Answers
Step-by-step explanation:
A. Right
B. Acute
C. Right
D. Obtuse
Related Questions
6. Consider the trigonometric equation sin x + 2 = 0. Explain why this equation would have no solutions. [C-2]
Answers
The given trigonometric equation is sin x + 2 = 0.
It is important to note that sine values range from -1 to 1 and never exceed those bounds. Thus, it can be determined that sin x + 2 will never equal zero.This is because the lowest possible value of sine is -1, which is not equal to zero. When 2 is added to that value, the sum is still negative. Therefore, the equation sin x + 2 = 0 has no solutions.
A trigonometric equation is one that has a variable and a trigonometric function. For instance, sin x + 2 = 1 is an illustration of a mathematical condition. The equations can be as straightforward as this or more complicated than that, such as sin2 x – 2 cos x – 2 = 0.
The six mathematical capabilities are sine, secant, cosine, cosecant, digression, and cotangent. The trigonometric functions and identities are derived by referencing a right-angled triangle as a reference: Sin is the opposite side or the hypotenuse.
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pls pls pls, I beg you to answer this question only if you know the correct answer please please please I beg you
Answers
Answer:
36 cubic feet
Step-by-step explanation:
Answer:
To find volume use the solution of l x w x h
Length x Width x Height
Step-by-step explanation:
16. 2 x 3 x 6 = 36 cubic feet
???. 5/8 x 3/4 x 2 = 15/16
???. 2 x 1 1/4 x 1 1/2 = 3 3/4 cubic inches
25.
1 = 4.89 / 3 = 1.63
3 = $4.89
9 = 4.89 + 4.89 + 4.89 = 14.67
10 = 14.67 + 1.63 = $16.30
???. Interquartile Range = Q3 - Q1
65 - 62 = 3
If i remember correctly
Ling measured a shopping center and made a scale drawing. The scale of the drawing was 1 millimeter: 3 meters. The actual width of the parking lot is 42 meters. How wide is the parking lot in the drawing?
Answers
Answer:
14
Step-by-step explanation:
42:?
3:1
42/3=14
42:14
3:1
A shipment of 5 boxes of toys weighed
34.53 pounds. Each box measures the same
weight, so what's the weight of 1 box?
Answers
Answer:
Each box weighs 6.906 pounds.
Step-by-step explanation:
We know the total weight of the boxes. We also know how many boxes we have. To find the weight of one box, we want to divide the total weight by the total number of boxes.
34.53 / 5
= 6.906
Each box weighs 6.906 pounds.
Hope this helps!
Which values of N and p define a random graph ensemble G(N, p) with average degree (k) = 40 and variance of the degree distribution o2 = 50? = Select one: = a. p = 0.25, N = 501 b. p = 1/10, N = 401 p = 1/5, N = 501 = = C. = d. None of the above.
Answers
The values of N and p that define the random graph ensemble G(N, p) with an average degree (k) of 40 and a variance of the degree distribution (σ²) of 50 are N = 201 and p = 0.2.
The values of N and p that define a random graph ensemble G(N, p) with an average degree (k) of 40 and a variance of the degree distribution (σ²) of 50, we can use the following formulas:
k = (N-1) × p
σ² = (N-1) × p × (1-p)
Plugging in the given values:
k = 40
σ² = 50
We can solve these equations to find the values of N and p:
From the first equation:
40 = (N-1) × p
From the second equation:
50 = (N-1) × p × (1-p)
By substituting the value of (N-1) × p from the first equation into the second equation, we can solve for p.
40 = 50 × (1-p)
1-p = 40/50
1-p = 0.8
p = 1 - 0.8
p = 0.2
Now, we can substitute the value of p back into the first equation to solve for N:
40 = (N-1) × 0.2
200 = N-1
N = 200 + 1
N = 201
Therefore, the correct values of N and p that define the random graph ensemble G(N, p) with an average degree (k) of 40 and a variance of the degree distribution (σ²) of 50 are N = 201 and p = 0.2.
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Please help 6th grade math please please help i will give brainliest
Answers
Median:26
Greatest value:34
liam is making chocolate chip cookies. The recipe calls for 1 cup of sugar for every 3 cups of flour. Liam only has 2 cups of flour. How much sugar does liam use?
Answers
1/5 cups of sugar.
have a nice day
As sales manager for Montevideo Productions, Inc., you are planning to review the prices you charge clients for television advertisement development.
You currently charge each client an hourly development fee of $2,500. With this pricing structure, the demand, measured by the number of contracts.
Montevideo signs per month, is 15 contracts. This is down 5 contracts from the figure last year, when your company charged only $2,000.
Construct a linear demand equation in the form q= ap + b where the number of contracts q is given as a function of the hourly fee p Montevideo charges for development.
Give a formula for the total revenue obtained by charging $p per hour.
The costs to Montevideo Productions are estimated as follows:
Fixed costs: $120,000 per month and Variable costs: $80,000 per contract
Express Montevideo Productions’ monthly cost as a function of the hourly production charge p.
Express Montevideo Productions’ monthly profit as a function of the
hourly development fee p and hence the price it should charge to maximize the profit.
The XYZ Clothing Company manufactures football boots for sale to
College/University bookstores, in Trinidad. Football boots are in runs of up to 500. It cost (in dollars) for a run of x football boots is:
(x) = 3,000 + 8x + 0.1x2 0 ≤ x ≤ 500
XYZ Clothing sells the football boots at $ 120 each.
(a) How many football boots does XYZ have to sell to breakeven?
(b) How many football boots does XZY have to sell to make maximum profit?
(c) On a labelled pair of axes, sketch the XYZ’s profit function. Label all important points.
Answers
(a) The number of football boots XYZ Clothing needs to sell to break even can be found by solving the quadratic equation.
(b) To maximize profit, XYZ Clothing needs to sell 1120 football boots.
(c) Plotting relevant points on a graph, we can sketch XYZ Clothing's profit function, indicating important points.
(a) To find the number of football boots XYZ Clothing needs to sell to break even, we set the profit function equal to zero:
Profit = Revenue - Cost
Since the revenue is given as $120 per football boot and the cost function is provided as C(x) = 3,000 + 8x + 0.1x^2, the profit function is:
Profit = 120x - (3,000 + 8x + 0.1x^2)
Setting the profit function equal to zero, we have:
0 = 120x - (3,000 + 8x + 0.1x^2)
Simplifying the equation, we get:
0 = 112x - 0.1x^2 - 3,000
To find the number of football boots needed to break even, we solve the quadratic equation:
0.1x^2 - 112x + 3,000 = 0
Solving this equation will give us the value of x, which represents the number of football boots XYZ Clothing needs to sell to break even.
(b) To find the number of football boots XYZ Clothing needs to sell to maximize profit, we need to determine the vertex of the profit function. The profit function is the same as in part (a):
Profit = 120x - (3,000 + 8x + 0.1x^2)
To find the vertex, we can use the formula x = -b/2a, where a = 0.1 and b = 112.
x = -(-112) / (2 * 0.1)
x = 1120
So, XYZ Clothing needs to sell 1120 football boots to maximize profit.
(c) To sketch the profit function on a pair of axes, we can plot the points that are relevant to the problem. We know that the profit function is given by:
Profit = 120x - (3,000 + 8x + 0.1x^2)
We can plot the following points:
Breakeven point: (x, 0) where x is the number of football boots needed to break even.
Maximum profit point: (1120, Profit(1120)) where Profit(1120) is the maximum profit obtained when 1120 football boots are sold.
Additionally, we can plot a few more points to get an idea of the shape of the profit function.
By connecting these points, we can sketch the profit function on the axes, indicating the relevant points and labeling them accordingly.
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Find the perimeter and thank
Answers
Answer:
12
..................................................
Step-by-step explanation:
2 + 2 + 1 + 2 + 1 + 4 = 12
Answer:
12 centimeters
Step-by-step explanation:
The double dashes are both the same number. Since the base is 4cm and they have to be two of the same number, the double dashes are each 2cm long. Meanwhile, the height is 2cm, and the single dashes are two of the same numbers (but not 2cm, because that's what the double dashes are). So, each dash must be 1cm long. When you add them up: [tex]4+2+2+2+1+1=12[/tex]
The dot plot shown displays the amount of money,
in millions of dollars, that different companies
spend on television advertising in one year. Which
of the following statements describe the data set?
Answers
Answer:
44
Step-by-step explanation:
because it a millon and not a bilolon
Which number represents the probability of an event that is impossible to
occur?
Answers
Answer:
An impossible event has a probability of 0. A certain event has a probability of 1.
can someone please help me with this question?? (no links!!!)
Answers
Answer:
960
Step-by-step explanation:
it says use 3 for pi, so you plug the values into given equation
V = 3·(4)²·20
r is 4 because it is half of 8.
Then multiply
3·16·20 = 48·20 = 960
Answer:
960 cm
Step-by-step explanation:
8/2 = 4 (radius)
3 (since we're using 3 for pi) * 4^2 * 20
3 * 16 * 20
48 * 20
= 960 cm
I hope this helps heh :)
Assume that a sample is used to estimate a population proportion p. Find the 99.9% confidence interval for a sample of size 176 with 118 successes. Enter your answer as an open-interval (i.e., parentheses) using decimals (not percents) accurate to three decimal places. 99.9% C.1. =
Answers
The 99.9% confidence interval for a sample of size 176 with 118 successes is (0.558, 0.778).
The formula for finding the confidence interval for a sample proportion is given as follows:
Confidence interval = sample proportion ± zα/2 * √(sample proportion * (1 - sample proportion) / n)
Where,
zα/2 is the z-value for the level of confidence α/2,
n is the sample size,
sample proportion = successes / n
Here, level of confidence, α = 99.9%, so α/2 = 0.4995. The value of zα/2 for 0.4995 can be found from the z-table or calculator and it comes out to be 3.291.
Putting all the values in the formula, we get:
Confidence interval = 0.670 ± 3.291 * √(0.670 * 0.330 / 176)
= (0.558, 0.778) (rounded to three decimal places and put in parentheses)
Thus, the 99.9% confidence interval for a sample of size 176 with 118 successes is (0.558, 0.778).
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A person draws balls twice, with replacement, from a urn which contains only one red ball and one black ball. (a) (5 pt) Using the notations R and B to denote the red balls and black balls, respectively, list all the elements of the sample space S. (b) (5 pt) Which elements are in the event A that the person draws one red ball and one black? (c) (5 pt) What is P(A)?
Answers
(a) All the elements of S are : S = {RR, RB, BR, BB},
(b) Elements in event A are {RB and BR},
(c) The probability of "event-A", which is the person drawing one "red-ball" and one "black-ball", is 1/2.
Part (a) : The "sample-space" (S) consists of all possible outcomes when drawing two balls with replacement. In this case, there are four possible outcomes : S = {RR, RB, BR, BB},
Part (b) : The "event-A" represents the person drawing one red-ball and one black-ball. The elements in event A are {RB and BR},
Part (c) : To calculate the probability P(A), we need to determine the ratio of favorable outcomes (elements in A) to the total number of possible outcomes (elements in S).
P(A) can be written as : (Number of favorable outcomes)/(Total number of possible outcomes),
Substituting the values from part(a) and part(b),
We get,
P(A) = 2/4
P(A) = 1/2
Therefore, the required probability is 1/2.
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Determine a condition on |x - 4| that will assure that:
(a)∣∣x−2∣∣<21,
(b)∣∣x−2∣∣<10−2.
Answers
Given the expression |x - 4|, condition on |x - 4| that will assure that:(a)|x - 2| < 2/1(b)|x - 2| < 0.01
Given expression |x - 4|, the two possible values are: x - 4 if x > 4 -(x - 4) if x < 4Let us solve each part of the question separately:
(a)Part (a) can be expressed as follows:|x - 2| < 2/1Subtracting 2 from both sides of the in equality |x - 2| - 2 < 0Adding 4 to both sides of the inequality. |x - 2| - 2 + 4 < 0|x - 2| - 2 + 4 = |x - 4| < 0Since it is impossible to have an absolute value less than 0, therefore there is no solution.
(b)Part (b) can be expressed as follows:|x - 2| < 0.01 Subtracting 2 from both sides of the inequality |x - 2| - 2 < -0.01Adding 4 to both sides of the inequality. |x - 2| - 2 + 4 < -0.01|x - 2| - 2 + 4 = |x - 4| < -0.01Since it is impossible to have an absolute value less than 0, therefore there is no solution.
Thus, there are no solutions for the given conditions.
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Solve for x. (log510 - log52) (log61296) = log₂(x - 5)²
Answers
The solution to the equation (log510 - log52) (log61296) = log₂(x - 5)² is x = 9. To solve the given equation, let's break it down step by step.
First, we simplify the left side of the equation using logarithmic properties. Using the property log(a) - log(b) = log(a/b), we can rewrite (log510 - log52) as log5(10/2), which simplifies to log5(5) or 1.
Next, we simplify the right side of the equation. Using the property logₐ(b²) = 2logₐ(b), we can rewrite log₂(x - 5)² as 2log₂(x - 5).
Now our equation becomes 1 * log61296 = 2log₂(x - 5).
Since log61296 is the logarithm base 6 of 1296, which is 4, we can simplify the equation further to 4 = 2log₂(x - 5).
Dividing both sides by 2, we have 2 = log₂(x - 5).
Now we can rewrite this equation in exponential form: 2² = x - 5.
Simplifying, we get 4 = x - 5.
Adding 5 to both sides, we find x = 9.
Therefore, the solution to the equation (log510 - log52) (log61296) = log₂(x - 5)² is x = 9.
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What is the difference, in meters, between the length of the longest line and the length of the shortest line?
Answers
Answer:
[tex]Range = 3.169m[/tex]
Step-by-step explanation:
Given
See attachment for complete question
Required
Determine the difference between the shortest and the longest
This question implies that we calculate the range.
[tex]Range = Longest - Shortest[/tex]
From the table, we have:
[tex]Longest = 8.7m[/tex]
[tex]Shortest = 5.531m[/tex]
So, we have:
[tex]Range = 8.7m- 5.531m[/tex]
[tex]Range = 3.169m[/tex]
Consider the case of equally likely transmission of multilevel signaling over AWGN channel with Variance , and mean . The signaling used is M=4 with the data rate of . The symbols are assigned the pulse values: -2v -1v +2v +1v. Develop the expression for the optimum threshold values. Write the expression of correct transmission when symbol level -2v was transmitted. Write the expression for the total probability of error. Evaluate the total probability of error when =0.40 and =0.
Answers
In multilevel signaling over an Additive White Gaussian Noise (AWGN) channel, the received signal can be represented as:
Y = X + N
where Y is the received signal, X is the transmitted signal, and N is the AWGN with zero mean and variance σ^2.
In this case, M = 4, which means we have 4 symbols: -2v, -v, +2v, +v. The pulse values assigned to these symbols are: -2v, -v, +2v, +v.
To find the optimum threshold values, we need to consider the decision regions between adjacent symbols. Let's denote the threshold values as T1, T2, and T3, corresponding to the decision boundaries between -2v and -v, -v and +2v, and +2v and +v, respectively.
For correct transmission of the symbol -2v, the received signal Y should be greater than T1 and less than or equal to T2. Mathematically, this can be expressed as:
T1 < Y ≤ T2
The probability of correct transmission for the symbol -2v can be obtained by integrating the probability density function (PDF) of the received signal Y over the region T1 < Y ≤ T2.
Now, let's find the expression for the total probability of error. The total probability of error (P_e) can be obtained by summing the probabilities of error for each symbol. In this case, we have four symbols, so the expression for P_e is:
[tex]P_e = P_error(-2v) + P_error(-v) + P_error(+2v) + P_error(+v)[/tex]
where P_error(-2v) is the probability of error for symbol -2v, P_error(-v) is the probability of error for symbol -v, and so on.
Finally, to evaluate the total probability of error when σ^2 = 0.40 and v = 0, we need more information. Specifically, we need the signal-to-noise ratio (SNR) or the value of σ^2 in order to proceed with the calculation.
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98 = x + 55 what is it????
Answers
Answer:
x = 43
Step-by-step explanation:
x = 98 - 55
x = 43
Answer:
x=43
Step-by-step explanation:
have a nice day and stay safe:)
The box-and-whisker plots below show the test scores for Mr. Scott's three math classes.
Based on this information, with which class or classes does Mr. Scott most need to review the material covered on the test?
A. third period
B. first period
C. third and fourth periods
D. fourth period
Answers
Answer:
A. Third period
Step-by-step explanation:
- hope this helped!
Answer:
3rd
Step-by-step explanation:
There are 30 students in Mrs. Rodriguez’s class. 20% got an A on the test. How many students got an A?
Answers
Answer
6.
Steps
30/100×30
=6
Answer:
6
Step-by-step explanation:
There are 30 students
So,
20÷100×30=6
Sales by Quarter A company made sales of $1,254,000 last year. Quarter 1 Quarter 2 Produced 13 more sales than in quarter 1 Quartor 3 Quarter 4 Produced 17% of total sales for the year Sales increased 100% ovor tho provious quarter. Question: Adjust the ple chart to represent the sales each quarter.
Answers
Quarter 1: $300,000Quarter 2: $300,013Quarter 3: $250,000Quarter 4: $500,000 the adjusted chart representing the Sales .
The given chart to represent the sales each quarter, we need to find out the sales of each quarter first and then represent them in the chart. Let's calculate the sales of each quarter one by one:
Sales of Quarter 1Let the sales of Quarter 1 be xSales of Quarter 2As per the given data, Quarter 2 produced 13 more sales than Quarter 1Therefore, sales of Quarter 2 = x + 13Sales of Quarter 3Let the sales of Quarter 3 be sales of Quarter 4As per the given data, Quarter 4 produced 17% of total sales for the year
therefore, 17% of $1,254,000 = (17/100) x 1,254,000= 213,180Sales of Quarter 4 = 213,180Sales increased 100% over the previous quarter
Therefore, sales of Quarter 4 = 2 x sales of Quarter 3= 2yNow, we can form the equation as follows: Total Sales = Sales of Quarter 1 + Sales of Quarter 2 + Sales of Quarter 3 + Sales of Quarter 4$1,254,000 = x + (x + 13) + y + 2y + 213,180$1,254,000 = 4x + 3y + 213,193or 4x + 3y = $1,040,807
Now, we can assume some values of x and y and then calculate the values of other variables. Let's assume x = $300,000 and y = $250,000Therefore, sales of Quarter 1 = $300,000Sales of Quarter 2 = $300,000 + $13 = $300,013Sales of Quarter 3 = $250,000Sales of Quarter 4 = 2 x $250,000 = $500,000Now, we can represent these sales in the chart as follows:
Quarter 1: $300,000Quarter 2: $300,013Quarter 3: $250,000Quarter 4: $500,000
Therefore, the adjusted chart representing the sales each quarter is shown above.
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et J2= {0, 1}. Find three functions f, g and h such that f : J2→
J2, g : J2→ J2, and h : J2→ J2, and f = g = h
Answers
There are many possible solutions, but one example in the case of three functions f, g, and h would be: f(0) = 0, f(1) = 1g(0) = 1, g(1) = 0h(0) = 0, h(1) = 1
We have the set J2 = {0,1} and we need to estimate three functions f, g, and h such that f:
J2→ J2, g: J2→ J2, and h:
J2→ J2, and f = g = h.
To do this, we can simply assign values to each element of the set J2 for each of the three functions. For example, we can let f(0) = 0 and f(1) = 1, which means that the function f maps 0 to 0 and 1 to 1. We can also let g(0) = 1 and g(1) = 0, which means that the function g maps 0 to 1 and 1 to 0.
Finally, we can let h(0) = 0 and h(1) = 1, which means that the function h maps 0 to 0 and 1 to 1. Note that all three functions have the same values for each element in J2, so we can say that f = g = h.
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Rhonda and Laura are planning to watch two movies over the weekend from Laura's collection of 35 DVDs. Rhonda has two favorites among the collection. What is the probability that the girls would randomly choose those two movies to watch? Enter a fraction or round your answer to 4 decimal places, if necessary.
Answers
The probability that the girls would randomly choose their two favorite movies to watch is 0.0034.
The number of favorable outcomes is 2 (since Rhonda has two favorite movies).
Using the combination formula:
C(n, r) = n! / (r! * (n - r)!)
In this case, n = 35 (total number of movies) and r = 2 (number of movies to be chosen).
C(35, 2) = 35! / (2! x (35 - 2)!)
C(35, 2) = 35! / (2! x 33!)
= (35 x 34 x 33!) / (2! x 33!)
= (35 x 34) / 2
= 595
Therefore, there are 595 possible outcomes when choosing any two movies from Laura's collection.
Now, Probability = Favorable Outcomes / Total Outcomes
Probability = 2 / 595
Therefore, the probability that the girls would randomly choose their two favorite movies to watch is 0.0034.
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: 1. Two equilateral triangles are always similar. 2. The diagonals of a rhombus are perpendicular to each other. 3. For any event, 0
Answers
Both the given statements are true
1. Two equilateral triangles are always similar: True.
An equilateral triangle is a triangle in which all three sides are equal. Since two equilateral triangles have the same shape and size, they are always similar. Similarity means that the corresponding angles are equal, and the corresponding sides are in proportion.
2. The diagonals of a rhombus are perpendicular to each other: True.
In a rhombus, opposite sides are parallel, and all sides have equal length. The diagonals of a rhombus bisect each other at right angles, which means they are perpendicular to each other. This property holds true for all rhombuses, regardless of their size or orientation.
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Given question is incomplete, the complete question is below
State true or false
1. Two equilateral triangles are always similar.
2. The diagonals of a rhombus are perpendicular to each other.
The algebra question is in the image
Answers
Answer:
B
Step-by-step explanation:
A or constant is the answer
Find inverse of f(x) = 2 + 3x / x - 2
Know answer is 2x + 2 / x - 3 I just want someone to explain the steps for my revision!
Thank you.
Answers
Now just interchange x and y -> x=2+3y/y-2, now multiply both sides with (y-2), we get (y-2)x=2+3y.
Multiply x with the brackets, getting xy-2x=2+3y. Move 3y to the left changing it’s sign, also for -2x to the right: xy-3y=2+2x.
Factor out y from (xy-3y) and get (3-x)y=2+2x - now divide both sides by (3-x) resulting in y=2+2x/3-x .
what is the axis of symmetry for the graph shown?
Answers
Answer:
x=2
Step-by-step explanation:
The axis of symmetry goes through the vertex and is the line that makes the image the same on one side as the other
Since this is a vertical parabola, the axis is symmetry is of the form x=
The vertex is at x=2 so the axis of symmetry is x=2
A cuboid made from metal plates with the dimensions x, 3x and y cm has a surface area 450 cm. Find the volume of the cuboid as a function of x.
Answers
The volume of the cuboid as a function of x is V(x) = x * 3x * y.
the volume of the cuboid as a function of x is V(x) = 675/4 - (9/4)x^2.
The surface area of the cuboid is given as 450 cm, which can be expressed as:
2(x * 3x) + 2(x * y) + 2(3x * y) = 450.
Simplifying this equation, we get:
6x^2 + 2xy + 6xy = 450,
6x^2 + 8xy = 450,
3x^2 + 4xy = 225.
Now, we need to express y in terms of x. From the given dimensions, we know that the surface area is formed by six rectangular faces of the cuboid. Therefore, the length of one face is x, the width is 3x, and the remaining face (height) is y.
To find y, we can use the equation for the surface area. Rearranging the equation above, we have:
3x^2 + 4xy = 225,
y(4x) = 225 - 3x^2,
y = (225 - 3x^2) / (4x).
Now we can substitute the value of y into the expression for the volume:
V(x) = x * 3x * [(225 - 3x^2) / (4x)].
Simplifying further:
V(x) = (3/4) * (225 - 3x^2),
V(x) = 675/4 - (9/4)x^2.
Therefore, the volume of the cuboid as a function of x is V(x) = 675/4 - (9/4)x^2.
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let a be a square matrix. prove an alternate form of the polar decomposition for a: there exists a unitary matrix w and a positive semidefinite matrix p such that a = pw.
Answers
The alternate form of the polar decomposition of `A` is given by `A = PW`, where `W` is a positive semidefinite Hermitian matrix and `P` is a positive semidefinite Hermitian matrix.
Let `A` be a square matrix. Prove an alternate form of the polar decomposition for `A`.For a given square matrix `A`, the polar decomposition of `A` is a factorization of `A` into the product of a unitary matrix `U` and a positive semi-definite Hermitian matrix `P`. This polar decomposition of `A` can be given by `A = UP` or `A = PU*`, where `U` is the unitary matrix and `P` is a positive semidefinite matrix such that `P = (AA*)^(1/2)` or `P = (A*A)^(1/2)`.
The alternate form of the polar decomposition of `A` is given by `A = PW`, where `W = P^(1/2)U P^(1/2)` and `P` is a positive semidefinite matrix.Let `A = UP` be the polar decomposition of `A`, where `U` is unitary and `P` is positive semi-definite Hermitian. Then `P = A(A*)^(1/2)` and `U = P^(-1)A`. Let `W = P^(1/2)U P^(1/2)` and `W* = P^(1/2)U* P^(1/2)`. Since `U` is unitary, we have `U* = U^(-1)`. Hence `W* = P^(1/2)U^(-1) P^(1/2)`.Multiplying `UP` by `P^(1/2)`, we get `UP^(1/2) = P^(1/2)U P`. Multiplying both sides of the equation by `P^(1/2)` on the right, we get `UP^(1/2)P^(1/2) = P^(1/2)U P P^(1/2)` or `UP = P^(1/2)U P^(1/2)P^(1/2)` or `UP = P^(1/2)U P^(1/2)` or `U = P^(-1/2)W P^(1/2)`.
Substituting the value of `U` in `A = UP`, we get `A = P^(-1/2)W P^(1/2)P`. Since `P` is positive semi-definite, `P = (P^(1/2))^2` is a Hermitian matrix. Therefore, `W = P^(1/2)U P^(1/2)` is a Hermitian matrix and is positive semi-definite. Thus, we have `A = PW` where `W = P^(1/2)U P^(1/2)` is a positive semidefinite Hermitian matrix. Hence, the alternate form of the polar decomposition of `A` is given by `A = PW`, where `W` is a positive semidefinite Hermitian matrix and `P` is a positive semidefinite Hermitian matrix.
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