Answer:
if sample of 30 bulbs has 2 defective bulbs so,
there is one defective bulb every
[tex] \frac{30}{2} = 15[/tex]
bulbs.
total defective bulbs in 450 bulbs =
[tex] \frac{450}{15} = 30[/tex]
A garden center sells a certain grass seed in 5-pound bags at $13.85 per bag, 10-pound bags at $20.43 per bag, and 25-pound bags $32.25 per bag. If a customer is to buy at least 65 pounds of the grass seed, but no more than 80 pounds, what is the least possible cost of the grass seed that the customer will buy?
A) $94.03
B) $96.75
C) $98.78
D) $102.07
E) $105.3
Answer:
B)$96.75
Step-by-step explanation:
5pound bag cost = $13.85
10pound bag cost= $20.43
25pound bag cost = $32.25
The least quantity of bag the customer can by is 65 pounds = (2*25 pound)+ 10 pound + 5pound = $98.78
But careful examination will show actually that since the most the customer can buy is 80 pound, then buying 25pound in three places give him even a lot cheaper than buying the least amount
75 pound = 3*25 pound = 32.25*3 = $96.75.
Therefore the least amount in cost the customer can buy is $96.75
The recipe makes 20 portions of potato soup. Richard follows the recipe but wants to make 4 portions. Complete the amounts of each ingredient that he needs. 120 ml oil 280 g onion 1.8kg potatoes 2.2 l milk
Answer:
Oil: 24 ml; Onion: 56 g; Potatoes: 0.36 kg; Milk: 0.44 l
Step-by-step explanation:
First, we see how much of the soup recipe he wants make.
4 portions out of 20 portions = 4/20 = 1/5 = 0.2
He wants to make 0.2 of the amount of the recipe.
Now we multiply every amount by 0.2
Oil: 0.2 * 120 ml = 24 ml
Onion: 0.2 * 280 g = 56 g
Potatoes: 0.2 * 1.8 kg = 0.36 kg
Milk: 0.2 * 2.2 l = 0.44 l
Richard needs to use a scaling factor to reduce the amounts of each ingredient to make 4 portions of potato soup, resulting in 24 ml oil, 56 g onion, 360 g potatoes, and 0.44 liters of milk. Devon will require 37.5 liters of soup for a party of 100 guests.
To scale down the recipe from 20 portions to 4 portions of potato soup, Richard needs to multiply each ingredient by the scaling factor, which is 4/20 or 1/5. Here's how to calculate the amounts needed:
Oil: 120 mlTherefore, for 4 portions, Richard needs 24 ml of oil, 56 g of onion, 360 g of potatoes, and 0.44 liters of milk.
A woman drives 169/4 miles to work each day . She stops for coffee at a shop that is 2/5 of the way to her job . How far does the woman drive before she stops for coffee?
Answer: the woman drives 16.9 miles before she stops for coffee
Step-by-step explanation:
A woman drives 169/4 miles to work each day. She stops for coffee at a shop that is 2/5 of the way to her job.
To determine how far the woman drives before she stops for coffee, we will multiply the total miles to her job each day by the fraction of the total miles that she drives before stopping at the coffee shop. It becomes
2/5 × 169/4 = 16.9 miles
Complete the square to determine the minimum or maximum value of the function defined by the expression.
−x² + 10x + 5
A. maximum value at 30
B. minimum value at 30
C. maximum value at −30
D. minimum value at −30
Please provide a full explanation, thank you!
Answer:
A. maximum value at 30
Step-by-step explanation:
−x² + 10x + 5
First, factor out the leading coefficient from the first two terms:
-1 (x² − 10x) + 5
Take half of the next coefficient, square it, then add and subtract the result.
(-10/2)² = 25
-1 (x² − 10x + 25 − 25) + 5
-1 (x² − 10x + 25) + 25 + 5
-1 (x² − 10x + 25) + 30
Factor the perfect square.
-1 (x − 5)² + 30
The equation is now in vertex form. This is a downwards parabola with a vertex at (5, 30). Since the parabola points down, the vertex is a maximum.
Two researchers select a sample for a population with a mean of 12.4 and a standard deviation of 9. Researcher A selects a sample of 30 participants. Researcher B selects a sample of 40 participants. Which sample is associated with a smaller standard error?
a. Researcher A's, because the sample size was smaller.
b. Researcher B's, because the sample size was smaller.
c. Researcher A's, because the sample size was larger.
d. Researcher B's, because the sample size was larger.
Answer: Option 'd' is correct.
Step-by-step explanation:
Since we have given that
Researcher A :
Mean = 12.4
Standard deviation = 9
sample size = 30
So, the standard error is given by
[tex]\dfrac{\sigma}{\sqrt{n}}\\\\=\dfrac{9}{\sqrt{30}}\\\\=1.643[/tex]
Researcher B:
Mean = 12.4
Standard deviation = 9
Sample size = 40
So, the standard error is given by
[tex]\dfrac{\sigma}{\sqrt{n}}\\\\=\dfrac{9}{\sqrt{40}}\\\\=1.423[/tex]
Sample B has smaller standard error than sample A because the sample size was larger than A.
Hence, Option 'd' is correct.
Suppose that Bob places a value of $10 on a movie ticket and that Lisa places a value of $7 on a movie ticket. In addition, suppose the price of a movie ticket is $5. Refer to Scenario 12-2. Suppose the government levies a tax of $1 on each movie ticket and that, as a result, the price of a movie ticket increases to $6.00. If Bob and Lisa both purchase a movie ticket, what is total consumer surplus for Bob and Lisa?
Final answer:
Bob's consumer surplus after a tax is $4, and Lisa's is $1, making the total consumer surplus for both after the tax $5.
Explanation:
Consumer surplus is the difference between the value a consumer places on a good and what they actually pay. Before the government levies a tax, Bob's consumer surplus for a movie ticket is the difference between his valuation of $10 and the market price of $5, which is $5. Lisa's consumer surplus is the difference between her valuation of $7 and the market price of $5, which is $2.
After the government implements a $1 tax on movie tickets, increasing the price to $6, Bob's consumer surplus becomes $4 ($10 - $6), and Lisa's consumer surplus is now $1 ($7 - $6). Thus, the total consumer surplus for Bob and Lisa after the tax is implemented is $5 ($4 for Bob and $1 for Lisa).
Find the y-intercept of the line on the graph. HELP PLEASE !!!!
Ethan is saving money in his piggy bank for his upcoming trip to Disney world on the first day he put in $12 and plans to add seven more dollars each day write an explicit formula that can be used to find the amount of money saved on any given day
Answer:
S=12+7D
Step-by-step explanation:
Linear relationships.
The initial amount of money Ethan has is $12. Each day, he adds up $7 to his savings. At a given day D after his initial funding, he will have added $7D, and he will have in his piggy bank
S=12+7D
For example, on the day D=30 he will have
S=12+7(30)=$222
The probability of observing the experiment result, a sample mean, for example, or something more unusual just by chance if the null hypothesis is true is the definition of____________.
a) the test statistic
b) a confidence interval.
c) a p-value
d) the alternative hypothesis
The probability of observing the experiment result, a sample mean, for example, or something more unusual just by chance if the null hypothesis is true is the definition of p value
hence option (c) is correct
Test statistic is a way to check the authenticity of null hypothesis that is considered. Test statistic is extremely important to accept and reject the null hypothesis.
The data from the experiment is fed into the equation and compares your result with the expected results of null hypothesis.
Test statistic is a number calculated from statistical test of a hypothesis.
It shows how closely our data matches with the distribution expected under null hypothesis of the distribution.
Confidence interval
Confidence interval is the range of plausible values of a random variable with a certain percentage of confidence level. Confidence level shows that how much certainty or uncertainty in test statistic considering the null hypothesis to be true. It is expressed in percentage. 98% , 95% and 90% Confidence intervals are a few examples.
p- value is the probability of a happening of a particular event is a random chance or some other event with similar probability or any rarer event considering null hypothesis to be true.
Alternate hypothesis
Alternate hypothesis is the opposite of Null hypothesis. Null hypothesis is the generally or by default accepted hypothesis which is tested by various statistical tests.
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Michelle rents a movie for a flat fee of $1.50 plus an additional $1.25 for each night she keeps the movie. Choose the cost function that represents this scenario if x equals the number of nights Michelle has the movie.
A) c(x) = 1.50 + 1.25x
B) c(x) = 1.50x + 1.25
C) c(x) = 2.75
D) c(x) = (1.50 + 1.25)x
Answer:
A) c(x)=1.50+1.25x
Step-by-step explanation:
The fixed rate (constant) is 1.50 and 1.25 (variable) depending on the number of additional nights, that is, c (x) = 1.25 (x) +1.50 =1.50+1.25x
the answer would be A
Find the inverse function of
(Show work)
f(x)=x^2-4
Answer:
The answer to your question is [tex]f(x) = \sqrt{x+ 4}[/tex]
Step-by-step explanation:
f(x) = x² - 4
Process
1.- Change f(x) for y
y = x² - 4
2.- Change "x" for "y" and "y" for "x".
x = y² - 4
3.- Make "y" the object of the equation
y² = x + 4
[tex]y = \sqrt{x+ 4}[/tex]
4.- Change "y" for f(x)
[tex]f(x) = \sqrt{x+ 4}[/tex]
A statistical procedure used to determine whether observed frequencies at each level of one categorical variable are similar to or different from frequencies expected, is called the chi-square:
Answer:
This statement is true.In statistics, the chi-square test is used to prove a specific hypothesis, accepting or rejecting the null one. In order to find enough evidence to prove the hypothesis, we compare two group of frequencies, which belongs to two different groups (like a quasi-experimental design, with a control and experimental group). The researcher have set an expected frequency, based on the hypothesis, and then he/she will observe a frequency from the data recollected.
Therefore, by comparing this two frequencies (the expected with the observed), the researcher is able to demonstrate the hypothesis.
There are statistical analyses beyond simple descriptive measures, statistical inference, and differences tests including ________, which determine whether a stable relationship exists between two variables.
A) associative analyses
B) analysis of variance analyses
C) regression analyses
D) predictive analyses
Answer:
Associative analysis
Step-by-step explanation:
Associative analysis is an approach that is used to analyses the peoples mental representation , focusing on meaning and similarities and differences across the culture.It determined that relationship that is hidden in the large data set.It determine the relationship in between two variable as well.
A physics exam consists of 9 multiple-choice questions and 6 open-ended problems in which all work must be shown. If an examinee must answer 6 of the multiple-choice questions and 4 of the open-ended problems, in how many ways can the questions and problems be chosen?
A) 1260
B) 1296
C) 261,273,600
D) 21,772,800
Answer: A) 1260
Step-by-step explanation:
We know that the number of combinations of n things taking r at a time is given by :-
[tex]^nC_r=\dfrac{n!}{(n-r)!r!}[/tex]
Given : Total multiple-choice questions = 9
Total open-ended problems=6
If an examine must answer 6 of the multiple-choice questions and 4 of the open-ended problems ,
No. of ways to answer 6 multiple-choice questions
= [tex]^9C_6=\dfrac{9!}{6!(9-6)!}=\dfrac{9\times8\times7\times6!}{6!3!}=84[/tex]
No. of ways to answer 4 open-ended problems
= [tex]^6C_4=\dfrac{6!}{4!(6-4)!}=\dfrac{6\times5\times4!}{4!2!}=15[/tex]
Then by using the Fundamental principal of counting the number of ways can the questions and problems be chosen = No. of ways to answer 6 multiple-choice questions x No. of ways to answer 4 open-ended problems
= [tex]84\times15=1260[/tex]
Hence, the correct answer is option A) 1260
To solve the problem, use the combination formula to find the number of ways to choose 6 multiple-choice questions from 9 and 4 open-ended problems from 6, then multiply the results. The answer is Option(D) 1260.
Solution to the Question
To determine the number of ways to choose 6 multiple-choice questions out of 9, and 4 open-ended problems out of 6, we can use combinations.
The number of ways to choose 6 multiple-choice questions out of 9 is given by the combination formula:
C(9, 6) = 9! / (6! * (9-6)!) = 84.
Similarly, the number of ways to choose 4 open-ended problems out of 6 is given by:
C(6, 4) = 6! / (4! * (6-4)!) = 15.
Now, multiply these two results to get the total number of ways to choose the questions:
84 * 15 = 1260.
Therefore, the answer is A) 1260.
I really need help with this, I got 13.7 yards as the altitude by using the law of cosines and I found the area of the triangle by using the formula 1/2(side)(side)cos
Answer:
The length of the altitude is 9.3 yards and
The area of the triangle Δ UVW is 139.3 yd².
Step-by-step explanation:
Given
WU = 22 yd
WV = 30 yd
∠ UWV = 25°
To Find:
Altitude, UM = ?
area of the Δ UVW = ?
Construction:
Draw UM perpendicular to WV, that is altitude UM to WV.
Solution:
In right triangle Δ UWM if we apply Sine to angle W we get
[tex]\sin W = \frac{\textrm{side opposite to angle W}}{Hypotenuse}\\ \sin W=\frac{UM}{UW} \\[/tex]
substituting the values we get
[tex]\sin 25 = \frac{UM}{22}\\0.422 = \frac{UM}{22} \\UM = 0.422\times 22\\UM = 9.284\ yd[/tex]
Therefore, the altitude from U to WV is UM = 9.3 yd.(rounded to nearest tenth)
Now for area we have formula
[tex]\textrm{area of the triangle UVW} = \frac{1}{2}\times Base\times Altitude \\\textrm{area of the triangle UVW} = \frac{1}{2}\times VW \times UM\\=\frac{1}{2}\times 30\times 9.284\\ =139.26\ yd^{2}[/tex]
The area of the triangle Δ UVW is 139.3 yd². (rounded to nearest tenth)
Of the 36 students in a certain class, 10 are in the chess club and 13 are in the bridge club. If 20 of the students are not in either club, how many of the students are in only one of the two clubs?A. 7B. 9C. 14D. 16E. 23
There are 9 students in only one of the two clubs.
Step-by-step explanation:
Since we have given that
Number of students = 36
Number of students are in chess club = 10
Number of students are in bridge club = 13
Number of students are not in either club = 20
So, Number of students in both the club is given by
[tex]Total=n(chess)+n(bridge)-n(both)+n(neither)\\\\36=10+13-n(both)+20\\\\36=43-n(both)\\\\36-43=n(both)\\\\-7=-n(both)\\\\n(both)=7[/tex]
Number of students only in chess = 10-7 =3
Number of students only in bridge = 13-7=6
Hence, there are 3+6=9 students in only one of the two clubs.
There are two cookie jars: jar 1 contains two chocolate chip cookies and three plain cookies, and jar 2 contains one chocolate chip cookie and one plain cookie. Blind- folded Fred chooses a jar at random and then a cookie at random from that jar. What is the probability of him getting a chocolate chip cookie?
Answer:
P = 0.55 or 55 %
Step-by-step explanation:
First step: Fred has probability of 0,5 when chossing jar 1 or jar 2
Second step : The probability of chossing one chocolate chp cookie in jar 1 is 3/5 and from the jar 2 is 1/2
Then the probability of Fred to get a chocolate chip cookie is
P ( get a chocolate chip cookie ) =( 0.5 * 3/5) +( 0.5* 1/2)
P = 0.3 + 0.25
P = 0.55 or 55 %
A system of inequalities is shown:
y ≤ x + 6
y ≥ 9x − 9
Which point is in the solution set of the system of inequalities shown? Explain your answer.
Question 5 options:
(3, 10) because it lies below the boundary line y=9x−9 and above the boundary line y=x+6.
(-1, 7) because it lies below the boundary line y=x+6.
(3, 2) because it lies above the boundary line y=9x−9.
(-2, 2) because it lies above the boundary line y=9x−9 and below the boundary line y=x+6.
Answer:
(-2, 2)
Step-by-step explanation:
Well first you would have to graph both inequalities.
y = mx + b
b is the y-axis intercept
m is the slope
In the first equation in the system of equations, y represents the money collected from selling sweatshirts. In the second equation, y represents the money spent to produce x sweatshirts with team logos on them for a professional sports league.
y=35x; y=-0.05(x-400)^2+9,492
What does the solution of the system represent in this context?
The solution to the system of equations represents the break-even point for the production and sale of sweatshirts, indicating the number of units that must be sold to cover production costs.
Explanation:The solution of the system of equations y=35x and y=-0.05(x-400)^2+9,492 represents the break-even point where the money collected from selling x number of sweatshirts equals the money spent to produce those sweatshirts. In this context, solving the system means finding the number of sweatshirts (x) that need to be sold at $35 each to exactly cover the production costs described by the second equation.
This involves first substituting the expression for y from the first equation into the second equation, then solving for x to find the exact point where income equals expenses. Once x is found, it can be plugged back into either equation to verify the break-even amount of money (y).
The length of the escalator is 30 feet and the distance between the floors is 12 feet. Find the distance from the base of the escalator to the point on the first floor directly below the top of the escalator.
Answer:
Distance from the base of the escalator to the point on the first floor directly below the top of the escalator = 27.5 feet
Step-by-step explanation:
Given:
Length of escalator = 30 feet
Distance between the floors = 12 feet
To find the distance from base of escalator to the point on the first floor directly below the top of the escalator we will create the figure for the situation.
From the figure we see that a triangle ABC is formed.
We see that the Δ ABC is a right triangle.
Applying Pythagorean theorem for right triangle ABC to find the missing side.
[tex]AB^2=BC^2+AC^2[/tex]
AB = 30 feet
BC = 12 feet
Plugging in values in the theorem.
[tex]30^2=12^2+AC^2[/tex]
Solving for AC.
[tex]900=144+AC^2[/tex]
Subtracting both sides by 144.
[tex]900-144=144+AC^2-144[/tex]
[tex]756=AC^2[/tex]
Taking square root both sides.
[tex]\sqrt{756}=\sqrt{AC^2}[/tex]
[tex]27.5=AC[/tex]
∴ [tex]AC=27.5[/tex] feet.
∴ Distance from the base of the escalator to the point on the first floor directly below the top of the escalator = 27.5 feet
To find the distance from the base of the escalator to the point on the first floor directly below the top of the escalator, we can use the Pythagorean theorem. The distance is approximately 27 feet.
Explanation:To find the distance from the base of the escalator to the point on the first floor directly below the top of the escalator, we can use the Pythagorean theorem. The length of the escalator represents the hypotenuse of a right triangle, and the distance between the floors represents one of the legs.
We can use the formula a^2 + b^2 = c^2, where a and b are the legs of the triangle and c is the hypotenuse. Substituting the given values, we have 12^2 + b^2 = 30^2. Simplifying the equation, we get b^2 = 30^2 - 12^2. Calculating the value, we find that b^2 = 756, which means b is equal to the square root of 756. Rounding to the nearest foot, the distance from the base of the escalator to the point on the first floor directly below the top of the escalator is approximately 27 feet.
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A truck driver drives from Chicago to Cincinnati in 14 hours. The distance traveled is 840 miles. Write the average speed as a unit rate in fraction form
Answer:
60 miles/hour
Step-by-step explanation:
840 miles divided by 14 hours
840/14=60 miles per hour
A plumber charges $45 per hour, plus a one-time fee for making a house call. The total fee for 3 hours of service is $285. Write the point-slope form of an equation to find the total fee y for any mumber if hours
Answer:
y - 285 = 45(x - 3)
Step-by-step explanation:
The given point is (hours, fee) = (3, 285), and the slope is given as 45 per hour.
The point-slope form of the equation for a line is ...
y - k = m(x - h) . . . . . . . for a slope m and a point (h, k)
Using the given values, and letting x stand for the number of hours, the equation is ...
y - 285 = 45(x - 3)
Answer:
y - 285 = 45(x - 3)
Step-by-step explanation:
What is the interest earned on $3000 at a rate of 0.04 for three years? The formula is interest equals (principal) (rate) (time) Substitute and multiply
Answer:
The interest after 3 years is $360
Explanation:
Given the principal amount (P) = $3000
Rate of interest (R) = 0.04
Time period (T) is given as 3 years
The Simple Interest is calculated by the formula;
[tex]SI = Principal \times Rate of Interest \times Time[/tex]
Substituting the values in the above formula,
[tex]SI = 3000 \times 0.04 \times 3[/tex]
SI = $360
Therefore, the interest after 3 years is $360
Answer:
$360
Step-by-step explanation:
Assume that John Smith is a salesperson employed by McCrackin Company. Smith's regular rate of pay is $36 per hour, and any hours worked in excess of 40 hours per week are paid at 1½ times the regular rate. Smith worked 42 hours for the week ended October 27. What are his total earnings for the week?
Answer: $1548
Step-by-step explanation:
We are told the normal rate of payment is $36 per hour
and with an excess of 40 hours the pay will be 1 and a half the normal rate(1.5)
And John works for 42hours
For first we know John worked for an excess of 2 hours
And calculating his pay for 40hours of the normal rate that week, we multiply $36 by 40 which will give $1440
Then the extra 2 hours, the new pay rate will be $36 multiplied by 1.5 which will give $54 per hour
And for the extra 2 hours, John will get extra $54 multiplied by 2 which will give $108
Adding both $1440 and $108, we will get $1548
Based on data from Bloodjournal.org, 10% of women 65 years of age and older have anemia, which is a deficiency of red blood cells. In tests for anemia, blood samples from 8 women 65 and older are combined. What is the probability that the combined sample tests positive for anemia? Is it likely for such a combined sample to test positive?
Answer:
the probability that the combined sample tests positive for anemia is ≈ 0,38.
Thus it is 38% likely that such a combined sample to test is positive.
Step-by-step explanation:
The combined sample tests positive if at least one of the 8 women has anemia.
Let p be the probability that a women 65 years of age and older have anemia
Then p=0.1
The probability that one of the 8 women has anemia and others does not is:
p×[tex]p^{7}[/tex] .
Since there are 8 combinations of this probability is possible, the probability that at least one of the 8 woman has anemia is:
8×p×[tex]p^{7}[/tex] =8×0.1×[tex]0.9^{7}[/tex] ≈ 0,3826
The probability that a combined sample of blood from 8 women aged 65 and older tests positive for anemia is approximately 56.95%, making it likely for the sample to test positive. This probability is calculated using the complement rule in probability.
To determine the probability that a combined sample of blood from 8 women aged 65 and older tests positive for anemia, we need to use the complement rule and properties of probability.
Given:
- Probability that one woman has anemia (success): 10% or 0.10
- Probability that one woman does not have anemia (failure): 90% or 0.90
- Number of women sampled: 8
The probability that all 8 women do not have anemia can be calculated as:
[tex](0.90)^8[/tex]
Now let's compute this:
[tex](0.90)^8[/tex] ≈ 0.4305
This means there is an approximately 43.05% probability that none of the 8 women have anemia. Therefore, the probability that at least one woman in the sample has anemia is:
1 - 0.4305 ≈ 0.5695 or 56.95%
Since the probability is greater than 50%, it's quite likely that such a combined sample will test positive for anemia.
Which fraction is equivalent to 0.65?
A) 5/13
B) 13/20
C) 19/25
D) 27/35
Anthony purchases two bags. The price of all bags is $5.20. Anthony purchases one school bag and one hand bag. Write an expression that represents the total cost,T, of the bag if s represents the number of school bags and h represents the number of hand bags
Answer:
t=5.20s
Step-by-step explanation:
I think that is the expression because it say he purchases two bags, all the bags are 5.20 so he brought two bags which are school bag and one hand bag the formula I used was y=mx+b but in these case I had to put t=5.20x. I hope this really helped you..
Answer:
Step-by-step explanation:
Anthony purchases two bags. The price of all bags is $5.20
Anthony purchases one school bag and one hand bag. It means that he purchased one handbag and one school bag for a total cost of $5.20
Let s represent the number of school bags and
Let h represent the number of hand bags.
An expression that represents the total cost,T, of the bag will be
T = s + h
Since total cost = $5.20
Then,
5.20 = s + h
Find the smallest relation containing the relation {(1, 2), (1, 4), (3, 3), (4, 1)} that is:
a. reflexive and transitive
b. symmetric and transitive
c. reflexive, symmetric, and transitive.
Answer:
Remember, if B is a set, R is a relation in B and a is related with b (aRb or (a,b))
1. R is reflexive if for each element a∈B, aRa.
2. R is symmetric if satisfies that if aRb then bRa.
3. R is transitive if satisfies that if aRb and bRc then aRc.
Then, our set B is [tex]\{1,2,3,4\}[/tex].
a) We need to find a relation R reflexive and transitive that contain the relation [tex]R1=\{(1, 2), (1, 4), (3, 3), (4, 1)\}[/tex]
Then, we need:
1. That 1R1, 2R2, 3R3, 4R4 to the relation be reflexive and,
2. Observe that
1R4 and 4R1, then 1 must be related with itself.4R1 and 1R4, then 4 must be related with itself.4R1 and 1R2, then 4 must be related with 2.Therefore [tex]\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,4),(4,1),(4,2)\}[/tex] is the smallest relation containing the relation R1.
b) We need a new relation symmetric and transitive, then
since 1R2, then 2 must be related with 1.since 1R4, 4 must be related with 1.and the analysis for be transitive is the same that we did in a).
Observe that
1R2 and 2R1, then 1 must be related with itself.4R1 and 1R4, then 4 must be related with itself.2R1 and 1R4, then 2 must be related with 4.4R1 and 1R2, then 4 must be related with 2.2R4 and 4R2, then 2 must be related with itselfTherefore, the smallest relation containing R1 that is symmetric and transitive is
[tex]\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,4),(2,1),(2,4),(3,3),(4,1),(4,2),(4,4)\}[/tex]
c) We need a new relation reflexive, symmetric and transitive containing R1.
For be reflexive
1 must be related with 1,2 must be related with 2,3 must be related with 3,4 must be related with 4For be symmetric
since 1R2, 2 must be related with 1,since 1R4, 4 must be related with 1.For be transitive
Since 4R1 and 1R2, 4 must be related with 2,since 2R1 and 1R4, 2 must be related with 4.Then, the smallest relation reflexive, symmetric and transitive containing R1 is
[tex]\{(1,1),(2,2),(3,3),(4,4),(1,2),(1,4),(2,1),(2,4),(3,3),(4,1),(4,2),(4,4)\}[/tex]
To find the smallest relation containing given pairs with specific properties, we add missing pairs and ensure all existing pairs satisfy the required properties.
Explanation:a. To find the smallest relation that is reflexive and transitive, we need to add any missing pairs that would make the relation reflexive and ensure that all existing pairs satisfy the transitive property. In this case, the relation already contains (1, 2), (1, 4), (3, 3), and (4, 1). To make it reflexive, we add (2, 2) and (4, 4). To satisfy the transitive property, we need to add (1, 1), (2, 4), (3, 1), and (4, 2). Therefore, the smallest relation that is reflexive and transitive is {(1, 1), (1, 2), (1, 4), (2, 2), (2, 4), (3, 1), (3, 3), (4, 1), (4, 2), (4, 4)}.
b. To find the smallest relation that is symmetric and transitive, we need to add any missing pairs that would make the relation symmetric and ensure that all existing pairs satisfy the transitive property. In this case, the relation already contains (1, 2), (1, 4), (3, 3), and (4, 1). To make it symmetric, we need to add (2, 1) and (4, 4). To satisfy the transitive property, we need to add (1, 1), (2, 4), (3, 1), and (4, 2). Therefore, the smallest relation that is symmetric and transitive is {(1, 1), (1, 2), (1, 4), (2, 1), (2, 4), (3, 1), (3, 3), (4, 1), (4, 2), (4, 4)}.
c. To find the smallest relation that is reflexive, symmetric, and transitive, we need to add any missing pairs that would make the relation reflexive, symmetric, and ensure that all existing pairs satisfy the transitive property. In this case, the relation already contains (1, 2), (1, 4), (3, 3), and (4, 1). To make it reflexive, we add (2, 2) and (4, 4). To make it symmetric, we need to add (2, 1). To satisfy the transitive property, we need to add (1, 1), (2, 4), (3, 1), and (4, 2). Therefore, the smallest relation that is reflexive, symmetric, and transitive is {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (2, 4), (3, 1), (3, 3), (4, 1), (4, 2), (4, 4)}.
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explain with words how you find the area of the figure. then find the area.
image attached
Answer:
13x² -3x
Step-by-step explanation:
A horizontal line from the corner of the "notch" will divide the figure into two rectangles whose dimensions are given. The total area is the sum of the areas of those rectangles. Each area is the product of length and width.
A = A1 + A2
= x(3x-7) + 2x(5x+2)
= 3x² -7x +10x² +4x
= (3+10)x² +(-7+4)x
= 13x² -3x
The area is 13x² -3x.
Suppose that f(x)=14x−6x3. (A) Find the average of the x values of all local maxima of f. Note: If there are no local maxima, enter -1000.
Answer:
Maximum at [tex]x =\frac{\sqrt{7}}{3}[/tex]
Step-by-step explanation:
Given function,
[tex]f(x) = 14x - 6x^3[/tex]
Differentiating with respect to x,
[tex]f'(x) = 14 - 18x^2----(1)[/tex]
For critical values :
[tex]f'(x) = 0[/tex]
[tex]14 - 18x^2 =0[/tex]
[tex]14 = 18x^2[/tex]
[tex]x^2 = \frac{14}{18}[/tex]
[tex]x^2=\frac{7}{9}[/tex]
[tex]x = \pm \frac{\sqrt{7}}{3}[/tex]
Now, differentiating equation (1) again with respect to x,
[tex]f''(x) = -36x[/tex]
Since,
[tex]f''(\frac{\sqrt{7}}{3}) = -36(\frac{\sqrt{7}}{3}) < 0[/tex]
This means that the function is maximum at [tex]x=\frac{\sqrt{7}}{3}[/tex]
While,
[tex]f''(-\frac{\sqrt{7}}{3}) = 36(\frac{\sqrt{7}}{3}) > 0[/tex]
This means that the function is minimum at [tex]x=-\frac{\sqrt{7}}{3}[/tex]