The relation "a-b a mod 4 = b mod 4" means that for any two numbers a and b, if their difference is divisible by 4, then they belong to the same equivalence class. To find the equivalence class of -, we need to find all the numbers that have the same modulus as - when divided by 4.
We can start by listing out some numbers with the same modulus as -. For example, we have {-9, -5, -1, 3, 7, ...}, since these numbers are all congruent to -1 mod 4. Similarly, we have {0, 4, 8, 12, ...} for numbers that are congruent to 0 mod 4, and {1, 5, 9, 13, ...} for numbers that are congruent to 1 mod 4.
Therefore, the equivalence class of - is {-9, -5, -1, 3, 7, ...}, which contains all the negative numbers that are congruent to -1 mod 4.
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Suppose a bag contains 4 white chips and 6 black chips. What is the probability of randomly choosing a black chip, not replacing it, and then randomly choosing another black chip?
The probability of choosing a black chip and then another black chip is,
⇒ 1/3
Since, There are 4 + 6 = 10 chips in the bag.
And, 6 of them are black .
Hence, The probability that the first chip chosen will be black is,
⇒ 6/10
⇒ 3/5.
After that, there is one black chip less in the bag, so there are 9 chips in the bag, 5 of them are black.
Hence, The probability that the second chip chosen will be black is,
⇒ 5/9
Now, Multiply the probabilities to find the probability that the first chip will be black and the second chip will be black:
⇒ 3/5 × 5/9
⇒ 1/3
Thus, The probability of choosing a black chip and then another black chip is,
⇒ 1/3
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If the slope of a line is 5/8 and the run of a triangle connecting two points on the line is 16, what is the rise?
The rise of the line that has a slope of 5/8 and a run of 16 is calculated as: 10.
What is the Slope of a Line?The slope of a line is defined as the ratio of the rise of the line to the run of the line. This can also be defined as change in y over the change in x of a line.
Given the following:
Slope of a line (m) = 5/8
Run of the triangle = 16
Rise = x
Using the slope formula, we have:
Slope of a line (m) = rise/run
5/8 = x/16
Solve for x:
x = (5 * 16) / 8
x = 80/8
x = 10
Therefore, we can conclude that the rise is 10.
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The functions f(x)=−34x+214 and g(x)=(12)x+1 are shown in the graph. What are the solutions to −34x+214=(12)x+1? Select each correct answer.
The graphs cross at x=-1 and x=1. Those are the solutions to to the equation
How to explain the graphWe know that, If two functions are equal then there solution is the intersection point of the curves.
When we determine the graph the intersection points are (0,2) and (1,1.25).
The values of x of the intersection points are the solutions of the system
Using a graphing tool, there are two intersection points and therefore the solutions are x = -1 and x [ 1.
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calculate the moment of inertia when an object's mass is 12 kg and the mass is distributed 4 meters from the axis of rotation.
To calculate the moment of inertia of an object, you need to know its mass and the distance it is from the axis of rotation. In this case, the object has a mass of 12 kg and is distributed 4 meters from the axis of rotation. The formula to calculate the moment of inertia is I = mr^2, where the moment of inertia, m is the mass, and r is the distance from the axis of rotation.
Using this formula, we can calculate the moment of inertia of the object:
I = 12 kg x (4 m)^2
I = 192 kgm^2
Therefore, the moment of inertia of the object is 192 kgm^2.
To calculate the moment of inertia for an object, you can use the following formula:
Moment of Inertia (I) = Mass (m) × Distance² (r²)
Given the object's mass is 12 kg and the mass is distributed 4 meters from the axis of rotation, we can plug these values into the formula:
I = 12 kg × (4 m)²
Now, we'll square the distance:
I = 12 kg × 16 m²
Finally, multiply the mass and the squared distance:
I = 192 kg·m²
So, the moment of inertia of the object is 192 kg·m².
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3(n + 5) is equivalent to (n + p)3.
Answer:
[tex]3(n + 5) = (n + 5)3[/tex]
So p = 5.
PLEASE HELP ME!!! + points
the gas station and hotel are both on a highway, and the distance between them is about 100 miles. john has to drive to the gas station or hotel, which are both 60 miles away from his farmhouse, to get on the highway. he wants to build a road to the highway using the shortest distance possible from his farmhouse. enter the shortest distance possible from his farmhouse. enter the shortest distance, in miles, from the farmhouse, to the highway
The shortest distance from John's Farm house to the high way is 116.6miles. This is solved using Pythagorean theorem.
What is the explanation?When triangulated, we find three possible distances:
D - the Gas Station to the Hotel = 100miles
P - The gas station to the farm house = 60 miles
x - shortest distance between farm ouse to the highway
In Pythagorean format:
x² = 60² + 100²
x² = 3600 +10000
x = √13600
x [tex]\approx[/tex] 116.6 Miles.
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[1] Find the probabilities of the followings. (a) toss five coins and find three heads and two tails. (b) the face ‘6’ turns up 2 times in 3 rolls of a die as (6 + other + 6). (c) 46% of the population approve of the president’s performance. What is the probability that all four individuals in a telephone toll disapprove of his performance? (d) take five cards from a card deck and find ‘full house.’
The total number of ways to choose five cards from a deck of 52 cards is (52 choose 5) = 2,598,960. Therefore, the probability of getting a full house is 3,744/2,598,960 = 0.00144.
(a) The total number of possible outcomes when tossing five coins is 2^5 = 32. The number of ways to get three heads and two tails is the number of ways to choose three heads out of five times the number of ways to choose two tails out of five, which is (5 choose 3) x (5 choose 2) = 10 x 10 = 100. Therefore, the probability of getting three heads and two tails is 100/32 = 0.3125.
(b) The probability of getting a '6' on a single roll of a die is 1/6. The probability of not getting a '6' on a single roll of a die is 5/6. The probability of getting '6 + other + 6' in three rolls of a die is (1/6) x (5/6) x (1/6) x 3 = 5/216. Therefore, the probability of getting '6 + other + 6' two times in three rolls of a die is (5/216)^2 x (211/216)^1 x (3 choose 2) = 0.0029.
(c) The probability of an individual disapproving of the president's performance is 1 - 0.46 = 0.54. The probability that all four individuals in a telephone poll disapprove of his performance is 0.54^4 = 0.054.
(d) A full house consists of three cards of one rank and two cards of another rank. The number of ways to choose the rank for the three cards is 13, and the number of ways to choose the three cards of that rank is (4 choose 3) = 4. The number of ways to choose the rank for the two cards is 12 (since one rank has already been chosen), and the number of ways to choose the two cards of that rank is (4 choose 2) = 6. Therefore, the number of ways to get a full house is 13 x 4 x 12 x 6 = 3,744. The total number of ways to choose five cards from a deck of 52 cards is (52 choose 5) = 2,598,960. Therefore, the probability of getting a full house is 3,744/2,598,960 = 0.00144.
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Both of these groups started with 22, 6-sided dice and followed the same procedure for removing dice until they had no dice left. How could they end up with such different scatterplots? Does it make sense that one set of data could look so possibly linear while the other does not?
It is possible for one set of data to have a scatterplot that appears linear while the other does not, even if both groups started with the same number of dice and followed the same removal procedure.
This is because the way the dice were removed could have been different between the two groups, leading to different patterns of results. Additionally, other factors such as the order in which the dice were removed or the number of trials conducted could also affect the resulting scatterplot.
Ultimately, the scatterplot is a visual representation of the relationship between the variables being measured, and it can take on many different forms depending on the specific data and conditions being analyzed.
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Is the number of sit-ups Anna does proportional to the time she spends doing them?
No, the number of sit-ups Anna does, is not proportional to the time she spends doing them.
When Anna starts doing sit-ups for her first triathlon, she does a sit-up every 22 seconds. But we know that as she gets tired, each sit-up takes longer and longer to do. The situps may take 40 seconds or 75 seconds as she gets more tired.
As we can see that there is no constant rate at which she gets tired and take more seconds to do sit-ups. In order to be proportional, the increasing or decreasing rate should be constant. We can see that the time she spends doing them is not increasing or decreasing at a constant rate along with the number of sit-ups she is doing.
Therefore, the number of sit-ups Anna does is not proportional to the time she spends doing them.
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The complete question is "Anna does sit-ups to get ready for her first triathlon. When she starts, she does a sit-up every 22 seconds. But, as she gets tired, each sit-up takes longer and longer to do. Is the number of sit-ups Anna does proportional to the time she spends doing them? "
Classifiy the triangle by using its side lengths
Answer: Where is the triangle?
Step-by-step explanation: Brainliest pls:)
"needed in 20 mins i will rateyour answerQuestion 1 dy Find for the following functions, simp dx 9 (i) y = 2x5 – 4x3 + 9/4x^(3)
The derivative of the given function is:
y' = 10x^4 - 12x^2 + (27/4)x^2
To find the derivative of the given function, y = 2x^5 - 4x^3 + (9/4)x^3, we can use the power rule and the sum rule of differentiation.
The power rule states that the derivative of x^n is nx^(n-1), where n is any real number.
Using the power rule, we can find the derivative of each term in the function:
dy/dx (2x^5) = 10x^4
dy/dx (-4x^3) = -12x^2
dy/dx ((9/4)x^3) = (27/4)x^2
Using the sum rule, we can add the derivatives of each term to find the derivative of the function:
dy/dx (y) = dy/dx (2x^5) + dy/dx (-4x^3) + dy/dx ((9/4)x^3)
dy/dx (y) = 10x^4 - 12x^2 + (27/4)x^2
Therefore, the derivative of the given function is y' = 10x^4 - 12x^2 + (27/4)x^2.
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Find the area of each triangle. Round answers to the nearest tenth.
7)
8)
9)
3.2 mi
8.7 yd
12 yd
square yards
6 mi
square miles
10)
4.1 ft
9.4 in
8.3 ft
square feet
6.8 in
square inches
7. The area of the triangle is 52.2 yd².
8. The area of the triangle is 17.02 ft².
9. The area of the triangle is 9.6 mi².
10. The area of the triangle is 31.96 in².
What is the area of each of the triangle?
The area of each triangle is calculated by applying the following formula as shown below;
Area = ¹/₂bh
where;
b is the base of the triangleh is the height of the triangle7. The area of the triangle is calculated as
A = ¹/₂ x 12 yd x 8.7 yd
A = 52.2 yd²
8. The area of the triangle is calculated as
A = ¹/₂ x 8.3 ft x 4.1 ft
A = 17.02 ft²
9. The area of the triangle is calculated as
A = ¹/₂ x 6 mi x 3.2 mi
A = 9.6 mi²
10. The area of the triangle is calculated as
A = ¹/₂ x 9.4 in x 6.8 in
A = 31.96 in²
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Which one of the following is true regarding the use of the mode, mean, and median for different levels of measurement?
The mode, mean, and median are all valid measures of central tendency for interval and ratio level data. For nominal level data, only the mode is appropriate, while for ordinal level data, both the mode and median can be used, but the mean is not recommended as it assumes equal intervals between categories.
The correct statement regarding the use of the mode, mean, and median for different levels of measurement is:
The mode can be used for nominal and ordinal levels of measurement, the median is used for ordinal, interval, and ratio levels, while the mean is used for interval and ratio levels of measurement.
Let's break it down:
1. Mode: applicable to nominal and ordinal levels as it represents the most frequently occurring value in the data.
2. Median: applicable to ordinal, interval, and ratio levels as it represents the middle value when data is arranged in order.
3. Mean: applicable to interval and ratio levels as it represents the average value by summing all data points and dividing by the number of data points.
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Suppose that it is known that on any given day in the month ofmarch there is a 0.3 probability of rain. Find the standarddeviation of rainy days in March.
The standard deviation of rainy days in March is approximately 2.55 days.
To find the standard deviation of rainy days in March, we first need to determine the expected value or the mean number of rainy days in March.
The expected value of a binomial distribution can be found using the formula: E(X) = np, where X is the random variable representing the number of rainy days in March, n is the number of trials (days in March), and p is the probability of success (rain) on a given day.
In this case, n = 31 (number of days in March) and p = 0.3 (probability of rain on any given day in March). Therefore, the expected value of rainy days in March is
E(X) = np = 31 × 0.3 = 9.3
Next, we need to find the variance of the binomial distribution, which is given by the formula: Var(X) = np(1 - p).
Var(X) = 31 × 0.3 × (1 - 0.3) = 6.51
Finally, the standard deviation of rainy days in March is the square root of the variance:
SD(X) = √Var(X) = √6.51 ≈ 2.55
Therefore, the standard deviation of rainy days in March is approximately 2.55 days.
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The point A is shown below.
Reflect A across the x-axis.
Then reflect the result across the y-axis.
Plot the final point.
Important: Only plot the final point in your answer.
8-1
X
5
The coordinate of final point of A after transformation is,
A'' = (0, - 7)
We have to given that;
Coordinate of A = (0, 7)
We know that;
Rule for the across the x - axis is,
(x, y) = (x , - y)
Hence, We get;
Point after transformation is,
A' = (0, - 7)
And, Rule for the across the y - axis is,
(x, y) = (-x , y)
Hence, Point after transformation is,
A'' = (0, - 7)
Thus, The coordinate of final point of A after transformation is,
A'' = (0, - 7)
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in 2020, the population of a city was 1,596,000 . this was a 3.5% increase from 2019. the population is expected to continue to increase at the same rate per year. write a function, f(x) , to model the population, where x is the number of years since 2020.
To model the population of a city, we can use the formula:
f(x) = 1,596,000 * (1 + 0.035)^x
Where x is the number of years since 2020 and 0.035 represents the annual increase rate of 3.5%.
To model the population of a city, we need to take into account the initial population in 2020 and the annual increase rate of 3.5%. The annual increase rate of 3.5% means that the population is expected to grow by 3.5% every year from the previous year's population. To incorporate this growth rate into our model, we can use the formula for compound interest:
A = P * (1 + r)^t
Where A is the final amount, P is the initial amount, r is the annual interest rate, and t is the number of years. In our case, the final amount is the population after x years since 2020, the initial amount is the population in 2020 (1,596,000), the annual interest rate is 3.5%, and the number of years is x.
By substituting the values into the formula, we get:
f(x) = 1,596,000 * (1 + 0.035)^x
This formula can be used to calculate the expected population of the city for any year in the future. For example, if we want to know the population in 2025, we can substitute x = 5 into the formula:
f(5) = 1,596,000 * (1 + 0.035)^5
= 1,825,854
Therefore, we can expect the population of the city to be around 1,825,854 in 2025, assuming the same annual increase rate of 3.5%.
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You invest $2,000 for 3 years at interest rate 6%, compounded every 6 months. What is the value of your investment at the end of the period?
If you invest $2,000 for 3 years at an interest rate of 6%, compounded every 6 months. The value of your investment at the end of the period is $2,397.39.
The interest rate is 6% and it is compounded every 6 months, so the period is 6 months. To calculate the value of the investment at the end of the period, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount.
P = the principal amount (initial investment)
r = the annual interest rate (6%).
n = the number of times the interest is compounded per year (2, since it's compounded every 6 months).
t = the time period in years (3)
Plugging in the numbers, we get:
A = 2,000(1 + 0.06/2)^(2*3)
A = 2,000(1 + 0.03)^6
A = 2,000(1.03)^6
A = $2,397.39
Therefore, the value of your investment at the end of the period is $2,397.39.
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Please help for question 9
a) The linear function giving the cost after x months is given as follows: C(x) = 88 - 8x.
b) The cost of the shoes after 8 months is given as follows: $24.
How to define a linear function?The slope-intercept representation of a linear function is given by the equation presented as follows:
y = mx + b
The coefficients of the function and their meaning are described as follows:
m is the slope of the function, representing the change in the output variable y when the input variable x is increased by one.b is the y-intercept of the function, which is the initial value of the function, i.e., the numeric value of the function when the input variable x assumes a value of 0. On a graph, it is the value of y when the graph of the function crosses the y-axis.Each month, the balance decays by $8.00, hence the slope m is given as follows:
m = -8.
Hence:
y = -8x + b.
When x = 1, y = 80, hence the intercept b is given as follows:
80 = -8 + b
b = 88.
Hence the function is:
C(x) = 88 - 8x.
The cost after 8 months is given as follows:
C(8) = 88 - 8(8) = 88 - 64 = $24.
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What is the equation of a line that is perpendicular to the line y = –23 x – 7 and passes through the point (–4, 2)?
The equation of a line that is perpendicular to the line y = –23x – 7 and passes through the point (–4, 2) is y = x/23 + 50/23.
How to determine an equation of this line?In Mathematics and Geometry, the point-slope form of a straight line can be calculated by using the following mathematical equation (formula):
y - y₁ = m(x - x₁)
Where:
x and y represent the data points.m represent the slope.Since the equation of this line is perpendicular to the line y = –23x – 7, the slope is given by;
Slope, m = -23
m₁ × m₂ = -1
-23 × m₂ = -1
m₂ = -1/-23
Slope, m₂ = 1/23
At data point (-4, 2) and a slope of 1/23, a linear equation for this line can be calculated by using the point-slope form as follows:
y - y₁ = m(x - x₁)
y - 2 = 1/23(x - (-4))
y - 2 = 1/23(x + 4)
y = x/23 + 4/23 + 2
y = x/23 + 50/23
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Sales associates at an electronics store earn different commission
percentages based on the items they sell. The table shows the total
sales and commission earnings for four sales associates at the
electronics store last month.
GIFTING EXTRA POINTS
The required model is c = 0.03d+1.81.
Given are 4 entries, but we just need 2 to plot the relation lets pick the first two, we would be using the equation of a line in the two-point form,
c - c₁ / d - d₁ = c₂ - c₁ / d₂ - d₁
We, put in the points, (673,22) and (3277,101), we get,
c - 22 / d-673 = 0.03
c-22 = 0.03 (d-673)
c-22 = 0.03d-20.19
c-22 / d-673 = 101-22 / 3277-673 = 79 / 2604 = 0.03
c = 0.03d+1.81
Hence, the required model is c = 0.03d+1.81.
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A circular region has a population of about 175,000 people and a population density of about 1318 people per square mile. Find the radius of the region. Round your answer to the nearest tenth.
The radius of the region is 11.55 mile.
We have,
Population = 175,000
So, population density
= 175,000 / 1318
= 132.77
and, the radius using from the population density
Radius = √area / (22/7)
= √1318 x 7/22
= 11.55 mile
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One way to measure a person’s fitness is to measure their body fat percentage. Average body fat percentages vary by age, but according to some guidelines, the normal range for men is 15-20% body fat, and the normal range for women is 20-25% body fat.
The body fat of 25 gym goers was measured by a trainer and the Mean and standard deviation for each group is summarized in table below.
Group Sample Size (n) Average (X-bar) Standard deviation (s)
Women 10 22.29 5.32
Men 15 14.95 6.84
A) What should the Null hypothesis say about the mean body fat percentage of women compared to the mean body fat percentage of males? B) What should the Alternative hypothesis say about the mean body fat percentage of women compared to the mean body fat percentage of males? C) Is the p-value for your test less than 0.05? "yes" or "no" D) At the 0.05 significance level, is there enough evidence to conclude that the mean body fat percentage for women is more than 3% greater than men? "yes" or "no" E) At the 0.01 significance level, is there enough evidence to conclude that the mean body fat percentage for women is more than 3% greater than men? "yes" or "no" F) Does the 95% confidence interval support the alternative hypothesis? "yes" or "no" G) Why or Why not does the 95% confidence interval support the alternative hypothesis?
A) The null hypothesis should say that the mean body fat percentage of women is equal to the mean body fat percentage of men.
B) The alternative hypothesis should say that the mean body fat percentage of women is greater than the mean body fat percentage of men.
C) The p-value for the test cannot be determined without knowing the results of the actual test.
D) Yes, there is enough evidence to conclude that the mean body fat percentage for women is more than 3% greater than men at the 0.05 significance level, because the difference between the means is 7.34% (22.29% - 14.95%) which is greater than 3%.
E) No, there is not enough evidence to conclude that the mean body fat percentage for women is more than 3% greater than men at the 0.01 significance level, because the difference between the means is not significant enough to reject the null hypothesis.
F) Yes, the 95% confidence interval supports the alternative hypothesis because it does not include the null value of 0. The confidence interval for the difference between the means is (1.63%, 12.05%).
G) The 95% confidence interval supports the alternative hypothesis because it provides a range of plausible values for the difference between the means that do not include 0. This means that we can be 95% confident that the true difference between the means is somewhere within the interval, and that the mean body fat percentage for women is likely to be higher than men.
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The probability distribution for a game is shown in the table below.
What is the probability of getting more than 1 point if the game is played one time?
Answer:
3/8
Step-by-step explanatio
the person shows the answer and explanation nice!
A square with sides measuring 5 millimeters each is drawn within the figure shown. A point within the figure is randomly selected.
What is the approximate probability that the randomly selected point will lie inside the square?
5.3%
8.4%
13.3%
18.1%
Answer:
C) 13.3%-------------------------
Area of square with side of 5 mm is:
A = a² = (5 mm)² = 25 mm²Find total area of the figure:
A(total) = A(trapezoid) + A(triangle)A(total) = (b₁ + b₂)h/2 + bh/2A(total) = (14 + 18)(17 - 12)/2 + 18*12/2 = 80 + 108 = 188Find the percent value of the ratio of areas of the square and full figure, which determines the probability we are looking for:
25/188*100% = 13.2978723404 % ≈ 13.3%This is matching the choice C.
The relationship between marketing expenditures (x) and sales (y) is given by the following formula, y = 9x − 0.20x2 + 8. (Hint: Use the Nonlinear Solver tool). What level of marketing expenditure will maximize sales? (Round your answer to 2 decimal places.) What is the maximum sales value? (Round your answer to 2 decimal places.)
Hi! To find the level of marketing expenditure that will maximize sales and the maximum sales value, we can follow these steps:
1. The relationship between marketing expenditure (x) and sales (y) is given by the formula: y = 9x - 0.20x^2 + 8.
2. To maximize sales, we need to find the maximum point of this quadratic function, which can be done by finding the vertex.
3. The vertex formula for a quadratic function is: x = -b / (2a), where a and b are coefficients in the equation (in this case, a = -0.20 and b = 9).
4. Calculate x (marketing expenditure) for the vertex: x = -9 / (2 * -0.20) = -9 / -0.40 = 22.50.
5. Round the marketing expenditure to 2 decimal places: 22.50.
6. Plug the marketing expenditure value (x) back into the sales formula to find the maximum sales value (y): y = 9(22.50) - 0.20(22.50)^2 + 8.
7. Calculate y: y = 202.50 - 0.20(506.25) + 8 = 202.50 - 101.25 + 8 = 109.25.
8. Round the maximum sales value to 2 decimal places: 109.25.
So, the level of marketing expenditure that will maximize sales is $22.50, and the maximum sales value is $109.25.
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how can i prove 1/xy = 1/x * 1/y
To prove that 1/xy = 1/x * 1/y, we can start by multiplying both sides of the equation by xy.
Multiplying both sides of the equation by xy gives us:
1 = xy * 1/x * 1/y
Next, we can simplify the right-hand side by canceling out the x and y terms that appear in both the numerator and denominator:
1 = y/x + x/y
To further simplify this expression, we can multiply both sides by xy:
xy = y^2 + x^2
This equation can be rearranged to get:
x^2 + y^2 = xy
Finally, we can use the formula for the sum of squares:
x^2 + y^2 = (x+y)^2 - 2xy
Substituting this into the previous equation, we get:
(x+y)^2 - 2xy = xy
Simplifying, we get:
(x+y)^2 = 3xy
Taking the square root of both sides, we get:
x+y = sqrt(3xy)
Dividing both sides by xy, we get:
1/xy = 1/x * 1/y
Therefore, we have proven that 1/xy = 1/x * 1/y.
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whats the answer to 12ab x 3cd
1Find 58% of 5.26
2 find 35% of 4.19
3 find 22% of 3.27
4 find 2% of 5.83
5 find 38% of 8.92
1:. 3.0508
2:. 1.4665
3:. 0.7194
4:. 0.1166
5:. 3.3896
Algibra 1, unit 5! Help
Answer: -15
Step-by-step explanation:
x+y=10
y=-x+10
2x+3(-x+10)=45
2x-3x+30=45
-x=15
x=-15
The Fourier transform of the function: f(t) = sin 3t,k≤|t|≤2k 0, |t|
a.-i[(sin2k(w-3)-sink(w-3)/w-3 )-(sin2k(w+3)-sink(w+3)/w-3)]
b.-1/2[(sin2k(w-3)-sink(w-3)/w-3 )-(sin2k(w+3)-sink(w+3)/w-3)]
c.i[(sin2k(w-3)-sink(w-3)/w-3 )-(sin2k(w+3)-sink(w+3)/w-3)]
d.none of the above
Its Fourier transform is also 0.
The Fourier transform of a function f(t) is defined as:
F(w) = (1/√(2π)) ∫[from -∞ to +∞] f(t) e^(-iwt) dt
Let's find the Fourier transform of the given function f(t) = sin 3t, k≤|t|≤2k and 0, |t|>2k.
For k≤|t|≤2k, we can write:
f(t) = sin 3t
= (1/2i) (e^(i3t) - e^(-i3t))
Using the Fourier transform properties, we can write:
F(w) = (1/2i) [∫[from -2k to -k] e^(i3t) e^(-iwt) dt + ∫[from k to 2k] e^(i3t) e^(-iwt) dt]
Applying the integral formula ∫ e^(ax) dx = (1/a) e^(ax) + C, we get:
F(w) = (1/2i) [(1/i(3-w))(e^(i(3-w)2k) - e^(i(3-w)k)) + (1/i(3+w))(e^(i(3+w)k) - e^(i(3+w)2k))]
Simplifying the above expression, we get:
F(w) = (1/2) [(sin(2kw-3) - sin(kw-3))/(kw-3) + (sin(kw+3) - sin(2kw+3))/(kw+3)]
For |t|>2k, f(t) = 0. Thus, its Fourier transform is also 0.
Therefore, the correct option is d. none of the above.
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