Answer:
First we write y and its derivatives as power series:
y=∑n=0∞anxn⟹y′=∑n=1∞nanxn−1⟹y′′=∑n=2∞n(n−1)anxn−2
Next, plug into differential equation:
(x+2)y′′+xy′−y=0
(x+2)∑n=2∞n(n−1)anxn−2+x∑n=1∞nanxn−1−∑n=0∞anxn=0
x∑n=2∞n(n−1)anxn−2+2∑n=2∞n(n−1)anxn−2+x∑n=1∞nanxn−1−∑n=0∞anxn=0
Move constants inside of summations:
∑n=2∞x⋅n(n−1)anxn−2+∑n=2∞2⋅n(n−1)anxn−2+∑n=1∞x⋅nanxn−1−∑n=0∞anxn=0
∑n=2∞n(n−1)anxn−1+∑n=2∞2n(n−1)anxn−2+∑n=1∞nanxn−∑n=0∞anxn=0
Change limits so that the exponents for x are the same in each summation:
∑n=1∞(n+1)nan+1xn+∑n=0∞2(n+2)(n+1)an+2xn+∑n=1∞nanxn−∑n=0∞anxn=0
Pull out any terms from sums, so that each sum starts at same lower limit (n=1)
∑n=1∞(n+1)nan+1xn+4a2+∑n=1∞2(n+2)(n+1)an+2xn+∑n=1∞nanxn−a0−∑n=1∞anxn=0
Combine all sums into a single sum:
4a2−a0+∑n=1∞(2(n+2)(n+1)an+2+(n+1)nan+1+(n−1)an)xn=0
Now we must set each coefficient, including constant term =0 :
4a2−a0=0⟹4a2=a0
2(n+2)(n+1)an+2+(n+1)nan+1+(n−1)an=0
We would usually let a0 and a1 be arbitrary constants. Then all other constants can be expressed in terms of these two constants, giving us two linearly independent solutions. However, since a0=4a2 , I’ll choose a1 and a2 as the two arbitrary constants. We can still express all other constants in terms of a1 and/or a2 .
an+2=−(n+1)nan+1+(n−1)an2(n+2)(n+1)
a3=−(2⋅1)a2+0a12(3⋅2)=−16a2=−13!a2
a4=−(3⋅2)a3+1a22(4⋅3)=0=04!a2
a5=−(4⋅3)a4+2a32(5⋅4)=15!a2
a6=−(5⋅4)a5+3a42(6⋅5)=−26!a2
We see a pattern emerging here:
an=(−1)(n+1)n−4n!a2
This can be proven by mathematical induction. In fact, this is true for all n≥0 , except for n=1 , since a1 is an arbitrary constant independent of a0 (and therefore independent of a2 ).
Plugging back into original power series for y , we get:
y=a0+a1x+a2x2+a3x3+a4x4+a5x5+⋯
y=4a2+a1x+a2x2−13!a2x3+04!a2x4+15!a2x5−⋯
y=a1x+a2(4+x2−13!x3+04!x4+15!x5−⋯)
Notice that the expression following constant a2 is =4+ a power series (starting at n=2 ). However, if we had the appropriate x -term, we would have a power series starting at n=0 . Since the other independent solution is simply y1=x, then we can let a1=c1−3c2, a2=c2 , and we get:
y=(c1−3c2)x+c2(4+x2−13!x3+04!x4+15!x5−⋯)
y=c1x+c2(4−3x+x2−13!x3+04!x4+15!x5−⋯)
y=c1x+c2(−0−40!+0−31!x−2−42!x2+3−43!x3−4−44!x4+5−45!x5−⋯)
y=c1x+c2∑n=0∞(−1)n+1n−4n!xn
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Determine the range of f(x) = |x + 5|.
Answer:
[0,+♾)
Step-by-step explanation:
|x| function is kinda like a half opened book from (0, 0) and |x+5| is from (–5, 0)
So we can't find any changes for y
so the domain is
[0, +♾)
helpp assignment due at 3:40
Question 17: B. A’ (1,2), B’ (3,2), C’ (3,5)
Translating 4 to the right means that you should add 4 to the x value of your coordinate point.
Question 18: C. (3,-5)
This is because reflecting over the x-axis would mean to move down the same amount of units above the x-axis the original point is (the original point is five units above the x-axis so you just move down five units). Look at the attached picture for more help.
Question 19: A. C’ (0,-2)
We can see that the triangle has been translated 1 unit to the right and 2 units up. This would mean you add 1 to the x-value of the coordinate and add 2 to the y-value of the coordinate. I can’t see point C clearly, but I’m guessing that it’s at (-1,-4). This would give point C’ as (0,-2)
Hope this helps! Let me know if you need more help! Have a fantastic day!
Find the measure of the third angle of a triangle if the measures of the other two angles are given. 35.5 and 82.6
Answer:
61.9°
Step-by-step explanation:
The angles of a triangle always add up to 180°. Using this information along with the values of two of the three angles, we can set up an equation to find the measure of the third angle.
Set up an EquationLet the measure of the third angle be represented as "x". The first angle + the second angle + the third angle must equal 180 therefore...
[tex]35.5+82.6+x=180[/tex]
Solve the EquationStart by adding the like terms on the left side of the equation:
[tex]118.1+x=180[/tex]
Then, subtract both sides by 118.1:
[tex]x=61.9[/tex]
Therefore the measure of the third angle of the triangle is 61.9°.
Assume that the custodian of a $705 petty cash fund has $147. 50 in coins and currency plus $535. 50 in receipts at the end of the month. the entry to replenish the petty cash fund will include:______.
Answer:
$528
If not then it might be $385
What is the value of the variable intresult, given: int a = 3, b = 2, c = 10, intresult; intresult = c *b;
The integer value of the given integers is 20.
According to the statement
we have given that the some integers values like a = 3, b = 2, c = 10 and we have to find the intresult = c *b it means integer result = c *b
So, For this purpose we know that the
An integer is a whole number (not a fractional number) that can be positive, negative, or zero.
And here integer a = 3 and b = 2 and c = 10 and we have to find the value of c *b then
intresult = c *b
Substitute the values in it then
intresult = 10 *2
intresult = 20.
So, the intresults is 20 from the given numbers.
Hence, The integer value of the given integers is 20.
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If one earthquake is 20 times as intense as another how much latger is its magnitude on the richter scale?
The magnitude of the larger earthquake is about 1.3 greater than the magnitude of the smaller earthquake.
In physics, significance is defined actually as “distance or quantity.” It depicts the absolute or relative course or size wherein an object moves within the sense of motion. it's miles used to express the scale or scope of something. In physics, importance normally refers to distance or quantity.
Magnitude is described as big in size or very vital. An example of value is the depth of the Grand Canyon. An instance of importance is the dimensions of the trouble of world hunger. (geology) A measure of the amount of energy launched by using an earthquake, as indicated on the Richter scale.
The magnitude is more than a few that characterizes the relative size of an earthquake. value is primarily based on the measurement of the maximum movement recorded with the aid of a seismograph.
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What is the unit rate of the proprrional relationship represented by the equation y=3x
The unit rate of the proportional relationship y = 3x is 3
How to determine the unit rate?The proportional relationship is given as:
y = 3x
Divide both sides by x
y/x = 3
For a proportional relationship y/x = k;
The unit rate is 3
This means that the unit rate of the proportional relationship y = 3x is 3
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Area=
Help Me asap pleaseee thanks
The area of the shaded region shown in the image is 14π unit²
What is an equation?An equation is an expression that shows the relationship between two or more numbers and variables.
Area of semicircle = π*diameter² / 4
The area of the shaded region is the sum of the area of the individual semi circles shown in the diagram.
Area of shaded region = area of SR + area of QR + area of PQ
Area of shaded region = (1/2) * π(4²/4) + (1/2) * π(2*4²/4) + (1/2) * π(2 * 2 * 4²/4) = 14π
The area of the shaded region shown in the image is 14π unit²
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If two 12-sided dice are rolled what is the probability that both numbers will be even?
Answer:
probability result of sample space would result in 12/144
Step-by-step explanation:
as result there is 12 numbers on each dice, blth have exact same numbers, resulting in 144 combinations ti be thrown as sample space event outcome wluld be;
1/1 1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10 1/11 1/12
2/1 2/2 2/3 2/4 2/5 2/6 2/7 2/8 2/9 2/10 2/11 2/12
3/1 3/2 3/3 3/4 3/5 3/6 3/7 3/8 3/9 3/10 3/11 3/12
4/1 4/2 4/3 4/4 4/5 4/6 4/7 4/8 3/9 4/10 4/11 4/12
5/1 5/2 5/3 5/4 5/5 5/6 5/7 5/8 5/9 5/10 5/11 5/12
6/1 6/2 6/3 6/4 6/5 6/6 6/7 6/8 6/9 6/10 6/11 6/12
7/1 7/2 7/3 7/4 7/5 7/6 7/7 7/8 7/9 7/10 7/11 7/12
8/1 8/2 8/3 8/4 8/5 8/6 8/7 8/8 8/9 8/10 8/11 8/12
9/1 9/2 9/3 9/4 9/5 9/6 9/7 9/8 9/9 9/10 9/11 9/12
10/1 10/2 10/3 10/4 10/5 10/6 10/7 10/8 10/9 10/10 10/11 10/12
11/1 11/2 11/3 11/4 11/5 11/6 11/7 11/8 11/9 11/10 11/11 11/12
12/1 12/2 12/3 12/4 12/5 12/6 12/7 12/8 12/9 12/10 12/11 12/12
giving 12 matching numbers in a 144 possible rolled outcome
mathematic formula would be P(A) = n(A)/n(S)
As P(A)= probabolity of event A
n(A)= number of favouravle outcome
n(S)= total number of outcome in sample space
giving
12=2/144=0,0138888889
please help on this geometry question
Angles are said to be produced whenever two or more straight lines intersect. Some types of angles are right angle, obtuse angle, acute angle, etc. Thus, the required proof is stated below:
Prove: <A > <C
<B > <D
An obtuse angle is a given angle that is greater than [tex]90^{o}[/tex] but less than [tex]180^{o}[/tex]. While an acute angle is a given angle that is less than [tex]90^{o}[/tex].
Thus extend AB to F, such that BF = CD.
Draw a perpendicular from a to intersect CD at point E.
So that;
<BAD = <BAE + <DAE (addition property of an angle)
Such that;
<A is an obtuse angle, while <C is an acute angle. Thu,
<A > <C (different types of angles)
Also,
<ADC = <CDF - <ADF (subtraction property of angles)
But both angles B and D are acute angles.
So that;
<B > <C (comparison of measures of two angles)
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find the value of n: [tex]3n (\frac{1}{4} ) = 16 \frac{1}{2}[/tex]
[tex]\boldsymbol{\sf{3n\left(\dfrac{1}{4}\right)=16\dfrac{1}{2} } }[/tex]
Multiply both sides of the equation by 4, the lowest common denominator of 4.2.
[tex]\boldsymbol{\sf{ 12n\times\left(\dfrac{1 }{4}\right)=2(16\times2+1) }}[/tex]
Multiply 12 and 1/4, because once multiplied, we get the following 12/4.
[tex]\boldsymbol{\sf{\dfrac{12}{4}n=2(16\times2+1) \ \ \longmapsto \ \ [Divide \ 12 \div 4.] }}[/tex]
[tex]\boldsymbol{\sf{3n=2(16\times2+1) \ \ \longmapsto \ \ [ Multiply \ 16\times 2] }}[/tex]
[tex]\boldsymbol{\sf{3n=2(32+1) \ \longmapsto \ \ [Add] }}[/tex]
[tex]\boldsymbol{\sf{3n=2\times33 \ \ \longmapsto \ \ [Multiply] }}[/tex]
[tex]\boldsymbol{\sf{3n=66}}[/tex]
Divide both sides by 3.
[tex]\boldsymbol{\sf{n=\dfrac{66}{3} \ \longmapsto \ \ [Split] } }[/tex]
[tex]\boxed{\boldsymbol{\sf{n=22}}}[/tex]
Once the exercise is solved, we find that the value of n is 22.
prove:- sin^2A-cos^2B=sin^2B-cos^2A
Step-by-step explanation:
1: change 'sin²A' and 'cos²B' into (1-cos²A)and (1-sin²A) respectively.
2:open the brackets and put signs accordingly.
3: cancel (-1) and (+1).
4: remaining is your answer.
Help on this question
Answer:
11.3 miles more than
Step-by-step explanation:
Looking at the graph, when x is 10 the y is 308. For the equation, if you make x 10, then y is 421. 421 is 113 more than 308. Jace travels 113 more than Jeriel on 10 gallons of gas. If you divide 113 by 10 (to get 1 gallon of gas), you would end up with 11.3 miles more than :)
what is the sum of (6x^2-x+8) - (x^2+2)
Answer:
5x^2-x+6
Step-by-step explanation:
That should work
A rotating sprinkler head sprays water as far as 20 feet. the head is set to cover a central angle of . what area of grass will be watered?
The area of the grass that will be watered is given by 800/9 pi ft^2.
What is the area of a circle?
Area of a circle is the region occupied by the circle in a two-dimensional plane. It can be determined easily using a formula, A = πr2, (Pi r-squared) where r is the radius of the circle. The unit of area is the square unit, such as m2, cm2, etc. Area of Circle = πr2 or πd2/4, square units. where π = 22/7 or 3.14.The area of a circle of radius r is given by:
A = πr²
In this problem, a rotating sprinkler head sprays water as far as 20 feet, hence the radius is of r = 20 ft.
It covers a central angle of 80º, while an entire circle has 360º, hence the area of grass that will be watered corresponds to 80/360 = 2/9 of the area of the circle.
Hence the watered area in ft² is given by-
A = 2/9 π × (20)²
= 800π/ 9
Therefore, the area of the grass that will be watered is given by 800/9 pi ft^2.
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Using the same functions given above, find (h−f)(x).
The results of subtraction of functions are listed below:
x - 8m(x) = x² + 10 · x + 5m(3) = 44How to use the subtract function operatorThere three fundamental binary operators (addition, multiplication, composition) and two secondary binary operators (subtraction, division) for functions. The subtraction is operation derived from addition, defined by the following expression:
(h - f) (x) = h(x) - f(x) (1)
Now we proceed to develop each expression:
i) (h - f) (x):
h(x) = 2 · x - 3, f(x) = x + 5 Given(h - f)(x) = (2 · x - 3) - (x + 5) Definition of subtraction between functions(2 · x - x) + [(- 3) + (- 5)] Associative and commutative properties / (- 1) · a = - ax - 8 Distributive property / Definitions of addition and subtraction / Resultii) m(x) = (g - j) (x):
g(x) = x² + 6 · x + 5, j(x) = - 4 · x Givenm(x) = (x² + 6 · x + 5) - (- 4 · x) Definition of subtraction between functionsm(x) = x² + (6 · x + 4 · x) + 5 Associative, commutative and distributive property / (- a) · (- b) = a · bm(x) = x² + 10 · x + 5 Distributive property / Definition of adition / Resultiii) m(3):
m(3) = 3² + 10 · 3 + 5
m(3) = 9 + 30 + 5
m(3) = 44
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A(n) Blank______ is a two-dimensional representation of data in which values are represented by colors. Multiple choice question. disintermediation long tail heat map interactivity
Answer:
heat map is the answer
The Math Club has 23 members and needs to elect officers. They will need a President, Vice President, Secretary, and Treasurer. How many ways can a 4-member committee be formed?
92
8,655
212,520
1,573,210
There are 8855 ways in which a 4-member committee be formed from 23 members
What is an equation?
An equation is an expression that shows the relationship between two or more numbers and variables.
An independent variable is a variable that does not depend on any other variable for its value while a dependent variable is a variable that depends on other variable.
The number of ways that a 4-member committee be formed from 23 members is:
Number of ways = 23! / [(23 - 4)!4!] = 8855 ways
There are 8855 ways in which a 4-member committee be formed from 23 members
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Answer:
The answer is 212,520.
Step-by-step explanation:
This is a permutation, not a combination. So, we need to follow the formula
[tex]nP_{r} =\frac{n!}{(n-r)!}[/tex]
Plug in the numbers and solve:
[tex]nP_{r}= \frac{23!}{(23-4)!}[/tex]
[tex]nP_{r}= \frac{23!}{19!}\\[/tex]
[tex]nP_{r}= \frac{25,852,016,738,884,976,640,000}{121,645,100,408,832,000}[/tex]
[tex]nP_{r}= 212,520[/tex]
Because 212,520 is an answer choice, this is the correct answer.
What is the product of (5 m superscript negative 2 baseline) (2 m superscript negative 3 baseline)?
Answer:
A
Step-by-step explanation:
Mrs. gomes found that 40% of students at her high school take chemistry. she randomly surveys 12 students. what is the probability that exactly 4 students have taken chemistry? round the answer to the nearest thousandth. p (k successes) = subscript n baseline c subscript k baseline p superscript k baseline (1 minus p) superscript n minus k. subscript n baseline c subscript k baseline = startfraction n factorial over (n minus k) factorial times k factorial endfraction
Answer:
0.213
Step-by-step explanation:
→ Convert into binomial information
x ~ B ( 12 , 0.4 )
→ Write down probability required
p ( x = 4 )
→ Evaluate
0.2128
At Eva’s restaurant, 2/3 of the dishes on the menu are vegetarian. Of the vegetarian dishes, 1/10 are pasta dishes. What fraction of the dishes on the menu are vegetarian pasta?
Answer:
1/15 of the dishes on the menu are vegetarian pasta.
Step-by-step explanation:
1/10 of 2/3 of all dishes are vegetarian pasta.
1/10 of 2/3 of all = 1/10 * 2/3 * 1 = 2/30 = 1/15
When calculating a nation’s digital power, what term is used to refer to the demonstrated capabilities of a nation-state in cyber offense or defense?
When calculating a nation’s digital power, the term that is used to refer to the demonstrated capabilities of a nation-state in cyber offense or defense is Cyberwarfare.
What is Cyberwarfare?Cyberwarfare can be regarded as the use of cyber attacks against a nation-state so as to produce significant harm.
It should be noted that Cyber Warfare is set of actions which is been carried out by nation or organization so as to be able to launch an attack on institutions' computer network systems.
The interests or reasons for cyberwarfares based on the Cybersecurity as well as Infrastructure Security Agency are;
To weaken the system of another country.To disrupt or destroy the sytem of another nation.In order to achieve the goals behind the cyberwarfare programs, it is usually designed to focus on spectrum of objectives which will bring harm to national interests.
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Use an appropriate half-angle formula to find the exact value of the expression. cos(22. 5°)
The value of given trigonometric value cos(22. 5°) is 0.387.
According to the statement
we have given that the trigonometric value and we have to find the exact value of that given value with the help of the half angle formula.
So, For this purpose, we know that the
The half angle formulas are used to find the exact values of the trigonometric ratios of the angles like 22.5°.
So,
The given value is :
cos(22. 5°)
Then the value of cos half angle formula is
[tex]cos (theta/2) = +- sqrt((1-cos theta) / 2)[/tex]
then
The value of θ/2 is 22.5 degree.
And θ is 22.5*2
θ = 45 degree
and cos 45 degree is 1/squareroot 2
And by this the value of given value become is
[tex]cos 22.5 = +\sqrt{((\sqrt{2} -1)/(2 \sqrt{2} )) } = + 0.3827[/tex]
So, The value of given trigonometric value cos(22. 5°) is 0.387.
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PLEASE HELP ME !!!
The height of the pyramid in the diagram is three times the radius of the cone. The base area of the pyramid is the same as the base area of the
cone. What is the expression for the volume of the pyramid in terms of the radius r of the cone?
base area for cone= B
base area for pyramid= B, 3r heigh
A. V = pi r³
B. V = 1/3 pi r²
C. V = 2 pi r²
D. V = 1/9 pi r³
E. V = 3 pi r³
Answer:
The answer is A. V=pi r³
Step-by-step explanation:
:)
The expression for the volume of the pyramid in terms of the radius r of the cone is [tex]\pi r^3[/tex]. So, option A is correct.
The cone is a three-dimensional figure with a vertex and a flat base.
The volume of a pyramid is given by the formula:
[tex]V = \dfrac{1}{3} \times base\ area \times height.[/tex]
Given that:
The base area of the pyramid = B
The height is 3r
The base area of the pyramid is the same as that of the cone.
The base of the cone = [tex]\pi r^2[/tex]
The volume of the pyramid is calculated as:
[tex]V = \dfrac{1}{3} \times \pi r^2 \times 3r[/tex]
V =[tex]\pi r^3[/tex]
Therefore, the volume of the pyramid is [tex]\pi r^3[/tex]. So, option A is correct.
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Jane gained 25 points in the first round of a game. In the second round she lost 40 points and
on the third round she gained 10 points. Write and evaluate an expression to find how many
points Jane gained or lost.
Jane lost 5 points in the game.
Given that, Jane gained 25 points in the first round of a game. In the second round, she lost 40 points and in the third round, she gained 10 points.
We need to write and evaluate an expression to find how many points Jane gained or lost.
What are integers?An integer is a number with no decimal or fractional part and it includes negative and positive numbers, including zero.
Now, for the given situation we can write an expression as follows:
25 – 40 + 10
=25 + 10 + ( -40 )
=35 -40= -5
Therefore, Jane lost 5 points in the game.
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show that xcosec^2x = cotx - d/dx xcotx
The given equation, x.cosec²x = cot x - d/dx x.cot x, is proved using the product rule of differentials.
In the question, we are asked to show that x.cosec²x = cot x - d/dx x.cot x.
To prove, we go by the right-hand side of the equation:
cot x - d/dx x.cot x.
We solve the differential d/dx using the product rule, according to which, d/dx uv = u. d/dx(v) + v. d/dx(u), where u and v are functions of x.
cot x - {x. d/dx(cot x) + cot x. d/dx(x)}
= cot x - {x. (-cosec²x) + cot x} {Since, d/dx(cot x) = -cosec²x, and d/dx(x) = 1}
= cot x + x. cosec²x - cot x
= x. cosec²x
= The left-hand side of the equation.
Thus, the given equation, x.cosec²x = cot x - d/dx x.cot x, is proved using the product rule of differentials.
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Fatimah is x years old and nadia is 3 years older than fatmah. find expression, in it's simplest form in terms of x, for the sum of the girls ages in two years time and in y years time
The algebraic formula that represents the situation of the age difference between Fatimah and Nadia is x + 3 = y, where x, y > 0 and y > x.
How to derive an algebraic expression from a word problem
Herein we have a situation where two people have different ages, Fatimah has an age such that she is 3 years younger than Nadia. Let be x and y variables that respesent the ages of Fatimah and Nadia, respectively. In summary, the word problem can be reduced into the following algebraic expression:
y - x = 3 (Expression that represents age difference between Fatimah and Nadia)
x + 3 = y, where x, y > 0 and y > x. (1)
The algebraic formula that represents the situation of the age difference between Fatimah and Nadia is x + 3 = y, where x, y > 0 and y > x.
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please help me this has to be just a fake question 98 points
[tex] \frac{7}{4} - 9 \div .08 + \frac{4}{5} - 7 \times .8[/tex]
Let's try solving the problem using the order of operations method, otherwise known as PEMDAS.
[tex]\begin{tabular}{c | l}Order of Operation & Required Operation \\\cline{1-2}1 & Parentheses () \\2 & Exponents e^{x} \\3 & Multiplication \times \\4 & Division \div \\5 & Addition + \\6 & Subtraction -\end{tabular}[/tex]
We are given this question :
[tex]\mathrm {\frac{7}{4} - 9 \div .08 + \frac{4}{5} - 7 \times .8}[/tex]
Since there are no parentheses or exponents, let's multiply.
[tex]\mathrm {\frac{7}{4} - 9 \div .08 + \frac{4}{5} - 5.6}[/tex]
The next step is to divide.
[tex]\mathrm {\frac{7}{4} - 112.5 + \frac{4}{5} - 5.6}[/tex]
After that, add.
[tex]\mathrm {\frac{7}{4} - 112.5 + 0.8 - 5.6}[/tex]
[tex]\mathrm {\frac{7}{4} - 111.7 - 5.6}[/tex]
Finally, subtract the remaining terms.
[tex]\mathrm {1.75 - 111.7 - 5.6}[/tex]
[tex]\mathrm {-109.95 - 5.6}[/tex]
[tex]\mathrm {-115.55}[/tex]
The final answer is -115.55.
I hope this helped solve the problem.
Good luck on your studies!
Answer:
-115.55
Step-by-step explanation:
7/4 -9÷ .08 + 4/5 -7 * .8
I would change fractions to decimals
1.75 - 9 ÷ .08 +.8 -7 * .8
Now using pemdas, multiply and divide from left to right
1.75 -112.5 +.8 -7 * .8
1.75 -112.5 +.8 -5.6
Now add and subtract from left to right
-110.75+.8 -5.6
-109.95-5.6
-115.55
Find the rectangular coordinates of the point with spherical coordinates (rho,θ,ϕ) : (4,π6,2π3)
The spherical coordinate (4,6π,2π/3) is (5,2π3,0.927) in spherical coordinates.
I’m going to assume that your values are in the form of:
(r,θ,z)=(4,6π,2π/3)
where
(r,θ)are the polar coordinates of the point’s projection in the xy-plane
z is the usual z-coordinate in the Cartesian coordinate system
Now solving for Cartesian coordinates:
x=rcosθ=4cos(6π)=4(0.86)=3.44
y=rsinθ=4sin(6π)=2
Now for spherical transformation, note that:
(ρ,θ,ϕ)
ρ is the distance between P and the origin
θ is the same angle used to describe the location in cylindrical coordinates
ϕ is the angle formed by the positive z-axis and line segment from the origin and point in space, note that0≤ϕ≤π
OK, now that we have that out of the way, let’s compute:
ρ=r2+z2 = 5
θ=θ=6π
ϕ=arccos(zr2+z2−−−−−−√)=arccos(35)≈0.927rad
The spherical coordinate (4,6π,2π/3) is (5,2π3,0.927) in spherical coordinates.
For more information about coordinates, visit
https://brainly.com/question/17206319
How many solutions does the equation a+b+c+d+e+f = 2006 have where a b c d e and f are all positive integers?
The number of solutions of a+b+c+d+e+f = 2006 is 9.12 * 10^16
How to determine the number of solutions?The equation is given as:
a+b+c+d+e+f = 2006
In the above equation, we have:
Result = 2006
Variables = 6
This means that
n = 2006
r = 6
The number of solutions is then calculated as:
(n + r - 1)Cr
This gives
(2006 + 6 - 1)C6
Evaluate the sum and difference
2011C6
Apply the combination formula:
2011C6 = 2011!/((2011-6)! * 6!)
Evaluate the difference
2011C6 = 2011!/(2005! * 6!)
Expand the expression
2011C6 = 2011 * 2010 * 2009 * 2008 * 2007 * 2006 * 2005!/(2005! * 6!)
Cancel out the common factors
2011C6 = 2011 * 2010 * 2009 * 2008 * 2007 * 2006/6!
Expand the denominator
2011C6 = 2011 * 2010 * 2009 * 2008 * 2007 * 2006/720
Evaluate the quotient
2011C6 = 9.12 * 10^16
Hence, the number of solutions of a+b+c+d+e+f = 2006 is 9.12 * 10^16
Read more about combinations at:
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