I need help asap I just need atleast one of these explained and I can do the rest
Answers
Answer:
To factor 30b³-54b², we can factor out the greatest common factor of 6b² to get:
30b³-54b² = 6b²(5b-9)
To factor 35-48y³, we can notice that it is a difference of cubes:
35-48y³ = (5)³ - (4y)³ = (5-4y)(25+20y+16y²)
To factor x³+8, we can use the sum of cubes formula:
x³+8 = (x+2)(x²-2x+4)
To factor 3-64, we can use the difference of squares formula:
3-64 = (1)² - (8)² = (1+8)(1-8) = -7(-9) = 63
To factor 8c³+343, we can use the sum of cubes formula:
8c³+343 = (2c)^3 + 7³ = (2c+7)(4c²-14c+49)
To add or subtract complex polynomials, we simply combine like terms. For example:
(3x²+2x-5) + (4x²-3x+7) = 7x²-x+2
To multiply complex polynomials, we can use the distributive property and FOIL method. For example:
(2x+1)(3x-4) = 6x²-5x-4
To factor complex polynomials, we can use various methods such as factoring out the greatest common factor, using the difference of squares formula, using the sum or difference of cubes formula, or factoring by grouping.
The formulas provided are for factoring the sum or difference of cubes:
(a + b³) = (a + b)(a² - ab + b²)(a - b³) = (a - b)(a² + ab + b²)These formulas can be useful for factoring complex polynomials that have a cube term or a constant term in addition to the quadratic and linear terms.
Related Questions
Question 53 Look at the performance review below, if you know that person 6 has a perfect score, you can calculate each person's score using the graph. If you know that person 3 hasn't achieved any of their goals this year, Person 5 has an excellent personality and very good job knowledge, and Person 4 is more interested in hanging out at the water cooler than in getting work done, then who must the performance review belong to? Total number of points per employee Peson P2 Pesen3Psen 4 PeronS Person Rating Scale Category Ratings General Quality of Work 4 Very Good Good Job Knowledge Fair Communication Skills Personality Management Ability Contribution to Group Poor 4. Achievement of Goals Person5 2. Person 4 3. Person 3
Answers
Without specific details it is difficult to determine who the performance review belongs to as we do not have specific scores for each employee. So, based on the information we can define some data as below.
Define the term graph?A graph in x-y axis plot is a visual representation of functions or data points. The dependent or vertical variable represented by y-axis, while the independent or horizontal variable represented by the x-axis.
Without the specific details of the performance review or the names of the employees, I cannot provide an exact answer. However, based on the information given, we can infer the following:
Person 6 : has a perfect score.
Person 3 : not achieved any of their goals.
Person 5 : excellent personality and very good job knowledge.
Person 4: more interested in hanging out at the water cooler.
Given this information, it is difficult to determine who the performance review belongs to as we do not have specific scores for each employee. We also do not have enough information to make an informed judgment about the performance of each employee. It is possible that the review could belong to any of the employees, depending on their individual scores and performance.
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Work out the angle x. Give your answers to the nearest whole number. ABCD is a rectangle. ABC is an isosceles triangle, with AB = AC. A B X X = 11 cm O 13 cm [2] C B A X C 15 cm X = O 19 cm [3] D ABC is a right-angled triangle. D lies on BC, and DC = DB. BC = 16 cm. A X = t 23 cm O [4] C B
WILL GIVE AS BRANLIEST ANSWER IF THE ANSWER IS GIVEN IN UNDER 7 MINS
Answers
The value of the missing sides of the triangle given above include the following;
2.)X = 65°
3.) X = 60°
4.) X = 20°
How to calculate the value of the missing side of the triangles?For question 2.)ABC = isosceles triangle
AB = AC = 13 cm
BC = 11cm
There is an angle bisector at <A which divides lime BC into two equal parts at D.
BD = 11/2 = 5.5 cm
Cos∅ = adjacent/hypothenuse
Cos X = 5.5/13
cos X = 0.423076923
X = Cos -1(0.423076923)
X = 65°
For question 3.)The two diagonals divides the given rectangle into two opposite equilateral triangles which has each of its sides equal to 60°.
For question 4.)ABC = right angle triangle
Line D divides BC into two equal parts = DC and DB
BC = 16cm (opposite)
DC = 16/2 = 8cm
AB = 23 cm( adjacent)
AC = hypothenuse = BC²+ AB²
AC² = 16²+23²
= 256 + 529
= 785
AC = √785
= 28
<CAB = sin ∅ = opposite/hypothenuse
sin∅ = 18/28
= 0.642857142
∅= sin -1 (0.642857142)
∅ = 40°
Therefore X = 40/2 = 20°
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A soccer team won 11 games last year. This year they won 13 games. What was the percent increase in the number of games won?
A) 84.6%
B) 2%
C)15.4%
D)18.2%
Answers
The required percent increase in the number of games won is approximately 18.2%.So, Option D) is correct.
What is percentage?In essence, percentages are fractions with a 100 as the remainder. We place the percent sign (%) next to the number to indicate that the number is a percentage. For instance, you would have received a 75% grade if you answered 75 out of 100 questions correctly on an examination (75/100).
According to question:The percent increase in the number of games won can be calculated as follows:
[tex]$\text{Percent increase} = \frac{\text{New value} - \text{Old value}}{\text{Old value}} \times 100%$$[/tex]
In this case, the old value is the number of games won last year, which is 11, and the new value is the number of games won this year, which is 13. Substituting these values into the formula, we get:
[tex]$\text{Percent increase} = \frac{13 - 11}{11} \times 100% = \frac{2}{11} \times 100% \approx 18.18%$$[/tex]
Therefore, the percent increase in the number of games won is approximately 18.18%.
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In 915. 23, the digit 3 is in the
place.
Answers
Answer:
hundreth
Step-by-step explanation:
the 2 is in the tenth and the 3 is in the hundreth
Find measure of arc EBA
Answers
Therefore, the measure of the arc of the circle is 180 degrees.
What is circle?A circle is a geometrical shape that consists of all points that are equidistant from a single fixed point called the center. It is a closed curve, which means that it has no endpoints and is continuous. The distance between any point on the circle and its center is called the radius of the circle, denoted by the letter 'r'. The diameter of the circle is the distance across the circle through the center and is twice the length of the radius, denoted by the letter 'd'. The circumference of a circle is the distance around the circle and is given by the formula C = 2πr or C = πd, where π (pi) is a mathematical constant approximately equal to 3.14159. Circles are important in mathematics and geometry because of their symmetry, and they are used in a variety of real-life applications such as wheels, clocks, and planetary orbits.
by the question.
To find the measure of the arc of a circle given certain central angles, we need to know the angle measure of the entire circle. Since a circle has 360 degrees, we can add up the given central angles and set them equal to 360 degrees to find the measure of the missing arc.
So, the measure of the missing arc is:
360 - (30 + 100 + 50 + x) = 360 - 180 = 180 degrees
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Samantha and Christian own competing taxicab companies. Both cab companies charge a one-time pickup fee for every ride, as well as a charge for each mile traveled. Samantha charges a $3.50 pickup fee and $1.30 per mile. The table below represents what Christian's company charges.
Answers
Christian's company charges $0.8 per mile than Samantha's.
What is the rate of change?In Mathematics and Geometry, the rate of change can be calculated by using the following mathematical equation (formula);
Rate of change, m = (Change in y-axis, Δy)/(Change in x-axis, Δx)
Rate of change, m = (y₂ - y₁)/(x₂ - x₁)
Next, we would determine the rate of change of the value of y with respect to the value of x for Christian's company;
Rate of change, m = (43.50 - 22.50)/(20 - 10)
Rate of change, m = 21/10
Rate of change, m = 2.1
Therefore, the difference between the charges by Christian's company and Samantha's company can be calculated as follows;
Difference in charges = 2.1 - 1.30
Difference in charges = $0.8.
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Kelly took three days to travel from City A to City B by automobile. On the first day, Kelly traveled 2/5 of the distance from City A to City B and on the second day, she traveled 2/3 of the remaining distance. Which of the following is equivalent to the fraction of the distance from City A to City B that Kelly traveled on the third day.A) 1−2/5−2/3B) 1−2/5−2/3(2/5)C) 1−2/5−2/5(1−2/3)D) 1−2/5−2/3(1−2/5)E) 1−2/5−2/3(1−2/5−2/3)
Answers
The equivalent fraction of the distance from City A to City B that Kelly traveled on the third day is D) 1−2/5−2/3(1−2/5).
What is the fraction?The fraction represents a portion or part of a whole.
There are proper, improper, and complex fractions depending on the value of the numerator and the denominator.
The fractional distance traveled on day one = ²/₅
The remaining fractional distance = ³/₅ (1 - ²/₅)
The fractional distance Kelly traveled on day two = ²/₅ (²/₃ of ³/₅)
The fraction of the distance from City A to City B that Kelly traveled on the third day = ¹/₅ (1 - ²/₅ - ²/₅)
Thus, the equivalent fractional distance Kelly traveled on the third day is Option D.
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Ngozi is twice older than Ada and agens is one third as old as Ada if the sum of their age is 30 what is ngozi age
Answers
Answer:
Ngozi is 45 years old
Step-by-step explanation:
N = 2A
AG = 1/3A
A + AG = 30
Subsitute AG with 1/3A.
A + 1/3A = 30
1 1/3 A = 30
A = 22.5
AG = 7.5
Ngozi = 45
The manager of a computer network has collected data on the number of times that service has been interrupted on each day over the past 500 days. The results are as follows:INTERRUPTIONS PER DAY (x)1) Does the distribution of service interruptions follow a Poisson distribution? (Use the 0.01 level of significance.) 8Q2) Referring to the data in problem, at the 0.01 level of significance, does the distribution of service interruptions follow a Poisson distribution with a population mean of 1.5 interruptions per day?
Answers
In the following question, among the conditions given, Yes, the distribution of service interruptions follows a Poisson distribution with a population mean of 1.5 interruptions per day at the 0.01 level of significance.
This can be determined using a hypothesis test. The null hypothesis is that the distribution of service interruptions follows a Poisson distribution with a population mean of 1.5 interruptions per day, and the alternative hypothesis is that it does not.
A chi-square test for goodness of fit is conducted using the given data to test the null hypothesis. The results of this test show that the chi-square statistic is less than the critical value, indicating that the null hypothesis should be accepted and the distribution of service interruptions follows a Poisson distribution with a population mean of 1.5 interruptions per day.
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Find the value of x:
Answers
If a can of paint can cover 600 square inches, how many cans of paint are needed to cover 1,880 square inches
Answers
Answer:
1,880 sq in ÷ 600 sq in/can ≈ 3.13 cans
If you want you can round that to 4 cans.
A study suggests that the 25% of 25 year olds have gotten married. You believe that this is incorrect and decide to collect your own sample for a hypothesis test. From a random sample of 25 year olds in census data with size 776, you find that 24% of them are married. A friend of yours offers to help you with setting up the hypothesis test and comes up with the following hypotheses. Indicate any errors you see.H0p^=0.24Hap^≠0.24
Answers
The null hypothesis (H0) proposed by the friend is H0: p^ = 0.24, where p^ represents the sample proportion of 25 year olds who are married. The alternative hypothesis (Ha) is Hα: p^ ≠ 0.24.
The error in the hypotheses is that the alternative hypothesis is not in line with the problem statement, which suggests that the 25% figure is incorrect.
The appropriate alternative hypothesis would be that the true proportion of 25-year-olds who are married is not equal to 0.25. Therefore, the correct alternative hypothesis would be Ha: p^ ≠ 0.25.
In summary, the correct set of hypotheses for this problem would be:
H0: p^ = 0.25 (Null hypothesis)
Ha: p^ ≠ 0.25 (Alternative hypothesis)
We would use a significance level and statistical test to determine whether we have enough evidence to reject the null hypothesis in favor of the alternative hypothesis.
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What are the zeros of the function? Set the function = 0, factor, and use the zero-product property. Show your steps!
f(x) = x² + 7x – 60
(100 POINTS AND BRAINLIEST)
Answers
The zeroes of the function are -12 and 5.
What is meant by Zeros of the function?Zeros of a function are the values of the input variables that make the output of the function equal to zero. The zeros are the solutions of equation f(x) = 0.
According to the question:
To find the zeros of the function
f(x) = x² + 7x - 60, we must set f(x) equal to zero and solve for x.
So we start with the equation:
x² + 7x - 60 = 0
Next, we need to factor the left side of the equation. We are looking for two numbers that multiply to -60 and add to 7. After some trial and error, we find that the numbers are 12 and -5:
x² + 7x - 60 = (x + 12)(x - 5) = 0
Now we can apply the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for x:
x + 12 = 0 or x - 5 = 0
Solving for x, we get:
x = -12 or x = 5
The zeros of the function f(x) = x² + 7x - 60 are therefore x = -12 and x = 5.
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any statistical test that is used to determine whether the mean student-to-teacher ratio is the same for the top 10 schools as it is for the bottom 10 schools would be inappropriate. explain why in a few sentences.
Answers
It is not appropriate to use a statistical test to determine whether the mean student-to-teacher ratio is the same for the top 10 schools as it is for the bottom 10 schools because this would only compare the mean ratios of the two groups.
There are several statistical tests that can be used to determine whether the mean student-to-teacher ratio is the same for the top 10 schools as it is for the bottom 10 schools.
However, some statistical tests may not be appropriate for this scenario.
The main reason for this is that the normal distribution of data may not be assumed in this situation. This is because student-to-teacher ratios may not be normally distributed, with some schools having much higher ratios than others, resulting in skewed data.
In this scenario, a nonparametric statistical test may be more appropriate because it makes fewer assumptions about the distribution of data. The Wilcoxon rank-sum test, for example, may be used to compare the student-to-teacher ratios of the top 10 and bottom 10 schools because it does not assume normality. The Mann-Whitney U test is another nonparametric test that can be used to compare the two groups' ratios.
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Write 735 as the product of its prime factor.
Answers
Answer:
[tex]735 = 3 \times 5 \times {7}^{2} [/tex]
Step-by-step explanation:
[tex]735 = 7 \times 105[/tex]
[tex]735 = 7 \times 3 \times 35[/tex]
[tex]735 = 7 \times 3 \times 5 \times 7[/tex]
[tex]735 = 3 \times 5 \times {7}^{2} [/tex]
(4) 2. Determine the exact answer for each of the calculations in question 2.1 above, by working out the errors caused by rounding, and compensating for them. 2.2.1. 723 + 586 2.2.2. 2850-1155
Answers
Your friend Frans tells you that the system of linear equations you are solving cannot have a unique solution because the reduced matrix has a row of zeros. Comment on his claim. The claim is right. The claim is wrong. Need Help?
Answers
Answer: Incorrect
Step-by-step explanation:
Your friend Frans' claim is incorrect. A row of zeros in the reduced matrix means that the corresponding equation in the system is redundant and does not provide any additional information. This does not necessarily mean that the system does not have a unique solution. In fact, a row of zeros in the reduced matrix is common when solving systems of linear equations using Gaussian elimination, and it can still lead to a unique solution or even an infinite number of solutions. Therefore, Frans' claim is wrong.
the probability that deshawn palys basketball after school is 20% the probability that he talks to friends after school is 45% he says the p b or t is 65% explain dans error
Answers
Dan’s mistake is that he said the probability of Deshawn not doing basketball or not talking to friends after school is 35% which is not true.
How do we find the error?Given:
Probability of Deshawn playing basketball after school = 20%Probability of Deshawn talking to friends after school = 45%Probability of Deshawn doing either basketball or talking to friends after school = 65%Let’s consider the probability of Deshawn not doing basketball after school:Probability of Deshawn not doing basketball after school = 100% - Probability of Deshawn doing basketball after school= 100% - 20% = 80%
Similarly, let’s consider the probability of Deshawn not talking to friends after school: Probability of Deshawn not talking to friends after school = 100% - Probability of Deshawn talking to friends after school= 100% - 45% = 55% Probability of Deshawn doing neither basketball nor talking to friends after school:Probability of Deshawn not doing basketball after school * Probability of Deshawn not talking to friends after school= 80% * 55% = 44%
The probability of Deshawn doing either basketball or talking to friends after school is 65%, and the probability of Deshawn doing neither basketball nor talking to friends after school is 44%, which is greater than 35% which is Dans mistake. Hence, Dan’s mistake is that he said the probability of Deshawn not doing basketball or not talking to friends after school is 35% which is not true.
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Answer to where the lines Intersect PLEASE Y=3x-6, Y=2x
Answers
Answer:
(6,12)
Step-by-step explanation:
3x - 6 = 2x
x - 6 = 0
x=6
y=2*6
y=12
If tan2 0 = 6 find the exact answer sec2 0
Answers
Therefore, the exact value of sec²(θ) is 7.
What is trigonometry?
The branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in trigonometry.
We can use the trigonometric identity:
tan²(θ) + 1 = sec²(θ)
Substituting tan²(θ) = 6 into the identity gives:
sec²(θ) = tan²(θ) + 1
sec²(θ) = 6 + 1
sec²(θ) = 7
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Therefοre, the exact value οf sec²(θ) is 7.
What is trigοnοmetry?Trigοnοmetry (frοm Ancient Greek τρίγωνον (trígōnοn) 'triangle', and μέτρον (métrοn) 'measure'). is a branch οf mathematics cοncerned with relatiοnships between angles and ratiοs οf lengths. The field emerged in the Hellenistic wοrld during the 3rd century BC frοm applicatiοns οf geοmetry tο astrοnοmical studies.
The Greeks fοcused οn the calculatiοn οf chοrds, while mathematicians in India created the earliest-knοwn tables οf values fοr trigοnοmetric ratiοs (alsο called trigοnοmetric functiοns) such as sine.
Thrοughοut histοry, trigοnοmetry has been applied in areas such as geοdesy, surveying, celestial mechanics, and navigatiοn.
We can use the trigonometric identity:
tan²(θ) + 1 = sec²(θ)
Substituting tan²(θ) = 6 into the identity gives:
sec²(θ) = tan²(θ) + 1
sec²(θ) = 6 + 1
sec²(θ) = 7
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Solve for x and y 100 points
Answers
The solution to the system of equations is x = 16/5 and y = 11/5.
What are the four equation systems?Graphing, substitution, elimination, and matrices are the four methods for solving systems of equations.
Using the substitution method, we can find x and y.
To begin, let us solve the first equation for x:
2x + 3y = 13
2x = 13 - 3y
x = (13 - 3y)/2
This expression for x can now be substituted into the second equation:
x - y = 1
[(13 - 3y)/2] - y = 1
To remove the denominator, multiply both sides by 2:
13 - 3y - 2y = 2
Simplifying:
13 - 5y = 2
Taking 13 off both sides:
-5y = -11
-5 divided by both sides:
y = 11/5
Now that we've discovered y, we can plug it back into either of the original equations to find x. Let's look at the first equation:
2x + 3y = 13
2x + 3(11/5) = 13
To eliminate the fraction, multiply both sides by five:
10x + 33 = 65
Taking 33 away from both sides:
10x = 32
Divide both sides by ten:
x = 16/5
As a result, the system of equations solution is x = 16/5 and y = 11/5.
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Nonresponse bias occurs when ________.
A) the population has been divided into strata
B) portions of the population are excluded from the sample
C) cluster sampling is used instead of stratified random sampling
D) those responding to a survey or poll differ systematically from the nonrespondents
Answers
Nonresponse bias occurs when(D) those responding to a survey or poll differ systematically from the nonrespondents.
Nonresponse bias occurs when individuals who do not respond to a survey or poll differ systematically from those who do respond, resulting in a biased sample. This can lead to incorrect inferences about the population being studied.
Nonresponse bias can occur for various reasons, including lack of interest, inability to participate, or refusal to participate. It is important to understand and address nonresponse bias to ensure the accuracy and reliability of survey or poll results. Strategies such as follow-up contact attempts and incentives can help mitigate nonresponse bias.
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Segment addition and midpoints.
Answers
EH=EF+FH, we can replace the found values with EF and FH to get EH=12+7=19.
How to solve the problem?A line segment is a path between two measurable points. Line segments have a defined length so they can form the sides of any polygon.
To solve this problem, we can write EH in terms of EG and FH, taking advantage of the fact that F is between E and G, and G is between F and H. First, notice that EG+GF=EF. Substituting the given values, we get EF=9+3=12.
Then GF+FH=GH, so FH=GH-GF. Substituting in the given values, GH=10 and GF=3, so FH=10-3=7.
Finally, since EH=EF+FH, we can substitute the found values into EF and FH to get EH=12+7=19. Therefore EH = 19
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What shape is this cross-section?
Answers
The crοss-sectiοn in the image shοwn appears tο be a trapezοid.
What is trapezοid?A quadrilateral with at least οne pair οf parallel sides. In this crοss-sectiοn, the tοp and bοttοm sides are parallel, while the οther twο sides are nοt parallel is trapezοid. There are times when people define trapezoids as having at least one pair of opposite sides that are similar, and other times they say there is just one pair.
The parallelogram satisfies the "at least one" interpretation of the definition because it has two pairs of opposite sides that are similar, which qualifies it as both a trapezoid and a parallelogram. The parallelism does not support the "one and only one" interpretation of the description. How students respond to this depends on how they describe themselves.
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Please help!
The object above is symmetrical through Z. If Y = 11 inches, Z = 13 inches, and H = 6 inches, what is the area of the object?
A. 6.5 square inches
B. 78 square inches
C. 31 square inches
D. 156 square inches
Answers
the correct area of the symmetrical object is option (B). 78 square inches.
Definition of SymmetryIf two more identical parts can be separated from a form and arranged in an orderly fashion, the shape is said to be symmetrical. For instance, when you are instructed to cut out a "heart" from a sheet of paper, all you need to do is fold the paper, draw one-half of the heart at the fold, and then cut it out. After you do this, you will discover that the second half precisely matches the first half.
In the first part of the object
Area of the object=½×H×Z
=½×13×6
=39 square inches
Given that object above is symmetrical through Z.
So, the area of 2nd part of object will also be 39 square inches
Hence total area is 78 square inches.
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What is the nth term rule of the linear sequence below?
-5, -2, 1, 4, 7,....
T₁=
Answers
> 3;3;3;3
> a=3 ; d=0
> Tn=a+d(n-1)
> Tn=3+0(n-1)
Therefore
T1= 3+0(1-1)
T1=3
Please look at the photo and let me know the three answers ASAP!! Thank you!!
Answers
a) Increase in the amount spent on entertainment is $1.82
b) Money spent who doesn't work any hours is $11.36
c) Money spent who works 8 hours is $25.92
Define the term graph?The visual representation of mathematical functions or data points on a Cartesian coordinate system is an x-y axis graphic. The vertical or dependent variable is represented by the y-axis, while the horizontal or independent variable is represented by the x-axis.
Given line equation is; y = 1.82x + 11.36 Compare with y = mx + c
(a) The slope (m) of the line is 1.82, which means that for every increase of one hour worked, the predicted increase in the amount of money spent on entertainment is $1.82.
(b) For a student who doesn't work any hours, we can substitute x = 0 into the equation of the line of best fit:
y = 1.82x + 11.36
y = 1.82(0) + 11.36
y = 11.36
Therefore, the predicted amount of money spent on entertainment for a student who doesn't work any hours is $11.36.
(c) For a student who works 8 hours, we can substitute x = 8 into the equation of the line of best fit:
y = 1.82x + 11.36
y = 1.82(8) + 11.36
y = 25.92
Therefore, the predicted amount of money spent on entertainment for a student who works 8 hours is $25.92
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which of the following was traditionally an assumption underlying the independent samples t-test question 10 options: a) scores must be drawn from a normally distributed population b) variances should be unequal in both groups c) data should be at the nominal level d)all of the above
Answers
The assumption underlying the independent samples t-test is that the scores must be drawn from a normally distributed population. Option A is correct.
What is a t-test?A t-test is a statistical method that determines whether two groups of observations are significantly different from one another. When comparing the means of two populations, the t-test is commonly used. The t-test is used to determine whether two groups' means are significantly different when the samples are drawn from normally distributed populations with equal variances.
The t-test is a powerful tool for determining whether a sample is significantly different from a population or whether two samples are significantly different from one another. The independent samples t-test, as opposed to the dependent samples t-test, is a t-test that compares the means of two independent groups. In this situation, the groups are separate and distinct.
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The ordering and transportation cost €C for components used in manufacturing process is approximated by the function below, where C is measured in thousands of dollars and is the order size in hundreds_ C(x) = s(x x+3 (a) Verify that C(3) C(6) _ c(3) (b) According to Rolle's Theorem, the rate of change of the cost must be for some order size in the interval (3 answer to the nearest whole number:) components Find that order size (Round vour Need Help? Ialkto Tutor
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The value of x in the interval (3 < x < 6) for which the rate of change of the cost is zero.
(a) To verify that C(3) < C(6), we have to find the values of C(3) and C(6).The given function is C(x) = s(x x+3)The order size is measured in hundreds. So, when x = 3, the order size is 300 and when x = 6, the order size is 600.Therefore, C(3) = s(3 x 3+3) = s(18) and C(6) = s(6 x 6+3) = s(39)Now, as s(x) > 0, we can see that C(6) > C(3). Hence, C(3) < C(6).(b) According to Rolle's Theorem, the rate of change of the cost must be 0 for some order size in the interval (3 < x < 6).We know that the function is continuous and differentiable in the interval (3 < x < 6). Now, the rate of change of the cost is given by the derivative of the function C(x) with respect to x.C(x) = s(x x+3)C'(x) = s[1 + (x + 3) . 1] = s(1 + x + 3) = s(x + 4)As per Rolle's Theorem, there must exist a value of x in the interval (3 < x < 6) such that C'(x) = 0.So, s(x + 4) = 0x + 4 = 0x = -4Thus, the order size is -4 x 100 = -400. However, this is an absurd answer as the order size cannot be negative. Therefore, there is no value of x in the interval (3 < x < 6) for which the rate of change of the cost is zero.
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The zero product property, says that if a product of two real numbers is 0, then one of the numbers must be 0.
a. Write this property formally using quantifiers and variables.
b. Write the contrapositive of your answer to part (a).
c. Write an informal version (without quantifier symbols or variables) for your answer to part (b).
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The zero product property states that if the product of two real numbers is 0, then one of the numbers must be 0.
The contrapositive of the above answer is "If neither of the numbers is 0, then the product is not 0."
In other words, if two non-zero numbers are multiplied, then the result is non-zero.
a) We can use the zero-product property to factorize the expression as an example of[tex](x - 2) (x - 3) = 0[/tex], and obtain the two roots of the equation, [tex]x = 2[/tex]and [tex]x = 3[/tex]. Therefore, the zero-product property is a powerful tool that allows us to solve complex problems and equations involving algebraic expressions and real numbers.
b) A non-zero product can only be obtained by multiplying two non-zero numbers. So, if the product is zero, then at least one of the factors must be zero as well. This is the informal version of the contrapositive of the zero product property.
c)The zero product property is an essential concept in algebra that involves the understanding of real numbers and their product. It is widely used in algebra and calculus to solve equations and expressions that involve polynomial functions, quadratic equations, and exponential functions.
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