¿Cuales son las propiedades de la Sustracción de Números Racionales Decimales?
The following characteristics of racional decimal number abstraction apply: Conmutative property: The order of the remaining rational decimal numbers has no bearing on the operation's outcome,
Proprietary property: The racional decimal numbers may remain in various groups without affecting the operation's ultimate outcome, i.e., (a - b) - c = a - (b - c). Distributive property: Subtracting one racional decimal number from a sum of racional decimal numbers equals the sum of the subtractions of each one of them, or a - (b + c) = a - b - c. Neutral element: If a racional decimal number is left at zero, the outcome is the same number, i.e., a - 0 = a. Estas propiedades son útiles para simplificar y realizar cálculos más complejos con números racionales decimales.
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Evaluate
(
3
7
)
−
2
Give your answer as an improper fraction in its simplest form
The value of (37)-2 is 1/1369, in its simplest form as an improper fraction.
An improper fraction is a fraction where the numerator is greater than or equal to the denominator. In other words, it is a fraction that is larger than a whole number.
When an expression is written in the form of [tex]x^{(-n)[/tex], it means the reciprocal of [tex]x^n.[/tex] In this case, we have the expression[tex](37)^{(-2)[/tex] which means the reciprocal of 37².
The expression (37)-2 means 37 raised to the power of -2, or 1/(37²). To simplify this fraction, we can multiply the numerator and denominator by 1,296 (37²):
1/(37²) = 1 * 1 / (37 * 37)
= 1/1369
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determine whether the set S spans R2. If the set does not span R2, then give a geometric description of the subspace that it does span. a, S = {(1, −1), (2, 1)} b, S = {(1, 1)} c, S = {(0, 2), (1, 4)}
a. S = {(1, -1), (2, 1)}Let's begin by calculating the determinant of the matrix composed of the vectors of S, and checking if it is equal to 0. Because the two vectors are not colinear, they should span R2.|1 -1||2 1| determinant is not 0, therefore S spans R2. No geometric description is required for this example.
b. S = {(1, 1)} The set S contains one vector. A set containing only one vector cannot span a plane because it only spans a line. Therefore, S does not span R2. Geometric description: S spans a line that passes through the origin (0, 0) and the point (1, 1).c. S = {(0, 2), (1, 4)} Let's again begin by calculating the determinant of the matrix composed of the vectors of S, and checking if it is equal to 0.|0 2||1 4| determinant is 0, thus S does not span R2. In this scenario, S only spans the line that contains both vectors, which is the line with the equation y = 2x.
Geometric description: S spans a line that passes through the origin (0, 0) and the point (1, 2).
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Which of the following are true statements? Check all that apply. A. F(x)= 2 square x has the same domain and range as f(x)= square x. B. The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will shrink it vertically by the factor of 1/2. C. The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will shrink it horizontally by a factor of 1/2. D. The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will stretch it vertically by factor of 2.
The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will shrink it vertically by the factor of 1/2.
The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will stretch it vertically by factor of 2.
Thus, Option B and Option D are correct.
What is function?A function is a relationship or expression involving one or more variables. It has a set of input and outputs.
A. F(x)= 2 square x has the same domain and range as f(x)= square x.
B. The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will shrink it vertically by the factor of 1/2.
D. The graph of f(x)= 2 square x will look like the graph of f(x)= square x but will stretch it vertically by factor of 2.
Option A is false because multiplying the function by 2 will change the range of the function to include all non-negative real numbers (since the square of any number is non-negative).
Option B is true because multiplying the function by 2 will vertically shrink the graph by a factor of 1/2 (since the output values will be half the size of the original function).
Option C is false because multiplying the function by 2 will not affect the horizontal scale of the graph.
Option D is true because multiplying the function by 2 will vertically stretch the graph by a factor of 2 (since the output values will be twice the size of the original function).
Therefore, Option B and Option D are correct.
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Set up iterated integrals for both orders of integration. Then evaluate the double integral using the easier order.
y dA, D is bounded by y = x − 6; x = y2
D
The value of the double integral using the easier order, ydA bounded by y = x − 6; x = y² is 125/12.
The double integral, indicated by ', is mostly used to calculate the surface area of a two-dimensional figure. By using double integration, we may quickly determine the area of a rectangular region. If we understand simple integration, we can easily tackle double integration difficulties. Hence, first and foremost, we will go over some fundamental integration guidelines.
Given, the double integral ∫∫yA and the region y = x-6 and x = y²
y = x-6
x = y²
y² = y +6
y² - y - 6 = 0
y² - 3y +2y - 6 = 0
(y-3) (y+2) = 0
y = 3 and y = -2
[tex]\int\int\limits_\triangle {y} \, dA\\ \\[/tex]
= [tex]\int\limits^3_2 {y(y+6-y^2)} \, dx \\\\\int\limits^3_2 {(y^2+6y-y^3)} \, dx \\\\(\frac{y^3}{3} + 3y^2-\frac{y^4}{4} )_-_2^3\\\\\frac{63}{4} -\frac{16}{3} \\\\\frac{125}{12}[/tex]
The value for the double integral is 125/12.
Integration is an important aspect of calculus, and there are many different forms of integrations, such as basic integration, double integration, and triple integration. We often utilise integral calculus to determine the area and volume on a very big scale that simple formulae or calculations cannot.
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In a candy factory, each bag of candy contains 300 pieces. The bag can be off by 10 pieces.
Write an absolute value inequality that displays the possible number of candy pieces that a bag contains.
Answer:
[tex] |x - 300| \leqslant 10[/tex]
A satellite TV company offers two plans. One plan costs $115 plus $30 per month. The other plan costs $60 per month. How many months must Alfia have the plan in order for the first plan to be the better buy?
3p^2 +7p=0 solve by factoring
Answer:
p = 0, p = -7/3
Step-by-step explanation:
Pre-SolvingWe are given the following equation:
3p² + 7p = 0
We want to solve the equation by factoring.
Solving
To factor, we want to look for a common term that we can pull out.
You may notice that both terms have 'p' in common, so we can pull out p from both terms.
This will then make the equation:
p(3p + 7) = 0
Now, we can use zero product property to solve the equation.
p = 0
3p + 7 = 0
Subtract.
3p = -7
Divide.
p = -7/3
Our answers are p = 0 and p = -7/3
Which points satisfy both inequalities?
The pοint that satisfies bοth inequalities is the pοint inside this triangular regiοn.
What is inequality?An inequality is a mathematical statement that cοmpares twο values οr expressiοns and indicates whether they are equal οr nοt, οr which οne is greater οr smaller.
Since the shading is nοt included, we will need tο use the lines themselves tο determine the cοrrect regiοn οf the cοοrdinate plane.
The first inequality y > (3/2)x - 5 has a slοpe οf 3/2 and a y-intercept οf -5. This means the line will have a pοsitive slοpe and will be lοcated belοw the pοint (0,-5).
The secοnd inequality y < (-1/6)x - 6 has a negative slοpe οf -1/6 and a y-intercept οf -6. This means the line will have a negative slοpe and will be lοcated abοve the pοint (0,-6).
Tο find the pοint that satisfies BOTH inequalities, we need tο lοοk fοr the regiοn οf the cοοrdinate plane that is belοw the line y = (3/2)x - 5 AND abοve the line y = (-1/6)x - 6. This regiοn is the triangular-shaped area that is bοunded by the twο lines and the x-axis.
The pοint that satisfies bοth inequalities is the pοint inside this triangular regiοn.
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Aaron sampled 101 students and calculated an average of 6.5 hours of sleep each night with a standard deviation of 2.14. Using a 96% confidence level, he also found that t* = 2.081.confidence intervat = x±s/√n A 96% confidence interval calculates that the average number of hours of sleep for working college students is between __________.
The average number of hours of sleep for working college students is between 6.28 and 6.72 hours of sleep each night
According to the given data,
Sample size n = 101
Sample mean x = 6.5
Standard deviation s = 2.14
Level of confidence C = 96%
Using a 96% confidence level, the value of t* for 100 degrees of freedom is 2.081, as given in the question.
Now, the formula for the confidence interval is:x ± (t* × s/√n)Here, x = 6.5, s = 2.14, n = 101, and t* = 2.081
Substituting the values in the above formula, we get:
Lower limit = x - (t* × s/√n) = 6.5 - (2.081 × 2.14/√101) = 6.28
Upper limit = x + (t* × s/√n) = 6.5 + (2.081 × 2.14/√101) = 6.72
Therefore, the 96% confidence interval for the average number of hours of sleep for working college students is between 6.28 and 6.72 hours of sleep each night.
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Find the outer perimeter.
6 ft
4 ft
15 ft
10 ft
P = [?] ft
Round to the nearest
hundredth.
Answer:
P= 40 ft
Step-by-step explanation:
Perimeter is the sum of all the lengths
So,
Perimeter= 6+4+15+10ft
= 35ft
Nearest ten can be 40ft or 30ft
If you succeed In understanding then kindly mark my answer the brainliest. Thank you :)
Help me please I need to show my work
Answer:
x=33
Step-by-step explanation:
all angles in a triangle sum to 180 degrees
x+2x+(2x+15) = 180 <---- simplify this
5x+15 = 180
5x=165
x = 33
according to a census, 3.3% of all births in a country are twins. if there are 2,500 births in one month, calculate the probability that more than 90 births in one month would result in twins. use a ti-83, ti-83 plus, or ti-84 calculator to find the probability. round your answer to four decimal places. provide your answer below:
According to a census, 3.3% of all births in a country are twins. In a month, there are 2,500 births. The census reports that 3.3% of all births result in twins, and the probability of having more than 90 twins in a month is "0.4351."
We will solve this problem using the binomial distribution formula, which is as follows:P (X > 90) = 1 - P (X ≤ 90)where P represents the probability, X represents the number of twins born in a month, and X is a binomial random variable with a sample size of n = 2,500 and a probability of success (having twins) of p = 0.033. Using the TI-83 calculator, TI-83 Plus, or TI-84 calculator, the following steps can be followed:
Press the "2nd" button followed by the "VARS" button (DISTR) to access the distribution menu. Scroll down and select "binomcdf (" from the list of options (use the arrow keys to navigate). The binomcdf ( menu will appear on the screen. The first number in the parentheses is the number of trials, n, and the second number is the probability of success, p. We want to find the probability of having more than 90 twins, so we need to use the "compliment" option. Therefore, we will subtract the probability of having 90 twins or less from 1 (using the "1 -" key). Type in "binomcdf (2500,0.033,90)" and press the "ENTER" button on your calculator.
This will give you the probability of having 90 twins or fewer in a month. Subtract this value from 1 to obtain the probability of having more than 90 twins in a month, which is the answer to our question. P(X>90) = 1 - binomcdf (2500,0.033,90)P(X>90) = 1 - 0.5649P(X>90) = 0.4351Therefore, the probability of having more than 90 twins in a month is 0.4351.
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With the information given, can you prove
that this quadrilateral is a parallelogram?
A. Yes
B. No
AB = DC
We cannot prove that the quadrilateral is a parallelogram with only the given information that AB = DC.
What is quadrilateral and parallelogram ?
A quadrilateral is a four-sided polygon, which means it is a closed shape with four straight sides. Some examples of quadrilaterals include rectangles, squares, trapezoids, and rhombuses.
A parallelogram is a special type of quadrilateral where both pairs of opposite sides are parallel. This means that the opposite sides never intersect, and they have the same slope. Additionally, the opposite sides of a parallelogram are congruent (i.e., have the same length), and the opposite angles are also congruent. Some examples of parallelograms include rectangles, squares, and rhombuses.
To prove that a quadrilateral is a parallelogram, we need to show that both pairs of opposite sides are parallel. Knowing that AB = DC only gives us information about the lengths of the sides, but it doesn't tell us anything about their orientation or whether they are parallel.
We would need additional information, such as the measures of angles or the lengths of other sides, to determine whether the quadrilateral is a parallelogram.
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A square is inscribed in a right triangle with leg lengths 6 and 8 so that they have a common right angle. FInd the square's side length.
Answer:
10 units
Step-by-step explanation:
Here, legs = base and perpendiculars.
So, Clearly given Base = 6 units Perpendicular = 8 cm
Square's Side = Hypotenuse.
By Pythagoras theorem,
H² = B²+P²
H ² = 6²+8²
H² = 36+64 = (10)²
H = 10 units.
Square's Side length = 10 units
This question has two parts. First, answer Part A. Thenanswer Part B
Part A
BAKERY Aisha can work up to 20 hours per week Working at a bakery, she earns $7 per hour most of the time and $ 8.50 per hour during the early morning shift. Aisha needs to earn at least $150 this week to pay for a trip with her friends. Determine the number of regular and early morning hours that Aisha could work
Part A Select the correct system and graph. Let r=regular hours and m = early morning hours
R<20
7r+8.5m>=150
R+m <=20
r+m<=150
r+m<= 20
7r+ 8.5m >= 150
7r+8.5m>20
7r+8.5m>= 150
Part B
Drag every viable solution to the bin.
The other solutions are not viable because either they exceed the maximum number of hours Aisha can work (20 hours) or they do not meet the minimum amount Aisha needs to earn ($150).
What is an illustration of a workable solution?If the ongoing research is successful, this approach might be an effective remedy. The only real way to resolve the problem is through negotiations between the military administration and the various opposition movements.
Part A: The correct system and graph to represent Aisha's situation is:
r + m ≤ 20 (maximum number of hours Aisha can work)
7r + 8.5m ≥ 150 (minimum amount Aisha needs to earn)
Part B: The viable solutions are:
r = 20, m = 0 (Aisha works only regular hours for 20 hours at $7 per hour)
r = 14, m = 6 (Aisha works 14 regular hours and 6 early morning hours at $7 per hour and $8.50 per hour, respectively)
r = 0, m = 18 (Aisha works only early morning hours for 18 hours at $8.50 per hour)
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the amount of bacteria present in a medium after t hours is given by a (t )equals 16 e to the power of 0.32 t end exponent. at what rate is the amount of bacteria changing after 12 hours?
The rate at which the amount of bacteria is changing after 12 hours is 21.698 units/hour.
What is the rate at which the amount of bacteria changing after 12 hours?The given formula for the amount of bacteria present in a medium after t hours is:
[tex]a(t) = 16e^ (0.32t)[/tex]
Now, we need to find out at what rate the amount of bacteria is changing after 12 hours.
This means we need to find out the derivative of a(t) with respect to t and then substitute t = 12 in the derivative formula to get the rate of change of bacteria after 12 hours.
Differentiating the given formula for a(t) with respect to t, we get:
a'(t) = [tex]16(0.32)e^(0.32t)[/tex]
On substituting, t = 12, we get
a'(12) = [tex]16(0.32)e^ (0.32X12)[/tex]
On solving this, we get a'(12) = 21.698.
Therefore, the rate at which the amount of bacteria is changing after 12 hours is 21.698 units/hour.
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The exponential 12 (3) 2x-12 has been converted to 12(k)*-6, what is the value of k?
Answer:
The solution set is (13,− 32). A quadratic equation of the form x 2= k can be solved by factoring with the following sequence of equivalent equations.
Step-by-step explanation:
1 Find the value of x.
i’m like struggling
Answer: 23 degrees
Step-by-step explanation:
Assuming that 117 is the entire angle we can find that:
94+x = 117
Subtract 94 from both sides:
x = 117-94
x = 23 degrees
Student A can solve 75% of problems, student B can solve 70%. What is the probability that A or B can solve a problem chosen at random?
The probability that student A or B can solve a problem chosen at random is 0.95.
Probability is calculated by dividing the number of favourable outcomes by the number of possible outcomes.
Random: An event is referred to as random when it is not possible to predict it with certainty. The probability that either student A or B will be able to solve a problem chosen at random can be calculated as follows:
P(A or B) = P(A) + P(B) - P(A and B) where: P(A) = probability of A solving a problem = 0.75, P(B) = probability of B solving a problem = 0.7, P(A and B) = probability of both A and B solving a problem. Since A and B are independent, the probability of both solving the problem is:
P(A and B) = P(A) x P(B) = 0.75 x 0.7 = 0.525
Now, using the above formula: P(A or B) = P(A) + P(B) - P(A and B) = 0.75 + 0.7 - 0.525 = 0.925
Therefore, the probability that student A or B can solve a problem chosen at random is 0.95 (or 95%).
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three cards are drawn with replacement from a standard deck of 52 cards. find the the probability that the first card will be a club, the second card will be a red card, and the third card will be the six of hearts.
The probability of drawing a club, a red card, and the six of hearts in that order from a standard deck of 52 cards is [tex]1/13,552.[/tex]
This is because the probability of drawing a club is 1/4, and the probability of drawing a red card is 1/2, and the probability of drawing the six of hearts is 1/52.
Since the cards are drawn with replacement, the total probability is the product of the individual probabilities, which is equal to [tex]1/4 * 1/2 * 1/52 = 1/13,552[/tex].
It is important to note that if the cards were not drawn with replacement, then the probability of drawing the three cards would be slightly different. The total probability would be equal to [tex]1/4 * 1/2 * 1/51 = 1/12,600.[/tex]
It is also important to note that since this is a probability question, the answer can be expressed as a decimal or percentage. In decimal form, the probability of drawing the three cards is 0.000074, and in percentage form, the probability of drawing the three cards is 0.0074%.
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Verify that W is a subspace of V. Assume that V has the standard operations.
W is the set of all 3x2 matrices of the form [a,b;(a+b),0;0,c] and V=M[-subscript-(3,2)]
The zero vector: The zero vector 0 = [0,0;0,0;0,0] is also a member of W. Thus, the third criterion is satisfied.
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Goods with a cost price of R200 are sold at a mark-up of 100%. The selling price is:
If the cost price of the goods is R200 and they are sold at a mark-up of 100%, then the selling price is equal to the cost price plus the mark-up, or:
Selling price = Cost price + Mark-up
Mark-up = 100% x Cost price
= 100% x R200
= R200
So the mark-up is R200.
Selling price = Cost price + Mark-up
= R200 + R200
= R400
Therefore, the selling price of the goods is R400.
what is the as surface area of the rectangular prism
Answer:
142 sq cm
Step-by-step explanation:
A= 2(lh + wh + lw)
2(7*3+5*3+7*5)
2(21+15+35)
2(71)
A= 142 sq cm
use the trapezoidal rule and simpson's rule to approximate the value of the definite integral for the given value of n. round your answer to four decimal places and compare the results with the exact value of the definite integral. 4 x x2 1 0 dx, n
The Trapezoidal rule and Simpson's rule are two methods used to approximate the value of a definite integral. The Trapezoidal rule approximates the integral by dividing the region between the lower and upper limits of the integral into n trapezoids, each with a width h. The approximate value of the integral is then calculated as the sum of the areas of the trapezoids. The Simpson's rule is similar, except the region is divided into n/2 trapezoids and then the integral is approximated using the weighted sum of the area of the trapezoids.
For the given integral 4 x x2 1 0 dx, with n = 200, the Trapezoidal rule and Simpson's rule approximate the integral to be 7.4528 and 7.4485 respectively, rounded to four decimal places. The exact value of the integral is 7.4527. The difference between the exact and approximate values is very small, thus indicating that both the Trapezoidal rule and Simpson's rule are accurate approximations.
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NEED HELP DUE TODAY!!!! GIVE GOOD ANSWER
2. How do the sizes of the circles compare?
3. Are triangles ABC and DEF similar? Explain your reasoning.
4. How can you use the coordinates of A to find the coordinates of D?
The triangles ABC and DEF are similar triangles, but DEF is twice as big as ABC.
What does it signify when two triangles are similar?
Congruent triangles are triangles that share similarity in shape but not necessarily in size. All equilateral triangles and squares of any side length serve as illustrations of related objects.
Or to put it another way, the corresponding angles and sides of two triangles that are similar to one another will be congruent and proportionate, respectively.
How do the sizes of the circles compare?
Given the triangles ABC and DEF
From the figure, we have
AB = 1
DE = 2
This means that the triangle DEF is twice the size of the triangle ABC
Are triangles ABC and DEF similar?
Yes, the triangles ABC and DEF are similar triangles
This is because the corresponding sides of DEF is twice the corresponding sides of triangle ABC
How can you use the coordinates of A to find the coordinates of D?
Multipliying the coordinates of A by 2 gives coordinates of D.
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Five cars start out on a cross-country race. The probability that a car breaks down and drops out of the race is 0.2. Cars break down independently of each other.
(a) What is the probability that exactly two cars finish the race?
(b) What is the probability that at most two cars finish the race?
(c) What is the probability that at least three cars finish the race?
(a) The probability that exactly two cars finish the race is 0.0512.
(b) The probability that at most two cars finish the race is 0.05792.
(c) The probability that at least three cars finish the race is 0.94208.
(a) To determine the probability that exactly two cars finish the race, we have to use binomial distribution. In this case, we have n = 5 trials, and p = 0.8 is the probability that a car finishes the race (1 - 0.2). Using the binomial distribution formula:
P(X = k) = (nCk)(p^k)(1 - p)^(n - k)
Where X is the number of cars that finish the race, we get:
P(X = 2) = (5C2)(0.8²)(0.2)³= (10)(0.64)(0.008)= 0.0512
Therefore, the probability that exactly two cars finish the race is 0.0512.
(b) To determine the probability that at most two cars finish the race, we have to calculate the probabilities of 0, 1, and 2 cars finishing the race and add them up.
P(X ≤ 2) = P(X = 0) + P(X = 1) + P(X = 2)= (5C0)(0.8⁰)(0.2)⁵ + (5C1)(0.8¹)(0.2)⁴ + (5C2)(0.8²)(0.2)³= 0.00032 + 0.0064 + 0.0512= 0.05792
Therefore, the probability that at most two cars finish the race is 0.05792.
(c) To determine the probability that at least three cars finish the race, we can calculate the probability of 0, 1, and 2 cars finishing the race and subtract it from 1, which gives us the probability of at least three cars finishing the race.
P(X ≥ 3) = 1 - [P(X = 0) + P(X = 1) + P(X = 2)]= 1 - (0.00032 + 0.0064 + 0.0512)= 0.94208
Therefore, the probability that at least three cars finish the race is 0.94208.
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there exists a complex number $c$ such that we can get $z 2$ from $z 0$ by rotating around $c$ by $\pi/2$ counter-clockwise. find the sum of the real and imaginary parts of $c$.
The sum of the real and imaginary parts of $c$ is$$\operatorname{Re}(c) + \operatorname{Im}(c) = \frac{\operatorname{Re}(2c)}{2} + \frac{\operatorname{Im}(2c)}{2}$$$$= \frac{\operatorname{Re}(z_0+z_2)}{2} - \frac{\operatorname{Im}(z_0)}{2}(1-\cos(\theta/2)) - \frac{\operatorname{Re}(z_0)}{2}\sin(\theta/2)$$$$+ \frac{\operatorname{Im}(z_0+z_2)}{2} - \frac{\operatorname{Re}(z_0)}{2}(1-\cos(\theta/2)) + \frac{\operatorname{Im}(z_0)}{2}\sin(\theta/2).$$
The given problem can be solved using algebraic and geometric methods. We can use algebraic methods, such as the equations given in the problem, and we can use geometric methods by visualizing what the problem is asking. To start, let's translate the given problem into mathematical equations. Let $z_0$ be the original complex number. We want to rotate this point by 90 degrees counter-clockwise about some complex number $c$ to get $z_2$. Thus,$$z_2 = c + i(z_0 - c)$$$$=c + iz_0 - ic$$$$= (1-i)c + iz_0.$$We also know that this transformation will rotate the point $z_1 = (z_0 + z_2)/2$ by 45 degrees. Thus, using similar logic,$$z_1 = (1-i/2)c + iz_0/2.$$Now let's use the formula for rotating a point about the origin by $\theta$ degrees (where $\theta$ is measured in radians) to find a relationship between $z_1$ and $z_0$.$$z_1 = z_0 e^{i\theta/2}$$$$\implies (1-i/2)c + iz_0/2 = z_0 e^{i\theta/2}$$$$\implies (1-i/2)c = (e^{i\theta/2} - 1)z_0/2.$$We can solve for $c$ by dividing both sides by $1-i/2$.$$c = \frac{e^{i\theta/2} - 1}{1-i/2}\cdot\frac{z_0}{2}.$$We can now use the information given in the problem to solve for the sum of the real and imaginary parts of $c$. We know that rotating $z_0$ by 90 degrees counter-clockwise will result in the complex number $z_2$. Visually, this means that $c$ is located at the midpoint between $z_0$ and $z_2$ on the line that is perpendicular to the line segment connecting $z_0$ and $z_2$. We can use this geometric interpretation to solve for $c$. The midpoint of the line segment connecting $z_0$ and $z_2$ is$$\frac{z_0+z_2}{2} = c + i\frac{z_0-c}{2}.$$Solving for $c$, we get$$c = \frac{z_0+z_2}{2} - \frac{i}{2}(z_0-c)$$$$\implies 2c = z_0+z_2 - i(z_0-c)$$$$\implies 2c = z_0+z_2 - i(z_0- (e^{i\theta/2} - 1)(z_0/2)/(1-i/2)).$$We can now find the real and imaginary parts of $c$ and add them together to get the desired answer. Let's first simplify the expression for $c$.$$2c = z_0+z_2 - i(z_0 - (e^{i\theta/2} - 1)\cdot(z_0/2)\cdot(1+i)/2)$$$$= z_0 + z_2 - i(z_0 - z_0(e^{i\theta/2} - 1)(1+i)/4)$$$$= z_0 + z_2 - i(z_0 - z_0e^{i\theta/2}(1+i)/4 + z_0(1-i)/4)$$$$= z_0 + z_2 - i(z_0(1-e^{i\theta/2})/4 + z_0(1-i)/4)$$$$= z_0 + z_2 - i(z_0/4(1-e^{i\theta/2} + 1 - i))$$$$= z_0 + z_2 - i(z_0/2(1-\cos(\theta/2) - i\sin(\theta/2)))$$$$= z_0 + z_2 - i(z_0(1-\cos(\theta/2)) + z_0\sin(\theta/2) - i(z_0\cos(\theta/2))/2.$$Now we can find the real and imaginary parts of $2c$ and divide by 2 to get the real and imaginary parts of $c$. We have$$\operatorname{Re}(2c) = \operatorname{Re}(z_0+z_2) - \operatorname{Im}(z_0)(1-\cos(\theta/2)) - \operatorname{Re}(z_0)\sin(\theta/2)$$$$\operatorname{Im}(2c) = \operatorname{Im}(z_0+z_2) - \operatorname{Re}(z_0)(1-\cos(\theta/2)) + \operatorname{Im}(z_0)\sin(\theta/2).$$Thus, the sum of the real and imaginary parts of $c$ is$$\operatorname{Re}(c) + \operatorname{Im}(c) = \frac{\operatorname{Re}(2c)}{2} + \frac{\operatorname{Im}(2c)}{2}$$$$= \frac{\operatorname{Re}(z_0+z_2)}{2} - \frac{\operatorname{Im}(z_0)}{2}(1-\cos(\theta/2)) - \frac{\operatorname{Re}(z_0)}{2}\sin(\theta/2)$$$$+ \frac{\operatorname{Im}(z_0+z_2)}{2} - \frac{\operatorname{Re}(z_0)}{2}(1-\cos(\theta/2)) + \frac{\operatorname{Im}(z_0)}{2}\sin(\theta/2).$$
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Four pipes can fill a tank in 16 hours. How long will it take to fill the tank if twelve
pipes of the same dimensions are used ?
Answer:
5.333 hours
Step-by-step explanation:
We know
4 Pipes fill a tank in 16 hours.
How long will it take to fill the tank if 12 pipes of the same dimensions are used?
We Take
16 x 1/3 = 5.333 hours
So, it takes about 5.333 hours to fill the tank.
if the circumference of the moon is 6783 miles what is its diameter in miles
Answer:
C = 21,309.4
Step-by-step explanation:
Diameter of moon is miles is,
d = 2159.8 miles
We have,
The circumference of the moon is, 6783 miles
Since, We know that,
the circumference of circle is,
C = 2πr
Substitute given values,
6783 miles = 2 × 3.14 × r
6783 = 6.28 × r
r = 6783 / 6.28
r = 1079.9 miles
Therefore, Diameter of moon is miles is,
d = 2 x r
d = 2 x 1079.9
d = 2159.8 miles
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