Answer:
R = {(10,2), (10,4), (10,6), (10,8), (2,10), (2,2), (2,4), (2,6), (2,4), (4,10), (4,2), (4,6), (6,10), (6,2), (6,4), (6,8)}
Step-by-step explanation:
A = (10, 1, 2, 3, 4, 5, 6)
B = (10, 1, 2, 3, 4, 5, 6, 7, 8)
R is a relation and defined as
R: A→B
R: The greatest common divisor of a and b is 2
To find the elements of R we need to find pair from A and B respectively.
R = {(10,2), (10,4), (10,6), (10,8), (2,10), (2,2), (2,4), (2,6), (2,4), (4,10), (4,2), (4,6), (6,10), (6,2), (6,4), (6,8)}
Now, these pairs were found by picking one element from A and checking with elements of B if they have a gcd of 2. If yes, they fall under the defined relation.
The Eco Pulse survey from the marketing communications firm Shelton Group asked individuals to indicate things they do that make them feel guilty (Los Angeles Times, August 15, 2012). Based on the survey results, there is a .39 probability that a randomly selected person will feel guilty about wasting food and a .27 probability that a randomly selected person will feel guilty about leaving lights on when not in a room. Moreover, there is a .12 probability that a randomly selected person will feel guilty for both of these reasons. a. What is the probability that a randomly selected person will feel guilty for either wasting food or leaving lights on when not in a room (to 2 decimals)? b. What is the probability that a randomly selected person will not feel guilty for either of these reasons (to 2 decimals)?
Answer:
a) There is a probability of 42% that the person will feel guilty for only one of those things.
b)There is a probability of 46% that a randomly selected person will not feel guilty for either of these reasons
Step-by-step explanation:
This probability problem can be solved by building a Venn like diagram for each probability.
I say that we have two sets:
-Set A, for those people that will feel guilty about wasting food.
-Set B, for those people that will feel guilty about leaving lights on when not in a room.
The most important information is that there is a .12 probability that a randomly selected person will feel guilty for both of these reasons. It means that [tex]P(A \cap B) = .12.[/tex]
The problem also states that there is a .39 probability that a randomly selected person will feel guilty about wasting food. It means that P(A) = 0.39. The probability of a person feeling guilty for only wasting food is PO(A) = .39-.12 = .27.
Also, there is a .27 probability that a randomly selected person will feel guilty about leaving lights on when not in a room. So, the probability of a person feeling guilty for only leaving the lights on is PO(B) = 0.27-0.12 = 0.15.
a) What is the probability that a randomly selected person will feel guilty for either wasting food or leaving lights on when not in a room?
This is the probability that the person feels guilt for only one of those things, so:
P = PO(A) + PO(B) = 0.27 + 0.15 = 0.42 = 42%
b) What is the probability that a randomly selected person will not feel guilty for either of these reasons
The sum of all the probabilities is always 1. In this problem, we have the following probabilies
- The person will not feel guilty for either of these reasons: P
- The person will feel guilty for only one of those things: PO(A) + PO(B) = 0.42
- The person will feel guilty for both reasons: PB = 0.12
So
`P + 0.42 + 0.12 = 1
P = 1-0.54
P = 0.46
There is a probability of 46% that a randomly selected person will not feel guilty for either of these reasons
how many numbers are there that have distinct digits and are greater than 4500? how many 5 digit odd numbers are there that have distinct digits?how many 5 digit even number are there that have distinct digits?
Answer:
4-digit numbers with distinct digits and greater than 4500: 2800 numbers
5-digit numbers with distinct digits: 27216 numbers.
Step-by-step explanation:
If we represent a 4 number digit by ABCD, we have 9 posibilities for A (1,2,3,4,5,6,7,8 and 9, all but 0).
If every digit has to be different, we have 9 posibilities for B: ten digits (0,1,2,3,4,5,6,7,8 and 9 minus the one already used in A).
Int he same way, we have 8 posibilities for C and 7 for D.
Considering all 4-digits numbers, we have 9*9*8*7 = 4536 numers with distinct digits.
To know how many of these numbers are greater than 4500, we can substracte first the numbers that are smaller than 4000: A can take 3 digits (1,2 and 3) and B, C and D the same as before.
3*9*8*7 = 1512 numbers smaller than 4000
Then we can substrat the ones that are between 4000 and 4500
1*4*8*7 = 224 numbers between 4000 and 4500
So, if we substract from the total the numbers that are smaller than 4500 we have the results:
4-digit numbers with distinct digits greater than 4500 = 4536-(1512+224) = 2800
For 5-digit numbers, we can call the number ABCDE.
For A we have 9 digits possible (all but 0).
For B, we also have 9 posibilities (all digits but the one used in A).
For C, we have 8 digits (all 10 but the ones used in A and B).
For D, we have 7 digits.
For E, we have 6 digits.
Multiplying the possible combinations, we have:
9*9*8*7*6 = 27,216 5-digit numbers with distinct digits.
Two thousand five hundred twenty numbers have distinct digits and are more significant than 4500. There are 15120 5-digit odd numbers and 15120 5-digit even numbers with different numerals.
Explanation:In Mathematics, to calculate how many numbers have distinct digits and are more significant than 4500, consider that any number greater than 4500 and less than 10000 is a 4-digit number. The thousands place can be filled by any number from 5 to 9, giving five options. Any ten digits can fill the hundreds place minus the one used in the thousands, giving nine options. The tens place, similarly, has eight votes. Similarly, the one's site has seven options, as the digit in that place cannot duplicate any of the prior digits. So, the answer is 5*9*8*7 = 2520 distinct numbers.
For the second part of your query, the 5-digit odd numbers with non-repeating digits, the tens place must be filled by five possible odd digits (1, 3, 5, 7, 9), and the first place can be filled by any number from 1 to 9, giving nine options. The other three areas have 8, 7, and 6 votes, respectively, leading to 9*8*7*6*5 = 15120 distinct numbers.
As for the 5-digit even numbers with non-repeating digits, the tens place can be filled by five possible even digits (0, 2, 4, 6, 8), and the first place can be filled by any number from 1 to 9, giving nine options. The other three areas have 8, 7, and 6 votes, respectively. So there are 9*8*7*6*5 = 15120 five-digit, distinct digit, even numbers.
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If we are performing a two-tailed test of whether mu = 100, the probability of detecting a shift of the mean to 105 will be ________ the probability of detecting a shift of the mean to 110.
Answer:
less than
Step-by-step explanation:
When we are performing a two-tailed test of whether μ = 100, the probability of detecting a shift of the mean to 105 will be ___Less__than___ the probability of detecting a shift of the mean to 110.
this means that
mean of 105 < mean of 110.
You have an outdoor swimming pool that is 5.0 m wide and 12.0 m long. If weekly evaporation is 2.35 in, how many gallons of water must be added to the pool each week (if it doesn't rain)?
Answer:
Water should be greater than 946.11 gallons per week to prevent the pool from drying.
Step-by-step explanation:
You have an outdoor swimming pool that is 5.0 m wide and 12.0 m long.
The height of water evaporated is = 2.35 inch
We will convert this to m.
1 inch = 0.0254 meter
So, 2.35 inch = [tex]2.35\times0.0254=0.05969[/tex] meter
Now, volume of water in the pool = [tex]5\times12\times0.05969=3.5814[/tex] cubic meter per week.
1 cubic meter = 264.172 gallons
So, 3.5814 cubic meter = [tex]3.5814\times264.172=946.11[/tex]gallons
Hence, the volume of the water that should be poured in the swimming pool should be greater than 946.11 gallons per week to prevent the pool from drying.
Use z scores to compare the given values.
Based on sample data, newborn males have weights with a mean of 3232.9 g and a standard deviation of 714.6 g. Newborn females have weights with a mean of 3094.9 g and a standard deviation of 586.3 g. Who has the weight that is more extreme relative to the group from which they came: a male who weighs 1600 g or a female who weighs 1600 g?
Since the z score for the male is z =__ and the z score for the female is z=__, the
female or male(which one) has the weight that is more extreme.
Answer:
Female
Step-by-step explanation:
The formula for finding z-score is;
z=(x-μ)/δ
where x= weight given=1600g
μ=mean weight given
δ=standard deviation given
For male
x=1600g
μ=3232.9g
δ=714.6g
z=(1600-3232.9)/714.6
z= -2.3
The weight for the male is 2.3 standard deviations below the mean
For female
x=1600g
μ=3094.9g
δ=586.3g
z=(1600-3094.9)/586.3
z= -2.5
The weight for the female is 2.5 standard deviations below the mean
⇒The female has the weight that is more extreme
Answer:
z= 1
Step-by-step explanation:
got it on egde
A certain test preparation course is designed to help students improve their scores on the LSAT exam. A mock exam is given at the beginning and end of the course to determine the effectiveness of the course. The following measurements are the net change in 6 students' scores on the exam after completing the course:23,18,23,12,13,23Using these data, construct a 80% confidence interval for the average net change in a student's score after completing the course. Assume the population is approximately normal.
We are 80% confident that the average net change in a student's score after completing the course lies between 7.88 and 21.12 points.
Explanation:Here's how to construct the 80% confidence interval:
1. Calculate the sample mean and standard deviation:
Sample mean (xbar) = (23 + 18 + 23 + 12 + 13 + 23) / 6 = 18.33 points
Sample standard deviation (s) = √[(23 - 18.33)^2 + (18 - 18.33)^2 + ... + (23 - 18.33)^2] / (6 - 1) ≈ 5.38 points
2. Find the critical value for a 80% confidence interval:
For a two-tailed 80% confidence interval with 5 degrees of freedom (n-1), the critical value from the t-distribution table is approximately 1.942.
3. Calculate the margin of error:
Margin of error (ME) = critical value * standard deviation / √n
ME = 1.942 * 5.38 / √6 ≈ 4.49 points
4. Construct the confidence interval:
Lower limit = sample mean - margin of error = 18.33 - 4.49 ≈ 13.84 points
Upper limit = sample mean + margin of error = 18.33 + 4.49 ≈ 22.82 points
Therefore, we can be 80% confident that the true average net change in a student's score after completing the course falls within the range of 13.84 to 22.82 points, or rounded to the nearest whole number, 7.88 to 21.12 points.
Note: This calculation assumes the population is approximately normal. If there is reason to believe the population is not normal, a different method like bootstrapping may be more appropriate.
Find the marginal and average revenue functions associated with the demand function P= -0.3Q + 221
Answer:
Marginal revenue = R'(Q) = -0.6 Q + 221
Average revenue = -0.3 Q + 221
Step-by-step explanation:
As per the question,
Functions associated with the demand function P= -0.3 Q + 221, where Q is the demand.
Now,
As we know that the,
Marginal revenue is the derivative of the revenue function, R(x), which is equals the number of items sold,
Therefore,
R(Q) = Q × ( -0.3Q + 221) = -0.3 Q² + 221 Q
∴ Marginal revenue = R'(Q) = -0.6 Q + 221
Now,
Average revenue (AR) is defined as the ratio of the total revenue by the number of units sold that is revenue per unit of output sold.
[tex]Average\ revenue\ = \frac{Total\ revenue}{number\ of\ units\ sold}[/tex]
Where Total Revenue (TR) equals quantity of output multiplied by price per unit.
TR = Price (P) × Total output (Q) = (-0.3Q + 221) × Q = -0.3 Q² + 221 Q
[tex]Average\ revenue\ = \frac{TR}{Q}[/tex]
[tex]Average\ revenue\ = \frac{-0.3Q^{2}+221Q}{Q}[/tex]
∴ Average revenue = -0.3Q + 221
Prove that the each element of a group G has a unique inverse. That is, if a, b, W E G satisfy then b
Step-by-step explanation:
Say [tex]a[/tex] is an element of [tex]G[/tex] which might have more than 1 inverse. Let's call them [tex]b[/tex], and [tex]c[/tex]. So that [tex]a[/tex] has apparently two inverses, [tex]b[/tex] and [tex]c[/tex].
This means that [tex]a*b = e[/tex] and that [tex]a*c=e[/tex](where [tex]e[/tex] is the identity element of the group, and * is the operation of the group)
But so we could merge those two equations into a single one, getting
[tex]a*b=a*c[/tex]
And operating both sides by b by the left, we'd get:
[tex]b*(a*b)=b*(a*c)[/tex]
Now, remember the operation on any group is associative, meaning we can rearrange the parenthesis to our liking, gettting then:
[tex](b*a)*b=(b*a)*c[/tex]
And since b is the inverse of a, [tex]b*a=e[/tex], and so:
[tex](e)*b=(e)*c[/tex]
[tex]b=c[/tex] (since e is the identity of the group)
So turns out that b and c, which we thought might be two different inverses of a, HAVE to be the same element. Therefore every element of a group has a unique inverse.
In American football, the playing field is 53.33 yards (yd) wide by 120 yards (yd) long. For a special game, the field staff want to paint the playing field orange. Of course, they will use biodegradable paint available for purchase in 25-gallon (gal) containers. If the paint is applied in a thickness of 1.2 millimeters (mm) in a uniform layer, how many containers of paint will they need to purchase?
Answer:
214
Step-by-step explanation:
The playing field is 53.33 yards wide and 120 yards long you would need to find the area so multiply 53.33 by 120 yards. That equals 6399.6 , the thickness they are applying is 1.2 millimeters. You would divide the area, 6399.6 by 1.2 which would equal 5333. Divide that by 25 gallons and it equals 213.32, you would need to purchase 214, rounded.
Answer:
The number of containers to purchase is [tex]N_V= 67.85[/tex]
Step-by-step explanation:
From the question we are told that
The playing field width is [tex]w_f = 53.33 \ yard = 53.33*0.9144 = 48.76m[/tex]
The playing field length is [tex]l_f = 120 \ yards = 120 * 0.9144 = 109.728m[/tex]
The volume of one container is [tex]V= 25 \ gallon = 25 * 0.00378541 = 0.094625m^3[/tex]
The thickness of the painting is [tex]t = 1.2 \ mm = 1.2 * 0.001 = 0.0012m[/tex]
The area of the playing field is [tex]A = 48.76 * 109.728[/tex]
[tex]=5350.337m^2[/tex]
The number of container of paint needed [tex]N_V[/tex] [tex]= \frac{area \ of \ playing \ field(A) * thickness \ of \ paint \ application(t) }{volume\ single \ container(V)}[/tex]
=> [tex]N_V = \frac{5350.337 * 0.0012}{0.094625}[/tex]
[tex]N_V= 67.85[/tex]
The Cabernet Cafe sold eight times as much Cabernet Sauvignon as Zinfandel. How many bottles of Cabernet Sauvignon were sold if the cafe sold 5 bottles of Zinfindel? This class is Food and Beverage Cost
Answer:
40 bottles
Step-by-step explanation:
In the problem it is said that there were sold eigth times as much Cabernet Sauvignon as Zinfandel, this menas that if for example the cafe sold 1 bottle of zinfandel it would be 8 bottles of Cabernet, so we have to multiply the number of bottles sold fon Zinfandel by 8, as there were 5 bottles of sauvingon sold we have to multiply
8x5=40 bottles.
The Cabernet Cafe sold 40 bottles of Cabernet Sauvignon, given that it sold 8 times as many as the 5 bottles of Zinfandel.
Explanation:The question involves a simple proportion calculation, typical of mathematics problems. Given the information that the Cabernet Cafe sold eight times as much Cabernet Sauvignon as Zinfandel, we can set up a ratio and solve for the unknown. In this case, we know that the number of Cabernet Sauvignon sold is 8 times the amount of Zinfandel sold. If the cafe sold 5 bottles of Zinfandel, using the said proportion, we multiply 5 (number of Zinfandel bottles sold) by 8 (the given ratio) to find the number of Cabernet Sauvignon bottles sold - which is 40 bottles.
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1) Solve the word problem for the portion, rate, or base.
A quality control process finds 46.8 defects for every 7,800 units of production. What percent of the production is defective?
2) Solve the word problem for the portion, rate, or base.
A medical insurance policy requires Ana to pay the first $100 of her hospital expense. The insurance company will then pay 90% of the remaining expense. Ana is expecting a short surgical stay in the hospital, for which she estimates the total bill to be about $4,800.
How much (in $) of the total bill will Ana owe?
Answer:
1) 0.6% of the production is defective.
2) Ana will owe $570.
Step-by-step explanation:
Both questions here are percentage problems
Percentage problems can be explained as a rule of three problem
In a rule of three problem, the first step is identifying the measures and how they are related, if their relationship is direct of inverse.
When the relationship between the measures is direct, as the value of one measure increases, the value of the other measure is going to increase too. In this case, the rule of three is a cross multiplication.
When the relationship between the measures is inverse, as the value of one measure increases, the value of the other measure will decrease. In this case, the rule of three is a line multiplication.
Percentage problems have a direct relationship between the measures.
1) The problem states that a quality control process finds 46.8 defects for every 7,800 units of production. And asks what percent of the production is defective? We have to answers how many defects are there for 100 units of production. So:
46.8 defects - 7,800 units
x defects - 100 units
7,800x = 4680
[tex]x = \frac{4680}{7800}[/tex]
x = 0.6
0.6% of the production is defective.
2) The problem states that the medical insurance policy requires Ana to pay the first $100 of her hospital expense. The insurance company will then pay 90% of the remaining expense. The total bill is expected to be about $4,800.
Ana has to pay:
P = P1 + P2
-P1 :The first $100
-P2: 10% of the remaining expense. The remaining expense is $4,800-$100 = $4,700. Ana has to pay 10% of this. So
4700 - 100%
P2 - 10%
100P2 = 47000
[tex]P2 = \frac{47000}{100}[/tex]
P2 = $470
Ana will owe P = P1 + P2 = $100 + $470 = $570.
To find the percentage of defective units in the production, set up a ratio and calculate the percentage. Approximate 0.6% of the production is defective. To find how much Ana will owe of the total bill, subtract $100 from the total bill and calculate 10% of the remaining expense. Ana will owe $470 of the total bill.
Explanation:To find the percent of the production that is defective, we need to find the ratio of the number of defective units to the total number of units produced. In this case, we have 46.8 defects for every 7,800 units of production. So, we can set up the ratio as:
Defective Units / Total Units = 46.8 / 7,800
To find the percentage, we can multiply the ratio by 100:
Percentage of Defective Units = (46.8 / 7,800) * 100
Now, calculate the value of the ratio and simplify to find the percentage:
Percentage of Defective Units = 0.006 * 100 = 0.6%
Therefore, approximately 0.6% of the production is defective.
For the second question, to find how much of the total bill Ana will owe, we need to calculate the 10% of the remaining expense after she pays the first $100. First, subtract $100 from the total bill:
Remaining Expense = $4,800 - $100 = $4,700
Next, calculate 10% of the remaining expense:
Amount Ana will owe = 10% of $4,700 = ($4,700 * 10) / 100 = $470
Therefore, Ana will owe $470 of the total bill.
Population mean = 80
standard deviation = 20
sample = 60
What is the probability that the sample mean will be between84
and 88?
Answer:
The answer is : 0.0597
Step-by-step explanation:
Population mean = μ = 80
Standard deviation = σ = 20
Sample = N = 60
σ_mean = σ/√N
= (20)/√(60) = 2.582
Now we will find z1 and z2.
z1 = {(84) - μ}/σ_mean = [tex]{(84)-(80)}/(2.582)= 1.549[/tex]
z2 = {(88) - μ}/σ_mean = [tex]{(88)-(80)}/(2.582)= 3.098[/tex]
Now probability that the sample mean will be between 84 and 88 is given by:
Prob{ (1.549) ≤ Z≤ (3.098) } = (0.9990) -(0.9393) = 0.0597
8 + 12 = 2x
HELPPPP GOOD ANSWERS
Answer:
x = 10
Step-by-step explanation:
8 + 12 = 2x
8 + 12 = 20
20 = 2x
---- -----
2 2
10 = x
x = 10
Hey!
----------------------------------------------------
Solution:
8 + 12 = 2x
~Divide 2 to both sides
20/2 = 2x/2
~Simplify
10 = x
----------------------------------------------------
Answer:
x = 10
----------------------------------------------------
Hope This Helped! Good Luck!
Kim uses two ropes to help support a newly planted tree. The ropes are placed on both sides of the tree and pinned to the ground, as shown in the diagram above. The length of each rope from the tree to the ground is 14 feet. The distance from the base of the tree to where it is pinned to the ground is 8 feet.
Enter the acute angle each rope makes with the ground, rounded to the nearest degree.
Answer:
Angle = 35°
Step-by-step explanation:
The figure would form a triangle and the tree will divide it into two 90 degree triangles.
Height of the triangles will be 8 feet which is opposite to the angle you have to find.
Hypotenuse of both 90 degree triangles will be 14 feet.
To find the angle, use the sin formula
sin (angle) = opposite/hypotenuse
angle = ?
opposite = 8
hypotenuse = 14
sin (angle) = 8/14
angle = sin inverse (0.57)
angle = 34.8° rounded off to 35°
The angle will be the same for both triangles.
Therefore, the acute angle each triangle makes with the ground is 35°.
For a certain event, 817 tickets were sold, for a total of $1919. If students paid $2 per ticket and nonstudents paid $3 per ticket, how many student tickets were sold?
Answer: 532
Step-by-step explanation:
Let x be the number of students and y be the number of non-students.
Then, by considering the given information, we have
[tex]x+y=817-----(1)\\\\2x+3y=1919-----------(2)[/tex]
Multiply 2 on the both sides of equation (1), we get
[tex]2x+2y=1634--------(3)[/tex]
Subtract (3) from (2), we get
[tex]y=285[/tex]
Put the value of y in (1), we get
[tex]x+285=817\\\\\Rightarrow\ x=817-285=532[/tex]
Hence, 532 student tickets were sold .
Exercise 4.x2 Make a reasonable conjecture about the nth term in the sequence. 1 3 16 125 1296...
Answer:
The nth term in the sequence is given by the equation:
[tex]n_{th} =(n+1)^{n-1}[/tex]
Step-by-step explanation:
Arranging a table for n and nth:
[tex]\left[\begin{array}{ccc}n&nth\\1&1\\2&3\\2&16\\4&125\\5&1296\end{array}\right][/tex]
It is easier to notice that 16 and 125 result from the second power of 4 and the third of 5, respectively, which are one number below their respective position. That is why we can deduce that the base of the power is n+1.
For n=2, the base n+1 results in 3, which matches the nth term for n=2. Since 3 is the result of 3 to the power of 1, 16 is 4 to the power of 2, and 125 is 5 to the power of 3, all the powers are one number behind n, so the power is given by n-1, giving the equation:
[tex]n_{th} =(n+1)^{n-1}[/tex]
pls tell me the
domain
range
and if its a function or not thx
Answer:
both domain and range are {-4, -3, -2, -1, 0, 1, 2, 3, 4}
it IS a function
Step-by-step explanation:
The domain is the list of x-values of the plotted points. It is all the integers from -4 to +4, inclusive.
The range is the list of y-values of the plotted points. It is all the integers from -4 to +4, inclusive. (domain and range are the same for this function)
No x-value has more than one y-value associated with it, so this relation IS A FUNCTION.
Show that the area of a right triangle of sides 5, 12 and 13 cannot be a square
Answer and Explanation:
Given : Sides of right triangle 5,12 and 13.
To find : Show that the area of a right triangle of sides 5, 12 and 13 cannot be a square ?
Solution :
If 5,12 and 13 are sides of a right angle triangle then
13 is the hypotenuse as it is largest side.
then we take perpendicular as 12 and base as 5.
The area of the right angle triangle is
[tex]A=\frac{1}{2}\times b\times h[/tex]
Here, h=12 and b=5
[tex]A=\frac{1}{2}\times 5\times 12[/tex]
[tex]A=5\times 6[/tex]
[tex]A=30[/tex]
The area of the right angle triangle is 30 units.
30 is not a perfect square as [tex]30=2\times 3\times 5[/tex]
There is no square pair formed.
4. Find the center and the radius of the circle which circumscribes the triangle with vertices ai, a, a3. Express the result in symmetric form.
[tex]\left[\begin{array}{ccc}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{array}\right]=\left[\begin{array}{ccc}-a_{1}^{2}-b_{1}^{2}\\-a_{2}^{2}-b_{2}^{2}\\-a_{3}^{2}-b_{3}^{2}\end{array}\right][/tex]
Step-by-step explanation:In the question,
We have to find out the circumcentre of the circle passing through the triangle with the vertices (a₁, b₁), (a₂, b₂) and (a₃, c₃).
So,
The circle is passing through these points the equation of the circle is given by,
[tex]x^{2}+y^{2}+ax+by+c=0[/tex]
On putting the points in the circle we get,
[tex]x^{2}+y^{2}+ax+by+c=0\\(a_{1})^{2}+(b_{1})^{2}+a(a_{1})+b(b_{1})+c=0\\and,\\(a_{2})^{2}+(b_{2})^{2}+a(a_{2})+b(b_{2})+c=0\\and,\\(a_{3})^{3}+(b_{3})^{3}+a(a_{3})+b(b_{3})+c=0\\[/tex]
So,
[tex](a_{1})^{2}+(b_{1})^{2}+a(a_{1})+b(b_{1})+c=0\\a(a_{1})+b(b_{1})+c=-(a_{1})^{2}-(b_{1})^{2}\,.........(1)\\and,\\a(a_{2})+b(b_{2})+c=-(a_{2})^{2}-(b_{2})^{2}\,.........(2)\\and,\\a(a_{3})+b(b_{3})+c=-(a_{3})^{3}-(b_{3})^{3}\,.........(3)\\[/tex]
On solving these equation using, Matrix method we can get the required equation of the circle,
[tex]\left[\begin{array}{ccc}a_{1}&b_{1}&c_{1}\\a_{2}&b_{2}&c_{2}\\a_{3}&b_{3}&c_{3}\end{array}\right]=\left[\begin{array}{ccc}-a_{1}^{2}-b_{1}^{2}\\-a_{2}^{2}-b_{2}^{2}\\-a_{3}^{2}-b_{3}^{2}\end{array}\right][/tex]
This is the required answer.
The center of the circumscribed circle of the symmetric isosceles triangle is at the origin, and the radius is equal to the length of the triangle's equal sides, denoted as r.
Explanation:To find the center and the radius of the circle which circumscribes the triangle with vertices at ai, a, and a3, we must first understand the nature of the triangle. Given that the triangle is described to be symmetric with equal sides AB = BC = r, it is an isosceles triangle. The perpendicular bisector of the base a will pass through the midpoint of the base and the opposite vertex, given it is symmetric about this bisector. This will also be the diameter of the circumscribed circle. Consequently, as the triangle is isosceles and symmetric, we can use the properties of similar isosceles triangles to solve for the center and radius of the circumscribed circle.
Since the base a will also be the diameter of the circumscribed circle, and we know from geometry that the diameter is twice the radius (a = 2r), the radius of the circumscribing circle is r. The center of this circle is at the midpoint of the base a in the given symmetric form. Therefore, the center of the circle is at the origin due to the symmetric property and the radius remains r.
Can someone help me solve this Statistics homeworl? Thanks!
Answer:
b. 5
Step-by-step explanation:
In a stem and leaf plot, each observation is split into a stem (the first digit or digits) and a leaf (usually the last digit).
In this example, the entries in the 20s, 30s, and 40s are:
20, 20, 24, 25, 26, 35, 36, 43, 43
We omit 20, 20, 43, and 43 because they are not between 20 and 40.
That leaves the five observations
24, 25, 26, 35, 36
A. One day Annie weighed 24 ounces more than Benjie, and Benjie weighed 3 1/4 pounds less than Carmen. How did Annie’s and Carmen’s weights compare on that day?
B. Why can’t you tell how much each person weighed?
Draw diagrams to support your answer.
A.
Let Annie's weight be = a
Let Benjie's weighs = b
Let Carmen's weight be = c
One day Annie weighed 24 ounces more than Benjie, equation forms:
[tex]a=b+24[/tex] ......(1)
Benjie weighed 3 1/4 pounds less than Carmen.
In ounces:
1 pound = 16 ounces
[tex]\frac{13}{4}[/tex] pounds = [tex]\frac{13}{4}\times16=52[/tex] ounces
[tex]b=c-52[/tex] or
[tex]c=b+52[/tex] ......(2)
Now adding (1) and (2), we get
a+b=b+24+c-52
=> [tex]a=c-28[/tex]
This gives Annie weighs 28 ounces less than Carmen.
B.
We cannot know anyone's actual weight, as we only know their relative weights.
Which function r or s is the inverse function for function q
And could you also help on my second picture
Answer:
In the first picture, correct answer is b: the inverse of function q is function r, because they are symmetrical about the line y = x.
In the second picture, correct answer is b: {y| 0<= y < 8}.
Step-by-step explanation:
The inverse function of f ( f⁻¹(x) ) must satisfy that: f(f⁻¹(x)) = f⁻¹(f(x)) = x; it returns every point x, transformed under function f, to its original place. In the graph, this property translates in the following statement: the inverse function of a function f is the reflection over the identity function (y = x).
In the graph shown in the question, the blue graph (r), is the one that corresponds to the reflection under the identity function. Therefore, the correct answer is b.
Regarding the second picture, first we need to understand what the range means. The range of a function corresponds to the set of all resulting values of the dependent variable. Since the values taken by the dependent variable span from 0 to 8 (including the 0 but not including the 8), then the answer is b: {y| 0<= y < 8}.
Consider the following. x'' + 2x' + x = 0, x(0) = 8, x'(0) = −8; family of solutions x = C1e−t + C2te−t Show that the family of solutions satisfies the equation for all values of the constants. (Enter your answers in terms of t.)
Answer:
Everything is verified in the step-by-step explanation.
Step-by-step explanation:
We have the following differential equation:
[tex]x'' + 2x' + x = 0[/tex]
This differential equation has the following characteristic polynomial:
[tex]r^{2} + 2r + 1 = 0[/tex]
This polynomial has two repeated roots of [tex]r = -1[/tex].
Since the roots are repeated, our solution has the following format:
[tex]x(t) = c_{1}e^{-t} + c_{2}te^{-t}[/tex]
This shows that the family of solutions satisfies the equation for all values of the constants. The values of the constants depends on the initial conditions.
Lets solve the system with the initial conditions given in the exercise.
[tex]x(0) = 8[/tex]
[tex]c_{1}e^{0} + c_{2}(0)e^{0} = 8[/tex]
[tex]c_{1} = 8[/tex]
--------------------
[tex]x'(0) = 8[/tex]
[tex]x(t) = c_{1}e^{-t} + c_{2}te^{-t}[/tex]
[tex]x'(t) = -c_{1}e^{-t} + c_{2}e^{-t} - c_{2}te^{-t}[/tex]
[tex]-c_{1}e^{0} + c_{2}e^{0} - c_{2}(0)e^{0} = 8[/tex]
[tex]-c_{1} + c_{2} = 8[/tex]
[tex]c_{2} = 8 + c_{1}[/tex]
[tex]c_{2} = 8 + 8[/tex]
[tex]c_{2} = 16[/tex]
With these initial conditions, we have the following solution
[tex]x(t) = 8e^{-t} + 16te^{-t}[/tex]
If AA and BB are countable sets, then so is A∪BA∪B.
Answer with Step-by-step explanation:
We are given that A and B are two countable sets
We have to show that if A and B are countable then [tex]A\cup B[/tex] is countable.
Countable means finite set or countably infinite.
Case 1: If A and B are two finite sets
Suppose A={1} and B={2}
[tex]A\cup B[/tex]={1,2}=Finite=Countable
Hence, [tex]A\cup B[/tex] is countable.
Case 2: If A finite and B is countably infinite
Suppose, A={1,2,3}
B=N={1,2,3,...}
Then, [tex]A\cup B[/tex]={1,2,3,....}=N
Hence,[tex]A\cup B[/tex] is countable.
Case 3:If A is countably infinite and B is finite set.
Suppose , A=Z={..,-2,-1,0,1,2,....}
B={-2,-3}
[tex]A\cup B[/tex]=Z=Countable
Hence, [tex]A\cup B[/tex] countable.
Case 4:If A and B are both countably infinite sets.
Suppose A=N and B=Z
Then,[tex]A\cup B[/tex]=[tex]N\cup Z[/tex]=Z
Hence,[tex]A\cup B[/tex] is countable.
Therefore, if A and B are countable sets, then [tex]A\cup B[/tex] is also countable.
Answer:
To remedy confusions like yours and to avoid the needless case analyses, I prefer to define X to be countable if there is a surjection from N to X.
This definition is equivalent to a few of the many definitions of countability, so we are not losing any generality.
It is a matter of convention whether we allow finite sets to be countable or not (though, amusingly, finite sets are the only ones whose elements you could ever finish counting).
So, if A and B be countable, let f:N→A and g:N→B be surjections. Then the two sequences (f(n):n⩾1)=(f(1),f(2),f(3),…) and (g(n):n⩾1)=(g(1),g(2),g(3),…) eventually cover all of A and B, respectively; we can interleave them to create a sequence that will surely cover A∪B:
(h(n):n⩾1):=(f(1),g(1),f(2),g(2),f(3),g(3),…).
An explicit formula for h is h(n)=f((n+1)/2) if n is odd, and h(n)=g(n/2) if n is even.
Hope it helps uh mate...✌
An automobile insurance company divides customers into three categories, good risks, medium risks, and poor risks. Assume that 78% of the customers are good risks, 20% are medium risks, and 2% are poor risks. Assume that during the course of a year, a good risk customer has probability 0.005 of filing an accident claim, a medium risk customer has probability 0.01, and a poor risk customer has probability 0.025. A customer is chosen at random. What is the probability that the customer has filed a claim? Round the answer to four decimal places.
Answer:
There is a 0.64% probability that the costumer has filed a claim.
Step-by-step explanation:
Probability:
What you want to happen is the desired outcome.
Everything that can happen iis the total outcomes.
The probability is the division of the number of possible outcomes by the number of total outcomes.
Our problem has these following probabilities:
-78% that a costumer is a good risk.
-20% that a costumer is a medium risk.
-2% that a costumer is a poor risk.
Also:
- 0.5% of a good risk costumer filling an accident claim
- 1% of a medium risk costumer filling an accident claim.
-2.5% of a poor risk costumer filling an accident claim.
The question is:
What is the probability that the customer has filed a claim?
[tex]P = P_[1} + P_{2} + P_{3}[/tex], in which:
-[tex]P_{1}[/tex] is the probability that a good risk costumer is chosen and files a claim. This probability is: the probability of a good risk costumer being chosen multiplied by the probability that a good risk costumer files a claim. So:
[tex]P_[1} = 0.78*0.005 = 0.0039[/tex]
-[tex]P_{2}[/tex] is the probability that a medium risk costumer is chosen and files a claim. This probability is: the probability of a medium risk costumer being chosen multiplied by the probability that a medium risk costumer files a claim. So:
[tex]P_[2} = 0.20*0.01 = 0.002[/tex]
-[tex]P_{3}[/tex] is the probability that a poor risk costumer is chosen and files a claim. This probability is: the probability of a poor risk costumer being chosen multiplied by the probability that a poor risk costumer files a claim. So:
[tex]P_[3} = 0.02*0.025 = 0.0005[/tex]
[tex]P = P_[1} + P_{2} + P_{3} = 0.0039 + 0.002 + 0.0005 = 0.0064[/tex]
There is a 0.64% probability that the costumer has filed a claim.
The probability that a randomly selected customer has filed a claim is calculated using the total law of probability, which yields a result of 0.0064 or 0.64%, after considering the probabilities of each risk group filing a claim.
Explanation:To calculate the probability that a randomly chosen customer has filed a claim, we need to use the total law of probability. This involves multiplying the probability of a customer being in each risk group by the probability that a customer in that risk group files a claim, and then summing these products.
For good risks, this probability is 0.78 (the percentage of good risk customers) multiplied by 0.005 (the probability a good risk customer files a claim): 0.78 * 0.005 = 0.0039.
For medium risks, the calculation is 0.20 * 0.01 = 0.0020.
For poor risks, the calculation is 0.02 * 0.025 = 0.0005.
Adding these probabilities together gives us the total probability that a customer has filed a claim: 0.0039 + 0.0020 + 0.0005 = 0.0064, or 0.64% if expressed as a percentage.
Therefore, the probability that a randomly selected customer from the insurance company has filed a claim is 0.0064 or 0.64%, rounded to four decimal places.
In the cost function below, C(x) is the cost of producing x items. Find the average cost per item when the required number of items is produced C(x)=7.6x + 10,800 a 200 items b. 2000 items c. 5000 items a. What is the average cost per item when 200 items are produced?
Answer:
The average cost per item when 200 items are produced is 61.6
Step-by-step explanation:
We start with the cost formula given by:
[tex]C(x)=7.6x+10,800[/tex]
Then we compute C(x) for x=200, 2000 and 5000 as follows:
[tex]C(200)=7.6*200+10,800=12,320\\C(2000)=7.6*2000+10,800=26,000\\C(5000)=7.6*5000+10,800=48,800[/tex]
Finally, to obtain the average cost per item when 200, 2,000 and 5,000 are produced (we will denote this by Av(200), Av(2000) and Av(5000) respectively) we just need to divide C(x) by the number of items produced. Then [tex]Av(x)=\frac{C(x)}{x}[/tex].
[tex]Av(200)=\frac{C(200)}{200}=\frac{12,320}{200}= 61.6\\Av(2000)=\frac{C(2000)}{2000}=\frac{26,000}{2000}= 13\\Av(5000)=\frac{C(5000)}{5000}=\frac{48,800}{5000}= 9.76\\[/tex]
It takes Kay 20 minutes to drive to work traveiling 45 mph. Two minutes after she left home this morning, her husband, Dan, started out with her briefcase, which she had forgotten. If Dan arrived at Kay's office just as she did, how fast did he drive?
Answer:
Kay's husband drove at a speed of 50 mph
Step-by-step explanation:
This is a problem of simple motion.
First of all we must calculate how far Kay traveled to her job, and then estimate the speed with which her husband traveled later.
d=vt
v=45 mph
t= 20 minutes/60 min/hour = 0.333 h (to be consistent with the units)
d= 45mph*0.333h= 15 miles
If Kay took 20 minutes to get to work and her husband left home two minutes after her and they both arrived at the same time, it means he took 18 minutes to travel the same distance.
To calculate the speed with which Kate's husband made the tour, we will use the same initial formula and isolate the value of "V"
d=vt; so
v=[tex]\frac{d}{t}[/tex]
d= 15 miles
t= 18 minutes/60 min/hour = 0.30 h (to be consistent with the units)
v=[tex]\frac{d}{t}=\frac{15 miles}{0.3 h}=50mph[/tex]
Kay's husband drove at a speed of 50 mph
If you had carried out the algebra using variables before plugging numbers into your expressions, you would have found that (vf)α=−2qαΔVmα−−−−−−−√, where ΔV is measured in volts. To verify that this expression for (vf)α has the correct units of velocity, you need to perform some unit analysis. Begin by finding the equivalent of a volt in terms of basic SI units. What is a volt in terms of meters (m), seconds (s), kilograms (kg), and coulombs (C)? Express your answer using the symb
A volt in terms of meters, seconds, kilograms, and coulombs is
[tex]$\frac{\text{kilogram} \times \text{meter}^2}{\text{second}^2 \times \text{coulomb}}$[/tex]
We are given that;
(vf)α=−2qαΔVmα
Now,
To find the equivalent of a volt in terms of basic SI units, we can use the definition of a volt as the potential difference that causes one joule of energy to be transferred per coulomb of charge.
A joule is the unit of energy, which is defined as the work done by a force of one newton over a distance of one meter.
A newton is the unit of force, which is defined as the product of mass and acceleration. Therefore, we can write:
[tex]$1 \text{ volt} = \frac{1 \text{ joule}}{1 \text{ coulomb}} = \frac{1 \text{ newton} \times 1 \text{ meter}}{1 \text{ coulomb}} = \frac{1 \text{ kilogram} \times 1 \text{ meter} \times 1 \text{ meter}}{1 \text{ second}^2 \times 1 \text{ coulomb}}$[/tex]
Simplifying, we get:
[tex]$1 \text{ volt} = \frac{\text{kilogram} \times \text{meter}^2}{\text{second}^2 \times \text{coulomb}}$[/tex]
Therefore, by volt answer will be [tex]$\frac{\text{kilogram} \times \text{meter}^2}{\text{second}^2 \times \text{coulomb}}$[/tex].
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A volt is a unit of potential difference, which is measured in joules per coulomb (V). 1 volt is equal to 1 joule per coulomb (1 V = 1 J/C).
Explanation:The potential difference between two points A and B, VB - VA, is defined as the change in potential energy divided by the charge q. This potential difference is measured in joules per coulomb, which is called a volt (V). To express a volt in terms of the basic SI units of meters (m), seconds (s), kilograms (kg), and coulombs (C), we need to use the equation J = V × C, where J represents the unit of energy, the joule.
We can rewrite the equation J = V × C as V = J / C. Since 1 V is equivalent to 1 J/C, we can express the volt in terms of meters (m), seconds (s), kilograms (kg), and coulombs (C) as:
1 V = 1 J/C
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At a certain school, twenty-five percent of the students wear a watch and thirty percent wear a bracelet. Sixty percent of the students wear neither a watch nor a bracelet. (a) One of the students is chosen at random. What is the probability that this student is wearing a watch or a bracelet? (b) What is the probability that this student is wearing both a watch and a bracelet?
Answer: a) 0.40 b) 0.15
Step-by-step explanation:
Let A denotes the event that students wear a watch and B denotes the event that students wear a bracelet.
Given : P(A)=0.25 ; P(B)=0.30
[tex]P(A'\cup B')=0.60[/tex]
Since, [tex]P(A\cup B)=1-P(A'\cup B')=1-0.60=0.40[/tex]
Thus, the probability that this student is wearing a watch or a bracelet = 0.40
Also, [tex]P(A\cup B)=P(A)+P(B)-P(A\cap B)[/tex]
[tex]P(A\cap B)=P(A)+P(B)-P(A\cup B)[/tex]
[tex]P(A\cap B)=0.25+0.30-0.40\\\\\Rightarrow\ P(A\cap B)=0.15[/tex]
Thus, the probability that this student is wearing both a watch and a bracelet= 0.15
Answer:
Step-by-step explanation:
Given that at a certain school, twenty-five percent of the students wear a watch and thirty percent wear a bracelet.
A- people who wear watch = 25%
B - people who wear bracelet = 30%
(AUB)' - People who wear neither a watch nor a bracelet=60%
[tex]A \bigcap B[/tex] - People who wear both =100%-60% = 40%
a) [tex]P(AUB) = P(A)+P(B)-P(AB) = 25%+30%-40%\\= 15%[/tex]
b) the probability that this student is wearing both a watch and a bracelet
= [tex]P(A \bigcap B) = 40%[/tex]
Two angles of a triangle measure 30 and 45 degrees. If the side of the triangle opposite the 30-degree angle measures 6√2 units, what is the sum of the lengths of the two remaining sides? Express your answer as a decimal to the nearest tenth.
Answer:
28.4 units
Step-by-step explanation:
If we call the given angles C and A, then the given side is c, and the other two sides can be found from the Law of Sines.
Angle B is the remaining angle of the triangle:
180° -C -A = 180° -30° -45° = 105°
The remaining sides are ...
b = sin(B)/sin(C)·c = sin(105°)/sin(30°)·6√2 ≈ 16.4
a = sin(A)/sin(C)·c = sin(45°)/sin(30°)·6√2 = 12
Then the sum of the lengths of the remaining sides is ...
a + b = 12 + 16.4 = 28.4 . . . units
By applying the principles of trigonometry and the Pythagorean theorem, we can determine that the sum of the lengths of the two remaining sides is approximately 25.5 units.
Explanation:This question is about the application of trigonometric principles and the Pythagorean theorem. In a triangle, the sine of an angle is defined as the ratio of the side opposite that angle to the hypotenuse. Given a 30-degree angle and its opposite side of length 6√2, we can find the hypotenuse (h) using the fact that sin(30) = 1/2. So, 1/2 = 6√2 / h. Solving this, we get h = 2 * 6√2 = 12√2.
The angle 45 degrees helps determine the length of the remaining side. Using the fact that cos(45) = s/h, where s is the remaining side, we get cos(45) = s / 12√2. Solving for s, s = cos(45) * 12√2, s = √2/2 * 12√2 = 6√2.
So, the sum of the lengths of the two remaining sides is 12√2 + 6√2 = 18√2, which is approximately 25.5 when rounded to the nearest tenth.
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