For each of the problems, we will start by identifying the values of r and θ from the given complex number in rectangular form (a + bi).
1) (1 + i)
r = sqrt(1^2 + 1^2) = sqrt(2)
θ = tan^-1(1/1) = π/4
Therefore, the polar form of (1 + i) is:
sqrt(2) * (cos(π/4) + i sin(π/4)) = sqrt(2) * e^(iπ/4)
2) (-3 + 3i)
r = sqrt((-3)^2 + 3^2) = 3sqrt(2)
θ = tan^-1(3/-3) = -π/4 or 7π/4
Note that we have two possible values for θ because the point (-3, 3) falls in the second and fourth quadrants. We will use the value 7π/4 because it is the standard angle in the fourth quadrant.
Therefore, the polar form of (-3 + 3i) is:
3sqrt(2) * (cos(7π/4) + i sin(7π/4)) = -3sqrt(2) * e^(i7π/4)
3) (-2 - 2i)
r = sqrt((-2)^2 + (-2)^2) = 2sqrt(2)
θ = tan^-1(-2/-2) = π/4
Therefore, the polar form of (-2 - 2i) is:
2sqrt(2) * (cos(π/4) - i sin(π/4)) = 2sqrt(2) * e^(-iπ/4)
4) (4 - 4i)
r = sqrt(4^2 + (-4)^2) = 4sqrt(2)
θ = tan^-1(-4/4) = -π/4 or 7π/4
Again, we have two possible values for θ. We will use 7π/4 because it is the standard angle in the fourth quadrant.
Therefore, the polar form of (4 - 4i) is:
4sqrt(2) * (cos(7π/4) - i sin(7π/4)) = -4sqrt(2) * e^(i7π/4).
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The combination for opening a safe is a four-digit number made up of different digits. How many different comninations can you make, using only add digits?
There are 625 different 4-digit codes only made with odd numbers.
How many different combinations can you make?To find the total number of combinations, we need to find the number of options for each one of the digits.
There are 4 digits, such that each digit can only be an odd number.
For the first digit, there are 5 options {1, 3, 5, 7, 9}For the second digit, there are 5 options {1, 3, 5, 7, 9}For the third digit, there are 5 options {1, 3, 5, 7, 9}For the fourth digit, there are 5 options {1, 3, 5, 7, 9}The total number of different combinations is given by the product between the numbers of options, so we have:
C = 5*5*5*5 = 625.
There are 625 different 4-digit codes only made with odd numbers.
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Which of the following are like terms?
3x and 5x²
4xy and 2x²y
6x and -3x
7x and 7y
A knight is placed at the origin of the cartesian plane it moves in an L shape what is the expected distance from the origin after 2016 moves
Answer:
Below in bold.
Step-by-step explanation:
Using the Pythagoras theorem:
Distance from the origin after 1 move = √(1^2 + 2^2) = √5.
So after 2016 distance is 2016√5 units.
This = 4507.9 units to the nearest tenth,
65. Find the perimeter & area:
13 cm
5 cm
12 cm
66.
Given, that the sides of a triangle are [tex]$5 \mathrm{~cm}, 12 \mathrm{~cm}$[/tex] and [tex]$13 \mathrm{~cm}$[/tex] semi-perimeter [tex]$=\frac{\mathrm{a}+\mathrm{b}+\mathrm{c}}{2}$[/tex]
[tex]$$\begin{aligned}&\mathrm{s}=\frac{(5+12+13)}{2} \\&\mathrm{~s}=\frac{30}{2} \\&\mathrm{~s}=15 \mathrm{~cm}\end{aligned}$$[/tex]
Area of the triangle (according to Heron's Formula)
[tex]$$\begin{aligned}&=\sqrt{\mathrm{s}(\mathrm{s}-\mathrm{a})(\mathrm{s}-\mathrm{b})(\mathrm{s}-\mathrm{c})} \\&=\sqrt{15(15-5)(15-12)(15-13)} \\&=\sqrt{15(10)(3)(2)} \\&=\sqrt{900} \\&=30 \mathrm{~cm}^{2}\end{aligned}$$[/tex]
What is Heron's Formula?
Heron's formula was first given by Heron of Alexandria. It is used to calculate the area of various triangles such as equilateral, isosceles, and scalene triangles or quadrilaterals. When we know the sides of a triangle, we can use Heron's formula to find its area. Using Heron's formula, we find the area of a triangle by taking its semi-perimeter and side lengths.Heron's formula is used to calculate the area of triangles given the lengths of all their sides, as well as the area of quadrilaterals. It's also known as Hero's formula. This formula for calculating the area does not rely on the angles of a triangle. It is solely determined by the lengths of all triangle sides. It contains the term "s," which stands for semi-perimeter, which is obtained by halving a triangle's perimeter. Similarly, the concept of determining the area is extended to determine the area of quadrilaterals.To learn more about Hero's formula visit:
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uan recently hired a roofer to do some necessary work. On the final bill, Juan was charged a total of $1131.5. $450 was listed for parts and the rest for labor. If the hourly rate for labor was $47, how many hours of labor was needed to complete the job?
Based on the total cost that Juan incurred to hire the roofer including the charge for parts and labor, the number of hours that was needed to complete the job was 14.50 hours.
How many hours did the roofer use?The first thing to do is to find out the amount that went to labor:
= Final bill - cost of materials
= 1,131.5 - 450
= $681.50
Now that you have the cost of labor, you can find out the number of labor hours that were used based on the hourly rate:
= Cost of labor / Hourly rate
Solving gives:
= 681.50 / 47
= 14.5 hours
In conclusion, the number of labor hours that the roofer used to complete the work, based on the total amount charged to Juan is 14.50 hours.
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which whole number is equal to the fraction 42/6
Answer:
7.
Step-by-step explanation:
Think of 42/6 as a division question.
42 ÷ 6.
The answer to this equation is 7.
URGENT WILL GIVE BRAINLIEST
Answer:
626.7 [tex]cm^{2}[/tex]
Step-by-step explanation:
Consider the figure as a circle with radius 7.5 and a rectangle with width 30 and length 15.
CIRCLE
A = pi*r*r = pi*7.5*7.5 = 56.25pi =176.7
RECTANGLE
A = l*w = 15*30 = 450
176.7 + 450 = 626.7 [tex]cm^{2}[/tex]
Answer: 626.7cm^2
Step-by-step explanation:
30 * 15 = 450
area of circle = πr^2
r = 15/2 = 7.5
[tex]7.5^2=56.25*pi=176.7[/tex]
450+176.7 = 626.7
The graph of the function f(x)=x2+12 is shown which stamen te describe the graph Check all that apply
James and Simon have a reading assignment to complete. James has read r
rr pages, and Simon has read 75 pages. Together they have read a total of 200 pages. Select the equation that matches this situation.
Choose 1 answer:
Answer:
Step-by-step explanation:
125 I think
State the transformations being applied to each quadratic function.
a) y = -1/2(x+2)^2+4
b) y= -(x-1)^2-2
c) y=(4x)^2
d) y= 4x^2
The function transformations are:
The function (a) y = -1/2(x + 2)^2 + 4 is translated to the left by 2 units, reflected across the x-axis, compressed vertically by a factor of 1/2 and translated up by 4 unitsThe function (b) y = -(x - 1)^2 - 2 is translated right by 1 unit, reflected across the x-axis, and translated down by 2 unitsThe function (c) y = (4x)^2 is stretched horizontally by a factor of 1/4The function (d) y = 4x^2 is stretched vertically by a factor of 4What are transformations?Transformations involve translating, reflecting, rotating and dilating a function across the coordinate plane
How to determine the transformations?The parent function of a quadratic function is represented as:
y = x^2
When the function is stretched horizontally by a factor of k, where k is between 0 and 1, we have:
y = (x/k)^2
Assume k = 1/4. we have:
y = (4x)^2
This means that the function (c) y = (4x)^2 is stretched horizontally by a factor of 1/4
When the function is stretched vertically by a factor of k, where k is greater than 1, we have:
y = k(x)^2
Assume k = 4. we have:
y = 4x^2
This means that the function (d) y = 4x^2 is stretched vertically by a factor of 4
Translating the function left is represented as:
y = (x + k)^2
Assume k = 2. we have:
y = (x + 2)^2
Reflecting the function across the x-axis is represented as
y = -(x + 2)^2
When the function is compressed vertically by a factor of k, where k is between 0 and 1, we have:
y = -k(x + 2)^2
Assume k = 1/2. we have:
y = -1/2(x + 2)^2
Translating the function up is represented as:
y = -1/2(x + 2)^2 + k
Assume k = 4. we have:
y = -1/2(x + 2)^2 + 4
Hence, the function (a) y = -1/2(x + 2)^2 + 4 is translated to the left by 2 units, reflected across the x-axis, compressed vertically by a factor of 1/2 and translated up by 4 units
Translating the function right is represented as:
y = (x - k)^2
Assume k = 1. we have:
y = (x - 1)^2
Reflecting the function across the x-axis is represented as
y = -(x - 1)^2
Translating the function down is represented as:
y = -(x - 1)^2 - k
Assume k = 2. we have:
y = -(x - 1)^2 - 2
Hence, the function (b) y = -(x - 1)^2 - 2 is translated right by 1 unit, reflected across the x-axis, and translated down by 2 units
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A pattern has 77 yellow triangles to every 33 green triangles. What is the ratios of green triangles to yellow triangles?
What is the parameter of interest to compare the proportions from two populations?.
Answer:
also known as the population parameter of interest the parameter of interest is a practical value that gives you more information about the research sample or population being studied in other words these parameter define the describe the search population
what is parameter answer is given above
What is the value of the discriminant for the quadratic equation zero equals 2x^2+x-3
Discriminant for the quadratic equation [tex]2x^2+x-3=0[/tex] is [tex]d=25[/tex]
How to find the d Discriminant of the quadratic equation ?
We know that the standard quadratic equation
[tex]ax^2+bx+c=0[/tex]
Discriminant of the equation is [tex]d=b^2-4ac[/tex]
Equation given in the question is
[tex]2x^2+x-3=0[/tex]
So we can find the values of [tex]a,b,c[/tex]
[tex]a=2\\b=1\\c=-3[/tex]
Substitute the values
[tex]d=1^2-4*2*(-3)\\d=1+24\\d=25[/tex]
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I need help!!! I don't get it.
The ship's horizontal distance from the lighthouse is 1930.59 feet. Using trigonometric ratio 'tanθ' the distance is calculated.
What are trigonometric ratios?The trigonometric ratios are used for determining the lengths of the right-angled triangle. There are six basic trigonometric ratios. they are:
sin θ = opposite/hypotenuse
cos θ = adjacent/hypotenuse
tan θ = opposite/adjacent
sec θ = hypotenuse/adjacent
cosec θ = hypotenuse/opposite
cot θ = adjacent/opposite
Calculation:It is given that,
The height of the bacon-light is 135 feet above the water
Consider the horizontal distance between the boat and the lighthouse = x feet
The angle of elevation is given as 4°
Constructing a model as below in the figure.
From the trigonometric ratios, we have
tan θ = 135/x
⇒ tan 4° = 135/x
⇒ x = 135/tan 4°
∴ x = 1930.589 feet ≅ 1930.59 feet (rounding to the nearest hundredth of a foot)
Therefore, the ship's horizontal distance from the lighthouse is 1930.59 feet.
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A marble has a mass of 5 grams. juan has 17 marbles in his bag.
what is the total mass of the marbles?
o a. 22 grams
o b. 55 grams
o c. 57 grams
o d. 85 grams
Answer:
85 grams
Step-by-step explanation:
Marble: 5 g/marble
17 marbles in a bag: (17 marbles)*(5 g/marble) = 85 grams/bag of 17 marbles
Annual starting salaries for college graduates with degrees in business administration are generally expected to be between and . Assume that a confidence interval estimate of the population mean annual starting salary is desired. a. What is the planning value for the population standard deviation
The planning value for the population standard deviation is 4330.
Given the confidence interval 20000 and 35000.
We have to find the planning value of standard deviation.
Standard deviation is measuring dispersion of data. Uniform probability distribution has two bounds a and b. The standard deviation is given by :
s=[tex]\sqrt{(b-a)^{2}/12 }[/tex].
Annual starting salaries for college graduates be between 20000 and 35000.
It is uniform in the interval so, a=20000, b=35000.
Now we have to just put the values of a and b in the above formula to get the value of standard deviation.
s=[tex]\sqrt{(35000-20000)^{2} /12}[/tex]
=[tex]\sqrt{(15000)^{2} /12}[/tex]
=[tex]\sqrt{225000000/12}[/tex]
=[tex]\sqrt{18750000}[/tex]
=4330
Hence the planning values for the population standard deviation is 4330.
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Question is incomplete as it should includes the confidence interval of $20000 and $35000.
The likelihood that a patient with a heart attack dies of the attack is 0.04 (i.e., 4 of 100 die of the attack). Suppose we have 50 patients who suffer a heart attack. What is the probability that all will survive
The probability solved by binomial distribution that all will survive is 0.8154.
What is Binomial distribution?When each trial has the same probability of achieving a given value, the number of trials or observations is summarized using the binomial distribution. The likelihood of observing a specific number of successful outcomes in a specific number of trials is determined by the binomial distribution.
Computation of probability of all survived people from heart attack;
A heart attack patient has a 0.04 percent chance of dying from the attack (i.e., 4 of 100 die of the attack).
What is the likelihood that each of the five patients who experience a heart attack will survive?
We'll refer to a victory in this scenario as a heart attack (p = 0.04). In other words, we are interested in the likelihood that none of our n=5 patients will die (0 successes).
Each attack has a chance of being fatal or not, with a probability of 4 percent for all patients, and each patient's result is independent.
Assume for the purposes of this example that the five individuals being examined are unrelated, of the same age, and free of any concomitant disorders.
By binomial distribution,
[tex]\begin{gathered}P(0 \text { successes })=\frac{5 !}{0 !(5-0) !} 0.04^{0}(1-0.04)^{5-0} \\P(0 \text { successes })=\frac{5 !}{5 !}(1)(0.96)^{5}=(1)(1)(0.8154)=0.8154\end{gathered}[/tex]
Therefore, with a 4 % chance that anyone will die, there is an 81.54% chance that every patient will survive the onslaught. The outcomes in this example could be 0, 1, 2, 3, 4 or 5 successes (fatalities).
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Quick algebra 1 question for 50 points!
Only answer if you know the answer, quick shout-out to Dinofish32, tysm for the help!
Answer:
y is equal to 24
Step-by-step explanation:
If y is directly proportional to x when x is equal to 5 and y is 12, then when you multiply 5 by 2 which is 10, y will be proportional to that when you multiply 12 by 2 which is 24
Answer:
b: 25
Step-by-step explanation:
The previous answer wowwubbzy13 uploaded is absolutely correct!
I'll set up a proportion.
If y = 12, then x = 5
--> 12:5
We want to find the y when x is 10
--> y:10
12:5 = y:10
5y = 120
y = 25
So B: 25 is the correct answer!
Quick algebra 1 question for 50 points!
Only answer if you know the answer, quick shout-out to Dinofish32, tysm for the
Answer:
10 points
Step-by-step explanation:
From the passage, we know:
1. Jay earned 15 points after spending $60
2. We want to know the number of points he can earn if he spends $40
We can set this up into proportions.
60:15 = 40:x
The x is the number of points he will get after spending $40.
15(40) = 60x
600 = 60x
x = 10
Jay will earn 10 points if he uses $40.
A combination lock uses three integers in the combination, and the dial is numbered with the integers 0, 1, 2 and 3. If adjacent numbers in the combination cannot be the same, how many possible combinations are there
There can be 36 combinations.
How to find the total number of combinations?The total number of combinations of the dial can be found by using permutations without any number repeating.
There are four integers on the dial. They are 0, 1, 2, and 3.
It is also given that the numbers shouldn't repeat on the adjacent dial.
Therefore, we can say that there are 4 possible numbers on the first.
Similarly, on the second dial, only 3 numbers are possible since no two adjacent dials can have the same.
This is also the case for the third dial. It can also have 3 possible numbers.
Therefore, the total number of combinations is given by 4*3*3 = 36 combinations.
Therefore, we have found that there can be 36 combinations.
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The volume, V, of a cylinder is V=r²h, where r is the
radius of the cylinder and h is the height. Using rounding
to the nearest whole number, which of the following is an
estimate of the volume of a cylinder with a radius of
3.75 inches and height of 6.21 inches?
Answer:
~87.33 in³
Step-by-step explanation:
All you have to to is pkug the numbers in the formula. You start by doing 3.75 as the r² and you get around 14.06. Then you multiply that by the height (6.21) which gets you around 87.33. I hope this helps!
I need help on this please!
Answer:
Step-by-step explanation:
Part A:
CIRCLE A : Circumference is (pi)(diameter) so if they give you the circumference (21.98), divide it by 7 which gives you 3.14.
CIRCLE B: 18.84/6 = 3.14
Part B:
CIRCLE A: Area is (pi)(radius^2) so if they give you the area (38.465), divide it by radius^2 (7/2 = 3.5^2 = 12.25) = 3.14
CIRCLE B: 28.26/(3^2) = 3.14
Part C:
The value of pi stays the same for circle A and B.
Hope this helps :)
Which type of function describes f(x)?
Exponential
Logarithmic
Rational
Polynomial
Which geometric solids would model the tent?
cone and sphere
cylinder and cone
pyramid and rectangular prism
triangular prism and rectangular prism
The interior angles of a polygon are; (3x + 30), x, 2x, (x + 20) and 3x. find the value of x
Answer: 49
Step-by-step explanation:
This polygon has 5 sides, meaning its interior angles add to [tex]180(5-2)=540^{\circ}[/tex].
[tex]3x+30+x+2x+x+20+3x=540\\\\10x+50=540\\\\10x=490\\\\x=49[/tex]
[tex] \color{red} \sf Solve \: for \: x : \\ \sf \: \: 4x + 2 = -8?[/tex]
Thank uh !
Answer:
x = -5/2
Step-by-step explanation:
4x+2 = -8
We are solving for x
The first step in isolating x is to subtract 2 from each side
4x+2-2 = -8-2
4x = -10
Divide each side by 4
4x/4 = -10/4
x = -10/4
Simplify the fraction
x = -5/2
Answer:
x= -5/2Step-by-step explanation:
4x+2 = -8
4x = -10
4x/4 = -10/4
x = -5/2
The diameter of a varicocele measures more than _____ millimeters (mm). * 5 points 5 3 4 2
Answer:
A varicocele measures more than 2 mm in diameter.
Step-by-step explanation:
pampiniform plexus veins measuring more than. 2 mm in diameter at rest and which increased in diameter by 1 mm, and subjective color Doppler.
Points A and B are 200 miles apart. A cyclist starting at point A and a motorcyclist starting at point B move toward each other. The speed of the cyclist is 17 mph and the speed of the motorcyclist is 83 mph. what distance from point A will they meet.
Answer:
34 miles from A
Step-by-step explanation:
Find time to cover 200 miles at ( 17 + 83 mph) = 200/100 = 2hours
cyclist will be at 2 * 17 = 34 miles from A
A bookstore sells books for $2, $3, $5, and $10. Let random variable X = "amount of
money for one book."
Look at the relative-frequency table below representing the amount of money spent on
one item and the relative frequencies with which customers purchase them
If the expected amount of money spent by a customer is $3.23 what is the standard deviation?
The value of the standard deviation is σ = 2.20. Using probability distribution, the required standard deviation is calculated.
How to calculate the standard deviation?The formula for the standard deviation of the given probability distribution is
σ = √∑([tex]x_i^2[/tex] × [tex]P(X_i)[/tex]) - μₓ²
Where the mean μₓ = ∑[[tex]x_i[/tex] × [tex]P(X_i)[/tex]]
Calculation:It is given that,
x: $2, $3, $5, $10
P(X=x): 0.55, 0.26, 0.11, 0.08
Step 1: Calculating the mean:
we have μₓ = ∑[[tex]x_i[/tex] × [tex]P(X_i)[/tex]]
⇒ μₓ = 2 × 0.55 + 3 × 0.26 + 5 × 0.11 + 10 × 0.08
∴ μₓ = 3.23
Step 2: Calculating the standard deviation:
x: 2, 3, 5, 10
x²: 4, 9, 25, 100
P(X=x): 0.55, 0.26, 0.11, 0.08
([tex]x_i^2[/tex]) × [tex]P(X_i)[/tex]: 4 × 0.55 = 2.2; 9 × 0.26 = 2.34; 25 × 0.11 = 2.75; 100 × 0.08 =8
∑[([tex]x_i^2[/tex]) × [tex]P(X_i)[/tex]]: 2.2 + 2.34 + 2.75 + 8 = 15.29
Therefore,
The standard deviation, σ = √∑([tex]x_i^2[/tex] × [tex]P(X_i)[/tex]) - μₓ²
⇒ σ = [tex]\sqrt{15.29-(3.23)^2}[/tex]
= [tex]\sqrt{15.29-10.43}[/tex]
∴ σ = 2.20
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A 12 foot ladder is leaning against a building. If the bottom of the ladder is sliding along the pavement directly away from the building at 2 feet/second, how fast is the top of the ladder moving down when the foot of the ladder is 3 feet from the wall
Using Pythagoras theorem, the top of the ladder moving down when the foot of the ladder is 3 feet from the wall is of -0.518 feet/sec.
Let distance from the wall to the foot of the ladder is 'x' feet and the height of the top of the ladder is 'y' feet.
Pythagoras theorem, [tex]x^{2} + y^{2} = (12)^{2}[/tex] --->(1)
Given,[tex]\frac{dx}{dt}= 2feet/second[/tex] at x=3
Put x=3 in Pythagoras theorem equation (1)
[tex](3)^{2} + y^{2} = 144[/tex]
[tex]y^{2} = 144 - 9[/tex]
[tex]y^{2}[/tex] = 135
y = 11.61
Derive equation (1) w.r.t to 't'
[tex]2x\frac{dx}{dt} + 2y\frac{dy}{dt} = 0[/tex] ---->(2)
substitute the value of 'x', 'dx/dt' and 'y' in equation (2), we get the fast of the top of the ladder moving down when the foot of the ladder is 3 feet from the wall
[tex]2(3)(2) + 2 (11.61)\frac{dy}{dt} = 0[/tex]
12 + 23.22 [tex]\frac{dy}{dt}[/tex] = 0
[tex]\frac{dy}{dt}= \frac{-12}{23.22}[/tex]
[tex]\frac{dy}{dt} = -0.518[/tex]
Hence, using Pythagoras theorem the top of the ladder moving down when the foot of the ladder is 3 feet from the wall is of -0.518 feet/sec.
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