Answer:
75.5 cm
Step-by-step explanation:
S = (a+b)*h/2
h = 2 * S / (a+b) = 2 * 7701 / 204 = 75.5 cm
Realiza las siguientes potencias
Answer:
-5/8
Step-by-step explanation:
5.(-2^-3)
-5. 1/2³
-5. 1/8
-5/8
How do I answer this assignment?
I do not understand
Answer:
number of caps is the ratio
blue + white + green = 96
5x + 4x + 7x = 96
16x = 96
x = 6
Blue caps : 5x = 5(6) = 30
white caps : 4x = 4(6) = 24
green caps : 7x = 7(6) = 42
total of : 96 caps
Therefore, a caps in the picture represent 6 capsStep-by-step explanation:
heart and star pls <3 brainliest will be appreciated <3(っ◔◡◔)っ -{ elyna s }-A rectangular park is 4a m and 3a m broad, find it's area in m2
Answer:
Area = 12a m²
Step-by-step explanation:
Given information,
→ Length = 4a m
→ Width = 3a m
Now we have to,
→ find the area of rectangular park.
Formula we use,
→ Area = L × W
Then the area of rectangle is,
→ L × W
→ 4a × 3a
→ (4 × 3)a
→ 12a m²
Therefore, the area is 12a m².
Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed with true standard deviation 0.72.
(a) Compute a 95% CI for the true average porosity of a certain seam if the average porosity for 23 specimens from the seam was 4.85. (Round your answers to two decimal places.)
(b) Compute a 98% CI for true average porosity of another seam based on 14 specimens with a sample average porosity of 4.56. (Round your answers to two decimal places.)
(c) How large a sample size is necessary if the width of the 95% interval is to be 0.42? (Round your answer up to the nearest whole number.)
(d) What sample size is necessary to estimate true average porosity to within 0.24 with 99% confidence? (Round your answer up to the nearest whole number.)
For the normal distribution of helium porosity (in percentage) of coal samples,
a) 95% CI for the true average porosity of a certain seam is equals to ( 4.56 , 5.14 ).
b) A 98% CI for true average porosity of another seam based on 14 specimens is equals to (4.11 , 5.01).
c) Sample size with width of confidence interval 0.42 is 45.
d) Necessary Sample size to estimate true average porosity to within 0.24 is 60.
We have Assume that the helium porosity (in percentage) of coal samples taken from any particular seam is normally distributed.
Standard deviations, σ = 0.72
a)Average porosity of specimens , X-bar = 4.85
Sample size, n = 23
Confidence level , alpha = 95% = 0.95
From Normal Distribution table
Z for 95% Confidence Interval = 1.96
so,95% Confidence interval = (x-bar - Z× σ/sqrt(n) , x-bar + Z×σ/sqrt(n) )
plugging all known values in above formula,
= ( 4.85-1.96×0.72/sqrt(23) , 4.85+1.96×0.72/sqrt(23) )
So, 95% Confidence interval = ( 4.56 , 5.14 )
b)Now, Sample size, n = 14
sample mean, X-bar = 4.56
from normal distribution Z- table
Z-score for 98% CI is equal to 2.33
so,98% CI = (x-bar-Z×σ/sqrt(n) , x-bar+Z×σ/sqrt(n) )
= ( 4.56-2.33×0.72/√14 , 4.56+2.33×0.72/√14 )
98% CI = ( 4.11 , 5.01 )
c)Using the normal Distribution Z- table
Z for 95% CI = 1.96
Width of confidence interval = 0.42
we have to determine the sample size, n .
Using formula, Width = 2×Z×σ/sqrt(n)
=> n = (2×Z×sd/width)²
=> n= (2×1.96×0.72/0.42)²
=> n = 45.15
=> n = 45 ( whole number)
d) From normal distribution table
Z for 99% CI = 2.58
Margin of error, ME = 0.24
margin of error = ME = Z×σ/sqrt(n)
=> n = (Z× σ/ME)²
=> n= (2.58×0.72/0.24)²
=> n = 59.9
=> n = 60
Hence, required sample size is 60.
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Solve.(Refer to the picture)
Answer:
[tex]y=2.182[/tex]
Step-by-step explanation:
Solve for y.
[tex]3.082=y+\frac{9}{10}[/tex]
Move all terms not containing [tex]y[/tex] to the right side of the equation.
[tex]y=3.082-\frac{9}{10}[/tex]
Write [tex]\frac{9}{10}[/tex] as a decimal.
[tex]y=3.082-0.9[/tex]
[tex]y=2.182[/tex]
two tangents $\overline{pa}$ and $\overline{pb}$ are drawn to a circle, where $p$ lies outside the circle, and $a$ and $b$ lie on the circle. the length of $\overline{pa}$ is $12,$ and the circle has a radius of $9.$ find the length $ab.$
The value of tangent AB as calculated from the given data is 14.4.
The tangents to the circle as given are PA and PB.
Let the point of intersection of PO and AB be X.
PA = 12
OA = 9
As given that the triangle is right angled we can use hypotenuse theorem to calculate PO,
PO = √ PA² + OA²
= √ 144 + 81
= 15
Now , the given triangles are similar because,
AX / AO = PA / PO
AX / 9 = 12 / 15
Therefore, the value of X is,
X = 7.2
Now we know that AB's midpoint is X,
Hence , AB = 2X = 14.4
Tangent is known as a point which passes through a circle only at one single point A circle can have infinite tangents.
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A hot air balloon travels 2016 miles in 96 hours. The balloon travels the same number of miles each hour. How many miles does the balloon travel in 1 hour
Answer:
21 m/ph
Step-by-step explanation:
2016 divided by 96
The balloon travels 21 miles in one hour.
What is division?In mathematics, division is one kind of operation. The phrases or numbers in this process are split into the same number of parts.
Given, a hot air balloon travels 2016 miles in 96 hours.
And the balloon travels the same number of miles each hour.
To find the same numbers of miles per hour:
We use the division of 2016 by 96.
That means,
2016 / 96
= 21
Therefore, the travel time of the balloon is 21 miles per hour.
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small regional carrier accepted 22 reservations for a particular flight with 18 seats. 16 reservations went to regular customers who will arrive for the flight. Each of the remaining passengers will arrive for the flight with a 58% chance, independently of each other.
(Report answers accurate to 4 decimal places.)
Find the probability that overbooking occurs. Find the probability that the flight has empty seats.
The probability that overbooking occurs is 0.7920 and the probability that the flight has empty seats is 0.0510 respectively.
How to calculate probability?We remove the 16 regular customers and their seats from consideration. Thus, we consider the 22 - 16 remaining customers = 6 and the 18-16 = 2 seats overbooking means 3 or more of the 6 remaining passengers; each has a 58% chance remaining. We can either use the binomial distribution formula P(x) = C(6, x) .58^x .42^(6-x) or use packages that provide the result, including cumulative probabilities.
Calculation:Overbooking = 1 - P(X <= 2) In Excel, we use =1 - binom.dist(2,6,.58,1) ; the 1 is for cumulative.
The solution is 0.7920
Probability of empty seats = P(x <= 1) = binom.dist(1,6,.58,1) = 0.0510
The probability that overbooking occurs is 0.7920 and the probability that the flight has empty seats is 0.0510 respectively.
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Between which values does lie?
Answer:
lies between 10 and 11.
Step-by-step explanation:
We are asked to find the values between which lies.
We know that square root of 100 is 10 and square root of 121 is 11.
Since is between and , therefore, lies between 10 and 11.
Sam invests 1700 in a savings account that pays a nominal annual interest rate of 2.74% Sam makes no further payments or withdrawals from this account
David also invests 1700$ in a savings account that pays an annual rate of r% compounded yearly David makes no further payments or withdrawals from this account fund the value of r so that the amount in David’s account after 10 years will be equal to sams account
a.) The amount that Sammy will have in his account after 10 years will be= $2,165.8
$2,165.8b.) The value of 'r' required would be = 0.27%
What is interest rate?Interest rate is defined as the rate at which an individual receives an amount of money from an investment made over a period of time.
The amount invested by Sam(P) = $1700
The interest rate(R) = 2.74%
The time of investment (T) = 10 years
Simple interest = P×T×R/100
= 1700×10×2.74/100
= 46580/100
= $465.80
Therefore, the amount that Sammy will have in his account after 10 years will be = $1700 + $465.80
= $2,165.8.
For David's account interest rate the following is carried out:
The amount invested by Sam(P) = $1700
The interest rate(R) = r%
The time of investment (T) = 10 years
Simple interest (SI) = $465.80
Using the formula for simple interest;
SI = P×T×R/100
Make R the subject of formula;
R = SI×100/P×T
R = 465.80×100/1700×100
R= 46580/170000
R= 0.27%
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Complete question:
Sam invests $1700 in a saving account that pays a nominal annual rate of interest of 2.74% compounded half-yearly. Sam makes no further payments to, or withdrawals from, this account. (a) Find the amount that Sam will have in his account after 10 years.
David also invests $1700 in a savings account that plays an annual rate of interest of r%, compounded yearly. David makes no further payments or withdrawals from this account. (b) Find the value of 'r' required so that the amount in David's account after 10 years will be equal to the amount in Sam's account.
The diagonal of a rectangular room is 13 ft long. One wall measures 7ft longer than the adjacent wall. Find the dimensions of the room.
Let x be the length of the shorter wall of the rectangular room and y be the length of the longer wall of the rectangular room. Since the longer wall is 7 ft longer than the shorter wall, we can write the equation y = x + 7. We can also express the diagonal of the rectangular room using the Pythagorean theorem as x^2 + y^2 = 13^2. We can solve for x and y by substituting the equation y = x + 7 into the equation x^2 + y^2 = 13^2 and then solving for x. Doing this, we get x^2 + (x + 7)^2 = 13^2. Expanding the square on the right side of the equation and then rearranging the terms, we get x^2 + 2x^2 + 14x + 49 = 169. Combining like terms, we get 3x^2 + 14x - 120 = 0. This quadratic equation can be factored as (x - 8)(3x + 15) = 0. Since the length of a side of a rectangle must be positive, we can ignore the solution x = -15/3. So, the length of the shorter wall of the rectangular room is x = 8 ft. The length of the longer wall can be found by substituting this value into the equation y = x + 7, giving us y = 8 + 7 = 15 ft. Therefore, the dimensions of the rectangular room are 8 ft by 15 ft.
Hummingbird feeders are filled with hummingbird nectar. Hummingbird
nectar is made using a ratio of 1 part sugar to 3 parts water.
How much sugar, in ounces, is needed to make 28 ounces of
hummingbird nectar? Show your work in the provided space.
The amount of sugar, in ounces, that is needed to make 28 ounces of hummingbird nectar is 7 ounces.
How to find the ounces of sugar needed ?Hummingbird nectar is said to be made by using a ratio of 1 part sugar to 3 parts water. For every 1 ounce of sugar therefore, there would be 3 ounces of water.
Put together, this would give 4 ounces. And the percentage of sugar in this 4 ounces would be:
= 1 / 4
= 25 %
If 28 ounces of hummingbird nectar needed to be made, the sugar needed is :
= Percentage of sugar x Ounces of hummingbird nectar
= 25 % x 28
= 7 ounces of sugar
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Howard Cunningham has a $90,000.00 whole life policy. What will the cash value of her policy be at the following anniversary dates?
(Include decimal places for cents in each answer, even when cents are not calculated. Use a thousands comma where applicable.)
Answer:
5 years = 2,700.00
25 years = 30,600.00
Step-by-step explanation:
Hey, these are difficult, I know.
I'll explain how you solve them.
In this case, $90,000.00 needs to first be divided by 1,000.
That gives us 90.
You then multiply 90 by 30, which is the 5 yr cash value.
That gives you 2,700.00.
Since you have 90, you next need to multiply that by 340 (25 yr cash value), which gives you 30,600.00.
Hope this helps. Feel free to mark Brainliest.
Q. An arithmetic series has first term a and common difference d, where d is a prime number. The sum of the first n terms of the series is S, and Sm=39 S2m = 320 Find the value of d and the value of m Show clear algebraic working. (Total for question = 5 marks)
Finding the values of d and the value of m with the algebraic working will give us the the value of d to be 7 and the value of m to be 3.
How do we calculate the values using the algebraic expression?Finding the value of d, we know that the sum of the first n terms of an arithmetic series is given by:
S = n/2 * (2a + (n-1)d)
Since the sum of the first n terms is S and the sum of the first 2m terms is S2m, we can set up the following equation:
S = m/2 * (2a + (m-1)d)
S2m = 2m/2 * (2a + (2m-1)d)
Substituting the given values for S and S2m into these equations, we get:
39 = m/2 * (2a + (m-1)d)
320 = 2m/2 * (2a + (2m-1)d)
Solving for d in each equation, we find that d = -7 in the first equation and d = 7 in the second equation. Since d must be a prime number, the only possible value for d is 7.
Now that we know the value of d, we can solve for m. Substituting the value of d back into one of the equations and solving for m, we get:
39 = m/2 * (2a + (m-1)7)
78 = m * (2a + (m-1)7)
78 = m * 2a + 7m^2 - 7m
7m^2 - m - 78 = 0
We can solve for m using the quadratic formula:
m = (-1 +/- sqrt(1^2 - 4*7*(-78)))/(2*7)
= (-1 +/- sqrt(2521))/14
= (-1 + 49)/14 = 3
= (-1 - 49)/14 = -7
Since m must be a positive integer, the only possible value for m is 3.
Therefore, the value of d is 7 and the value of m is 3.
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Can I get help with this quick check
The common difference of the sequence is 70, the slope of the 1st and 2nd lines is 70, the slope of the 3rd and 4th points is 70, and the slope is constant.
What is the slope?A line's slope is determined by how its y coordinate changes in relation to how its x coordinate changes. Y and x are the net changes in the y and x coordinates, respectively.
Given:
Time in minutes and jumping jack table, coordinates of lines (1, 70) and (2, 140)
(a)
Consider 70, 140, 210, and 280 as an arithmetic sequence then the common difference will be,
Common difference = 140 - 70 = 70
(b)
Calculate the slope of the line as shown below,
[tex]m = (y_2-y_1) / (x_2 - x_1)[/tex]
m = (140 - 70) / (2 - 1)
m = 70 / 1
m = 70
(c)
Calculate the slope for 3rd and 4th points,
m = (280 - 210) / (4 - 3)
m = 70 / 1
m = 70
(d)
As you can, the slope is constant, and the common difference is also constant.
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Find the y-intercept and the slope of the line.
y=-1/2-1/4x
y-intercept: ?
slope: ?
Answer: slope = -1/4 y-intercept = -1/2
Step-by-step explanation:
Given an equation in the form y = mx + b, m = slope and b = y-intercept.
We can rearrange y= -1/2 - 1/4x into that form.
y = -1/4x - 1/2
We can now obviously see that -1/4x is the slope, and -1/2 is the y-intercept.
What is the y-intercept
What is the product of the 4 x and the 3 x cubed y squared minus 2 x y cubed?
Answer:
4xy ×3xy -2xy
12xy-2xy
10xy
Could I get some help with this? Thanks! Here’s the question to go along with it.
For each diagram, calculate the value of x. Show your work and include an explanation of what you used (definitions and theorems) to solve the problem. If not possible, state why.
Answer:
x = 16 1/3°
Step-by-step explanation:
You have parallel lines with consecutive interior angles marked 7x-28° and 5x+12°, and you want to find the value of x.
Consecutive interior anglesAt parallel lines, consecutive interior angles, or same-side interior angles, are supplementary. This means the total of the two marked angles is 180°.
7x -28° +5x +12° = 180°
12x -16° = 180° . . . . . . . . . . simplify
12x = 196° . . . . . . . . . add 16°
x = (49/3)° = 16 1/3° . . . . . . . divide by 12
__
Additional comment
∠3 = 86 1/3°
∠5 = 93 2/3°
If X and Y are independent exponential random variables with respective parameters λ1 and λ2, how do I find the distribution of Z = X/Y ?
The distribution of Z = X/Y is λ₁ / (λ₁+λ₂).
What is a cumulative distribution function?The probability distribution of random variables is described using the cumulative distribution function. The probability for a discrete, continuous, or mixed variable may be described using it. The cumulative probability for a random variable is calculated by adding the probability density function.
Given:
X and Y are independent exponential random variables with respective parameters λ₁ and λ₂.
To find the distribution of Z = X/Y:
First, we have
[tex]f_x_y[/tex](x,y) = λ₁λ₂[tex]e^{-\lambda_1x[/tex][tex]e^{-\lambda_2y[/tex]
First, we find the cumulative distribution function (CDF) for Z = X/Y.
Derivative of Z, f(z) and put a = x/y
[tex]F_z[/tex](a) = P (X/Y ≤ a)
= P (X ≤ aY)
= [tex]\int\limits^{\infty}_ {x} \ \int\limits^{ay}_0 {\lambda_1\lambda_2e^{-\lambda_1x}e^{-\lambda_2y}\ dx dy[/tex]
[tex]= \int\limits^{\infty}_ {0} {\lambda_1\lambda_2e^{-\lambda_2y}\ dy [ -1/{\lambda_1}{e^{-\lambda_1x}]\limits^{ay}_ {0}[/tex]
[tex]= \lambda_2\int\limits^{\infty}_ {0} {e^{-\lambda_2y} - e^{-y(\lambda_2 + \lambda_1a)} \ dy[/tex]
[tex]= \lambda_2[ [{1/{\lambda_2}+{\lambda_1a]-{e^{-\lambda_2y} + e^{-y(\lambda_2 + \lambda_1a)}]\limits^{\infty}_ {0}[/tex]
[tex]= -[({\lambda_2}/{\lambda_2 + \lambda_1) - 1][/tex]
[tex]= -[({\lambda_2}/{\lambda_2 + \lambda_1a) - 1][/tex]
[tex]= [({\lambda_1a}/{\lambda_2 + \lambda_1a)][/tex]
So, P(X<Y) = P (X/Y <1)
= λ₁ / (λ₁+λ₂)
Therefore, distribution is λ₁ / (λ₁+λ₂).
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6,328 divide by 56 does anyone know
Answer:
113
Step-by-step explanation:
An airline that wants to assess customer satisfaction chooses a random sample of 10 of its flights during a single month and asks all of the passengers on those flights to fill out a survey. This is an example of a
A) Multistage sample.
B) Stratified sample.
C) Cluster sample.
D) Simple random sample.
E) Convenience sample
An airline that wishes to measure customer satisfaction chooses ten of its flights at random over the course of a single month and requests that each passenger fill out a survey. A Cluster sample looks like this.
Why do we utilize cluster sampling?Cluster sampling should be used to study large, dispersed populations because interviewing every person would be costly, time-consuming, and occasionally impossible.Smaller, more comparable groups that represent the population under study can be produced via cluster sampling.You divide a population into groups, such as districts or schools, and then randomly select a sample from among these groupings when using the probability sampling approach known as cluster sampling. Each cluster should ideally be a miniature representation of the total population.An airline that wants to gauge customer happiness selects a haphazard sample of 10 of its flights over the course of a single month and asks each passenger to complete a survey. An illustration of a Cluster sample is this.To learn more about Cluster sample refer to:
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determine the fifth percentile of the standard normal distribution as a decimal rounded to the nearest hundredth.
The fifth percentile of the standard normal distribution is -1.64.
In the given question, we have to determine the fifth percentile of the standard normal distribution as a decimal rounded to the nearest hundredth.
The standard normal distribution, commonly known as the z-distribution, is a unique type of normal distribution in which the mean and standard deviation are both equal to 1. Any normal distribution's values can be transformed into z scores to normalise it. Z score shows the amount of standard deviations from the mean that each consists of numbers.
Since, X is normally distributed.
P(X< c) = 0.05
Using Z table
c = -1.645
Fifth percentile = -1.64
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Solve the equation -128 = 4x for x
Answer: -32
Step-by-step explanation:
To solve the equation -128 = 4x for x, we need to isolate the x variable on one side of the equation. We can do this by dividing both sides of the equation by 4. This gives us:
-128/4 = 4x/4
Dividing -128 by 4 gives us -32, so we have:
-32 = 4x/4
We can simplify this by dividing both sides by 4/4 to get:
-32 = x
Thus, the solution to the equation -128 = 4x is x = -32.
Kadeesha bought snacks for her team's practice. She bought a bag of popcorn for $ 1.59 and a 5-pack of juice bottles. The total cost before tax was $13.34. Which equation or tape diagram could be used to represent the context if j represents each bottle of juice costs?
A
5 j=1.59-13.34
C
1.59 j=13.34-5
The equation or tape diagram could be used to represent the context if j represents each bottle of juice costs is; Option D tape diagram
How to Represent Algebraic equations?We are told that;
1 bag of popcorn costs $1.59
She bought a 5-pack of juice bottles
Total cost before tax = $13.34
Now, we are told that j represents each bottle of juice costs. Thus;
Cost of the 5 packs of juice = 5j
Thus, the equation of the total cost before tax is;
1.59 + 5j = 13.34
Looking at the options, only the tape diagram in option D correctly represents the required equation.
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What is the greatest common factor of 20 and 40?
What is the greatest common factor of 20 and 40?
20
Can somebody please tell me how do i solves these. Thank you.
2. It is false that the differential equation is non-linear.
The statement regarding the differential equation y''y + (y')² = sin(x) is false.
3. Statement 3: fourth order and nonlinear differential equation.
How to classify the differential equations?For a linear differential equation, the coefficients of y or the derivatives of y are either:
Constant terms.Other variables that are not y.Hence the differential equation for question 2 is linear, as the coefficients of the derivatives are x and -1, neither of which is y, hence the statement is false.
For the differential equation y''y + (y')² = sin(x), we have that:
It is non-linear, as y multiplies y''.It is of the second order, as the order of the highest derivative is of 2.For the differential equation in item 3, we have that:
It is of the fourth order, as the derivative of the third order multiplies the derivative of the first order.It is non linear, as the variable y multiplies the derivative of the third order of y.Hence statement 3 is correct.
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Kate Drew has been hand-painting wooden Christmas ornaments for several years. Recently, she has hired some friends to help her increase the volume of her business. In checking the quality of the work, she notices that some slight blemishes occasionally are apparent. A sample of 24 pieces of work resulted in the following number of blemishes on each piece: 1,1,2,3,1,0,2,0,0,1,3,1,0,1,2,0,0,2,1,2,3,2,1,1. Develop upper and lower control limits (99.73%) for the number of blemishes on each piece. Is the process in control? Why? **Please do this in excel*
A sample of 24 pieces of work is considered for the checking the quality of the work,
a) Upper control limit of 99.73% confidence interval is 4
Lower control limit of 99.73% confidence interval is Zero.
The process is in control because all the values of the blemishes per unit are lie within the control limits .
What is Control Limits?The Control limits consist of two limits lower control limit and upper control limit within which the value of the statistical observations of a product are expected to lie. If the values lie outside these limits, then the process is called out of statistical control. We have,
Kate Drew has been hand-painting wooden Christmas ornaments for several years.
Sample size = 24 pieces
confidence level = 99.73% = 0.9973
The number of blemishes per unit is given as:
1,1,2,3,1,0,2,0,0,1,3,1,0,1,2,0,0,2,1,2,3,2,1,1.
The control limits are to be computed for the number of defects per piece . Let us consider the number of defects per piece be denoted by c. In that case, the control limits for c chart are used. The upper control limit (UCL) and lower control limit (LCL) for c chart is given by,
UCL = c-bar + 3 s.d
LCL = c-bar - 3 s.d
where c-bar = 1/n(€c) average number of defects
So, c-bar is computed as,
c-bar = (1+1+2+3+1+0+2+0+0+1+3+1+0+ 1+2+0 +0+2 + 1+2 +3+ 2+1+1) /24 = 30/24 = 1.25
put the data in Excel sheet and calculate required limits using Excel Commands.
Excel Command for Average, c-bar
=Average( Range of numbers who's average you want to calculate )
In my excel sheet, = Average ( H2 : H25)
= 1.25
Excel Command for Standard deviations,
=STDEV( numbers who's standard deviations you want to calculate)
here, =STD(H2:H25) , output= 0.98907
UCL = 1.25 + 3( 0.98907)
= 4.217
LCL = 1.25 - 3( 0.98907)
= - 1.717
but defective item can never be negative so, Lower control limit is Zero.
Hence, for the given data, the upper control limit for the number of defects is 4 and the lower control limit is 0.
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The best approximation, in cubic inches, to the increase in volume of a sphere when the radius is increased from 3 to 3.1 inches, is Use V = Tr3 (a) 0.04n/3 (b) 3.61 n'
(c) 0.04 n'
(d) 1.2 n'
(e)36 n'
The change in volume of the sphere or best approximation to the increase in the volume of the sphere is 3.6π cubic inches.
How is volume of a sphere calculated?In simple words, when a sphere is placed in the space or three-dimension space, then the total space acquired by the sphere generally shows the volume of the sphere. The mathematical formula needed to determine the volume of a sphere is shown below,
[tex]V = \frac{4}{3} \pi r^3[/tex]
Calculation:Given- The initial radius of the sphere is r1 = 3 inch and the final radius of the sphere is r2 = 3.1 inch.
Now , the change in the radius of the sphere is [tex]dR =[/tex][tex]r2 - r1 =\\[/tex] 3.1 - 3 = 0.1 inch.
The mathematical expression of the volume of a sphere is shown below,
[tex]V = \frac{4}{3}\pi r^3[/tex] (here, [tex]r[/tex] is the radius of sphere)
Differentiate the above shown expression with respect to radius ([tex]r[/tex]) as shown below,
[tex]\frac{d}{dr} (V) = \frac{d}{dr} (\frac{4}{3} ) \pi r^3[/tex]
∴
dV/dR = 4/3 π × d/dR(r^3)
= 4/3 π × 3r^2
dV/dR = 4π r^2
∴ dV = 4π r^2 dR ................(1)
Substitute all values in equation (1) and calculate the approximation as shown below,
dV = 4π r^2 dR
= 4π (3)^2 × (0.1)
∴ dV = 3.6π cubic inches.
The change in volume of the sphere or best approximation to the increase in the volume of the sphere is 3.6π cubic inches.
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