Answers
By polar form properties, the product of complex numbers (5 · cos 225° + i 5 · sin 225°) and (2 · cos 135° + i r₂ · sin 135°).
How to multiply complex numbers
In this question we find the case of the product of two complex numbers in rectangular form but including features of polar form (magnitude, direction). Then, we have the following situation:
(r₁ · cos θ₁ + i r₁ · sin θ₁) · (r₂ · cos θ₂ + i r₂ · sin θ₂)
Result can be found easily by transforming the expression into polar form:
(r₁, θ₁) · (r₂, θ₂) = (r₁ · r₂, θ₁ + θ₂)
[r₁ · r₂ · cos (θ₁ + θ₂) + i r₁ · r₂ · sin (θ₁ + θ₂)]
If we know that r₁ = 5, r₂ = 2, θ₁ = 225° and θ₂ = 135°, then the product of the complex number is:
10 · [cos 360° + i sin 360°]
10 · (1 + i 0)
10
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Related Questions
what is the domain of the function shown in the graph below?
Answers
The domain of the function shown in the graph above include the following: D. {x|x ≥ -4}.
What is a domain?In Mathematics, a domain can be defined as the set of all real numbers for which a particular function is defined.
In Mathematics, the horizontal extent of any graph of a function represents all domain values and they are read and written from smaller to larger numerical values, and from the left of a graph to the right.
By using set builder notation, the domain of this function shown in the graph above can be written as follows;
Domain = {-4, ∞}
Domain = {x|x ≥ -4}
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Brown's Tree Farm Company is going to plant pine trees and birch trees this planting season. On Farm A, the land type allows for them to plant 25 pine trees per acre and 75 birch trees per acre. On Farm B, the land type allows for them to plant 50 pine trees per acre and 75 birch trees per acre. There are a total of 200 pine trees and 450 birch trees available for planting.
1. Write and solve a system of equations to determine how many of each tree the company will plant on each farm.
Answers
Let x be the number of acres of Farm A planted with pine trees and birch trees, and let y be the number of acres of Farm B planted with pine trees and birch trees.
The system of equations would be as follows:
25x + 50y = 200 (the total number of pine trees planted)
75x + 75y = 450 (the total number of birch trees planted)
To solve this system of equations, we can use the method of substitution. Solving the first equation for x in terms of y:
x = (200 - 50y) / 25
We can substitute this expression for x into the second equation:
75((200 - 50y) / 25) + 75y = 450
Simplifying and solving for y:
(200 - 50y) + 75y = 450
25y = 250
y = 10
So 10 acres of Farm B are planted with pine and birch trees. To find the number of acres of Farm A that are planted with pine and birch trees, we can substitute this value of y back into the first equation:
25x + 50(10) = 200
25x = 50
x = 2
So 2 acres of Farm A are planted with pine and birch trees.
Therefore, the company will plant 25x = 50 pine trees and 75x = 225 birch trees on Farm A, and 50y = 500 pine trees and 75y = 750 birch trees on Farm B.
The line passes through the points (3,5) and (6,11).
Algebraic rule (slope-intercept form or point-slope
form):
Answers
When the line to be examined's slope is known, and the provided point also serves as the y intercept, the slope intercept formula, y = mx + b, is utilized (0, b).
When should you use point-slope form?
When the slope of the line being studied is known, and the provided point is also the y intercept, the slope intercept formula, y = mx + b, is utilized (0, b). The y value of the y intercept point is represented by b in the equation.
One of the three ways we can express a straight line is using the point slope form, also known as the point-gradient form. By merely knowing one point on the line and the slope of the line, we may use this form to get the equation of the line.
The slope and y-intercept of the matching line can be rapidly determined when we have a linear equation in slope-intercept form. This enables us to graph it as well.
The equation of a line can be represented in either slope-intercept form or point-slope form.
Slope-intercept form:
The slope-intercept form of a line is given by y = mx + b, where m is the slope of the line and b is the y-intercept.
To find the equation of the line passing through the points (3,5) and (6,11) in slope-intercept form, we can use the point-slope formula to find the slope and then use one of the points to find the y-intercept.
Point-slope form:
The point-slope form of a line is given by y - y1 = m(x - x1), where m is the slope of the line and (x1, y1) is a point on the line.
To find the equation of the line passing through the points (3,5) and (6,11) in point-slope form, we can use the point-slope formula and one of the points.
Using the point-slope formula, the slope of the line is (11 - 5) / (6 - 3) = 6/3 = 2.
So, the equation of the line in slope-intercept form is:
y = 2x + b
We can use the point (3,5) to find the y-intercept:
5 = 2 * 3 + b
Solving for b, we get b = -1.
So the equation of the line in slope-intercept form is:
y = 2x - 1
In point-slope form, using the point (3,5), the equation of the line is:
y - 5 = 2(x - 3)
Both forms represent the same line, just in different ways.
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Need help solving this question
Answers
Answer: Billy drives 171 miles in 3 hours, so he could drive 256.5 miles in 4.5 hours.
Step-by-step explanation:
Speed = Distance/Time
The number of miles driven varies directly with the number of hours in the car. Billy drives 171 miles in 3 hours. Then the number of miles he can drive in 4.5 hours will be. This meas that x will be the number of miles covered in 4.5 hours.
x / 4.5 = 171 / 3
x = 256.5 miles
I hope this helps!
an online company sells and ships T-shirts. The company has to pay extra to ship a package that weighs over 13 oz. Before putting any T-shirts in the box, the package weighs 1.5 oz. Each shirt weighs 5 oz. Which number line best represents the possible number of shirts the company can ship in a package without playing extra for shipping?
Answers
The number line inequality that best represents the possible number of shirts the company can ship in a package without playing extra for shipping is given by the image presented at the end of the answer.
How to solve the inequality?Before putting any T-shirts in the box, the package weighs 1.5 oz. Each shirt weighs 5 oz, hence the total weight function for x shirts is given as follows:
y = 1.5 + 5x.
The maximum weight without needing to pay extra is of 13 ounces, hence the inequality is given as follows:
5x + 1.5 <= 13
x <= 11.5/5
x <= 2.3.
Which can be simplified as:
x <= 2.
(as the number of shirts is a countable amount).
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Vanessa and Jo share £128 in the
ratio 5: 3. Work out how much money
Vanessa receives.
Answers
Answer:
vanessa gest 80
Step-by-step explanation:
128/8 =16
5 * 16 =80
3*16. =48
Answer: £80
Step-by-step explanation:
Step 1:
Make a proportion:
5 - 8
x - 128
Step 2:
Cross multiply:
5*128=8x
640=8x
Step 3:
Divide by 8:
80=x or x=80
Paul won a race with a time of 71. 3 seconds.
This time, t, is to the nearest tenth of a second.
Complete the error interval due to rounding.
____________________ ≤ t < ____________________
Answers
The error interval due to rounding:
71.25 ≤ t < 71.35
Now, According to the question:
Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.
"Information available from the question"
Paul won a race with a time of 71. 3 seconds.
The time, t, is to the nearest tenth of a second.
So, Rounding off to nearest tenth
Rounded to the nearest 0.1 or the Tenths Place will be
The error interval due to rounding:
So, 71.25 ≤ t < 71.35
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Help would be so appreciated!
Find the equation of the tangent line to the function f(x) = 3x3 − x2 + 2 at a = 0.
A.
y = 2
B.
y = 2x
C.
y = 2x + 2
D.
y = –2x + 2
Answers
Answer: To find the equation of the tangent line to a function at a specific point, we can use the following steps:
Find the derivative of the function f(x) = 3x3 − x2 + 2
f'(x) = 9x^2 - 2x
Substitute the point a = 0 into the derivative to find the slope of the tangent line at that point
f'(0) = 9(0)^2 - 2(0) = 0
Use the point-slope form of a linear equation, y - y1 = m(x - x1) to find the equation of the tangent line, where m is the slope and (x1, y1) is the point at which the tangent line touches the function.
y - f(a) = m(x - a)
y - (3*0^3 - 0^2 + 2) = 0(x - 0)
y - 2 = 0
So the equation of the tangent line to the function f(x) = 3x3 − x2 + 2 at a = 0 is:
y = 2
Therefore, the correct answer is A. y = 2
Step-by-step explanation:
I need help in this problem
Answers
Answer:
b. 20°
Step-by-step explanation:
Find the missing angle in the top triangle:
180 - 40 - 20 = 120
because, the sum of interior angles of a triangle is 180.
The top angle of the bottom triangle = 120 because it is a vertical angle to the one you just found.
Now find the missing angle (x) on the bottom:
x = 180 - 120 - 40 = 20
The solution to an inequality is graphed on the number
line.
-5-4-3 -2 -1 0 1 2 3 4 5
What is another way to represent this solution set?
Oxx<4.5}
O {x|x≤4.5}
Oxx>4.5)
Ox|x24.5}
Answers
The term "solution set" refers to the collection of all possible solutions to an equation thus {x|x≤4.5} holds it . The value is a solution because the equation holds true if both sides are equal.
Option B is correct.
What is set?The distinct items that make up a set are referred to as the set's members or elements. Set notation is a method for representing a set using a combination of symbols and numbers, and it can be used to define sets. "x | -5 x 4.5" is another way to show the solution set of the inequality -5 x 4.5 on a number line. Interval notation is the method by which all real numbers between two given values can be represented. The solution set can be read as "x such that -5 is less than x and x is less than or equal to 4.5" because "|" denotes "such that" and the vertical line denotes "and."
Why is the set of solutions infinite?When the lines coincide and have the same y-intercept, the number of solutions to an equation system is infinite. If the slope and y-intercept of the two lines are the same, they are in fact on the same exact line.
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Identify the Constant of Proportionality From a Graph
Answers
The graph of constant of proportionality is determined and attached with a green check on the mark box
How to find the graph of constant of proportionalityThe ratio between two quantities that are directly proportional is the constant of proportionality.
When two quantities grow and shrink at the same rate, they are directly proportional. When y and x are two values that are directly proportional to one another, the proportionality constant k is defined as k=y/x.
The equation shows that y = kx and the y intercept is zero, because of this the graph will start from the origin
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The solution set to 6 + 2n> 12 is n> 3. Which are correct representations of this solution? Select two options.
{n\n<3}
{n\n≥3)
-5-4-3-2-1 0 1 2 3 4 5
-5-4-3-
(3,0⁰0)
0
2 3 4
5
Answers
-5-4-3- is the solution for {n\n<3} and 2-1-0 is the correct set for this solution {n\n≥3).
What is the set?A set is a well-defined collection of distinct objects, known as elements or members of the set. The objects can be anything, such as numbers, letters, words, animals, or even other sets. The defining feature of a set is that its elements are unique and distinct, meaning that no element can appear more than once in the set.
The solution set to 6 + 2n > 12 is n > 3. The correct representations of this solution are:
{n | n ≥ 3}
(3, ∞)
Option 1 {n | n < 3} is incorrect, as it represents the solution set of n being less than 3, which is the opposite of the solution set provided.
Option 2 -5-4-3- is Correct, as it is not in the correct format to represent a solution set.
Option 3 (3,0⁰0) is correct because it represents a point, not a set.
Option 4 and 5 -5-4-3-2-1 0 1 2 3 4 5 and 0 2 3 4 5 are also incorrect as they are not in the correct format to represent a solution set, and does not include the value 3.
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*
The value of tan x is given. Find sin x and cos x if x lies in the specified
tan x=2, x 0,
xe
T
sinx=| (Type an exact answer, using radicals as needed.)
Answers
The value of sin and cosine of angle x are √2 and √2/2
How to determine the value of sin and cosinefrom the question, we have the following parameters that can be used in our computation:
tan(x) = 2
Also, we have the interval to be
x = (0, ¤/2)
This means that the value of the angle x is from 0 to 90 degrees
Mathematically, we have
tan(x) = sin(x)/cos(x)
This means that
sin(x)/cos(x) = 1/2
So, we have
cos(x) = 2sin(x)
Let cos(x) = √2
This means that
sin(x) = √2/2
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you go out to eat with your family. the subtotal on your bill is $45.50 you must pay a 8% sales tax. what is your bill INCLUDING sales tax? please help
Answers
45.50+ (0.08x45.50) = 49.14
$49.14 total
AOB is a line . Ray OP is perpendicular to AB OC is another ray between rays OA and OP . If ray OD is perpendicular to OC , prove ∠COP = ∠BOD.
Answers
The proof of ∠COP = ∠POD is below.
What is a ray?When viewed as a vector, a ray is a vector from a point to a point . In geometry, a ray is usually taken as a half-infinite line (also known as a half-line) with one of the two points and. taken to be at infinity.
Given is that AOB is a line. Ray OP is perpendicular to AB. OC is another ray between rays OA and OP. If ray OD is perpendicular to OC.
Refer to the figure attached.
In ΔOCP and ΔODP -
∠OPC = ∠OPD = 90° {right angles}
OP = OP {Common}
OC = OD
By SAS criteria, ΔOCP ≅ ΔODP.
So, we can write -
∠COP = ∠POD
Therefore, the proof of ∠COP = ∠POD is given above.
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Lin is running in an election to be president of the seventh grade. She wants to predict her chances of winning. She has the following ideas for surveying a sample of the students who will be voting: a. Ask everyone on her basketball team who they are voting for. B. Ask every third girl waiting in the lunch line who they are voting for. C. Ask the first 15 students to arrive at school one morning who they are voting for.
Answers
Answer:
C
Step-by-step explanation:
Lets use process of elimination.
A would result in biased results as Lin is on the basketball team herself and thus, her teammates would be inclined to vote for her.
B would result in biased results as well. It doesn't say every third person in the line but rather every third girl. The girls might be inclined to vote for a female candidate.
That leaves C as the only viable option.
i need answers asap or pls
Answers
Answer:
see attached
Step-by-step explanation:
Can someone help me find the missing angle measurements?
Answers
Subtract the given angle from 180° and the missing angle measurements is 42°.
What is meant by angle ?
An angle is a figure in plane geometry that is created by two rays or lines that have a shared endpoint. The Latin word "angulus," which meaning "corner," is the source of the English term "angle." The shared terminus of two rays is known as the vertex, and the two rays are referred to as sides of an angle.A pair of rays (half-lines) with a shared terminal are combined to form an angle. The latter is referred to as the angle's vertex, and the rays are sometimes referred to as the angle's legs and other times as its arms.A+B+C=180°
57°+81°+C°=180°
138°+C=180°
C=138°-180°
C=42°.
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Use your formula to determine the height of a trapezoid with an area of
24 cubic centimeters and base lengths of 9 cm and 7 cm
Here is the formula: 1/2(a+b)h
Answers
Answer:
Step-by-step explanation:
The length of Ms. Sheahan's office is
3x yards and the width is x + 4 yards.
Write a polynomial that represents the
area. Find the area if x = 1 yd.
Answers
Answer:
see explanation
Step-by-step explanation:
the area (A) of a rectangle is calculated as
A = length × width
= 3x(x + 4) ← distribute parenthesis by 3x
A = 3x² + 12x
when x = 1 , then
A = 3(1)² + 12(1) = 3 + 12 = 15 yd²
Trey is having a party. He'll have 2 tables for every 10 guests.
Complete the table below showing the number of tables and the number of guests.
number of guests: (10) (15) (25) _ (50)
number of tables: (2)( _) (_) (7) (_)
Answers
The number of tables for 15 guests will be 3 tables.
The number of tables for 25 guests will be 5 guests
The number of tables for 50 guests will be 10.
How to calculate the number of tables and guest?Ratio in Mathematics simply demonstrates how many times one number can be able to fit into another number. It should be noted that ratios contrast two numbers by dividing them. In this case, A/B will be the formula if one is comparing one variable (A) to another variable (B).
In this case, Trey is having a party. He'll have 2 tables for every 10 guests. This means that 1 table will be for 10/2 = 5 guest
The number of tables for 15 guests will be:
= 15 / 5.
= 3 tables.
The number of tables for 25 guests will be:
= 25 / 5
= 5 guests
The number of tables for 50 guests will be:
= 50 / 5
= 10
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The area of this rectangle is given by the quadratic function
A = 50W - W2
.
What is the reasonable domain for this function?
Answers
The reasonable domain for this function is 0 < x < 50
What is the domain and range of the function?The domain of a function is defined as the set of all the possible input values that are valid for the given function.
The range of a function is defined as the set of all the possible output values that are valid for the given function.
Given;
A = 50W - W2
The area of a rectangle cannot be negative;
The practical domain of A is determined when;
50W-W^2>0
Taking common factor W
W(50-W)>0
Since W must be positive W>0
50-W>0
Or equivalently
50>w
W<50
The total interval is
0<W<50
Therefore, the domain of the function will be 0 < x < 50
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the shortest leg of a triangle is 2 centimeters shorter than the other leg. the hypotenuse of this triangle is 10 centimeters. what are the lengths of the two legs of this triangle?
Answers
Lengths of the legs of triangle are 6 cm and 8 cm by Pythagoras theorem of right triangle.
Let One short leg = x
then other = x+2
And hypotenuse is = 10 cm
so x²+(x+2)²=10²
Pythagoras theorem states that “In a right-angled triangle, the square of the hypotenuse side is equal to the sum of squares of the other two sides“.
From Pythagoras theorem : a^2 + b^2 = c^2
Where c is the hypotenuse and a,b are the other sides of a right angle triangle.
x²+ x²+4x+4=100
[(a + b)^2 = a^2 +2.a.b +b^2]
2 x²+4x-96=0
x²+2x-48=0
(x+8)(x-6)=0
x= -8 or 6
Neglecting - value x=6 as the length can not be negative.
So legs of triangle are 6 cm and 8 cm.
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a square with integer side length is cut into 10 squares, all of which have integer side length and at least 8 of which have area 1. what is the smallest possible value of the length of the side of the original square?
Answers
Answer: I need help with this what is the answer
Step-by-step explanation:
1/2 x 2 please show your work .
Answers
The solution of an expression 1/2 x 2 is 1.
What is multiplication?Multiplication is a mathematical arithmetic operation. It is also a process of adding the same types of expression for some number of times.
Example - 2 × 3 means 2 is added three times, or 3 is added 2 times.
Given:
We have an expression 1/2 x 2.
We will apply the inverse property of multiplication.
= any number x its reciprocal
= 2 x 1/2
= 1
Therefore, 1 is the solution.
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true or false correlations are used both as a hypothesis test to check the relationship between two variables, and also as internal reliability check, again to check the relationship between two items
Answers
Correlations are used both as a hypothesis test to check the relationship between two variables, and also as internal reliability check, again to check the relationship between two items is true.
To evaluate a hypothesis about a population, hypothesis testing uses sample data. A hypothesis test evaluates the result's unusualness, determining if there is a reasonable amount of random variation or whether the result is too extreme to be accounted for by chance.
Reliability can be measured by internal consistency. When a measurement is said to be reliable, it means that it will always produce the same result, given all other factors being equal.
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Find the distance between the given points. Round your answer to the nearest tenth. ( 1 , 2 ) and (5, 9)
Answers
The distance between the given points is 8.1 units
How to find the distance between the given points?Distance between two points is the length of the line segment that connects the two points in a plane.
The formula to find the distance between the two points is usually given by d=√((x2 – x1)² + (y2 – y1)²)
From the given info, we have:
( 1 , 2 ) and (5, 9),
where x1 = 1, y1 = 2 and x2 = 5 and y2 = 9
d=√((x2 – x1)² + (y2 – y1)²)
d=√((5 – 1)² + (2 – 9)²)
d=√((4)² + (-7)²)
d = √65
d = 8.1 units
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Find the area of the shaded region.
13.
12
Geometry IF8763
12
2014
11
86
15, 1111717-74
27
12.5
27
10 mm
24
27
27
MCMXCIV Instructional Fair,
Answers
Answer: need photo
Step-by-step explanation:
PLEASE HELP E ASAP!!!!!!!!!!
Answers
Answer:
78.4 meters
Step-by-step explanation:
PLSSSS HELP IF YOU TURLY KNOW THISSS
Answers
[tex] \pmb{ \underline{\color{green} \huge{Αηςωεr : ∼}}}[/tex]
[tex] \large \frak{ \red{x } - 4 = 8}[/tex]
[tex] \: [/tex]
[tex] \large \frak{ \red{x } = 8 + 4}[/tex]
[tex] \: [/tex]
[tex] \underline{ \boxed{ \large \frak{ \red{x } = \green{12}}}}[/tex]
[tex] \: [/tex]
hope it helps!