The closed form of the sum f(x) is f(x) = sin(x) + (1/3)sin(2x) + (1/5)sin(3x)
What is the trigonometric ratio?
The trigonometric functions are real functions that relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others.
We can use the identity:
[tex]sin(nx) = Im(e^{(inx))}[/tex]
where Im(z) denotes the imaginary part of z. Applying this identity, we can rewrite f(x) as:
[tex]f(x) = Im(e^{(ix))} + 1/3 Im(e^{(i2x))} + 1/5 Im(e^{(i3x))}[/tex]
Using the fact that Im(z) = (1/2i)(z - conj(z)) where conj(z) denotes the complex conjugate of z, we can simplify this expression:
[tex]f(x) = (1/2i)(e^{(ix)} - conj(e^{(ix)))} + (1/2i)(1/3)(e^{(i2x)} - conj(e^{(i2x)))} + (1/2i)(1/5)(e^{(i3x)} - conj(e^{(i3x)))}[/tex]
Now we can use the fact that [tex]e^{(ix)} - conj(e^{(ix))} = 2i sin(x) and e^{(i2x)} - conj(e^{(i2x))} = 2i sin(2x)[/tex] to get:
f(x) = sin(x) + (1/3)sin(2x) + (1/5)sin(3x)
Thus, we have expressed f(x) in a simpler form that allows us to evaluate it directly.
Therefore, the closed form of the sum f(x) is:
f(x) = sin(x) + (1/3)sin(2x) + (1/5)sin(3x)
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Elena is making an open-top box by cutting squares out of the corners of a piece of paper that is 11 inches wide and 17 inches long, and then folding up the sides. If the side lengths of her square cutouts are inches, then the volume of the box is given by V(x)=(11-2x)(17-2x).
There are 4 questions in the bottom! Please solve and explain them for me!
Thank youuu
See below for the solution to each question
The reasonable domain of V(x)From the graph, we can see that:
V(x) have a positive value from x > 0 to x < 5.5
When x = 0 and ≥ 5.5, the values of V(x) are either 0 or negative
Hence, the reasonable domain of V(x) is 0 < x < 5.5
The value of x that gives the maximum volumeFrom the graph, the maximum value of V(x) is at
V(2) = 182
Hence, the approximate value of x that gives the maximum volume is 2
The values of x where the volume increasesThis is the interval where V(x) increase as x increase
From the graph, V(x) increase from x > 0 to x = 2
Hence, the approximate values of x where the volume increases are 0 < x ≤ 2
The interpretation of the point of intersectionThe other function is
B(x) = 140
So, the point of intersection represent where the volume is 140
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A safe has a 4-digit lock code that does not include zero as a digit and no digit is repeated. what is the probability that the lock code consists of all even digits?
Probability that the lock code consists of all even digits is [tex]1344[/tex] if we choose the [tex]4[/tex] digit lock is randomly from numbers [tex]1,2,3,4,5,6,7,8,9[/tex]
How can we find the probability that the lock code consists of all even digits?
Total numbers are [tex]1,2,3,4,5,6,7,8,9[/tex]
And we choose [tex]4[/tex] digit code
So probability of choosing [tex]4[/tex] numbers from [tex]9[/tex] numbers without reapitation is
[tex]9p_{4} =\frac{9!}{(9-4)!} \\=\frac{9*8*7*6*5!}{5!} \\=3024[/tex]
For All numbers are even , probability the last digit of number is even.
So we have [tex]4[/tex] even number and if we fix one on last digit and can change other [tex]3[/tex] then the probability is
[tex]4*8P_{3} =4*\frac{8!}{(8-3)!} \\=4*\frac{8*7*6*5!}{5!} \\=1344[/tex]
So the total probability of code lock having all even number is [tex]1344[/tex]
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Suppose the variable x is represented by a standard normal distribution. What value of x is at the 40th percentile of the distribution
The value value of x is at the 40th percentile of the distribution is 0.65.
The standard normal distribution is a normal distribution with a mean of zero and standard deviation of 1. The standard normal distribution is centered at zero and the degree to which a given measurement deviates from the mean is given by the standard deviation
Given:- μ[tex]_X[/tex]=0,σ[tex]_X[/tex]=1
We can standardize the random variable: Z=(X−μx)/σx,
Then the equation is P(X≤ x )=Φ(x−μx/σx)=Φ(x−0/1)=0.4
Inverse function
Ф[tex]\\^{-1}[/tex](0.4) = [tex]\frac{x-0}{1}[/tex]
From the table we get,
Ф[tex]^{-1}[/tex] = 0.65
Thus
[tex]\frac{x-40}{10}[/tex] = 0.65
x -0 = 0.65
x = 6.5
Thus the value value of x is at the 40th percentile of the distribution is 0.65
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Unoccupied seats on flights cause airlines to lose revenue. Suppose a large airline wants to estimate its mean number of unoccupied seats per flight over the past year. To accomplish this, the records of 225 flights are randomly selected and the number of unoccupied seats is noted for each of the sampled flights. The sample mean is 11.6 seats and the sample standard deviation is 4.1 seats. Assume the underlying population of unoccupied seats is normally distributed. Using a t- statistic, construct a 95% confidence interval for the population mean number of unoccupied seats per flight.
The given sample's 95% confidence interval is 11.6 ± 0.5357. I.e., from the lower bound 11.06 to the upper bound 12.14. Its margin error is 0.5357.
How to find the confidence interval?To find the confidence interval C.I,
Determine the mean(μ) of the sampleDetermine the standard deviation(σ)Determine the z-score for the given confidence level using the z-score tableUsing all these, the confidence interval is calculated by the formula, C. I = μ ± z(σ/√n) where n is the sample size and z(σ/√n) gives the margin error. So, we can also write C. I = mean ± margin error.Calculation:It is given that,
Sample size n = 225
Sample mean μ = 11.6
Standard deviation σ = 4.1
Since 95% confidence interval, z-score is 1.96
So, the required confidence interval is calculated by,
C. I = μ ± z(σ/√n) or mean ± margin error
On substituting,
C. I = 11.6 ± 1.96(4.1/√225)
= 11.6 ± 1.96 × 0.2733
= 11.6 ± 0.5357
So, the lower bound = 11.6 - 0.5357 = 11.065 and
the upper bound = 11.6 + 0.5357 = 12.135.
Thus, the confidence interval is from 11.06 to 12.14, and its margin error is 0.0537 for the population mean number of unoccupied seats per flight.
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Can you plssss help !!!
The parts of the circle shown in the image include, chord, diameter, radius, tangent and center point.
Parts of the circle shown in the imageThe following are the parts of the circle shown;
Chord: the length DC and DG represent the chord of the circleDiameter: the length HF represent the diameter of the circleRadius: the length HA and FA represents the radius of the circletangent: the line JB represents a tangent to the circleCenter: point A represents the center of the circleThus, the parts of the circle shown in the image include, chord, diameter, radius, tangent and center point.
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What is the measure of arc QR?
Answer:
answer is the third choice : 104
Step-by-step explanation:
Inscribed Angle Theorem
If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc.
so the intercepted arc is TWICE the measure of an angle inscribed in a circle
1. A central angle is an angle with endpoints located on a circle's circumference and vertex located at the circle's center. A central angle in a circle determines an arc.
2. An inscribed angle is an angle formed by three points on the circle's circumference.
Angle at the Center Theorem: An inscribed angle is half of the central angle (if they determine the same arc).
In your case angles:
1. ∠QSR is insribed (determines the arc QR);
2. ∠QTR is central (determines the arc QR).
Then by Angle at the Center Theorem, m∠QTR=2m∠QSR=2·52°=104°. Arc QR has the same measure as central angle QSR.
gathmath
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HELP FAST‼️‼️
Jessenia usually earns $25 each week.
Last week, she received an extra $20 in
bonus. What percent of her usual weekly
income was her total income last week?
Normal income: 25 $
Last week's income: 25 + 20: 45 $
% = Last week's/normal = 45/25 ×100 =180 %
A has the coordinates (-4, 3) and B has the
coordinates (4, 4). If Do,1/2(x, y) is a dilation of AABC,
what is true about the image AA'B'C'? Check all that
apply.
If [tex]$D_{o,\frac{1}{2}} \frac{1}{2} (x, y)[/tex] exists a dilation of △ ABC, the facts regarding the image
△A'B'C' exist AB exists parallel to A'B'.
What is a coordinate plane?Any point in the coordinate plane exists directed by a point (x, y), where the x value exists the position of the point with respect to the x-axis, and the y value exists the position of the point with respect to the y-axis.
A dilation exists a transformation [tex]$D_{o,k} \frac{1}{2} (x, y)[/tex], with center O and a scale factor of k that exists not zero, that maps O to itself and any other point P to P'. The center O exists as a fixed point, P' exists as the image of P, and points O, P, and P' exist on the exact line.
In a dilation of [tex]$D_{o,\frac{1}{2}} \frac{1}{2} (x, y)[/tex], the scale factor, 1/2 exists mapping the original figure to the image in such a manner that the distances from O to the vertices of the image exist half the distances from O to the original figure.
Also, the size of the image exists half the size of the original figure.
If [tex]$D_{o,\frac{1}{2}} \frac{1}{2} (x, y)[/tex] exists a dilation of △ ABC, the facts regarding the image △A'B'C' exist AB is parallel to A'B'.
[tex]$D_{o,k} \frac{1}{2} (x, y) = (\frac{1}{2} x, \frac{1}{2}y)[/tex]
The distance from A' to the origin exists half the distance from A to the origin.
Therefore, the correct answer is option A). △ A'B'C' exist AB is parallel to A'B'.
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Find the area of each of these rectangles
A construction company buys a certain number of boxes of nails depending on how many houses they plan to build in a year. They have 5 boxes left over from the previous year and predict that this year they will need 3 boxes per house. Write an equation where y represents the number of boxes they need to purchase and x the number of houses they plan to build.
Answer:
Step-by-step explanation:
43
I dont exactly know if this is correct sorry if its wrong
A group of students consists of four people in all; two women and two men. They agree to take turns taking notes in lecture. One person at a time will be selected at random from the group (without replacement) until everyone has had a turn. The expected value of the number of people selected before and including the first time a woman has a turn is (Q17)______ .
The standard error of the number of people selected before and including the first time a woman has a turn is (Q18)______ .
The expected value of the number of people selected before and including the first time a woman has a turn is 1.6667 and the standard error is 0.7454.
How to illustrate the information?It should be noted that the expected value will be:
1[p(x = 1)] + 2[p(x = 2)] + 3[p(x = 3)]
= 1(0.5) + 2(0.3333) + 3(0.1667)
= 1.6667
Var(X) = E(X²) - E(X)2
= 3.3335 & (1.6667)²
= 0.555611
The standard error will be:
= ✓0.555611
= 0.7454
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What is the area of a square with a perimeter of 44 mm?
Answer:
The area of the square is 121mm
Step-by-step explanation:
To solve this we need two basic formulas:
P = 4s
A = [tex]s^{2}[/tex]
P = perimeter
s = the length of the side
Since the shape is a square all the sides are the same length. To get the value of 's' we need to isolate:
P = 4s
44 = 4s
44 / 4 = 4s / 4
11 = s
Now plug in the answer for 's' to get the area:
A = [tex]s^{2}[/tex]
A = [tex]11^{2}[/tex]
A = 121
Hope this helps.
You and a friend are playing a game of chance. Every time you roll a 1 or 2 you are successful, and your friend will pay you $1. Every time you roll (using a fair die) a 3, 4, 5 or 6, you must pay your friend $2. If 71% of your rolls are successful and 29% of your rolls are unsuccessful, how much money do you expect to have earned/owed by the end of the game
Considering the mean of a discrete distribution, you are expected to earn $0.13 by the end of the game.
What is the mean of a discrete distribution?The expected value of a discrete distribution is given by the sum of each outcome multiplied by it's respective probability.
Since 71% of your rolls are successful and 29% of your rolls are unsuccessful, the distribution of your earnings is:
P(X = 1) = 0.71.P(X = -2) = 0.29Hence the expected value is:
E(X) = 1 x 0.71 - 2 x 0.29 = 0.71 - 0.58 = 0.13
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If sin (90° - α) = BC /CA,write down the ratio of sina.
Answer:
Step-by-step explanation:
sin(90-a)=cos
cos = angle adjacent
theta=BC/CA
The length of a rectangle is 3 less than twice the width. Determine how the area will change if the length of the rectangle is increased by 5 and the width is decreased by 2. Show your work.
The change in the area is represented by the expression ΔA = 9 · x + 4, where x is the original width of the rectangle.
What is the change in the area of the rectangle if side lengths are changed?
By geometry we know that the area of a rectangle is equal to the product of its width (x) and length (y). Based on the statement, the original rectangle is equal to:
A = x · y
A = x · (2 · x - 3)
And the change in the area is:
A + ΔA = (x + Δx) · (y + Δy)
A + ΔA = x · y + Δx · y + Δy · x + Δx · Δy
ΔA = Δx · y + Δy · x + Δx · Δy
ΔA = (2 · x - 3) · Δx + Δy · x + Δx · Δy
ΔA = 2 · (2 · x - 3) + 5 · x + 10
ΔA = 9 · x + 4
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Here is a rail way timetable
What time does the first train get to oakengats?
The time that the first train would get to Oakengats is 08:15.
How to find the time of arrivalThe time of arrival of the first train can be gotten from the table.
The first entry has a blank space. This would have been the time of arrival for the train.
After the blank space, the next time is when the fits train got to this destination. That is at 08:15
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Lowest common multiple of 12,18,56
Step-by-step explanation:
12 = 2 × 2 × 3
18 = 2 × 3 × 3
56 = 2 ×2 × 2 × 7
Now
Common factor = 2
Remaining factor = 2 × 3 × 2 × 3 × 7
LCM = RF × CF
= 504
hence the lCM of 12 , 18 and 56 is 504...
[tex]...[/tex]
Find the other endpoint of the line segment with the given endpoint -5, 2 and midpoint 2, -5
Answer:
( 9, -12 )
Step-by-step explanation:
Mario's Math Tutoring
you have the midpoint
2, -5
use the midpoint formula
(x1+x2)/2 , (y1+y2)/2
to get the endpoint
x :
(-5+x2)/2 = 2
(-5+x2) = 4
-5+x2 = 4
x2 = 9
y:
(2+y2)/2 = -5
(2+y2) = -10
2+y2 = -10
y2 = -12
Thank you Mario!
Brass is made up of copper and zinc. 100 grams of brass contains 20 grams of zinc.
A) Workout the weight of zinc in 60 grams of brass.
Brass contains 4 parts by weight of copper to 1 part by weight of zinc.
B) Work out the weight of copper in 350 grams of brass.
Answer:
A. 12 grams Zn
B. 280 g of copper.
Step-by-step explanation:
100 grams of brass
20 grams of zinc
100 grams of brass has 20 grams of Zn.
20/100 = 0.200 Zn 20% Zn and 80% copper.
A. (60 g brass)*(0.2) = 12 grams Zn
B. (350 g brass)*(0.80) = 280 g of copper.
someone please help me with question #14 !!!
Answer: a, c, e
Step-by-step explanation:
a) True. If x=0, f(0)=c.
b) False. Considering the quadratic formula, we know the solutions are not always x=c.
c) True. This is a standard fact.
d) False. If the discriminant is not positive, there are less than two x-intercepts.
e) True. If you plug into the formula [tex]x=-\frac{b}{2a}[/tex], if b=0, then obviously the x-coordinate of the vertex is also 0.
In apirl is rained 1/4 of a inch on 15 diffret day how many inches of rain fell
In [tex]15[/tex] diffret days [tex]15/4[/tex] inches of rain fell in April month if every day rained [tex]1/4[/tex] of a inch .
How to find the rain fell in [tex]15[/tex] diffret days ?
We know that rain in every day is [tex]1/4[/tex] of a inch
So for [tex]15[/tex] days
[tex]1/4+1/4+1/4+......15times\\=15/4[/tex]
So in [tex]15[/tex] diffret days in April rain fell is [tex]15/4[/tex] inches.
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Which algebraic expression has a term with a coefficient of 9?
• A. 6 + X- 9
• B. 6x - 9
C. 6(x + 5)
• D. 9x ÷ 6
[tex]\textbf{ Heya !}[/tex]
Your Answer Is:-
[tex]\sf{9x\div6}[/tex]
because,
it has a coefficient of 9 (a number that comes before a variable)
`hope that was helpful to u ~
Segment ab falls on line 6x 3y = 9. segment cd falls on line 4x 2y = 8. what is true about segments ab and cd? they are parallel because they have the same slope of −2. they are perpendicular because they have slopes that are opposite reciprocals of −2 and one half. they are parallel because they have the same slope of 2. they are perpendicular because they have opposite reciprocal slopes 2 and negative one half.
They are parallel because they have the same slope of 2.
How to determine the true statements about the segments?The segments are given as;
AB => 6x - 3y = 9
CD => 4x - 2y = 8
We start by making y the subject of both equations
AB => 6x - 3y = 9
-3y = -6x + 9
Divide through by -3
y = 2x - 3
CD => 4x - 2y = 8
-2y = -4x + 8
Divide through by -2
y = 2x - 2
A linear equation is represented as:
y = mx + b
Where m represents the slope of the line.
By comparing the three equations, we have:
m1 = 2
m2 = 2
Parallel lines have equal slopes.
This means that the lines AB and CD are parallel lines
Hence, they are parallel because they have the same slope of 2.
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Answer: They are parallel because they have the same slope of 2.
Step-by-step explanation:vvvvvvvvvvvvvvvvvvvvv
PLEASE HELP IM REALLY STUCK
Answer:
[tex]a_{18}=-70[/tex]
Step-by-step explanation:
[tex]a_n=a_1+(n-1)d[/tex]
[tex]a_1=2, d = -4[/tex]
[tex]a_{18}=-2+(18-1)*(-4)=-2-68=-70[/tex]
4x-6y=-20
-2x+4y=6
8th grade pre algebra
The values of x and y that are solutions to the simultaneous equations are; x = -11 and y = -4
How to Solve Simultaneous Equations?
We are given the simultaneous equations;
4x - 6y = -20 -----(eq 1)
-2x + 4y = 6 -----(eq 2)
Multiply equation 2 by 2 and eq 1 by 1 to get;
-4x + 8y = 12 -----(eq 3)
Add eq 3 to eq 1 to get;
2y = -8
y = -4
Thus;
4x - 6(-4) = -20
4x + 24 = - 20
4x = -20 - 24
4x = -44
x = -44/4
x = -11
Thus, the values of x and y that are solutions to the simultaneous equations are; x = -11 and y = -4
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The reciprocals of what three different positive integers have sum equal to 1?
Answer:
1/a + 1/b + 1/c = 1 , a,b,c is nonzero
Step-by-step explanation:
bc +ac +ab /abc = 1
bc + ac + ab = abc
a = b = c = 1
True or false, if a domain is restricted to x > 1, then there will be an open circle at x = 1.
Answer:
Step-by-step explanation:
true
Which describes Which is a perfect square?
72
81
90
99
Answer:
81
Step-by-step explanation:
A perfect square is multiplying a number with the same number in question. 81 is a perfect square because either 9 or -9 will go into it perfectly, like a square's length and height.
[tex]9^{2} = 81[/tex]
Answer: 81
Step-by-step explanation:
81 can be perfectly square rooted into 9. [tex]\sqrt{81}=9[/tex]
It cost Cai $1,80 to send 12 text messages, How much would it cost to send 90 text
messages?
Answer:
$13.5
Step-by-step explanation:
cost of 1 text message= $1.80÷12 =$0.15
cost of 90 messages = $0.15×90 = $13.5
Answer: $13.50 for 90 text messages
Step-by-step explanation:
$0.15×90 = $13.5
$0.15 is the unit rate for 1 text message cost.
Need help with explanation on solving this problem.
Answer:
x=27.80
Step-by-step explanation:
we know that cosA=base/hypotaneous
so,
[tex]cos44 = \frac{20}{x} \\ x = \frac{20}{cos(44)} \\ = \frac{20}{0.7193...} \\ = 27.80 [/tex]