The polar form of any complex number can be written as
[tex]z = |z| e^{i\arg(z)}[/tex]
where [tex]\arg(z)[/tex] is the argument of [tex]z[/tex], i.e. the angle it makes with the positive real axis in the complex plane.
If [tex]z=\sqrt3+i[/tex], then [tex]z[/tex] has modulus
[tex]|z| = \sqrt{\left(\sqrt3\right)^2 + 1^2} = \sqrt4 = 2[/tex]
and argument
[tex]\arg(z) = \tan^{-1}\left(\dfrac1{\sqrt3}\right) = \dfrac\pi6[/tex]
Then
[tex]\sqrt3 + i = 2e^{i\frac\pi6} = 2 \left(\cos\left(\dfrac\pi6\right) + i \sin\left(\dfrac\pi6\right)\right)[/tex]
In constructing 98% confidence intervals for means, what do we expect will be true over the long run?
In constructing 98% confidence intervals for means, it is true that the statistic is good enough, and is less uncertain.
What is a confidence interval?A confidence interval expresses the degree of uncertainty surrounding a given statistic. A margin of error is frequently used with confidence intervals. It reveals the degree to which you may be certain that the findings of a poll or survey correspond to what you would anticipate discovering if it were possible to poll the complete population. Levels of confidence are inextricably linked to confidence intervals.
Your level of confidence in your findings is shown by the confidence interval. You can never be certain that your results will hold for future surveys or experiments. In statistics, being 95% or 98.5% certain is typically seen as "good enough."
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AC is a tangent to the circle below at point B. Calculate the sizes of angle x, angle y and angle z. Justify each of your answers. .
[tex]y=78^{\circ}[/tex] (alternate interior angles theorem)
[tex]x=78^{\circ}[/tex] (alternate segment theorem)
[tex]z=24^{\circ}[/tex] (angles in a triangle add to 180 degrees)
The water works commission needs to know the mean household usage of water by the residents of a small town in gallons per day. Assume that the population standard deviation is 2.4 gallons. The mean water usage per family was found to be 19.5 gallons per day for a sample of 3034 families. Construct the 98% confidence interval for the mean usage of water. Round your answers to one decimal place.
The confidence interval for the mean usage of water is (18.7,20.5).
Given population standard deviation of 2.4, mean of 19.5 gallons per day and confidence interval of 98%.
We have to find the confidence interval for the mean usage of water.
To find out the confidence interval we have to first find margin of error.
μ=19.5
σ=2.4
α=0.98
α/2=0.49
We have to find the z value for the p value 0.49 which is z=2.33
Margin of error=z*μ/[tex]\sqrt{n}[/tex]
=2.33*19.5/[tex]\sqrt{3034}[/tex]
=0.82
lower level=mean -m
=19.5-0.82
=18.68
after rounding upto 1 decimal
=18.7
upper mean = mean+m
=19.5+0.82
=20.52
Hence the confidence interval for the usage of water is (18.7,20.52).
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A board is 90 inches long. How long is it in feet and inches?
Answer:
feet: 7.5
Inches: 90
Step-by-step explanation:
Whoever helps me with this problem will have brainlist.
By definition of quotient, the rational number 1/3 represents a periodic infinite decimal number. That is to say, 1/3 is equivalent to the decimal number [tex]0.\overline{333}[/tex]
How to find the quotient of a fraction
The quotient of a fraction is found by dividing the fraction. Results in decimal form can be finite and infinite, infinite forms can be periodic or no periodic. This fraction represents a periodic infinite decimal number.
[tex]\frac{1}{3} = 0.\overline{333}[/tex]
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The expression of the fraction 1/3 as a quotient is; 0 remainder 3 or 0 r 3.
How to find the quotient of a fraction?
Quotient is described as the quantity produced by the division of two numbers.
Now, we want to find the quotient of 1/3. It should be noted that this quotient should not be in decimal form and so it gives us;
0 remainder 3 since 3 as a denominator is bigger than 1 as numerator.
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URGENT!! WORTH 27 POINTS
Jace has consumed 1,500 calories so far today. He has also burned off 400 calories at the gym. He would like to keep his daily calorie total to 1,900 calories per day. How many calories does he have left to consume for the day? Is 1,100 a viable solution to this problem?
a. yes, 1,100 is less than 1,500.
b. no, 1,100 is more than the 400 he burned off at the gym.
c. no, 1,100 will cause him to exceed 1,900.
d. yes, 1,100 is less than 1,900.
Answer:
Step-by-step explanation:
1,500-400=1,100 he has 800 calories left so no if he ate 1,100 more calories he would go over his goal of 1,900 so C is the answer hope this helps
Ella needed to get her computer fixed. She took it to the repair store. The technician at the store worked on the computer for 3 hours and charged her $61 for parts. The total was $226. Write and solve an equation which can be used to determine x, the cost of the labor per hour
3 connecting lines are shown. Line D F is horizontal. Line D E is about half the length of line D F. Line F E is about one-third of the length of line D F.
Which inequality explains why these three segments cannot be used to construct a triangle?
EF + FD > DE
ED + EF < DF
ED + EF > DF
EF + FD < DE
The inequality that explains why the three segments cannot be used to construct a triangle is ED + EF < DF
InequalitiesFrom the question, we are to determine which of the given inequalities explains why the three segments cannot be used to construct a triangle
From the given information,
Line DE is about half the length of line DF
That is,
ED = 1/2 DF
Also,
Line FE is about one-third of the length of line DF
That is,
EF = 1/3 DF
Then, we can write that
ED + EF = 1/2DF + 1/3DF
ED + EF = 5/6 DF
Since,
5/6 DF < DF
Then,
ED + EF < DF
Hence, the inequality that explains why the three segments cannot be used to construct a triangle is ED + EF < DF
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PLEASE REFER THE ATTACHMENT. THANK YOU
The solution is given below:
What is price index?An index number expressing the level of a group of commodity prices relative to the level of the prices of the same commodities during an arbitrarily chosen base period and used to indicate changes in the level of prices from one period to another.
Given:
Comm. p0 q0 p1 q1 p0q0 p1q0 p1q1 p0q1
A 25 750 30 960 18750 750 28800 24000
B 30 450 25 550 13500 750 13750 16500
C 5 250 6 360 1250 30 2160 1800
D 6 90 7 210 540 42 1470 1260
E 10 140 10 190 1400 100 1900 1900
F 4 48 5 65 192 240 365 260
Now,
[tex]\sum[/tex]p0q0 = 35632
[tex]\sum[/tex]p1q0 = 1912
[tex]\sum[/tex]p1q1 = 48445
[tex]\sum[/tex]p0q1= 45720
1) P01(L) = [tex]\sum[/tex]p1q0/[tex]\sum[/tex]p0q0 * 100
= 5.36
2) P01(L) = [tex]\sum[/tex]p1q1/[tex]\sum[/tex]p0q1 * 100
= 105.96
Dorbish- Bowley
= 5.36+ 105.96/2
=55.66
Marshall- Edgeworth
= 1912+ 48445/35632 + 45720
= 50357/ 81352
= 0.619 *100= 61.9
Fisher's price index
=( 5.36 * 105.96 )^ 0.5
=23.831
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P is a rectangle with a length of 40cm and width of x cm
Q is a rectangle with a width of y cm
the length of Q is 25% more than the length of P
the area of Q is 10% less than the area of P
work out the ratio of x:y
The ratio x:y as required in the task content is; 25:18.
What is the ratio x:y as required?It follows from the information given in the task content that;
The area of rectangle P is; 40x.
The area of rectangle Q is; y × (1.25 × 40) = 50y.
Hence, since the area of Q is 10% less than the area of P, it follows that A(q) = 0.9A(p).
And, A(p)/A(q) = 10/9
Hence, 10/9 = 40x/50y
x/y= 50/36 = 25/18
Hence, the required ratio is; 25: 18.
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How many ways are there to fill either the pitcher or catcher field position (but not both) from among the 25 players (leaving the other field positions empty)
Using the permutation formula, it is found that there are 600 ways to fill the pitcher and catcher positions.
The order is important, as one player is the pitcher and the other is the catcher, hence the permutation formula is used to solve this question.
What is the permutation formula?The number of possible permutations of x elements from a set of n elements is given by:
[tex]P_{(n,x)} = \frac{n!}{(n-x)!}[/tex]
2 players are taken from a set of 25, hence the number of ways is given by:
[tex]P_{(25,2)} = \frac{25!}{23!} = 600[/tex].
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Arianna invests $6,000 into a brokerage
account which earns 3% per year. In 30 years,
how much will Arianna's investment be worth
if interest is compounded quarterly (4 times a
year)? Round to the nearest dollar.
Answer: $14708.14
Step-by-step explanation:
Amount = initial balance(1 + annual rate/periods per year)^(periods per year*years)
A = 6000(1+0.03/4)^4*30
A = 6000(1.0075)^120
A = 6000(2.451)
A = 14708.14
Hope this helps!
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The surface area of the spherical ball is given as follows:
[tex]635.04\pi[/tex] yd².
What is the surface area of a sphere?The surface area of a sphere of radius r is given as follows:
[tex]S = 4\pi r^2[/tex].
In this problem, the diameter is of 25.2 yd, hence the radius is given by:
r = 0.5 x 25.2 yd = 12.6 yd.
Then the surface area in yd² is:
[tex]S = 4\pi (12.6)^2 = 635.04\pi[/tex].
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Which of these tables represents a linear function?
Answer:
a
Step-by-step explanation:
a is the anwer because b/c it has a constant rate of change unlike the others
find the perimeter and area of a triangle whose sides are of length 2cm, 5cm and 5cm
Answer:
Perimeter:12cm
Area:5cm²
Step-by-step explanation:
Perimeter is the total distance around the figure ie sum of all sides of figure
∴Perimeter of triangle equals 2cm plus 5cm plus 5cm
∴The perimeter equals 12cm
Area of triangle equals 1/2×base×height,where base is 2cm & height is 5cm
(Inserting values)
∴Area of the triangle equals 1/2×2cm×5cm which equals 5cm²
Use the Algebra Tiles to construct the algebraic expression for this situation:
What is this expression in simplified form?
`-4x -3 + 5x -2`
The given expression in simplified form is x - 5
Simplifying an expressionFrom the question, we are to simplify the given expression
The given expression is
-4x -3 + 5x -2
To simplify the expression, we collect like terms and then simplify
That is,
-4x + 5x -3 -2
= x - 5
Hence, the given expression in simplified form is x - 5
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Select the equivalent expression:
Please explain why it is changed from a fraction to a multiplication problem!
Answer:
On opening the bracket, we have (x^-28)/(x^56)
This is the same as (x^-28) . 1 / (x^56)
Following the law of indices 1/(x^a) = x^-a, we have 1 / (x^56) = (x^-56)
Hence (x^-28) . (x^-56)
Can someone pls help with these geometry questions, it’s urgent.
Answer:
6. x + x + 14 = 90
2x + 14 = 90
2x = 76
x = 38
angle b = 38 + 14 = 52°
7. angles on a straight line add up to 180°
x + x - 32 = 180
2x = 212
x = 106°
angle d = 106 - 32 = 74°
8. vertically opposite angles are equal
7x - 80 = 3x
-80 = -4x
20 = x
angle 1 = 7(20) - 80 = 60°
angle 2 = 3(20) = 60°
Write the following as an equation. then solve. twice the difference of a number and 8 is equal to three times the sum of the number and 3. find the number. the equation is (type an equation using x as the
The equation representing the given case is 2(x - 8) = 3(x + 3), where x is the number.
The number satisfying this equation is -25.
In the question, we are asked to find the number when twice the difference of the number and 8 is equal to thrice the sum of the number and 3.
We assume the number to be x.
The expression representing twice the difference between the number and 8, can be shown as 2(x - 8).
The expression representing thrice the sum of the number and 3, can be shown as 3(x + 3).
We are informed that these two expressions are equal. This can be shown as the equation:
2(x - 8) = 3(x + 3).
Thus, the equation representing the given case is 2(x - 8) = 3(x + 3), where x is the number.
Now, to find the number, we solve the equation as follows:
2(x - 8) = 3(x + 3),
or, 2x - 16 = 3x + 9 {Simplifying},
or, 2x - 16 + (16 - 3x) = 3x + 9 + (16 - 3x) {Adding (16 - 3x) to both sides of the equation},
or, 2x - 16 + 16 - 3x = 3x + 9 + 16 - 3x {Simplifying},
or, -x = 25 {Simplifying},
or, -x(-1) = 25(-1) {Multiplying both sides of the equation with -1},
or, x = -25 {Simplifying}.
Thus, the number satisfying the equation is -25.
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What is the slope of a line that passes through the points ( -1, 9) and (3,6)?
Answer:
-¾
Step-by-step explanation:
Slope=m
(-1,9) (3,6)
m= Y2-Y1 / X2-X1
m=6-9/3-(-1)
m=-3/3+1
m=-3/4
Jevon Noda's home has a market value of $72,000, the rate of assessment is 30%, and the tax rate is 85.
mills. Find the a) assessed value, b) tax rate as a decimal, and c) real estate tax.
The required a) assessed value = $21600,
b) tax rate = 0.0855,
c) real estate tax = $1836.
Cost price is that price for buyer which he pays to seller for an object or product.
a) assessed value = 72000 x 30/100
= $21600
b) tax rate = 85.5/1000
= 0.0855
c) real estate tax = 0.855 x 21600
= $ 1836
Thus, the required values of a) assessed value, b) tax rate as a decimal, and c) real estate tax is $21600, 0.085 and $1836 respectively.
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Which expression is equivalent to 54 · (5-2)3 ?
Point A is the point of concurrency of the angle bisectors of ΔDEF.
Point A is the point of concurrency of triangle D E F. Lines are drawn from each point of the triangle to point A. Lines are drawn from point A to the sides of the triangle to form right angles and line segments A X, A Y, and A Z. The length of F A is 6 centimeters, the length of A D is 5 centimeters, the length of A X is 3 centimeters, and the length of Y D is 4 centimeters.
What is the length of ZA?
ZA = 3cm
ZA = 4cm
ZA = 5cm
ZA = 6cm
The correct answer is option A which is ZA = 3 cm
What is the triangle?Triangle is a shape made of three sides in a two-dimensional plane. the sum of the three angles is 180 degrees.
The triangle is shown below.
From the triangle, A is the concurrency point of angle bisectors of all vertices.
Consider ΔAYD,
Using the Pythagorean theorem,
AD² = AY² + YD²
5² = AY² + 4²
25 = AY² + 16
AY² = 25 - 16
AY² = 9
AY = √9 = 3 cm
Consider triangles ADY and ADZ.
∠AYD ≅∠AZD=90
∠ADY≅∠ADZ ( Angle bisector)
AD ≡ AD (common side)
The two triangles are congruent by the AAS postulate.
Therefore, by CPCTE, AY=ZA=3 cm
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If the point P(-3/5,y) lies on the unit circle and P is in the 2nd quandrant, what does y equal
The value of y by virtue of the circle description above is; 0.8.
What is the value of y in the task?By definition, a unit circle is a circle whose radius is 1 and has its center at the origin (0,0).
On this note, the value of y can be evaluated as;
1² = (-3/5-0)² + (y-0)²
1 -0.36 = y²
0.64 = y²
±0.8 = y.
However, since point P is in the 2nd quadrant where y is positive, the value of y is 0.8.
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1. Your friend really wants to buy the newest cell phone which will cost him $600. He gets a job detailing cars and SUVs and plans to save all the money he earns. If he gets paid $65 to detail a car
and $90 to detail an SUV, what linear inequality can be used to determine how many of each he must detail in order to earn at least $600?
A. 65c+90s > 600
B. 65c + 90s 2 600
C. 65c+90z ≤ 600
D. 65c+90z< 600
Answer:
I'm not sure if B. is incorrectly written but the answer would be
65c + 90s [tex]\geq[/tex]600
Step-by-step explanation:
since he needs AT LEAST 600$ then the sign should look like that. 65 dollars per car, and c represents the amount of cars, and 90 per SUV and s represents how many SUVS
B. 65c + 90s ≥ 600 is the correct option.
Our friend really wants to buy the newest cell phone which will cost him $600. He gets a job detailing cars and SUVs and plans to save all the money he earns.
What is inequality?Inequality can be defined as the relation of the equation contains the symbol of ( ≤, ≥, <, >) instead of the equal sign in an equation.
Here,
If he gets paid $65 to detail a car and $90 to detail an SUV
The inequality form for above condition so he can earn at least $600
= 65c + 90s ≥ 600, where c = cars and s = SUV
Thus, the required inequality is given by 65c + 90s ≥ 600.
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Please help!
On a certain Saturday, the number of adult tickets sold was 1/3 (one third) of the children tickets sold. What was the greatest number of adult tickets sold if the box office receipts was not more than $3,190?
Adult admission fee: $8.50
Children admission fee: $5.50
The greatest number of adult tickets sold if the box office receipts was not more than $3,190 is 328.
How to find the greatest number of adult tickets sold ?let
number of children tickets = x
number of adult ticket sold = 1 / 3 x
The greatest number of adult tickets sold if the box office receipts was not more than $3,190 can be calculated as follows:
The equations that represent the situation is as follows:
5.50x + 8.50 (1/3 x) = 3190
5.50x + 2.83333333333x = 3190
8.33333333333x = 3190
x = 3190 / 8.33333333333
x = 982.839313573
x ≈ 983
Therefore,
adult ticket = 983/3 = 328
Therefore, the greatest number of adult tickets sold if the box office receipts was not more than $3,190 is 328.
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During the sale, Carly pays $54 for a wheelbarrow. What was the original price?
The original price of the wheelbarrow is £75
How to determine the original price?Let the original price be x.
From the question, the amount paid is:
Amount paid = 54
The reduced price is calculated as:
Reduced = Original price * (1 - discount)
This gives
x * (1- 20%) = 54
Evaluate the difference
x * 0.8 = 54
Divide by 0.8
x = 67.5
Hence, the original price is $67.5
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Complete question
The price of a wheelbarrow is reduced by 20%. During the sale, Carly pays $54 for a wheelbarrow. What was the original price?
Please help, I can’t really figure out the answer..
Answer:
0.625
Step-by-step explanation:
To find the constant, you do y/x. For the first pair, if we divide 1.25/2, we get 0.625. We can do the same for all the pairs and get 0.625
Hope this helps :)
Answer:
1.6
Step-by-step explanation:
To find the constant of proportionality, we do [tex]\frac{x}{y}[/tex]. We see that 2/1.25=1.6, 4/2.5=1.6, 6/3.75=1.6, and 10/6.25=1.6. This means that the constant of proportionality is equal to 1.6.
please help with sin question
Answer: A
Step-by-step explanation:
The cosine of an angle is equal to the sine of its complement, and vice versa.
Which of the following data sets has the mean, median, mode, and range as the same number?
A. 6, 11,14,14,19,20
B. 10,12,14,16,18,20
C. 8,11,13,15,19,22
D. 3,4,5,6,7,8
Answer:
A.
Step-by-step explanation:
6, 11,14,14,19,20
Adding these 6 numbers we get 84.
84 / 6 = 14 = mean.
Median = middle number(s) = 14
Mode = number occurring most = 14.