The correct answer is: Median calculation and Threshold calculation.
Median calculation: The Column Statistics tool can calculate the median of a dataset. The median is the middle value in a sorted dataset. It is a measure of central tendency that divides the data into two equal halves. Calculating the median can provide insights into the central value of a dataset and is less sensitive to extreme values compared to other measures like the mean.
Threshold calculation: The Column Statistics tool can calculate thresholds or cutoff points based on the data. These thresholds define certain conditions or ranges in the data. For example, you can set a threshold to identify data points above a certain value as outliers or flag data points that fall within a specific range. Threshold calculations can be useful for data cleansing, anomaly detection, or creating categorical variables based on certain conditions.
The other options, Date analysis and Regression analysis, are not typically calculated by the Column Statistics tool. Date analysis involves examining patterns and trends related to dates or time, which may require specialized techniques such as time series analysis or date-specific functions. Regression analysis is a statistical technique used to model the relationship between variables and requires more advanced tools or methods beyond basic column statistics.
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Simplify 4(3n + 2b + k)
a. 12nbk
b. 12n + 2bk
c. 12n + 8b + 4k
d. 24bkn
Answer:
C
Step-by-step explanation:
4(3n+2b+k)
12n+8b+4k
So it would be C.
---
hope it helps
The odds against a certain football team winning the championship are 70 : 1. a) Determine the probability that the team wins the championship. b) Determine the probability that the team does not win the championship.
a) The probability that the team wins the championship is approximately 0.0141 or 1.41%. b) The probability that the team does not win the championship is approximately 0.9859 or 98.59%.
a) To determine the probability that the team wins the championship, we need to convert the odds against winning into a probability.
The odds against winning are given as 70:1. This means that for every 70 unfavorable outcomes (losing), there is 1 favorable outcome (winning).
To calculate the probability of winning, we divide 1 by the sum of the favorable and unfavorable outcomes:
Probability of winning = 1 / (70 + 1) = 1 / 71 ≈ 0.0141 (or 1.41%)
Therefore, the probability that the team wins the championship is approximately 0.0141 or 1.41%.
b) The probability of not winning the championship is equal to 1 minus the probability of winning:
Probability of not winning = 1 - 0.0141 = 0.9859 (or 98.59%)
Therefore, the probability that the team does not win the championship is approximately 0.9859 or 98.59%.
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A=4 B=7 C=11 D=13 Given this number line, cd=
Answer:
.
Step-by-step explanation:
i dont know this i really hard
Answer: 2
Step-by-step explanation:
Write the ratio of 8 apples to 4 oranges in 3 different ways.
Then, write the ratio in the simplest form.
Answer:
Step-by-step explanation:
The three ways to write a ratio is:
8/4
8 to 4
8:4
The ratio in simpliest form is 2/1
You would divide each "side" by 4
The diameter of this circle is 12 inches. The diameter of the smaller circle is 8 inches. find the area of the SHADED region.
Answer:
so you multiply 12 by 8 =96
Answer:
area of large circle=pi×r²
area of large circle =3.14×12²
area of large circle=452.16in²
area of small circle=pi×r²
area of small circle=3.14×8²
area pf small circle=200.96in²
area of shaded circle=452.16in²- 200.96in²
area of shaded circle =251.2in²
Identify the like terms in the expression 4x+5-x^3-3x
You are making freshly squeezed orange juice for a brunch you are catering. You need to make 3 liters of orange juice; Oranges are purchases by the case for $24.Each case contains 100 oranges. Each orange weighs 6 ounces and has a yield percent, for juicing, of 50%. what is the edible portion cost for the orange juice for this brunch?
Answer : The edible portion cost for the orange juice for this brunch is $9.6.
Explanation :
GivenData: Cost of each case = $24 Number of oranges in each case = 100 Weight of each orange = 6 ounces Yield percentage of each orange = 50% Amount of orange juice required = 3 liters Formula used:To find the edible portion cost of orange juice, we need to find the cost per liter of orange juice and then multiply it by the required amount of orange juice.
Edible portion cost = (Cost per liter of orange juice) × (Amount of orange juice required) Cost per liter of orange juice = (Cost of 100 oranges) / (Yield of 100 oranges)Cost of 100 oranges = Cost of each case = $24 Therefore, Cost per liter of orange juice = (24) / [(50/100) × 100 × (6/16)]{Converting 6 ounces into liters by multiplying with 0.0166667}Cost per liter of orange juice = $3.20 Edible portion cost = (Cost per liter of orange juice) × (Amount of orange juice required)Edible portion cost = (3.2) × (3) = $9.6 Therefore, the edible portion cost for the orange juice for this brunch is $9.6.
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A student earns $10 per hour for tutoring and 57 per hour as a teachers aide To have enough free time for studies, he can work no more than 20 hours per week. The storing center requires that each tutor spends at least three hours per week tutoring, but no more than eight hours per week How many hours should he work to maximize his earnings hours of tutoring hours as a teacher's aide What is the maximum profit An automotive plant makes the Quartz and the Pacer. The plant has a maximum production capacity of 1200 cars per week, and they can make at most 600 Quartz cars and 800 Pacers each week. If the profit on a Quartz is $500 and the profit on a Pacer is $800, find how many of each type of car the plant should produce. Quartz Pacers What is the maximum profit? A manufacturer of ski clothing makes ski pants and ski jackets. The profit on a pair of ski pants is $2.00 and the profit on a jacket is $1.50. Both pants and jackets require the work of sewing operators and cutters. There are 60 minutes of sewing operator time and 45 minutes of cutter time available. It takes 8 minutes to sew one pair of ski pants and 4 minutes to sew one jacket. Cutters take 4 minutes on pants and 8 minutes on a jacket Find the number of pants and jackets the manufacturer should make in order to maximize the profit pairs of pants Jackets. What is the maximum profits ?
To maximize his earnings, the student should work 8 hours tutoring and 12 hours as a teacher's aide. The plant should produce 600 Quartz cars and 600 Pacers to maximize the profit. The maximum profit is $660,000.The manufacturer should make 1 pair of ski pants and 7 ski jackets to maximize the profit. The maximum profit is $29.00.
Part A:
To maximize his earnings, the student should work 8 hours tutoring and 12 hours as a teacher's aide.
The maximum profit is $786.00.
Part B:
Let's say, the plant make x number of Quartz and y number of Pacers.
Therefore, x + y = 1200 ----(1)
and, 500x + 800y = Profit Maximize.
Let's multiply Equation (1) by -500 and add it to Equation (2) so that we can solve for
y. -500x - 500y = -600000500x + 800y = Profit Maximize-300y = -180000 ⇒ y = 600
Therefore, x = 600
Hence, the plant should produce 600 Quartz cars and 600 Pacers to maximize the profit. The maximum profit is $660,000.
Part C:
Let's say, the manufacturer should make x pairs of ski pants and y ski jackets.
Therefore, the system of linear equations are as follows:
8x + 4y ≤ 60 (sewing operator time)4x + 8y ≤ 45 (cutter time)
Let's plot the graph to solve the linear equations. The feasible region is shaded in the following graph:
To find the maximum profits, we need to check all the coordinates of the feasible region.
8(1) + 4(7) = 368(2) + 4(6) = 242(3) + 4(5) = 224(5) + 4(4) = 20
Profit for (1,7) = $29.00
Profit for (2,6) = $26.00
Profit for (3,5) = $23.00
Profit for (5,4) = $18.00
Therefore, the manufacturer should make 1 pair of ski pants and 7 ski jackets to maximize the profit. The maximum profit is $29.00.
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Look at photo for the question and answer choices... NO LINKS OR BLANK ANSWERS
(also i added a cute/funny question aswell)
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Answer:
Rhombus Square
Step-by-step explanation:
how many outfits could you make if you had 6 pants,7 shirts,and 4 pairs of shoes to choose from?
A. 66
B. 12376
C. 17
D. 168
Answer:
D
Step-by-step explanation:
6x7x4= 168
Which statement best describes how the author of “The Train to Somewhere” alters historical details to tell her story?
A.The author includes historical facts, such as when the Children’s Aid Society was formed.
B.The author describes the experiences and feelings of an imaginary orphan who was sent west on an orphan train.
C. The author provides specific details about families who agreed with Brace’s beliefs about ways to help needy orphan children.
D.The author explores the adventures of two young children as they start their new lives on a farm in rural America.
Answer:
The statement that best describes how the author of "The Train to Somewhere" alters historical details to tell her story is option B;
B. The author describes the experiences and feelings of an imaginary orphan who was sent west on an orphan train
Step-by-step explanation:
The author included the experiences of one of the children who according to the historical details in the historical account of the Orphan Train Riders were aided by the minister, Charles Loring Brace, to be taken from the New York City streets on the East to other states and Canada to the West
The child, a 5 year old girl, who had lost both parents had been living on the street with her brother for some months
The author also highlighted the taught by the child, of being separated from her brother at one of the destination towns of the train, and the manifestation of the beliefs of the minister, Brace, of the children finding loving families that would help them become productive members of society.
the acceleration of an oscillator undergoing simple harmonic motion is described by the equation ax(t)=−(18m/s2)cos(33t) , where the time t is measured in seconds.
The equation [tex]ax(t) = -(18 m/s^2)cos(33t)[/tex] describes the acceleration of an oscillator. The acceleration varies sinusoidally with time, following a cosine function, and has a maximum value of [tex]-18 m/s^2[/tex].
The given equation [tex]ax(t) = -(18 m/s^2)cos(33t)[/tex] represents the acceleration of an oscillator undergoing simple harmonic motion. In this equation, t represents time measured in seconds.
The term cos(33t) indicates that acceleration varies sinusoidally with time. The cosine function has a period of 2π, meaning it completes one full cycle over the interval [0, 2π]. The coefficient 33 in front of t determines the frequency of oscillation. In this case, the oscillator completes approximately 33 cycles per second.
The negative sign indicates that the acceleration is directed opposite to the displacement of the oscillator. As the oscillator moves in one direction, the acceleration pulls it back in the opposite direction, causing it to oscillate around a stable equilibrium position.
The maximum acceleration is given by [tex]-18 m/s^2[/tex], which represents the amplitude of the oscillation. The acceleration varies between [tex]-18 m/s^2[/tex] and [tex]18 m/s^2[/tex], with the maximum magnitude occurring when the cosine function is at its peak value of 1 or -1.
Overall, the equation describes the acceleration of an oscillator undergoing simple harmonic motion, providing information about its amplitude, frequency, and direction of motion.
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A problem with a telephone line that prevents a customer from receiving or making calls is upsetting to both the customer and the telecommunications company. The data set below contains samples of 20 problems reported to two different offices of a telecommunications company and the time to clear these problems (in minutes) from the customers' lines.
a. At the 0.05 level of significance, is there evidence of a difference in the variability of the time to clear problems between the two central offices?
b. Interpret the p-value.
c. What assumption do you need to make in (a) about the two populations in order to justify your use of the F test?
The F-test requires that the two populations must be normally distributed. Therefore, in order to justify the use of the F-test, it is assumed that both populations of time to clear problems from two central offices are normally distributed
a. At the 0.05 level of significance, is there evidence of a difference in the variability of the time to clear problems between the two central offices?
For comparing the variability of the time to clear problems between the two central offices, an F-test can be used. The null hypothesis is:H0: σ12 = σ22, where σ1^2 and σ2^2 are the population variances of two central offices, and the alternative hypothesis is Ha: σ12 ≠ σ22, which means two variances are different. For this study, the significance level is 0.05. As we want to find out whether there is any difference in the variance of the time to clear the problem between two offices, a two-sample F-test can be performed.F-test statistics is given by the formula:F = s12/s22where s12 is the sample variance of the first sample (first central office), and s22 is the sample variance of the second sample (second central office).We can use Excel to calculate the F statistic.Using the given dataset, the F statistic is calculated as: σ12 = 22.66666667, σ22 = 25.25, F = 0.897949853As the F statistic is less than the F-critical value, there is no significant difference in the variability of the time to clear problems between the two central offices.b. Interpret the p-value.The p-value is the probability of observing the sample data given that the null hypothesis is true. If the p-value is less than the level of significance (α = 0.05), the null hypothesis will be rejected, and we can say that there is sufficient evidence to conclude that there is a difference in the variability of the time to clear problems between the two central offices. The p-value of this F-test is 0.467. As the p-value is greater than the level of significance, the null hypothesis is not rejected.c. What assumption do you need to make in (a) about the two populations in order to justify your use of the F test?
The F-test requires that the two populations must be normally distributed. Therefore, in order to justify the use of the F-test, it is assumed that both populations of time to clear problems from two central offices are normally distributed.
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The data set contains samples of 20 problems.
Thus, there is no evidence of a significant difference in the variance of the time to clear problems between the two central offices.
The p-value is 0.17.
Assume that the two populations have a normal distribution with equal variances.
a. The null hypothesis is that there is no difference in the variance of the time to clear problems among the two offices, while the alternative hypothesis is that there is a significant difference between the variance of the two offices. Using the F-distribution, we can test whether or not there is a difference in variance of the time to clear problems. The formula for F-value is given below:
F-value = s1^2 / s2^2
Where s1^2 and s2^2 are the variances of the two samples. With the help of the provided data, we can calculate the variances for the two samples, which are as follows:
s1^2 = 42.08
s2^2 = 22.80
Then, we can calculate the F-value as follows:
F = s1^2 / s2^2
= 42.08 / 22.80
= 1.84
Using the F-distribution table, we can find the critical value of F as 2.17 (with 19 degrees of freedom for both the numerator and denominator).Since the calculated F-value (1.84) is less than the critical value of F (2.17), we can fail to reject the null hypothesis and conclude that there is no evidence of a significant difference in the variance of the time to clear problems between the two central offices.
b. The p-value represents the probability of observing a test statistic as extreme as the one calculated, assuming that the null hypothesis is true. The p-value of the F-test can be calculated by finding the area to the right of the calculated F-value in the F-distribution table. In this case, the p-value is 0.17 (using a two-tailed test).
c. In order to use the F-test, we need to assume that the two populations have a normal distribution with equal variances. Furthermore, the samples should be independent and randomly selected. These assumptions are required in order to ensure that the F-test is valid.
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What’s the answer ?
Jim , a meteorologist for local television XYZ, would like to report the rainfall. The following are the rainfall measurements (in inches) for to-day’s date for 14 randomly chosen past years. The data are as follows: 0.47 0.27 0.13 0.54 0.08 1.05 0.34 0.26 0.42 0.17 0.50 0.86 0.01 2.5 NOTE: answers should be given to 2 decimal places REQUIRED 1. Find the RANGE of the data provided and what measure does this show 2. Calculate the sample variance . Why is variance not written in? 3. Calculate the sample standard deviation. What information does this measure give to the users? Why is it necessary to compute standard deviation instead of stopping at the variance calculation? 4. Find the COEFFICIENT OF VARIATION 5. Use Z-score to find if there is any outliers in the series
Where the above is given,
Range -2.49 inchesSample variance - 0.5599Sample standard deviation - 0.7483Coefficient of variation - 7497.32% Z14 = 3.3999, which is greater than 2, suggesting that the data point 2.50 inches may be considered an outlier based on the Z-Score criterion.What is the explanation for the above ?Range
Range = Maximum value - Minimum value
Range = 2.50 - 0.01
= 2.49 inches
Sample Variance
Step 1 - Calculate the mean (average) of the data.
Mean = (0.47 + 0.27 +0.13 + 0.54 + 0.08 + 1.05 + 0.34 + 0.26 + 0.42 + 0.17 + 0.50 + 0.86 + 0.01 + 2.5) / 14
= 1.36 /14
= 0.0971
Step 2 - Calculate the squared deviation from the mean for each data point.
Deviation from mean = Data point - Mean
Squared deviation = (Deviation from mean)²
Sum of squared deviations = (0.3739 + 0.0091 + 0.7307 + 0.1638 + 0.9469 + 0.104 + 0.0145 + 0.0081 + 0.0971 + 0.3037 + 0.0601 + 0.2601 + 1.0679 + 3.1391)
= 7.2799
Sample Variance = Sum of squared deviations / (n - 1)
= 7.2799/ (14 - 1)
= 0.5599
Sample Standard Deviation:
Sample Standard Deviation = √(Sample Variance)
= √0.5599
= 0.7483
Coefficient of Variation:
Coefficient of Variation = (Sample Standard Deviation / Mean) * 100
= (7.2799/ 0.0971) * 100
= 7497.32
Z-Scores for each data point
Z1= (0.47 - 0.0971)/ 0.7212= 0.5318
Z2= (0.27 - 0.0971)/ 0.7212= 0.9983
Z3= (0.13 - 0.0971)/ 0.7212= 0.0456
Z4= (0.54 - 0.0971)/ 0.7212= 0.6189
Z5= (0.08 - 0.0971)/ 0.7212= -0.0251
Z6= (1.05 - 0.0971)/ 0.7212= 1.2914
Z7= (0.34 - 0.0971)/ 0.7212= 0.4574
Z8= (0.26 - 0.0971)/ 0.7212= 0.2251
Z9= (0.42 - 0.0971)/ 0.7212= 0.4969
Z10= (0.17 - 0.0971)/ 0.7212= 0.1002
Z11= (0.50 - 0.0971)/ 0.7212= 0.6506
Z12= (0.86 - 0.0971)/ 0.7212= 1.0242
Z13= (0.01 - 0.0971)/ 0.7212= -0.1208
Z14= (2.50 - 0.0971)/ 0.7212= 3.3999
To determine if there are any outliers, we can compare the absolute values of the Z-Scores to a certain threshold , commonly considered as 2.
If the absolute value of a Z-Score is greater than 2, it indicates that the corresponding data point is an outlier.
In this case, Z14 = 3.3999, which is greater than 2 , suggesting that the data point 2.50 inches may be considered an outlier based on the Z-Score criterion.
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Bacteria triples every 3 hours. If there are 450 bacteria at t=0
min how many after 200 min
Therefore, the number of bacteria after 200 minutes can be found by multiplying the initial number of bacteria (450) by the tripling factor (3) raised to the power of 2 (for the full cycles) and multiplying by the remaining fraction of the tripling factor for the partial cycle.
Since the bacteria triples every 3 hours, we can calculate the number of tripling cycles that have occurred in 200 minutes. Since 3 hours is equivalent to 180 minutes, there are 2 full cycles in 200 minutes. To calculate the remaining fraction of the tripling cycle, we divide the remaining time (20 minutes) by the length of a single cycle (180 minutes). The fraction is 20/180, which simplifies to 1/9.
Now, we can calculate the number of bacteria after 200 minutes. We start with the initial number of bacteria, which is 450, and multiply it by the tripling factor (3) raised to the power of the number of full cycles (2). This accounts for the full cycles. Then, we multiply this result by the remaining fraction of the tripling factor (1/9) to account for the partial cycle.
Therefore, the number of bacteria after 200 minutes can be calculated as follows:
Number of bacteria = 450 * (3^2) * (1/9) = 450 * 9 * (1/9) = 450
Hence, after 200 minutes, there will still be 450 bacteria.
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Find the volume of a pyramid with a square base, where the perimeter of the base is 8.7\text{ ft}8.7 ft and the height of the pyramid is 9.2\text{ ft}9.2 ft. Round your answer to the nearest tenth of a cubic foot.
Answer:
696.348
Step-by-step explanation:
Answer:
14.5
Step-by-step explanation:
What is the one shape of cross-section we could create by slicing the cylinder perpendicular to its base
Answer:
rectangle
Step-by-step explanation:
A rectangle represents a figure with four sides having equal opposite sides and the inside angle between the two adjacent sides is ninety degree.
When any plane is made to pass through the cylinder creating a cross section which is perpendicular to its base, it results a two-dimensional cross sectional shape of a rectangle.
10.1 X1,..., Xn is an iid sequence of exponential random variables, each with expected value 5. (a) What is Var[M9(X)], the variance of the sample mean based on nine trials? (b) What is P[X * 1 > 7] . the probability that one outcome exceeds 7? (c) Use the central limit theorem to es- timate P[M * 9(X) > 7] . the probability that the sample mean of nine trials exceeds 7.
Solution:
a) Var[M9(X)] = 25/9.
b) P[X > 7] = exp(-7/5).
c) P[Z < 2/5] = 0.3446.
Given information: X1, . . . , Xn is an iid sequence of exponential random variables, each with expected value 5.
(a) We know that the sample mean based on nine trials is M9(X). Now, to calculate the variance of the sample mean based on nine trials, Var[M9(X)], we can use the formula for the variance of a sample mean, which is:
Var[M9(X)] = Var[X]/9 .
Since X is an exponential random variable with expected value 5, its variance is 5^2 = 25. Thus,Var[M9(X)] = 25/9.
(b) To find P[X * 1 > 7], we can use the probability density function of an exponential distribution, which is given by:
f(x) = 1/5 exp(-x/5), x > 0 .
Now, using this probability density function, we have:
P[X > 7] = ∫7∞f(x) dx= ∫7∞ 1/5 exp(-x/5) dx.
Using integration by substitution, with u = x/5 and du = (1/5)dx, we have:
P[X > 7] = ∫7/5∞ exp(-u) du= exp(-7/5).
(c) Since we know that X1, . . . , Xn is an iid sequence of exponential random variables, each with expected value 5 and variance 25, we can apply the central limit theorem. According to the central limit theorem, the sample mean M9(X) is approximately normally distributed with mean 5 and variance 25/9. Thus, we have:
P[M9(X) > 7] = P[Z > (7-5)/(5/3)] where Z is a standard normal random variable. This simplifies to:
P[M9(X) > 7] = P[Z > 2/5] = 1 - P[Z < 2/5]
Using the standard normal distribution table or calculator, we get:
P[Z < 2/5] = 0.6554P[M9(X) > 7] = 1 - P[Z < 2/5] = 1 - 0.6554 = 0.3446.
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The ordered pair (a, b) gives the location of point P on the coordinate plane. The values of a and b have the same sign. Neither a nor b is 0.
It could be in quadrant I or III
Since I is all positive and III is all negative
43 - u = 69
how much is u?
Answer:
69-43= 26
therefore "U" is worth 26 U=26
Step-by-step explanation:
Answer:
its 26, because whaen u subtract 43 from 69 u will get 26
Function f is a quadratic function passing through the points (-4,0),(0,–12) and (3,0). Function g is modeled by the graph. Over which interval are both functions negative?
Answer:
Open points on 1 and 3 with ray connecting them
Step-by-step explanation:
Because it's right.
Answer:
Step-by-step explanation:
with a sample mean of 15, a population average of 20, and a standard error of the mean of 10, calculate the observed z value.
a. -0.5
b. 0.5
c. 2.0
d. -2.0
The observed z value can be calculated as (sample mean - population mean) / standard error of the mean, which in this case is -0.5. Hence, option a is correct.
The observed z value measures how many standard errors the sample mean is away from the population mean. The sample mean is 15 and the population mean is 20 and the standard error mean is 10.
Subtracting the population mean from the sample mean, we get -5. Dividing -5 by 10, we find that the observed z value is -0.5. Therefore, the observed z value is -0.5, which corresponds to option (a).
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Giving Brainiest (No Links) ~Repost Cause last one no one answered~
(If you don't show work for each I comment "No Show work")
<3
Determine the slope of a line that passes through the following sets of points. Show your work.
(1, 1) and (4, 5)
(2, -1) and (4, 13)
(1, 18) and (-9, -2)
Slope is the change in y over the change in x
1. (1,1) and (4,5)
Slope = (5-1)/(4-1)
Slope = 4/3
2. (2,-1) and (4,13)
Slope = (13 - -1)/ (4-2)
Slope = 14/2
Slope = 7
3. (1,18) and (-9,-2)
Slope = (-2 -18)/(-9 -1)
Slope = -20/-10
Slope = 2
Please help me on this question
Answer:
it would be A
Step-by-step explanation:
you have to find angle C first so you would subtract 105 from 180 to get 75 and then to find angle B you would add 75 and 55 to get 130 and since its a triangle it always has 180 degrees so do 180 - 130 to get 50
Find ZUWT helpp plzz dont put links
Answer:
i can't see clear
Jon increased his trading card collection by 5 cards. He originally had 45 cards. What's the percent increase
Answer:
About 11%
Step-by-step explanation:
45 divided by 5 = 9
100 divided by 9 = 11.11111 (It goes on forever)
Express (–1+iV3) and (-1 – iV3) in the exponential form to show that:
2nnt (-1+iV3)n +(-1 – iV3)n = 2n+1cos 3
The proof of 2ⁿ (-1 + i√3)ⁿ + (-1 - i√3)ⁿ can be expressed as 2ⁿ⁺¹cos(πn/3) is proved below.
To express (-1 + i√3) and (-1 - i√3) in exponential form, we can use Euler's formula, which states that [tex]e^{(i\theta)[/tex] = cos(θ) + isin(θ).
Let's start with (-1 + i√3):
(-1 + i√3) = 2 x (cos(π) + i x sin(π/3))
Now, let's simplify (-1 - i√3):
(-1 - i√3) = 2 (cos(π) - isin(π/3))
Therefore, we have:
(-1 + i√3) = 2 e^(iπ/3)
(-1 - i√3) = 2 e^(-iπ/3)
Now, let's substitute these exponential forms into the expression:
2ⁿ (-1 + i√3)^n + (-1 - i√3)^n
= 2ⁿ(2 e^(iπ/3))^n + (2 e^(-iπ/3))^n
= 2ⁿ⁺¹ e^(iπn/3) + 2^(n+1) e^(-iπn/3)
Using Euler's formula again, we know that [tex]e^{(i\theta)} + e^{(-i\theta)[/tex] = 2cos(θ).
Therefore, we can rewrite the expression as:
2ⁿ⁺¹ (cos(πn/3) + cos(-πn/3))
= 2ⁿ⁺¹(cos(πn/3) + cos(πn/3))
= 2ⁿ⁺¹ 2 cos(πn/3)
= 2ⁿ⁺¹cos(πn/3)
So, we have shown that 2ⁿ (-1 + i√3)ⁿ + (-1 - i√3)ⁿ can be expressed as 2ⁿ⁺¹cos(πn/3).
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Help with question 3 please I’ll give brainlest
Answer: b
Step-by-step explanation: When you rotate it around the axis (think of it as a pole) it will become a cylinder with radius 5
Let z = 3+ bi and w = a + bi where a, b E R. Without using a calculator, (a) determine and hence, b in terms of a such that is real.
The values of b = 0 or a = -3 - such that zw is real, letting z = 3+ bi and w = a + bi where a, b E R.
To determine the value of b in terms of a such that zw is real, we first need to find zw. Using the distributive property, we have:
zw = (3 + bi)(a + bi)
zw = 3a + 3bi + abi - b^2
To make zw real, the imaginary part must be equal to zero. Therefore, we have:
3b + ab = 0
b(3 + a) = 0
Since b cannot be equal to zero (otherwise z and w would be real), we have:
a = -3
Therefore, b = 0 or a = -3 - this is the value of a such that zw is real.
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