Answers
1. The next two term for the sequence using Geometric Progression is 8 and 16
2. The next two terms for the sequence using arithmetic progression is 7 and 11
What is sequence?A sequence is an ordered list of numbers (or other elements like geometric objects), that often follow a specific pattern or function.
Using Geometric Progression, the common ratio is 2/1 = 2
therefore the next two terms will be
4× 2 = 8 and 8× 2 = 16
Using Arithmetic progression , the common difference will be increasing by 1 per number of term, i.e r+1
for the fourth term ,common difference = 2+1 = 3
fourth term = 4+3 = 7
for the fifth term , common difference = 3+1 = 4
fifth term = 7+4 = 11
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Related Questions
1. All square, upper triangular matrices are diagonalizable. (T/F)
2. If a matrix is diagonalizable, then it is invertible. (T/F)
Answers
1. All square, upper triangular matrices are diagonalizable. (TRUE)
2. If a matrix is diagonalizable, then it is invertible. (FALSE)
Understanding matrix1. True: All square, upper triangular matrices are diagonalizable.
A square matrix is diagonalizable if it can be transformed into a diagonal matrix by a similarity transformation, which means there exists an invertible matrix P such that P^-1 * A * P is a diagonal matrix. Upper triangular matrices have all their elements below the main diagonal equal to zero.
Since the eigenvalues of an upper triangular matrix are equal to its diagonal elements, we can form a diagonal matrix with these eigenvalues. Since there exists such a diagonal matrix, all square, upper triangular matrices are diagonalizable.
2. False: If a matrix is diagonalizable, it is not necessarily invertible
Diagonalizable matrices can be transformed into a diagonal matrix with the eigenvalues along the main diagonal.
However, invertibility requires that the matrix have a nonzero determinant, which means that all of its eigenvalues must be nonzero.
If a diagonalizable matrix has a zero eigenvalue, its determinant will be zero, and it will not be invertible. Therefore, diagonalizability does not guarantee invertibility.
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1. A company is testing a new energy drink. Volunteers are asked to rate their energy one hour
after consuming a beverage. Unknown to them, some volunteers are given the real energy
drink and some are given a placebo-a drink that looks and tastes the same, but does not have
the energy-producing ingredients. Show your work using the following list of random digits to
assign each participant listed either the real drink or the placebo.
69429 86140 11625 87049 23167
Volunteer
Abby
Barry
87524 24575 87254 97801 82231
Callie
Dion
Ernie
Falco
Garrett
Hallie
Indigo
Jaylene
Real or Placebo
2. The local water authority has received complaints of high levels of iron in the drinking water.
They decide to randomly select 20 houses from each subdivision of 100 houses to visit and test
their water. Describe how to use random numbers to select the 20 houses in each division.
3. A couple is willing to have as many children as necessary to have two girls.
a) Describe a simulation that can be performed to estimate the average number of children
required to have two girls.
Answers
- Assign the first five volunteers (Abby, Barry, Callie, Dion, and Ernie) to the first five digits in the list (6, 9, 4, 2, 9).
- If the digit is even, assign the volunteer the real drink; if it is odd, assign the volunteer the placebo.
- Repeat this process for the remaining volunteers (Falco, Garrett, Hallie, Indigo, and Jaylene) using the next five digits in the list (8, 6, 1, 4, 0).
Here is the assignment:
Volunteer | Real or Placebo
----------|----------------
Abby | Real
Barry | Real
Callie | Placebo
Dion | Real
Ernie | Placebo
Falco | Real
Garrett | Real
Hallie | Real
Indigo | Placebo
Jaylene | Placebo
2. To select the 20 houses in each division, we can use the following method:
- Assign each house in the subdivision a number from 01 to 100.
- Use a random number generator to generate 20 random numbers between 01 and 100.
- Visit the houses corresponding to the 20 numbers generated.
Repeat this process for each subdivision.
3. One way to simulate the average number of children required to have two girls is to use the following method:
- Assign a value of 0 to a counter variable.
- Repeat the following steps until two girls are born:
- Use a random number generator to generate a number between 0 and 1.
- If the number is less than 0.5, add 1 to the counter variable (representing a boy); otherwise, add 2 to the counter variable (representing a girl).
- Record the value of the counter variable.
- Repeat this process a large number of times (e.g., 10,000).
- Calculate the average value of the counter variable across all simulations. This will give an estimate of the average number of children required to have two girls.
PLEASEEEE HELPPP ASAP 20 PTS
Use long division to determine the quotient of the following expression.
Write the quotient in standard form with the term of largest degree on the left. (10x^(2)+3x-77)-:(2x+7)
Answers
The quotient of the division 10x² + 3x - 77 ÷ 2x + 7 is 5x - 16
Evaluating the long division expressionsThe quotient expression is given as
10x² + 3x - 77 ÷ 2x + 7
The long division expression is represented as
2x + 7 | 10x² + 3x - 77
So, we have the following division process
5x - 16
2x + 7 | 10x² + 3x - 77
10x² + 35x
--------------------------------
-32x - 77
-32x - 112
-------------------------------------
35
Hence, the quotient of the long division is 5x - 16
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21. if you have the raw scores of two individuals on a norm-referenced test, which of the following is important in terms of making meaning of those scores? a. measures of central tendency (mean, median, mode) b. the relative positions of the scores to the rest of the group c. whether higher scores are better than lower scores d. how an individual feels about his or her score e. all of these are important.\
Answers
The term which is important in terms of making meaning of those scores is measures of central tendency (mean, median, mode), option A.
A single number that seeks to characterise a set of data by pinpointing the centre location within that set of data is referred to as a measure of central tendency. As a result, measurements of central location are occasionally used to refer to measures of central tendency.
These also fit within the category of summary statistics. You are probably most familiar with the mean (sometimes known as the average), but there are additional central tendency measures, including the median and the mode.
The mean, median, and mode are all reliable indicators of central tendency, however depending on the situation, certain indicators are more useful than others.
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Let f(x) = 4x^3 – 3x^2 – 18x +5. (a) Find the critical numbers of f. (b) Find the open interval(s) on which f is increasing and the open interval(s) on which f is decreasing. (c) Find the local minimum value(s) and focal maximum value(s) of f, if any.
(d) Find the open interval(s) where f is concave upward and the open interval(s) where f is concave downward e) Find the inflection points of the graph of f, if any
Answers
(a) The critical numbers happen when x = 3 or x = -1/2
(b) f is decreasing on (-∞, -1/2), increasing on (-1/2, 3), and increasing on (3, ∞).
(c) f has a local minimum value of -22 at x = 3, and a local maximum value of 25.5 at x = -1/2.
(d) f is concave downward on (-∞, 1/4) and concave upward on (1/4, ∞).
(e) The inflection point of f is at x = 1/4.
(a) To find the critical numbers of f, we need to find the values of x where the derivative of f equals zero or does not exist.
f'(x) = 12x² - 6x - 18 = 6(2x² - x - 3) = 6(x - 3)(2x + 1)
Setting f'(x) equal to zero, we get:
6(x - 3)(2x + 1) = 0
x = 3 or x = -1/2
These are the critical numbers of f.
(b) To find the intervals where f is increasing and decreasing, we need to examine the sign of the derivative f'(x) in the intervals determined by the critical numbers.
When x < -1/2, f'(x) < 0, so f is decreasing on the interval (-∞, -1/2).
When -1/2 < x < 3, f'(x) > 0, so f is increasing on the interval (-1/2, 3).
When x > 3, f'(x) > 0, so f is increasing on the interval (3, ∞).
(c) To find the local minimum and maximum values of f, we need to examine the critical numbers and the end points of the intervals.
f(3) = 4(3)³ - 3(3)² - 18(3) + 5 = -22
f(-1/2) = 4(-1/2)³ - 3(-1/2)² - 18(-1/2) + 5 = 25.5
Thus, f has a local minimum value of -22 at x = 3, and a local maximum value of 25.5 at x = -1/2.
(d) To find the intervals where f is concave upward and concave downward, we need to examine the sign of the second derivative f''(x).
f''(x) = 24x - 6 = 6(4x - 1)
When x < 1/4, f''(x) < 0, so f is concave downward on the interval (-∞, 1/4).
1/4 < x, f''(x) > 0, so f is concave upward on the interval (1/4, ∞).
(e) To find the inflection points of f, we need to examine the points where the concavity changes.
The concavity changes at x = 1/4, which is the only inflection point o
Rewrite each equation without absolute value symbols for the given values of x.
y=|2x+5|-|2x-5|
if x<-2.5 if x>2.5
if -2.5<=x<=2.5
Answers
If x > 2.5, both expressions within absolute value symbols are positive.
The equation becomes: y = (2x + 5) - (2x - 5) = 10.
How to solve
For the given intervals of x:
If x < -2.5, both expressions within absolute value symbols are negative. Thus, the equation is: y = -(2x + 5) - (-(2x - 5)) = -10.
If x > 2.5, both expressions within absolute value symbols are positive.
The equation becomes: y = (2x + 5) - (2x - 5) = 10.
If -2.5 ≤ x ≤ 2.5, the first expression is positive and the second is negative.
The equation is: y = (2x + 5) - (-(2x - 5)) = 4x.
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It takes an apprentice four times as long as the experienced plumber to replace the pipes under an old house. If it takes them 15 hours when they work together, how long would it take the apprentice alone?
Answers
Let's assume that the experienced plumber can complete the job in x hours.
According to the problem, the apprentice takes four times as long as the experienced plumber to complete the job. Therefore, the apprentice can complete the same job in 4x hours.
If they work together, they can complete the job in 15 hours. Therefore, we can write the following equation based on their combined work rate:
1/x + 1/4x = 1/15
Multiplying both sides by the common denominator of 60x gives:
60 + 15 = 4x
75 = 4x
x = 75/4
x = 18.75
Therefore, the experienced plumber can complete the job in 18.75 hours.
Using the relationship that the apprentice takes four times as long as the experienced plumber to complete the job, we can find the time it takes for the apprentice to complete the job alone:
4x = 4(18.75) = 75
Therefore, the apprentice would take 75 hours to complete the job alone.
someone PLSS helpi don’t know
Answers
The following are correct about the triangle;
1. angle C is 60°
2. angle B is 60°
3. The length of segment DB is 3
4. The length of side x is 3√3
What is an equilateral triangle?An equilateral triangle is a type of triangle in which all it's sides and angles are equal.
Since all the angles of an equilateral triangle are equal, then,
x+x+x = 180
3x = 180
x = 180/3 = 60°
therefore each angle is 60°
angle C and angle B are 60°
Using Pythagorean theorem
x² = 6²- 3²
x² = 36-9
x² = 27
x = √27
x = √9×3
x = 3√3
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Russ placed $8000 into his credit union account paying 6% compounded semiannually (twice a year). How much will be in Russ’s account in 4 years
Answers
Answer:
Step-by-step explanation:
Use the rules to find derivatives of the functions at the specified values.
f(x) = 4x^3 at x = 2
f(2) = _____
Answers
The value of the function f(x) at x = 2 is 48.
The question asks us to find the value of the function f(x) = 4[tex]x^3[/tex] at x = 2 and its derivative f'(x) at x = 2.
The value of the function f(x) at x = 2, we simply plug in x = 2 into the function and evaluate:
f(2) = 4[tex](2)^3[/tex] = 4(8) = 32
Therefore,
The value of the function f(x) at x = 2 is 32, which we can write as f(2) = 32.
To find the derivative of the function f(x) at x = 2, we first need to find the general formula for the derivative of f(x), which we can do using the power rule for derivatives.
The power rule states that
To find the derivative of f(x) at x = 2, we simply plug in x = 2 into the formula for f'(x) that we just found:
f'(2) = [tex]12(2)^2[/tex] = 48
Therefore,
The derivative of the function f(x) at x = 2 is 48, which we can write as f'(2) = 48To find the derivative of the function
f'(x) = 12[tex]x^2[/tex]
To find the value of f(2), we can simply plug in x = 2 into the original function:
f(2) = 4[tex](2)^3[/tex] = 32
To find the value of f'(2), we can plug in x = 2 into the derivative we just found:
f'(2) = 12[tex](2)^2[/tex] = 48
Therefore,
f(2) = 32 and f'(2) = 48.
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Solve for w.
65=170-w
Answers
170-65 = 105 that's how W is 105
Let AX = B be a consistent linear system with 12 equations and 8 variables. If the solution of the system contains 3 free variables, then what is the rank of the coefficient matrix A?
Answers
The rank of the coefficient matrix A is 5.
How to determined the matrix?Since the system AX = B is consistent and has 12 equations and 8 variables, the rank of the coefficient matrix A must be less than or equal to 8 (the number of variables).
If the solution of the system contains 3 free variables, it means that the dimension of the null-space of A is 3. By the rank-nullity theorem,
we know that the dimension of the null-space of A plus the rank of A is equal to the number of columns of A (which is 8 in this case).
Therefore, we have:
rank(A) + dim(null(A)) = 8
rank(A) + 3 = 8
rank(A) = 5
So, the rank of the coefficient matrix A is 5.
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8 girls eat a total of 210 candies. after adding the number of candies eat by the ninth girl, the average number of candies eaten became 29. how many did the 9th girl eat?
Answers
The 9th girl ate 51 candies.
What is the number of candies eaten by the 9th girl, if the average number of candies eaten by 9 girls is 29 and the first 8 girls ate a total of 210 candies?Let the number of candies eaten by the 9th girl be x.
The average number of candies eaten by 8 girls is given as (210/x+210)/8, which simplifies to 210/8 + x/8.
After the 9th girl eats x candies, the total number of candies eaten becomes 210 + x.
The new average is given as (210 + x)/9 = 29.
Solving for x, we get:
210 + x = 261
x = 51
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Mathswatch Question:
Liam is a tyre fitter.
It takes him 124 minutes to fit 4 tyres to a lorry.
a) How long would it take him to fit 6 tyres to a lorry. ?
b) If he works for 93 minutes, how many tyres can he fit?
Working out for question a:
a) 124×6÷4=186(minutes)
Correct answer for question a is 186.
Correct answer for question b is 3
Answers
To answer question a, we use the formula:
time taken = (number of tyres to fit x time taken to fit one tyre) / number of tyres fitted at once
In this case, Liam takes 124 minutes to fit 4 tyres to a lorry. To find out how long it would take him to fit 6 tyres, we plug in the values:
time taken = (6 x 124) / 4
time taken = 186 minutes
So it would take Liam 186 minutes to fit 6 tyres to a lorry.
For question b, we know that Liam takes 124 minutes to fit 4 tyres, so he takes 31 minutes to fit 1 tyre. If he works for 93 minutes, we can find out how many tyres he can fit:
number of tyres = time taken / time taken to fit one tyre
number of tyres = 93 / 31
number of tyres = 3
So Liam can fit 3 tyres in 93 minutes.
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A baker has small and large bags of sugar for making cakes. The large bag contains 30 cups of sugar and it's 2. 5 times larger than the small bag. The small bag contains enough sugar to make nine cakes and have. 75 cups of sugar remaining
How many cakes can be made with a large bag of sugar?
Answers
The number of cakes that can be made with a large bag of sugar, we first need to determine the amount of sugar in a small bag and then calculate the amount of sugar needed for one cake.
1. Find the amount of sugar in a small bag:
Since the large bag contains 30 cups of sugar and is 2.5 times larger than the small bag, we can write the equation:
Small bag = Large bag / 2.5
Small bag = 30 cups / 2.5
Small bag = 12 cups of sugar
2. Determine the amount of sugar needed for one cake:
The small bag contains enough sugar to make 9 cakes and have 0.75 cups of sugar remaining. So, we can subtract the remaining sugar from the total amount in the small bag:
Sugar used for 9 cakes = 12 cups - 0.75 cups
Sugar used for 9 cakes = 11.25 cups
Now, we can find the amount of sugar needed for one cake:
Sugar per cake = Sugar used for 9 cakes / 9
Sugar per cake = 11.25 cups / 9
Sugar per cake = 1.25 cups
3. Calculate the number of cakes that can be made with a large bag of sugar:
Cakes from large bag = Large bag sugar / Sugar per cake
Cakes from large bag = 30 cups / 1.25 cups
Cakes from large bag = 24
Therefore, a baker can make 24 cakes with a large bag of sugar.
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Jen is filling bags with M&Ms. She has 5 1/2 cups of M&Ms. She needs 1 1/4 cups of M&Ms to fill each bag. How many bags can Jen fill completely?
Answers
Jen can fill 4 bags completely with the 5 1/2 cups of M&Ms she has, given that each bag requires 1 1/4 cups of M&Ms.
First, we need to find the total number of cups of M&Ms Jen has
5 1/2 cups = 11/2 cups
Then, we divide the total number of cups by the number of cups needed to fill each bag
(11/2 cups) ÷ (1 1/4 cups/bag)
To divide by a fraction, we can multiply by its reciprocal
(11/2 cups) x (4/5 cups/bag)
= 44/10 cups
Simplifying, we get
= 4 2/10 cups
= 4 1/5 cups
So, Jen can fill 4 bags completely.
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Farmer jones raises only pigs and geese. he wants to raise at most 16 animals. he wants no more
than 12 geese. he spends $5 to raise a pig and $2 to raise a goose and has 550 available to spend. if he
makes a profit of s4 per pig and s8 per goose, how many of each does he need to raise in order to
maximize his profits?
write the objective function. let a represent the number of pigs and y represent the number of geese.
Answers
The profit per pig is $4 and the profit per goose is $8.
How can the farmer raise at most 16 animals in order to maximize his profits?Let's start by defining the variables:
a = number of pigs Farmer Jones raises
y = number of geese Farmer Jones raises
The problem tells us that he wants to raise at most 16 animals, so we can write:
a + y ≤ 16
It also tells us that he wants no more than 12 geese, so we can write:
y ≤ 12
We know that it costs $5 to raise a pig and $2 to raise a goose, and he has $550 available to spend. So the cost of raising the animals can be expressed as:
5a + 2y ≤ 550
Finally, we want to maximize his profits. The profit per pig is $4 and the profit per goose is $8. So the objective function for Farmer Jones' profits is:
Profit = 4a + 8y
To summarize, the linear programming model for this problem is:
Maximize: Profit = 4a + 8y
Subject to:
a + y ≤ 16
y ≤ 12
5a + 2y ≤ 550
where a and y are non-negative integers.
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Frank needs to find the area enclosed by the figure. The figure is made by
attaching semicircles to each side of a 54-m-by-54-m square. Frank says the area
is 1,662. 12 m2. Find the area enclosed by the figure. Use 3. 14 for it. What error
might Frank have made?
The area enclosed by the figure is
m2
(Round to the nearest hundredth as needed. )
Answers
To find the area enclosed by the figure, we first need to find the area of the square and the area of each semicircle.
The area of the square is simply the length of one of its sides squared, which is:
54 m x 54 m = 2,916 m²
The area of each semicircle is half the area of a full circle with the same radius as the side of the square. The radius of each semicircle is 54 m/2 = 27 m.
The area of each semicircle is:
1/2 x π x 27 m² = 1/2 x 3.14 x 27 m x 27 m ≈ 1,442.31 m²
Since there are four semicircles, the total area of the semicircles is:
4 x 1,442.31 m² = 5,769.24 m²
Therefore, the total area enclosed by the figure is:
2,916 m² + 5,769.24 m² ≈ 8,685.24 m²
Frank's answer of 1,662.12 m² is significantly less than the actual area. He may have made the mistake of only calculating the area of one of the semicircles instead of all four, or he may have forgotten to include the area of the square.
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Part e
what is the mean absolute deviation for doctor a's data set on corrective lenses? what is the mean absolute deviation
for doctor b's data set on corrective lenses? write a sentence comparing the variation of the two data sets using their
mean absolute deviations.
Answers
The MAD for Doctor A's data set is 0.67 and the MAD for Doctor B's data set is 0.83. Doctor A's data set has less variation than Doctor B's data set, as indicated by their respective MADs.
To calculate the mean absolute deviation (MAD) for a data set, we first find the mean of the data set, and then find the absolute deviation of each value from the mean. We then find the mean of these absolute deviations.
a) For Doctor A's data set on corrective lenses, the mean is:
Mean = (15+18+17+16+14)/5 = 16
The absolute deviations from the mean are:
|15-16| = 1
|18-16| = 2
|17-16| = 1
|16-16| = 0
|14-16| = 2
The mean of these absolute deviations is:
MAD = (1+2+1+0+2)/5 = 1.2
Therefore, the MAD for Doctor A's data set is 1.2.
b) For Doctor B's data set on corrective lenses, the mean is:
Mean = (17+19+20+16+18)/5 = 18
The absolute deviations from the mean are:
|17-18| = 1
|19-18| = 1
|20-18| = 2
|16-18| = 2
|18-18| = 0
The mean of these absolute deviations is:
MAD = (1+1+2+2+0)/5 = 1.2
Therefore, the MAD for Doctor B's data set is also 1.2.
Comparing the two data sets using their MAD, we can see that they have the same amount of variation or dispersion from the mean. Both sets have a MAD of 1.2, indicating that the average absolute deviation of each value from the mean is the same for both sets.
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Answer:
Doctor A MAD: 11.8
Doctor B MAD: 9.32
Step-by-step explanation:
This is what I got on the assignment.
Max has eight circular chips that are all the same size and shape in a bag.
(3 chips are square, and 5 are stars)
Max reaches into the bag and removes one circular chip. What is the theoretical probability that the circular chip has a star on it? Write your answer as a fraction, decimal, and percent
Answers
The probability of drawing a star-shaped chip is 5/8.
The theoretical probability of drawing a star-shaped circular chip from the bag is 5/8 or 0.625 or 62.5%. Out of the total of eight circular chips, five are stars, and three are squares.
Therefore, the probability of drawing a star-shaped chip is the ratio of the number of star-shaped chips to the total number of chips in the bag, which is 5/8.
To understand this conceptually, we can think of probability as a fraction where the numerator is the number of favorable outcomes (in this case, drawing a star-shaped chip) and the denominator is the total number of possible outcomes (all the circular chips in the bag).
Thus, the theoretical probability of drawing a star-shaped chip is 5/8 because there are five star-shaped chips out of the total eight circular chips in the bag.
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Zachary says that the equation 0 = (x – 4)(x – c) has one unique real number solution. What must be the value of c for his statement to be true?
c = 4
Answers
Answer:
c=4
Step-by-step explanation:
as there is only one solution for this quadratic equation which is supposedly have 2 or more answers, c can only be 4 in order to be the same equation as the the first one x-4=0
Weights of erasers produced by a certain factory are known to follow the uniform distribution between 31. 5 g and 32. 3 g.
(a) (10 points) erasers produced by this factory are sold in packs of 45. A retailer randomly bought 200 packs. Find the probability that, for at least 15 packs, the average weight of the erasers in the pack is at least 31. 95 g.
(b) (10 points) each day, a quality control unit examines the erasers produced by this factory. The unit randomly chooses an eraser from the outputs of this factory and weighs it. This process is repeated 50 times. The unit then records the total number of erasers that were found to weigh at least 31. 7 g. (erasers with weights at least 31. 7 g are called "good" erasers)suppose this unit works for 42 consecutive days. Find the probability that, on average, it finds at least 37. 2 "good" erasers per day
Answers
a) The probability that, for at least 15 packs, the average weight of the erasers in the pack is at least 31.95 g is approximately 0.0384.
b) The probability that, on average, the unit finds at least 37.2 "good" erasers per day is approximately 0.3133.
a) To solve this problem, we need to use the central limit theorem. According to this theorem, the distribution of sample means becomes approximately normal, regardless of the shape of the population distribution, when the sample size is sufficiently large (usually, n >= 30). In this case, since the sample size is 45, we can assume that the distribution of sample means will be approximately normal.
Now, we need to find the probability that the average weight of at least 15 packs is at least 31.95 g. We can use the normal distribution to calculate this probability. We first calculate the z-score for this value as follows:
z = (31.95 - 31.9) / (0.163 / √(45)) = 1.77
Using a standard normal table or calculator, we can find the probability that a z-score is greater than or equal to 1.77. This probability is approximately 0.0384.
b) To solve this problem, we need to use the normal approximation to the binomial distribution. Since each eraser is either "good" or "bad", the number of "good" erasers that the unit finds each day follows a binomial distribution with parameters n = 50 and p = probability of finding a "good" eraser = (32.3 - 31.7)/(32.3 - 31.5) = 0.5.
Now, we need to find the probability that, on average, the unit finds at least 37.2 "good" erasers per day. We can use the normal distribution to calculate this probability. We first calculate the z-score for this value as follows:
z = (37.2 - 25) / 25 = 0.488
Using a standard normal table or calculator, we can find the probability that a z-score is greater than or equal to 0.488. This probability is approximately 0.3133.
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calculate div(f) and curl(f). f = 5ey, 2 sin(x), 9 cos(x)
Answers
Div(f) = 2cos(x) + [tex]5e^y[/tex], and curl(f) = < 0, 9sin(x), 5e^y >.
To calculate div(f) and curl(f), we need to express f as a vector field:
f = < 2 sin(x), [tex]5e^y[/tex][tex]5e^y[/tex], 9 cos(x) >
Then, we can use the formulas for divergence and curl:
div(f) = ∂f₁/∂x + ∂f₂/∂y + ∂f₃/∂z
curl(f) = < ∂f₃/∂y - ∂f₂/∂z, ∂f₁/∂z - ∂f₃/∂x, ∂f₂/∂x - ∂f₁/∂y >
Let's compute these step by step:
div(f) = ∂f₁/∂x + ∂f₂/∂y + ∂f₃/∂z
= 2cos(x) + [tex]5e^y[/tex] + 0
= 2cos(x) + [tex]5e^y[/tex]
curl(f) = < ∂f₃/∂y - ∂f₂/∂z, ∂f₁/∂z - ∂f₃/∂x, ∂f₂/∂x - ∂f₁/∂y >
= < 0 - 0, 0 - (-9sin(x)), 5e^y - 0 >
= < 0, 9sin(x), 5e^y >
Therefore, div(f) = [tex]2cos(x) + 5e^y[/tex], and curl(f) = [tex]< 0, 9sin(x), 5e^y > .[/tex]
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If KN = 3 cm, MN = 7 cm, RS = 14 cm, and PS = 6 cm, what is the scale factor of figure KLMN to figure PQRS?
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The scale factor of figure KLMN to figure PQRS is 6 cm / 3 cm = 2.
To find the scale factor of figure KLMN to figure PQRS, given KN = 3 cm, MN = 7 cm, RS = 14 cm, and PS = 6 cm, follow these steps:
1. First, find the length of a side in figure KLMN. We can use KN since it's given: KN = 3 cm.
2. Next, find the corresponding side length in figure PQRS. Since KN corresponds to PS, we have: PS = 6 cm.
3. Now, find the scale factor by dividing the length of the side in figure PQRS by the length of the corresponding side in figure KLMN: scale factor = PS/KN = 6 cm / 3 cm.
The scale factor of figure KLMN to figure PQRS is 6 cm / 3 cm = 2.
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Find the area of the shaded region:
Answers
Answer:
approximately 42.85 of whatever unit
If 4:15=a:2 1/2(two and a half), what is the value of a
Answers
The value of 'a' is 2/3.
What is the value of 'a' if the ratio of 4 to 15 is equivalent to the ratio of 'a' to 2 1/2?The problem presents a ratio, 4:15, that is equal to a ratio involving 'a' and 2 1/2. To solve for 'a', we need to isolate it on one side of the equation by cross-multiplying.
In the first step, we convert 2 1/2 to an improper fraction, 5/2, so that we can use it in the equation. We then cross-multiply by multiplying both sides of the equation by 5/2.
This eliminates the denominator on the right-hand side and simplifies the left-hand side.
Solve for 'a'
To solve for 'a', we can use cross-multiplication.
First, we need to convert 2 1/2 to an improper fraction:
2 1/2 = 5/2
Now we can write the equation as:
4/15 = a/(5/2)
To solve for 'a', we cross-multiply:
4/15 * 5/2 = a
a = 2/3
Finally, we solve for 'a' by multiplying 4/15 by 5/2 and simplifying the result. The answer is 2/3, which represents the value of 'a'.
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Integrate the function. 「 dx ,X> 6. Give your answer in exact form. x2x² - 36
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To integrate the function ∫(x² - 36) dx, we first need to factor out the expression inside the parentheses:
∫(x² - 36) dx = ∫(x - 6)(x + 6) dx
We can then use the power rule of integration to find the antiderivative:
∫(x - 6)(x + 6) dx = (1/3)x³ - 6x + C, where C is the constant of integration.
Since the original problem states X > 6, we can evaluate the definite integral using these limits:
∫(x² - 36) dx from 6 to X = [(1/3)X³ - 6X] - [(1/3)(6)³ - 6(6)]
= (1/3)X³ - 6X - 68
Therefore, the answer in exact form is (1/3)X³ - 6X - 68.
To integrate the given function, first note the correct notation for the function: ∫(x^2)/(x^2 - 36) dx for x > 6.
To solve this, we can use partial fraction decomposition. The given function can be rewritten as:
∫(A(x - 6) + B(x + 6))/(x^2 - 36) dx
Solving for A and B, we find that A = 1/12 and B = -1/12. Now we rewrite the integral as:
∫[(1/12)(x - 6) - (1/12)(x + 6)]/(x^2 - 36) dx
Next, separate the two terms and integrate them individually:
(1/12)∫[(x - 6)]/(x^2 - 36) dx - (1/12)∫[(x + 6)]/(x^2 - 36) dx
Now, notice that the integrals are of the form ∫u'/u dx. The integral of this form is ln|u|. So we have:
(1/12)[ln|(x - 6)| - ln|(x + 6)|] + C
Using the logarithm property, we can rewrite the answer as:
(1/12)ln|((x - 6)/(x + 6))| + C
That is the exact form of the antiderivative for the given function.
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What is the max/min of the quadratic equation in factored form: f(x) = 0. 5(x+3)(x-7)? Is it a maximum or a minimum? Show your workor explain your reasoning.
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The quadratic equation in factored form: f(x) = 0. 5(x+3)(x-7) have a minimum point. The minimum value of the function is -11.
To find the maximum or minimum of the quadratic equation in factored form f(x) = 0.5(x+3)(x-7), we need to convert it to standard form by expanding the terms:
f(x) = 0.5(x² - 4x - 21)
f(x) = 0.5x² - 2x - 10.5
The coefficient of x² is positive, so the parabola opens upwards and we have a minimum point.
To find the x-coordinate of the minimum point, we can use the formula x = -b/2a, where a = 0.5 and b = -2:
x = -(-2)/2(0.5) = 2
So the minimum point is at x = 2. To find the y-coordinate, we can substitute x = 2 into the equation:
f(2) = 0.5(2)^2 - 2(2) - 10.5 = -11
Therefore, the minimum value of the function is -11.
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I’m confused as to what the solution is if I follow the first step
Answers
Answer:
x = 4, y = 1, z = 5
Step-by-step explanation:
z + x =9....(3) we can say it's x + z = 9
x - z = - 1
x + z = 9
______ -
- 2z= - 10
z = 10/2
z = 5
if z + x = 9
5 + x = 9
x = 9 - 5
x = 4
if x + y = 5
4 + y = 5
y = 5 - 4
y = 1
#CMIIWThe following data points represent the number of holes
that moths ate in each of grandma marion's dresses.
7,8,8, 5, 7,8
using this data, create a frequency table.
number of holes
number of dresses
5
6
7
8
Answers
A frequency table was created using data points representing the number of holes in each of Grandma Marion's dresses. The table shows the number of dresses with 5, 7, and 8 holes, respectively.
To create a frequency table for the given data, first, the unique values in the data set are identified, which are 5, 7, and 8. Then, the number of occurrences of each unique value is counted, resulting in the frequencies 1 for 5, 2 for 7, and 3 for 8.
Count the frequency of each data point
5: 1 dress
7: 2 dresses
8: 3 dresses
Finally, these values are organized into a table with two columns, one for the unique values and another for their corresponding frequencies. The resulting frequency table shows the number of dresses with each number of holes eaten by moths.
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