Answer:
The answer is C. x=In(5/3)-2
Step-by-step explanation:
Answer:
C
Step-by-step explanation:
Suppose shirts are one of 3 colors (red, blue, and purple) and pants are black, brown, or white. An outfit consists of a shirt and pants. What is the minimum number of people that need to be in a room together to guarantee that at least two of them are wearing same-colored outfits
Answer:
10 people
Step-by-step explanation:
Given:
Colors of shirts: 3 (red, blue, and purple)
Colors of pants: 3 (black, brown, or white)
Total number of outfits ( both shirts and pants) =
3 * 3 = 9
The minimum number of people that need to be in a room together to guarantee that at least two of them are wearing same-colored outfits will be:
Total number + 1
= 9 + 1
= 10 people
A particular solution and a fundamental solution set are given for the nonhomogeneous equation below and its corresponding homogeneous equation. (a) Find a general solution to the nonhomogeneous equation. (b) Find the solution that satisfies the specified initial conditions.
y(1)--2, y'(1)-1, and y'(1)--36
Answer:
A.y=2x^5 + c1+ c2x + c3x^5
B. Y = 2x² + 9+7x+2x^5
Step-by-step explanation:
See attached file
if a rabbit can move 4/5 of a mile every hour then how many hours would it take for a rabbit to go 8 Miles
A cell phone company wants to determine if the use of text messaging is independent of age. The following data has been collected from a random sample of customers. Regularly use text messaging Do not regularly use text messaging Under 21 82 38 21-39 57 34 40 and over 6 83 Based on the data above what is the expected value for the "under 21 and regularly use text messaging" cell? Question 5 options: 58 120 50 82
Answer:
Hi Gabbie,
The answer to your question that was what is the expected value for the "under 21 and regularly use text messaging" cell is 82.
Please rate me good :)
Step-by-step explanation:
The answer is 82 because the table that clearly defines the result was in the question.
If you are a student with no assets of any value and have liability insurance on an old car that pays a maximum of $50,000 per accident, what is most likely to happen if you cause an accident that results in $75,000 in damage to the passengers in another car?
A. They will not pursue any action against you or your insurance company
B. They will sue you, personally, for more than $100,000
C. They will accept the $50,000 maximum offered by your insurance company
D. They will sue for the entire $75,000
Answer:
They will accept the $50,000 maximum offered by your insurance company
In this scenario, with $50,000 liability insurance, if you cause a $75,000 accident, the other party is likely to accept the $50,000 from your insurance (Option C) but could also sue you for the remaining $25,000 (Option D).
In this scenario, if you cause an accident resulting in $75,000 in damage to the passengers in another car, your liability insurance has a maximum coverage limit of $50,000 per accident. Typically, insurance policies cover up to the policy limits, and the insurance company would pay out up to $50,000 to the injured parties.
The most likely outcome in this situation would be that the injured parties may initially pursue a claim with your insurance company, and the insurance company would pay up to its policy limit of $50,000. However, since the damages exceed the policy limit, the injured parties may still have the option to sue you personally for the remaining $25,000 to cover their damages.
So, the answer could be a combination of options C and D: They may accept the $50,000 maximum offered by your insurance company but could also potentially sue you for the remaining $25,000 if they believe it's necessary to cover their damages. The actual outcome may vary depending on the specific circumstances, local laws, and the decisions made by the injured parties and their legal advisors. It's crucial to notify your insurance company as soon as an accident occurs so they can handle the situation accordingly.
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Pat needs to determine the height of a tree before cutting it down to be sure that it will not fall on a nearby fence. The angle of elevation of the tree from one position on a flat path from the tree is Upper H equals 60 degrees comma and from a second position Upper L equals 60 feet farther along this path it is Upper B equals 50 degrees . What is the height of the tree?
Answer:
229.23 feet.
Step-by-step explanation:
The pictorial representation of the problem is attached herewith.
Our goal is to determine the height, h of the tree in the right triangle given.
In Triangle BOH
[tex]Tan 60^0=\dfrac{h}{x}\\h=xTan 60^0[/tex]
Similarly, In Triangle BOL
[tex]Tan 50^0=\dfrac{h}{x+60}\\h=(x+60)Tan 50^0[/tex]
Equating the Value of h
[tex]xTan 60^0=(x+60)Tan 50^0\\xTan 60^0=xTan 50^0+60Tan 50^0\\xTan 60^0-xTan 50^0=60Tan 50^0\\x(Tan 60^0-Tan 50^0)=60Tan 50^0\\x=\dfrac{60Tan 50^0}{Tan 60^0-Tan 50^0} ft[/tex]
Since we have found the value of x, we can now determine the height, h of the tree.
[tex]h=\left(\dfrac{60Tan 50^0}{Tan 60^0-Tan 50^0}\right)\cdotTan 60^0\\h=229.23 feet[/tex]
The height of the tree is 229.23 feet.
Which rational function models the time, y, in hours, that it takes the train to travel between the two cities at an average speed of x miles per hour?
y= x/6.4
y= x/160
y= 6.4/x
y= 160/x
Answer:
[tex]y=\frac{x}{6.4}[/tex]
Step-by-step explanation:
The complete question in the attached figure
we know that
A relationship between two variables, x, and y, represent a proportional variation if it can be expressed in the form [tex]k=\frac{y}{x}[/tex] or [tex]y=kx[/tex]
In this problem, we have a proportional relationship between the average speed (x) and the time (y)
Find the constant of proportionality k
Fox =32, y=5
[tex]k=\frac{5}{32}[/tex]
substitute
[tex]y=\frac{5}{32}x[/tex]
we have
[tex]\frac{5}{32}=\frac{1}{6.4}[/tex]
therefore
[tex]y=\frac{x}{6.4}[/tex]
Experts have predicted that approximately 1 in 12 tractor-trailer units will be involved in an accident this year. One reason for this is that 1 in 3 tractor-trailer units has an imminently hazardous mechanical condition, probably related to the braking systems on the vehicle.
1. If you wanted to calculate a 99% confidence interval that is no wider than 0.06, how many tractor-trailers would you need to sample?
2. Suppose a sample of 124 tractor-trailers is taken and that 45 of them are found to have a potentially serious braking system problem. Find a 90% confidence interval for the true proportion of all tractor-trailers that have this potentially serious problem.
3. Interpret your confidence interval from 2
Answer:
1. 564 tractor trailers
2. (0.2919, 0.4339)
3. It should be noted that there is 90% confidence that the true population proportion lies between 0.2919 and 0.4339.
Step-by-step explanation:
1)
Proportion= P = 0.8333333333333 (1/12)
Margin error= 0.06/2 = 0.03
Confidence level= 99
Significance level = α= (100 - 99)%= 1%= 0.01
α/2 = 0.01/2 = 0.0005
Sample size= n
= p(1 - p) (Z*/E)
=0.8333333333333 x (2.576/0.03)^2
=563.15
Approximately
=564
2) Sample size n= 124
Sample number of event x =45
Sample proportion = p= x/n
=45/124
= 0.3629
Standard error =√p(1 - p) /n
√(0.3629× (1 - 0.3629)/ 124
= 0.0432
Confidence level= 90
Significance level α= (100-90)% =0.01
Critical value Z* = 1.645
Margin of error = Z×Standard error= 1.645 × 0.0432= 0.07103
Lower limit = p- margin error= 0.3629 - 0.07103= 0.2919
Upper limit = p+margin error= 0.3629 + 0.07103= 0.4339
The answer is (0.2919, 0.4339)
3) It should be noted that there is 90% confidence that the true population proportion lies between 0.2919 and 0.4339.
What Is The Area Of The Trapezoid
Answer:
256 m2
Step-by-step explanation:
- First, know the formula A = (a+b/2)h
- Using this, we will fill the equation in with our variables. For example...
A = ((13+19)/2)16
A = (32/2)16
A = (16)(16)
A = 256 m2
- Hope this helps! If you need a further explanation or step by step practice please let me know.
A toilet manufacturer has decided to come out with a new and improved toilet. The fixed cost for the production of this new toilet line is $16,600 and the variable costs are $63 per toilet. The company expects to sell the toilets for $160. Formulate a function C(x) for the total cost of producing x new toilets and a function R(x) for the total revenue generated from the sales of x toilets. How many toilets need to be sold to break even?
Answer:
C(x)=16600+63x
R(x)=160x
Break-even Point, x=172
Step-by-step explanation:
Let x be the number of Toilets Produced.
Fixed cost = $16,600
Variable costs = $63 per toilet.
Total Cost, C(x)=16600+63x
The company expects to sell the toilets for $160.
Selling Price Per Toilet=160
Total Revenue for x Toilets, R(x)=160x
Next, we determine the break-even point.
The break-even point is the point where the Cost of Production equals Revenue generated.
i.e. C(x)=R(x)
16600+63x=160x
16600=160x-63x
16600=97x
x=171.13
The company needs to sell at least 172 Toilets to break even.
To formulate the functions C(x) and R(x) for the total cost and revenue of producing and selling x toilets, we can use the given information about fixed costs, variable costs, and selling price. By setting the total cost equal to the total revenue, we can find the number of toilets needed to break even, which is approximately 171.
Explanation:To formulate the function C(x) for the total cost of producing x new toilets, we need to consider the fixed cost and the variable cost. The fixed cost is $16,600, which remains constant regardless of the number of toilets produced. The variable cost is $63 per toilet, so we multiply it by x to account for the number of toilets produced. Therefore, the function C(x) can be expressed as:
C(x) = 16,600 + 63x
The function R(x) for the total revenue generated from the sales of x toilets can be found by multiplying the selling price of each toilet by the number of toilets sold. Since each toilet is sold for $160, the function is:
R(x) = 160x
To find the number of toilets needed to break even, we need to determine the value of x when the total cost is equal to the total revenue. In other words, we set C(x) = R(x) and solve for x:
16,600 + 63x = 160x
Subtracting 63x from both sides:
16,600 = 97x
Dividing both sides by 97:
x = 170.10
Therefore, the company needs to sell approximately 171 toilets to break even.
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A cognitive psychologist wants to investigate whether memory for a shopping list is affected by the strategy used to study the list. She selects 20 random students and assigns each of them randomly to one of four different groups. Each group is given a different strategy by which to study the list. After the students study the list, they try to recall as many items as they can. The number of items remembered from the list for each students is given below.
Is there sufficient evidence to conclude that different strategies lead to different memory performance? Use significance level 0.05
Answer:
There is no sufficient evidence
Step-by-step explanation:
See attached files
32) In one region, the September energy consumption levels for single-family homes are found to be normally distributed with a mean of 990 kWh and a standard deviation of 198 kWh. For a randomly selected home, find the probability that the September energy consumption level is between 1100 kWh and 1250 kWh.
Answer:
0.1946 is the probability that the September energy consumption level is between 1100 kWh and 1250 kWh.
Step-by-step explanation:
We are given the following information in the question:
Mean, μ = 990 kWh
Standard Deviation, σ = 198 kWh
We are given that the distribution of energy consumption levels is a bell shaped distribution that is a normal distribution.
Formula:
[tex]z_{score} = \displaystyle\frac{x-\mu}{\sigma}[/tex]
P(September energy consumption level is between 1100 kWh and 1250 kWh)
[tex]P(1100 \leq x \leq 1250)\\\\ = P(\displaystyle\frac{1100 - 990}{198} \leq z \leq \displaystyle\frac{1250 -990}{198}) \\\\= P(0.5556 \leq z \leq 1.3131)\\\\= P(z \leq 1.3131) - P(z < 0.5556)\\\\= 0.9054- 0.7108= 0.1946[/tex]
0.1946 is the probability that the September energy consumption level is between 1100 kWh and 1250 kWh.
Consider the following time series: t sales 1 6 2 11 3 9 4 16 5 17 Use simple linear regression analysis with t as the predictor variable to find the parameters for the line that minimizes MSE for this time series. Enter the data into Excel and use Excel for your calculations. Enter the exact answers. a) y-intercept, b0 = -0.67 b) slope, b1 = 0.31 What is the MSE if this model is used to forecast sales for time periods 1-5? c) MSE = What is the forecast for time period t = 6?\
Answer:
Check the explanation
Step-by-step explanation:
Period, X Actual , Y
1 6
2 11
3 9
4 16
5 17
Y= intercept + slope*x
Answer a and b:
INTERCEPT(known Y's,known X's) Slope(known Y's,known X's)
3.7 2.7
Simple linear regression equation is
Forecast, Y= 3.7+ 2.7*x
Period, X Actual , Y linear Absolute squared
trend deviation= deviation=
Forecast, |Forecast - (absolute
Y= 3.7+ 2.7*x Actual| deviation)^2
1 6 6.40 0.4 0.2
2 11 9.10 1.9 3.6
3 9 11.80 2.8 7.8
4 16 14.50 1.5 2.3
5 17 17.20 0.2 0.0
6 19.90
2.78
MSE
Answer c: MSE= 2.78
Answer d: Forecast for x= 6= 19.90
Convert 3/7 into a percent.
Answer:
42.86%
Step-by-step explanation:
3/7 as a decimal is 0.4286... and to convert that into a decimal, you move the decimal point right twice.
0.4286 = 42.86%
I hope this helped!
Final answer:
To convert the fraction 3/7 into a percent, divide 3 by 7 to get the decimal 0.4286, then multiply by 100 to get 42.86%.
Explanation:
Converting a Fraction to a Percent
To convert a fraction like 3/7 into a percent, follow these steps:
Convert the fraction to a decimal by dividing the numerator by the denominator. For 3/7, divide 3 by 7 to get approximately 0.4286.Convert the decimal to a percent by multiplying it by 100. So, 0.4286 times 100 equals 42.86.Therefore, 3/7 as a percent is 42.86%.
A percent (or per cent) is a way of expressing a number as a fraction of 100. The term "percent" comes from the Latin per centum, which means "by the hundred." It is often represented by the symbol "%."
What is the v?
-18 - 3/4v = 3
Step-by-step explanation:
[tex] - 18 - \frac{3}{4v} = 3[/tex]
[tex] - \frac{3}{4v} = 3 + 18[/tex]
[tex] - \frac{3}{4v} = 21[/tex]
Now doing cross multiply
-3 = 21 * 4v
[tex]v = - \frac{3}{21 \times 4} [/tex]
Therefore
[tex]v \: = \frac{1}{28} [/tex]
Hope this helps.
is 0 an irrational
number
Answer: no
Step-by-step explanation:
0 is a rational number.A rational number is a number that can be expressed as the quotient or fraction m/n of two integers, a numerator m and a non-zero denominator n.0 can be expressed as 0/n ;therefore 0 is a rational number.Spymore3
4
of a number is 21. What is the number?
Answer:
28
Step-by-step explanation:
3/4x= 21
x= 21(4/3)
x= 28
Check! (optional)
28 x 3/4 = 21
Check correct!!!!!!
Answer:
the number is 28
Step-by-step explanation:
I know this because 21 is 3/4 of a number. 21 divided by 3 is 7, you do this to find what 1/4 of the number is. so then you add 7 to 22 and you get 28!
In the first semester, Jonas took seven tests in his math class. His scores were: 88 81 94 84 100 94 96.
What is the Median of his scores?
Butterflies: • Alice, Bob, and Charlotte are looking for butterflies. They look in three separate parts of a field, so that their probabilities of success do not affect each other. • Alice finds 1 butterfly with probability 17%, and otherwise does not find one. • Bob finds 1 butterfly with probability 25%, and otherwise does not find one. • Charlotte finds 1 butterfly with probability 45%, and otherwise does not find one. Let X be the number of butterflies that they catch altogether. A) Find the expected value of X. B) Write X as the sum of three indicator random variables, X1,X2,X3 that indicate whether Alice, Bob, Charlotte (respectively) found a butterfly. Then X=X1+X2+X3. Find the expected value of X by finding the expected value of the sum of the indicator random variables.
Answer:
The expected value is 0.87.
Step-by-step explanation:
a) To calculate the expected value X we will first see the posible outcomes. So could take value of 0,1,2,3. We will calculate the probability of each outcome. To do so, we will introduce the following notation. Consider the following tuple (A,B,C) where A is the number of butterflies found by Alice, B the number found for by Bob and C the number found by C. To calculate the probability of the tuple (A,B,C) we will do as follows. If the entry of the tuple is 1, then we will multiply by the probability of the person that found the butterfly. So, if A =1, we will multiply by 0.17(Alice finds a butterfly with probability 0.17). On the other side, if the entry of the tuple is 0, we will multiply by (1-p) where p is the probability of the person that found the butterly. So, if A=0, we will multiply by 0.83. So, for example, consider the tuple (1,0,1). The probability of having this result is 0.17*0.75*0.45 (Alice and Charlotte found a butterfly, but Bob didn't). We can do this since we are said that their probabilities of success don't affect others' probabilities.
We will see the total number of butterflies and the tuples associated to that number. That is
X number of butterflies - tuples
0 butterflies - (0,0,0)
1 butterfly - (1,0,0) or (0,1,0) or (0,0,1)
2 butterflies - (1,1,0) or (1,0,1) or (0,1,1)
3 butterflies - (1,1,1)
To find the probability of the value of X, we will sum up the probability of the associated tuples. The values of the probabilities are as follows
(0, 0, 0) = 0.342375
(0, 0, 1 ) = 0.280125
(0, 1, 0) = 0.114125
(0, 1, 1 ) = 0.093375
(1, 0, 0) = 0.070125
(1, 0, 1 ) = 0.057375 = 0.17*0.75*0.45
(1, 1, 0) = 0.023375
(1, 1, 1) = 0.019125
In this case,
P(X=0) = 0.342375 ,
P(X=1) = 0.464375 = 0.280125 +0.114125+ 0.070125
P(X=2) = 0.174125
P(X =3 ) = 0.019125
So, the expected value of X is given by
0* 0.342375 +1 * 0.464375 +2* 0.174125+3*0.019125 = 0.87
b)Let X1 be the number of butterflies found by Alice, X2 the number found by Bob and X3 the number found by Charlotte. Then X = X1+X2+X3. Using the expected value properties and the independence of X1, X2 and X3 we have that E(X) = E(X1)+E(X2)+E(X3).
Recall that each variable is as follows. Xi is equal to 1 with probability p and it is 0 with probability (1-p). Then, the expected value of Xi is
[tex]1\cdot p + 0\codt (1-p)=p[/tex]. Note that the value of p for X1,X2 and X3 is 17%, 25% and 45% respectively.
Then E(X) = 17%+25%+45%= 0.87.
So the expected number of butterflies is 0.87.
Final answer:
The expected value of the total number of butterflies, X, that Alice, Bob, and Charlotte catch is 0.87. This is found by summing their independent probabilities of catching a butterfly (0.17 for Alice, 0.25 for Bob, and 0.45 for Charlotte). X is also represented as the sum of three indicator random variables X1, X2, and X3, leading to the same expected value.
Explanation:
Expected Value of the Number of Butterflies Caught
In this scenario with Alice, Bob, and Charlotte searching for butterflies in separate parts of a field, the random variable X represents the total number of butterflies they catch. The expected value of X, or E(X), is calculated by adding the individual probabilities of finding a butterfly, since their probabilities are independent.
To find the expected value of X:
Multiply the probability of each person finding a butterfly by the number of butterflies they would find in that event (which is 1 since each either finds 1 butterfly or none), and
Add these products together.
The expected value is thus 0.17 + 0.25 + 0.45 = 0.87 butterflies. We can also express X as X1 + X2 + X3, where each Xi is an indicator random variable for whether Alice (X1), Bob (X2), or Charlotte (X3) found a butterfly.
The expected value for each indicator variable is the same as the person's probability of success. So, E(X1) = 0.17, E(X2) = 0.25, and E(X3) = 0.45. By the linearity of expectation, E(X) = E(X1) + E(X2) + E(X3), which also equals 0.87 butterflies.
Explain what "statistical significance" means. Choose the correct explanation below. A. Statistical significance means that the tools used to measure the data introduce error that needs to be accounted for when considering whether or not to reject the null hypothesis. B. Statistical significance means that the sample standard deviation is unusually small, resulting in an unusually large test statistic. C. Statistical significance means that the null hypothesis claims the population proportion is equal to something other than 0.5. D. Statistical significance means that the result observed in a sample is unusual when the null hypothesis is assumed to be true. E. Statistical significance means that the scenario being analyzed will have a meaningful real-world impact.
Answer:
The correct explanation is:
D. Statistical significance means that the result observed in a sample is unusual when the null hypothesis is assumed to be true.
Step-by-step explanation:
The statistical significance gives a threshold to measure if a sample result or observed effect is due only to a sampling or it really reflects a characteristics of the population we are studying.
The level of statistical significance is usually 0.05 or 5%, depending on how strong the evidence needs to be and the consequences of the conclusions of the study, taking into account the Type I and Type II errors.
The statistical significance provides a cutoff point for determining whether a sampling outcome, as well as observable impact, is attributable mostly to screening whether accurately reflects features of the target population under investigation.The threshold of such significance is very often 0.05 as well as 5%, on whatever the documentation has to be because the implications of the study's results, recognize the importance of Type I as well as the Type II mistakes.
Thus the response above i.e., "option D" is appropriate.
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Anne is a tailor and decides to evaluate her business to target potential customers with advertisements. She found that for every 12 women's dresses she alters, she alters 54 men's suits. Write this as a ratio in its simplest form.
Answer:
2/9
Step-by-step explanation:
The ratio is 12/54 which can be simplified by dividing the numerator and denominator by 6 to get 2/9
How do I find A’?
Let U={a,b,c,d,e,f,g} and A={a,b,e,f}
Let U={a,b,c,d,e,f,g,h}
A={a,c,d}
B={b,c,d}
C={b,e,f,g,h}
4. For the vectors b = (1, 2, −2) and a = (−3, 0, 4) (a) Compute the projection of vector b onto the line along vector a as p = ˆxa. (b) Compute the projection of vector b onto the line along vector a as p = Pb. (c) Compute the error vector. (d) Compute the length of the projection vector and length of the error vector.
Answer:
A) (33/5, 0, 44/5)
B) - 11
C) (28/5, 2, - 54/5)
D) 11, 12.83
Step-by-step explanation:
A)
Given the vectors
b= (1, 2,-2)
a= (-3, 0, 4)
Projection of the vector b onto the line along vector a as p=ˆxa
Calculating ab,
ab= a1b1 + a2b2 + a3b3
a1= - 3, a2=0, a3=4, b1=1, b2=2, b3= - 2
ab= (-3)(1) + (0)(2) +(4)(-2)
ab= - 3 + 0 +(-8)
ab= - 11
Vector projection which is
( ab÷ /vector a/^2) × vector a
= - 11/√(-3)^2 + (0)^2 + (4)^2
= - 11/ √9 +16
=-11/√25
= - 11/5× (-3, 0, - 4)
= (33/5, 0, 44/5)
B) When p= pb
It will be a scalar projection and will be written as:
ab÷ /a/
-11/√1
= - 11
C) Given the vector that form P to b in
e= b - p
=b- ˆxa
e= (1, 2,-2) - (33/5, 0, 44/5)
= (1 - 33/5, 2- 0, -2 - 44/5)
=(5-33/5, 2, - 10- 44/5)
= (28/5, 2, - 54/5)
D.
Length of the projection vector:
/e/ = √(33/5^2 + 0 + 44/5^2)
/e/= √33^2/25 + 0 + 44^2/25
/e/= √33^2/25 +44^2/25
/e/= √121
/e/= 11
Length of error vector
/e/ = √(28/5)^2 + 2^2 + (-54/5)^2
/e/= √28^2/25 +4 +(-54^2/25)
/e/= 12.83
Answer:
a) and b) p= (33/25, 0, -44/25)
c) e = (-8/25, 2, -6/25)
d) p = 11/5 = 2.2
e) e = (2/5)√26 = 2.039
Step-by-step explanation:
Given
b = (1, 2, −2)
a = (−3, 0, 4)
a) and b) We use the formula
p = (at*b)/(at*a)*a
⇒ p = ((−3, 0, 4)*(1, 2, −2)/((−3, 0, 4)*(−3, 0, 4)))*(−3, 0, 4)
[tex]p=\frac{at*b}{at*a}*a\\ p=\frac{(-3, 0, 4)*(1, 2,-2)}{(-3, 0, 4)*(-3, 0, 4)} *(-3, 0, 4)\\ p=\frac{-3+0-8}{9+0+16}*(-3, 0, 4)\\ p=-\frac{11}{25} *(-3, 0, 4)\\ p=(\frac{33}{25} ,0,-\frac{44}{25})[/tex]
c)
[tex]e=b-p\\ e=(1, 2, -2)-(\frac{33}{25} ,0,-\frac{44}{25})\\ e=(-\frac{8}{25} ,2,-\frac{6}{25} )[/tex]
d) We use the formula
[tex]p=\sqrt{(\frac{33}{25} )^{2} +(0)^{2} +(\frac{-44}{25} )^{2}} =\frac{11}{5}[/tex]
e) Applying the same formula we have
[tex]e=\sqrt{(\frac{-8}{25} )^{2} +(2)^{2} +(\frac{-6}{25} )^{2}} =\frac{2}{5}\sqrt{26} =2.039[/tex]
Making handcrafted pottery usually takes two major steps:wheel throwing and firing. The time of wheel throwing and thetime of firing are normally distributed random variables with meansof 40 min and 60 min and standard deviations of 2 min. and 3 min,respectively.
(a) What is the probability that a piece of pottery will befinished within 95 minutes?
(b) What is the probability that it will take longer than 110minutes?
Answer:
a) [tex]P(R<95)=P(\frac{R-\mu}{\sigma}<\frac{95-\mu}{\sigma})=P(Z<\frac{95-100}{3.606})=P(Z<-1.387)[/tex]
And we can find this probability using the normal standard table or excel and we got:
[tex]P(z<-1.387)=0.0827[/tex]
b) [tex]P(R>110)=P(\frac{R-\mu}{\sigma}>\frac{110-\mu}{\sigma})=P(Z>\frac{110-100}{3.606})=P(Z>2.774)[/tex]
And we can find this probability using the complement rule and the normal standard table or excel and we got:
[tex]P(z>2.774)=1-P(Z<2.774) = 1-0.9972 = 0.0028[/tex]
Step-by-step explanation:
Previous concepts
Normal distribution, is a "probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean".
The Z-score is "a numerical measurement used in statistics of a value's relationship to the mean (average) of a group of values, measured in terms of standard deviations from the mean".
Part a
Let X the random variable that represent the time for the step 1 and Y the time for the step 2, we define the random variable R= X+Y for the total time and the distribution for R assuming independence between X and Y is:
[tex]R \sim N(40+60 = 100,\sqrt{2^2 +3^2}= 3.606 s)[/tex]
Where [tex]\mu=65.5[/tex] and [tex]\sigma=2.6[/tex]
We are interested on this probability
[tex]P(R<95)[/tex]
And the best way to solve this problem is using the normal standard distribution and the z score given by:
[tex]z=\frac{R-\mu}{\sigma}[/tex]
If we apply this formula to our probability we got this:
[tex]P(R<95)=P(\frac{R-\mu}{\sigma}<\frac{95-\mu}{\sigma})=P(Z<\frac{95-100}{3.606})=P(Z<-1.387)[/tex]
And we can find this probability using the normal standard table or excel and we got:
[tex]P(z<-1.387)=0.0827[/tex]
Part b
[tex]P(R>110)=P(\frac{R-\mu}{\sigma}>\frac{110-\mu}{\sigma})=P(Z>\frac{110-100}{3.606})=P(Z>2.774)[/tex]
And we can find this probability using the complement rule and the normal standard table or excel and we got:
[tex]P(z>2.774)=1-P(Z<2.774) = 1-0.9972 = 0.0028[/tex]
Given Information:
Mean = μ = 40 + 60 = 100 minutes
Standard deviation = σ = 2² + 3² = 13 minutes
Required Information:
a. P(X < 95) = ?
b. P(X > 110) = ?
Answer:
a. P(X < 95) = 0.0823
b. P(X > 110) = 0 .0028
Explanation:
a)
Let random variable X represents the time in minutes of wheel throwing and firing.
The probability that a piece of pottery will be finished within 95 minutes means,
P(X < 95) = P(Z < (x - μ)/√σ)
P(X < 95) = P(Z < (95 - 100)/√13)
P(X < 95) = P(Z < -1.39)
The z-score corresponding to -1.39 is 0.0823
P(X < 95) = 0.0823
Therefore, there is 8.23% probability that a piece of pottery will be finished within 95 minutes.
b)
P(X > 110) = 1 - P(X < 110)
P(X > 110) = 1 - P(X < (x - μ)/√σ)
P(X > 110) = 1 - P(X < (110 - 100)/√13)
P(X > 110) = 1 - P(X < 2.77)
The z-score corresponding to 2.77 is 0.9972
P(X > 110) = 1 - 0.9972
P(X > 110) = 0 .0028
Therefore, there is 0.28% probability that a piece of pottery will take longer than 110 minutes.
The manager of a fund that provides loans for college students has estimated that the average monthly loan repayment for students borrowing from the fund is $75.00. You are to test this estimate. You take a sample of 20 students and find that the mean monthly payment is $69.46 with a standard deviation of $9.78. Which of the following statements is true about this test?
a. The value of the test statistic is -.57; therefore, the null hypothesis is rejected for mean = 0.02.
b. The value of the test statistic is -2.53; therefore, the null hypothesis is rejected for mean = 0.02.
c. The value of the test statistic is -2.53; therefore, the null hypothesis is rejected for mean = 0.05 but not for mean = 0.02.
d. The value of the test statistic is -.57; therefore, the null hypothesis is rejected for mean = 0.05 but not for mean = 0.02.
Answer:
Step-by-step explanation:
We would set up the hypothesis test.
For the null hypothesis,
µ = 75
For the alternative hypothesis,
µ ≠ 75
Since the number of samples is 20 and no population standard deviation is given, the distribution is a student's t.
Since n = 20,
Degrees of freedom, df = n - 1 = 20 - 1 = 19
t = (x - µ)/(s/√n)
Where
x = sample mean = $69.46
µ = population mean = $75
s = samples standard deviation = $9.78
t = (69.46 - 75)/(9.78/√20) = - 2.53
We would determine the p value using the t test calculator. It becomes
p = 0.01
Since alpha, 0.05 > than the p value, 0.01, then the null hypothesis is rejected.
Therefore,
The value of the test statistic is -2.53; therefore, the null hypothesis is rejected for level of significance = 0.05
Write each vector as a linear combination of the vectors in S. (If not possible, enter IMPOSSIBLE.) S = {(6, −7, 8, 6), (4, 6, −4, 1)} (a) u = (2, 19, −16, −4) u = −1 s1 + 2 s2 (b) v = 43 2 , 113 4 , −18, 13 2
Answer:
a) Yes, it is a linear combination.
b) Impossible to write as a linear combination.
Step-by-step explanation:
Recall that given vectors u,v,w we say that w is a linear combination of u and v if there exists real numbers a,b such that
[tex]w=au+bv[/tex]
a) [tex] u = (2,19,-16, -4)[/tex]. So, we have the following
[tex] (2,19,-16, -4)=a(6,-7,8,6)+b(4,6,-4,1)[/tex]. Which give us the following equations
[tex]6a+4b = 2[/tex]
[tex]-7a+6b = 19[/tex]
[tex]8a-4b = -16[/tex]
[tex]6a+b =-4[/tex]
Note that if we add the first and the third equation, we get that [tex]14a = -14[/tex] which implies that a=-1. In the first equation, if a=-1, then [tex]4b=2+6[/tex] which implies that b=2. We must check that when (a,b) =(-1,2) the four equations are still valid.
So
[tex]6(-1)+4(2) = -6+8 = 2[/tex]
[tex]-7(-1)+6(2) = 7+12 = 19[/tex]
[tex]8(-1)-4(2) = -8-8 = -16[/tex]
[tex]6(-1)+(2) =-6+2 = -4[/tex]
Since all equations are met, we have written the desired vector u as the linear combination of the initial vectors.
b) Repeating the same analysis, we get
[tex](432 , 1134 , −18, 132)=a(6,-7,8,6)+b(4,6,-4,1)[/tex]
[tex]6a+4b = 432[/tex]
[tex]-7a+6b = 1134[/tex]
[tex]8a-4b = -18[/tex]
[tex]6a+b =132[/tex]
adding the first and third equation we get [tex]14a = 414[/tex] so a = 207/7. Replacing this value will give us that b=891/14.
However,
note that
[tex]-7\frac{207}{7}+6\frac{891}{14} = \frac{1124}{7}\neq 1134[/tex]. Then, it is impossible to write the linear combination.
The question seeks a linear combination of vectors from a given set 'S' that matches specified vectors. Using vector addition and scalar multiplication, it is possible to find a unique combination for each specified vector. If no combination can provide the required vector, 'IMPOSSIBLE' is the answer.
Explanation:In this problem, you're asked to express a given vector as a linear combination of other vectors from the set 'S'. This involves using the properties of vector addition and scalar multiplication to define a unique way to represent each given vector. A linear combination of vectors involves adding or subtracting multiples of these vectors.
For instance, if we have the set S = {(6, -7, 8, 6), (4, 6, -4, 1)}, and we're asked to write the vector u = (2, 19, -16, -4), the correct linear combination would be -1 times the first vector in S plus 2 times the second vector (-1*(6,-7,8,6) + 2*(4,6,-4,1) = (2, 19, -16, -4)). This means vector 'u' would be expressed as u = -1 * S1 + 2 * S2. If no such combination is possible, the answer would be 'IMPOSSIBLE'.
Other important concepts related to this problem include vector addition, scalar multiplication, and the distributive property. Vector addition is both associative and commutative and vector multiplication by a sum of scalars is distributive. The direction and magnitude of vectors are also significant elements to consider while solving such problems.
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If L || m, solve for x (9x+2) 119
Alternate angles in a transversal are congruent.
The value of x is 13
See attachment for the image of the transversal,
Where [tex](9x + 2)[/tex] and [tex]119[/tex] are alternate angles
This means that:
[tex]9x + 2 = 119[/tex] ---- alternate angles are equal
Collect like terms
[tex]9x = 119 - 2[/tex]
[tex]9x = 117[/tex]
Divide both sides by 9
[tex]x = 13[/tex]
Hence, the value of x is 13
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Write a real-world problem that can be represented by the equation 1/2x+6=20
Step-by-step explanation:
The given expression in word problem can be translated as:
Six more than half of a number is 20
A real-world problem for the equation 1/2x + 6 = 20 could be determing the number of days a person should work, earning a rate of half the square of the number of days, to achieve a total sum of $20. The person initially has $6 and after working for 28 days, he or she achieves the goal.
Explanation:Consider a real-world example represented by the equation 1/2x + 6 =20. Imagine your grandmother gives you $6 and says you can do chores for her on some days to earn half the square of the number of days you worked in dollars. If you want to accumulate $20 in total, how many days should you work? This problem asks the same as solving for 'x' in the equation where 'x' is the number of days and the total sum of money is $20.
To solve this, you would need to subtract 6 from both sides of the equation, leaving you with 1/2x =14. Then, you multiply both sides by 2 to get x = 28, so it takes 28 days of work.
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A triangle is drawn on the coordinate plane. It is translated 4 units right and 3 units down. Which rule describes the
translation?
(x, y) - (x + 3, y - 4)
(x,y) → (x + 3, y + 4)
(x, y) - (x +4, Y-3)
(x, y) (x + 4, y + 3)
Answer:
C
Step-by-step explanation:
The correct option is option C , that is the rule which describes the translation is (x, y) → (x + 4, y - 3).
What do you mean by translation ?
The modification of an existing diagram to create a different version of the following diagram is known as translation.
The translated triangle is moved along the x-axis by 4 units, to the right. The numbers get more positive as one moves to the right along the x-axis, hence an additional 4 units should be added to the x-coordinate. So , the coordinate would be x + 4.
Along the y-axis, the triangle is also pushed downward by 3 units. More negative numbers are produced as the axis moves below, hence 3 should be subtracted. So , the coordinate should be y - 3.
Based on the above information, we can conclude that another way to write the given rule by translation is (x, y) → (x + 4, y - 3) .
Therefore , the correct option is option C , that is the rule which describes the translation is (x, y) → (x + 4, y - 3).
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We want to find the maximum and minimum values of f(x,y)=12x2+13y2 on the disk D: x2+y2≤1. What is the critical point in D? (x,y)=( , ) Now focus on the boundary of D, and solve for y2. Restricting f(x,y) to this boundary, we can express f(x,y) as a function of a single variable x. What is this function and its closed interval domain? f(x,y)=f(x)= where ≤x≤ What are the absolute maximum and minimum values of the function along the BOUNDARY of D? maximum value: minimum value: What are the absolute maximum and minimum values of f(x,y) over all of D? maximum value: minimum value:
Answer:
Over the boundary: maximum:13, minimum:12
Over D: maximum:13, minimum:0
Step-by-step explanation:
We are given that [tex]f(x,y) = 12x^2+13y^2[/tex] and D is the disc of radius one. Namely, [tex]x^2+y^2\leq 1[/tex].
First, we want to find a critical point of the function f. To do so, we want to find the values(x,y) such that
[tex] \nabla f (x,y) =0[/tex].
Recall that [tex] \nabla f (x,y) = (\frac{\partial f}{\partial x}, \frac{\partial f}{\partial y})[/tex].
So, let us calculate [tex] \nabla f (x,y)[/tex] (the detailed calculation of the derivatives is beyond the scope of the answer.
[tex]\frac{\partial f}{\partial x} = 24x[/tex]
[tex]\frac{\partial f}{\partial y} = 26y[/tex]
When equalling it to 0, we get that the critical point is (0,0), which is in our region D. Note that the function f is the sum of the square of two real numbers multiplied by some constants. Hence, the function f fulfills that [tex]f(x,y)\geq 0[/tex]. Note that f(0,0)=0, so without further analysis we know that the point (0,0) is a minimum of f over D.
If we restrict to the boundary, we have the following equation [tex] x^2+y^2=1[/tex]. Main idea is to replace the value of one of the variables n the function f, so it becomes a function of a single variable. Then, we can find the critical values by using differential calculus:
Case 1:
Let us replace y. So, we have that [tex]y^2=1-x^2[/tex]. So, [tex]f(x,y) = 12x^2+13(1-x^2) = -x^2+13[/tex].
So, we will find the derivative with respect to x and find the critical values. That is
[tex] \frac{df}{dx} = -2x=0[/tex]
Which implies that x =0. Then, [tex] y =\pm 1[/tex]. So we have the following critical points (0,1), (0,-1). Notice that for both points, the value of f is f(0,1) = f(0,-1) = 13. If we calculate the second derivative, we have that at x=0
[tex] \frac{d^2f}{dx^2} = -2<0[/tex]. By the second derivative criteria, we know that this points are local maximums of the function f.
Case 2:
Let us replace x. So, we have that [tex]x^2=1-y^2[/tex]. So, [tex]f(x,y) = 12(1-y^2)+13y^2 = y^2+12[/tex].
So, we will find the derivative with respect to y and find the critical values. That is
[tex] \frac{df}{dx} = 2y=0[/tex]
Which implies that y =0. Then, [tex] x =\pm 1[/tex]. So we have the following critical points (-1,0), (1,0). Notice that for both points, the value of f is f(1,0) = f(-1,0) = 12. If we calculate the second derivative, we have that at y=0
[tex] \frac{d^2f}{dx^2} = 2>0[/tex]. By the second derivative criteria, we know that this points are local minimums of the function f.
So, over the boundary D, the maximum value of f is 13 and the minimum value is 12. Over all D, the maximum value of f is 13 and the minimum value is 0.
Final answer:
The critical point of f(x, y) on the disk D is (0, 0). f(x) restricted to the boundary of D is f(x) = 12x^2 + 13(1 - x^2) with the domain [-1, 1]. The absolute maximum and minimum values of f(x, y) over all of D are 25 and 0, respectively.
Explanation:
To find the critical point of the function f(x, y) = 12x2 + 13y2 on the disk D: x2+y2 ≤ 1, we need to set the partial derivatives of f with respect to x and y equal to zero.
The partial derivative with respect to x is 24x, and setting it to zero gives x = 0. Similarly, the partial derivative with respect to y is 26y, which implies y = 0. Therefore, the critical point is (0, 0).
On the boundary of D, where x2+y2 = 1, we can solve for y2 as y2 = 1 - x2. Substituting into f, we get a function of a single variable x: f(x) = 12x2 + 13(1 - x2) with the closed interval domain [-1, 1].
The maximum value on the boundary occurs at x = ±1, giving a maximum of f(±1) = 25. The minimum on the boundary is at x = 0, which gives f(0) = 13.
Across the entire disk D, the absolute minimum is at the critical point (0,0), with f(0, 0) = 0, and the absolute maximum is the same as the boundary maximum, f(x) = 25.