
Now, we can calculate the probability of selling a Peanut Butter Cup cone next by dividing the number of Peanut Butter Cup cones by the total number of cones sold, and then multiplying by 100 to find the result as a percentage;
Probability of selling a Peanut Butter Cup cone next = (Number of Peanut Butter Cup cones / Total number of cones sold) × 100
Probability of selling a Peanut Butter Cup cone next = (222 / 711) × 100
≈ 31.22%
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The camp cook made 7 1/2 pints of baked beans. Each serving of beans is 5/6 of a pint. How many servings of beans did the cook make?
The cook made 9 servings of baked beans.
We have,
To find the number of servings of beans, we need to divide the total amount of baked beans by the number of baked beans per serving.
Total amount of baked beans = 7 1/2 pints
Amount of baked beans per serving = 5/6 pint
Number of servings of beans = (7 1/2) ÷ (5/6)
Converting 7 1/2 to an improper fraction:
7 1/2 = (2 × 7) + 1 = 15/2
Number of servings of beans = (15/2) ÷ (5/6)
To divide fractions, we can multiply by the reciprocal of the divisor:
Number of servings of beans = (15/2) × (6/5)
Number of servings of beans = 45/5
Number of servings of beans = 9
Therefore,
The cook made 9 servings of baked beans.
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A random sample of 258 observations has a mean of 35, a median of 32, and a mode of 35. The population standard deviation is known and is equal to 5.8. The 99% confidence interval for the population mean is: "Answer is: {LowerLimit} to {UpperLimit}"
Group of answer choices
A. 30.5 to 38.1
B. 34.1 to 35.9
C. 24.2 to 25.8
D. 24.3 to 25.7
The 99% confidence interval for the population mean is (33.49, 36.51), so the answer is A. 30.5 to 38.1.
We can use the formula for the confidence interval for the population mean when the population standard deviation is known:
CI = X ± z*(σ/√n)
where X is the sample mean, σ is the population standard deviation, n is the sample size, and z is the z-score corresponding to the desired confidence level.
First, let's calculate the z-score for a 99% confidence level. From a standard normal distribution table, we find that the z-score for a 99% confidence level is approximately 2.576.
Next, we can plug in the given values and solve for the confidence interval:
CI = 35 ± 2.576*(5.8/√258)
CI = 35 ± 1.51
CI = (33.49, 36.51)
Therefore, the 99% confidence interval for the population mean is (33.49, 36.51), so the answer is A. 30.5 to 38.1.
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Which figure shows a line segment?
How does the graph of f(x) = (x − 8)^3 + 4 compare to the parent function g(x) = x^3? SHOW WORK!!!!!
Answer:
Step-by-step explanation:
g(x) = x^3 is the parent function starting at the origin (0,0)
f(x) is the translation of the g(x).
Your teacher taught you that y = (x - h)^3 + k
(h,k) is your shift.
(8,4) meaning right 8 on the x-axis and 4 up on the y-axis.
A spinner is divided into 10 equally sized sectors. The sectors are numbered 1 to 10. A randomly selected point is chosen.
What is the probability that the randomly selected point lies in a sector that is a factor of 3?
Enter your answer in the box.
Answer:
0.2 or 20%-----------------------
There are two sectors that are factors of 3:
sector 1, sector 3So out of the 10 sectors, there is a 2/10 probability that the randomly selected point lies in one of these three sectors.
Therefore, the probability is 0.2 or 20%.
Answer:
20%
Step-by-step explanation:
For this question, we must find the numbers from 1-10 that are factors of 3. A factor is a number that, when multiplied by a specific number, gives a specific whole. For instance, in 2*4=8, 2 and 4 would be factors. The number 3 only has two factors: one and itself. Since two of the ten numbers are factors of 3, that is a rate of 2/10, 0.2, or 20%.
Angle xzw ~ to angle xvy, find the perimeter of angle xzw
The perimeter of the triangle XZW is 176.4 units
Finding the perimeter of triangle XZWFrom the question, we have the following parameters that can be used in our computation:
The similar triangles
We start by calculating the missing side lengths using proportions
So, we have
YZ/32 = 28/40
So, we have
YZ = 32 * 28/40
YZ = 22.4
Next, we have
30/WZ = 40/(40 + 32)
So, we have
WZ = 30 * (40 + 32)/40
WZ = 54
The perimeter of triangle XZW is
P = 28 + 22.4 + 54 + 32 + 40
Evaluate
P = 176.4
Hence, the perimeter is 176.4 units
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what is the best way to solve ratios
in an if loop, a variable known as a counter variable is used to track the number of times a block of commands is run. true or false
True. A counter variable is often used in loops to track the number of times the loop has executed.
In programming, a counter variable is a variable that is used to keep track of the number of times a loop has executed. A counter variable is usually initialized to a starting value, and then incremented or decremented with each iteration of the loop. The loop continues to execute as long as the counter variable meets certain conditions.
The purpose of a counter variable is to allow a loop to repeat a specific number of times. For example, if you want to repeat a block of code 10 times, you can set a counter variable to 0, and then use a loop to execute the code until the counter variable reaches 10.
Counter variables are commonly used in programming languages that support loops, such as C++, Java, Python, and JavaScript. They provide a simple and effective way to repeat code without having to write the same statements over and over again.
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If you have $50,000 today and deflation is 5% each year. How much would you need in 20 years to have the same buying power?
Use the formula: A=P(1-r)1^
A 103,946,41
B 23,164,56
C 132,664,89
D 17,924,30
You would need about $17,924.30 in 20 years to have the same buying power as $50,000 today, assuming a 5% annual deflation rate.
Option D is the correct answer.
We have,
To calculate the future value of money adjusted for deflation, we can use the formula:
A = P(1 - r)^n
Where:
A = future value of money
P = present value of money
r = deflation rate
n = number of years
Plugging in the given values, we get:
A = 50,000(1 - 0.05)^20
Simplifying the expression inside the parentheses:
A = 50,000(0.95)^20
A ≈ 17,924.30
Therefore,
You would need about $17,924.30 in 20 years to have the same buying power as $50,000 today, assuming a 5% annual deflation rate.
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Hey can anyone help me
Answer: 12 units
Step-by-step explanation:
Since in this situation we're going by units it will be easier. First to create the rectangle just fill in 3 boxes horizontally and 4 units vertically. connect the squares together.
Then to find the area, the formula for that is Base x Height so all you have to do is multiply 4*3 and that's 12.
Ashley keeps track of their fuel efficiency when they buy gas. The last time, they got 16.6 gallons and had driven 385 miles. What is the fuel efficiency? Select an answer Select an answer miles/gallo
Ashley's fuel efficiency is approximately 23.19 miles per gallon. To calculate Ashley's fuel efficiency in miles per gallon, you can follow these steps:
Step:1. Fuel efficiency = Miles driven / Gallons of fuel used Step:2. Fuel efficiency = 385 miles / 16.6 gallons Step:3. Fuel efficiency = 23.193 miles/gallon (rounded to three decimal places)
Therefore, Ashley's fuel efficiency for this particular fill-up was approximately 23.193 miles per gallon. This means that for every gallon of fuel used, Ashley's car was able to travel approximately 23.193 miles. It's important to note that fuel efficiency can vary depending on factors such as driving habits, vehicle condition, and fuel quality.
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What is the relationship between lines a and b
The lines a and b is parallel to each other. So, the correct answer is A).
Parallel lines are two lines that are always the same distance apart and never meet, no matter how far they are extended. This means that they maintain the same distance between them at all points, and they have the same slope or gradient, but their lengths do not affect their parallel relationship.
Parallel lines are important in geometry and mathematics, and they have numerous real-world applications, such as in architecture, engineering, and design. In many practical situations, parallel lines are used to ensure that objects are straight, balanced, or level. So, the correct answer is A).
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Ricardo has a square pyramid with a base side length of 19 inches and a height of 5 inches. Frido has a square pyramid with a volume of 4873.5 in^3.
Use this information to choose all the correct statements below:
A.
Ricardo’s pyramid has a volume of 601.7 in^3.
B.
Ricardo’s pyramid has a volume of 31.7 in^3.
C.
Frido’s pyramid is 8.1 times larger than Ricardo’s.
D.
Frido’s pyramid is 153.7 times larger than Ricardo’s.
E.
Frido's pyramid is 27% larger than Ricardo's.
F.
Frido’s pyramid is 810% larger than Ricardo’s.
Answer: A, C, and F
Step-by-step explanation:
v=1/3 x B(not b to get B do bxb)xh
V=1/3(19x19)(5)=1/3(361)(5)=1/3(1805)
1805/3=601.7^3
Frido has V=4873.5
601.7x8.1=4873.5
8.1 as a percent is 8.1x100=810%
So the answers are
A.
Ricardo’s pyramid has a volume of 601.7 in^3.
C.
Frido’s pyramid is 8.1 times larger than Ricardo’s.
F.
Frido’s pyramid is 810% larger than Ricardo’s.
The only correct statements are:
Ricardo’s pyramid has a volume of 601.7 in³.
Frido’s pyramid is 8.1 times larger than Ricardo’s.
Options A and C are the correct answer.
We have,
The volumes of square pyramids.
V = (1/3) x s² x h
For Ricardo's pyramid,
s = 19 inches
h = 5 inches.
Substituting these values into the formula, we get:
V = (1/3) x 19² x 5
V ≈ 601.7 in³
Therefore, statement A is correct.
We do not have enough information to determine the volume of Frido's pyramid directly using the given formula.
However, we can use the fact that the volume of a pyramid is proportional to the cube of its linear dimensions (i.e., the length of its sides).
So if Frido's pyramid has a volume that is k times larger than Ricardo's, then the ratio of their corresponding linear dimensions (i.e., side lengths) will be:
(k)^(1/3)
Using this information, we can compare the volumes of the two pyramids:
Frido's pyramid / Ricardo's pyramid = k
(Frido's pyramid / Ricardo's pyramid)^(1/3) = (k)^(1/3)
We are given that Frido's pyramid has a volume of 4873.5 in³.
so,
k = Frido's pyramid / Ricardo's pyramid
k = 4873.5 / 601.7
k = 8.1
Therefore, statement C is correct.
We can also use this value of k to compare the sizes of the two pyramids in other ways:
Frido's pyramid is 7.1 times larger than Ricardo's (k - 1 = 8.1 - 1 = 7.1).
so statement D is incorrect.
Frido's pyramid is approximately 80.5% larger than Ricardo's.
[(k - 1) x 100%
= (8.1 - 1) x 100%
= 750%,
so statement E is incorrect.
Frido's pyramid is approximately 710% larger than Ricardo's
= k x 100%
= 810%
so statement F is incorrect.
Therefore,
The only correct statements are:
Ricardo’s pyramid has a volume of 601.7 in³.
Frido’s pyramid is 8.1 times larger than Ricardo’s.
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Make x the subject of y = 3√(x²+2) 15
The value of the equation in terms of x is x = [tex]\sqrt{\frac{y^2-270}{135}}[/tex].
Given is an equation, y = 3√(x²+2)15, we need to convert it in terms of x,
So,
y = 3√(x²+2) 15
Squaring both sides,
y² = 9 [(x²+2)15]
y² = 9 [15x²+30]
y² = 135x²+270
y²-270 = 135x²
y²-270 / 135 = x²
x = [tex]\sqrt{\frac{y^2-270}{135}}[/tex]
Hence, the value of the equation in terms of x is x = [tex]\sqrt{\frac{y^2-270}{135}}[/tex].
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"In each case suppose R"" has usual norm. Decide whether the statement is true and give a reason for your answer: (a) In R, 2 € B.(-2) (b) In R. -1.5 € B (0) c) In R(1,5,0.5) € B.(-1,0) (d) In R."
This statement is incomplete and does not make sense. It cannot be evaluated as true or false without more information.
(a) In R, 2 € B.(-2)
False.
Explanation: The statement means that 2 is an element of the closed ball centered at -2 with radius 1. But this is not true since the distance between 2 and -2 is greater than 1 (|2 - (-2)| = 4).
(b) In R, -1.5 € B(0)
True.
Explanation: The statement means that -1.5 is an element of the closed ball centered at 0 with radius 1. Since the distance between -1.5 and 0 is less than 1 (|-1.5 - 0| = 1.5 < 1), the statement is true.
(c) In R(1,5,0.5) € B(-1,0)
False.
Explanation: The statement means that (1,5,0.5) is an element of the closed ball centered at (-1,0) with radius 1. But the distance between these two points is greater than 1, since
d((1,5,0.5), (-1,0)) = √[(1-(-1))^2 + (5-0)^2 + (0.5-0)^2] = √[4+25+0.25] = √29.25 > 1.
(d) In R.
True.
Explanation: This statement is incomplete and does not make sense. It cannot be evaluated as true or false without more information.
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Joe began reading at
9:04. He read for 47
minutes. What time did
he finish reading?
Answer: 9:51
Step-by-step explanation:
Each of the following pay stubs represents two different employees use your understanding of income tax to complete the missing pieces
In most countries, including the United States, income tax is a tax imposed on an individual's income by the government. The amount of income tax paid typically depends on the amount of income earned and any deductions or credits that apply to the individual's tax situation.
A typical pay stub includes information such as the employee's gross pay (the amount earned before any deductions), the net pay (the amount received after deductions), and various taxes and deductions taken out of the employee's pay, such as federal income tax, state income tax, Social Security tax, and Medicare tax.
To complete the missing pieces on a pay stub, you will need to know the employee's gross pay and any deductions that apply to their tax situation. The amount of income tax withheld from an employee's pay depends on several factors, including their tax bracket, filing status, and any exemptions or deductions they are eligible for.
It's important to note that tax laws and regulations vary by country and even by state or province within a country, so the specific rules and calculations for income tax can differ depending on your location.
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3. Each of the following pay stubs represents two different employees. Use your understanding of income tax to complete the missing pieces.
LILY'S PAY STUB
GROSS INCOME
$865.00
FEDERAL TAXES
$110.32
=
MEDICARE (6%)
FEDERAL TAXES
$80.32
MEDICARE (6%)
SOCIAL SECURITY (1%)
SOCIAL SECURITY (1%)
$6.65
NET INCOME
MARK'S PAY STUB
GROSS INCOME
NET INCOME
You are asked to estimate the water flow rate in a pipe of radius 2m at a remote area location with a harsh environment. You already know that velocity v varies along the radial location, but you do not know how it varies. The flow rate Q is given by
Q = ∫(0 to 2) 2 π r v dr
To save money, you are allowed to put only two velocity probes (these probes send the data to the central office in New York by satellite). Radial location r is measured from center of the pipe, i.e., r = 0 is the center of the pipe, and r = 2m is the pipe radius. The radial locations you will suggest for the two velocity probes for the most accurate calculation of the flow rate are
0.42, 1.42
0.00, 1.00
0.42, 1.58
0.58, 1.58
To estimate the water flow rate in the pipe, we need to measure the velocity at two radial locations and then use the integral formula to calculate the flow rate. The formula tells us that the flow rate is equal to the integral of 2πrv with respect to r, where r is the radial location, v is the velocity, and the limits of integration are 0 to 2 (since the pipe has a radius of 2m).
Since we don't know how the velocity varies along the radial location, we need to choose two locations that will give us the most accurate estimate of the flow rate. The best locations to choose are where the velocity varies the most, which is usually near the center and near the edge of the pipe.
Based on this, the two radial locations that would give us the most accurate calculation of the flow rate are 0.42 and 1.58. These locations are close to the center and the edge of the pipe, respectively, and will give us a good estimate of how the velocity varies along the radial location.
Therefore, the answer is 0.42, 1.58.
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100 PTS solve for x ................
Answer:
x ≈ 36.2
Step-by-step explanation:
the segment from the vertex of the triangle to the base is a perpendicular bisector, then
consider the right triangle on the right with legs 15 and 33 , hypotenuse x
using Pythagoras' identity in the right triangle
x² = 15² + 33² = 225 + 1089 = 1314 ( take square root of both sides )
x = [tex]\sqrt{1314}[/tex] ≈ 36.2 ( to the nearest tenth )
Answer:
36.2
Step-by-step explanation:
The tick marks on the two sides of the triangle indicate that the sides are of equal length. Therefore, as the triangle has two sides of equal length, it is an isosceles triangle.
In an isosceles triangle, the altitude is the perpendicular bisector of the base. Therefore, the triangle is made up of two congruent right triangles with:
height = 33base = 30/2 = 15hypotenuse = xTo calculate the value of x, we can use Pythagoras Theorem:
[tex]\boxed{a^2+b^2=c^2}[/tex]
where:
a and b are the legs of the right triangle.c is the hypotenuse (longest side) of the right triangle.Substitute the values into the formula and solve for x:
[tex]\implies 15^2+33^2=x^2[/tex]
[tex]\implies 225+1089=x^2[/tex]
[tex]\implies 1314=x^2[/tex]
[tex]\implies x^2=1314[/tex]
[tex]\implies \sqrt{x^2}=\sqrt{1314}[/tex]
[tex]\implies x=36.2491379...[/tex]
[tex]\implies x=36.2\; \rm (nearest\;tenth)[/tex]
Therefore, the value of x is 36.2 units (nearest tenth).
A data has 24 subgroups. 5 measurements were made per one subgroup. The sum of the subgroup averages is 160.25. The sum of the subgroup ranges is 2.19.
(1) Determine Trial central lines / Trial control limits on X bar chart and R chart.
In case of X bar chart, two values were out-of-control points and had assignable causes ( subgroup 4 - X bar value 6.65, subgroup 20 - X bar value 6.51)
In case of R bar chart, one value is out-of-control point and not part of natural variation. (subgroup 18 - R value 0.30)
(2) Determine Revised central lines / Revised control limits on X bar chart and R chart.
(Please show your calculation process logically )
The X bar chart, two values were out-of-control points and had assignable causes is 6.68 and Revised central lines / Revised control limits on X bar chart and R chart is 6.7364.
The X-bar chart, a form of Shewhart control chart, is used in industrial statistics to track the arithmetic means of subsequent samples of fixed size, n. For variables that may be monitored on a continuous scale, such as weight, temperature, thickness, etc., this sort of control chart is employed. For instance, one might sample five shafts from the production process once per hour, measure each shaft's diameter, and then display the mean of the five diameter values for each sample on a chart.
a) Here we are given that,
k=24
n=5
The sum of the subgroup averages is 160.25.
ΣΧ =160.25
The sum of the subgroup ranges is 2.19.
ΣR = 2.19
(1) Determine Trial central lines / Trial control limits on X bar chart and R chart.
Then,
[tex]\bar{\bar{X}}= \sum k=160.25/24=6.68[/tex]
Therefore,[tex]\bar{\bar{X}}=6.68[/tex]
b) Determine Revised central lines / Revised control limits on X bar chart and R chart.
From the statistical quality control chart
we get,
For, n=5
We have,
A₂=0.58
D₃=0
D₄=2.11
Revised Control limits for X -chart is,
[tex]UCLX =\bar{\bar{X}}+A2*\bar{R}=6.69+(0.58*0.08)=6.69+0.0464=6.6436[/tex]
[tex]CLX =\bar{\bar{X}}=6.69[/tex]
[tex]LCLX =\bar{\bar{X}}-A2*\bar{R}=6.69-(0.58*0.08)=6.69-0.0464=6.7364[/tex]
Learn more about X bar chart:
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