Therefore, the amount that can be withdrawn quarterly from the account for 20 years, assuming ordinary annuity is $7,366.99. Option (d) is correct.
An account paying 4.6% interest compounded quarterly has a balance of $506,732.32.
The amount that can be withdrawn quarterly from the account for 20 years, assuming ordinary annuity is $7,366.99 (option D). Explanation: An ordinary annuity refers to a series of fixed cash payments made at the end of each period.
A typical example of an ordinary annuity is a quarterly payment of rent, such as apartment rent or lease payment, a car payment, or a student loan payment. It is important to understand that the cash flows from an ordinary annuity are identical and equal at the end of each period. If we observe the given problem,
we can find the present value of the investment and then the amount that can be withdrawn quarterly from the account for 20 years, assuming an ordinary annuity.
The formula for calculating ordinary annuity payments is: A = R * ((1 - (1 + i)^(-n)) / i) where A is the periodic payment amount, R is the payment amount per period i is the interest rate per period n is the total number of periods For this question, i = 4.6% / 4 = 1.15% or 0.0115, n = 20 * 4 = 80 periods and A = unknown.
Substituting the values in the formula: A = R * ((1 - (1 + i)^(-n)) / i)where R = $506,732.32A = $506,732.32 * ((1 - (1 + 0.0115)^(-80)) / 0.0115)A = $506,732.32 * ((1 - (1.0115)^(-80)) / 0.0115)A = $7,366.99
Therefore, the amount that can be withdrawn quarterly from the account for 20 years, assuming ordinary annuity is $7,366.99. Option (d) is correct.
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Use the graph that shows the solution to f(x)=g(x).
f(x)=1/x−2
g(x)=x−2
What is the solution to f(x)=g(x)?
Select each correct answer.
−1
1
2
3
The solution to the system of equations is given as follows:
x = 1 and x = 3.
What is a system of equations?A system of equations is when two or more variables are related, and equations are built to find the values of each variable.
In this problem, the two equations are:
[tex]\frac{1}{x - 2} = x - 2[/tex]
Applying cross multiplication:
(x - 2)(x - 2) = 1
x² - 4x + 4 = 1
x² - 4x + 3 = 0
(x - 1)(x - 3) = 0
Hence the solutions are:
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please provide the answer?
Using the given table:
a) the average rate of change is 32.5 jobs/year.
b) the average rate of change is 12.5 jobs/year.
How to find the average rate of change?
For a function f(x), the average rate of change on an interval [a, b] is:
[tex]\frac{f(b) - f(a)}{b - a}[/tex]
a) The average rate of change between 1997 and 1999 is:
[tex]A = \frac{695 - 630}{1999 - 1997} = 32.5[/tex]
So the average rate of change is 32.5 jobs/year.
b) Now the interval is 1999 to 2001.
The rate this time is:
[tex]A ' = \frac{720 - 695}{2001 - 1999} = 12.5[/tex]
So the average rate of change is 12.5 jobs/year.
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Is (x-4) a factor of f(x)=x^3- 2x^2 + 5x +1. use either the remainder theorem or the factor theorem to explain your reasoning
If x - 4 is a factor then x = 4 is a root:
f(4) = 4^3 - 2 * 4^2 + 5 * 4 + 1
= 64 - 32 + 20 + 1
= 53
since f(4) is not zero then 4 is not a root of the f(x)
Also, x - 4 is not a factor
In algebra, the remainder theorem, or Bezout's small theorem, applies the principle of polynomial division. It states that the remainder of dividing the polynomial f (x) by the linear polynomial {\ display style x-r} is equal to {\ display style f (r). }
In algebra, a factor set is a set of factors and roots of. Polynomial. This is a special case of the remainder theorem. The factor theorem shows that the polynomial f (x) has a factor only if f = 0.
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Surface=
Area=
Help please thanks
Surface area of the rectangular solid = 416 in.².
Volume = 480 in.³.
What is the Surface Area and Volume of a Rectangular Solid?Surface area = 2(wl+hl+hw)
Volume = (length)(width)(height).
Given the following:
Length (l) = 12 in.
Width (w) = 10 in.
Height (h) = 4 in.
Surface area = 2(wl+hl+hw) = =2·(10·12+4·12+4·10) = 416 in.².
Volume = (12)(10)(4) = 480 in.³.
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Ayudaaaaaaa es para hoy:
Necesito saber cuanto es 389 pesos mexicanos con un 12% de rebaja,ayuda plis
El precio resultante con descuento de 12 % es igual a 342.32 pesos.
¿Cómo hallar el precio resultante con descuento?
En este problema tenemos que determinar el precio resultante, el cual es igual al precio inicial menos el descuento. A continuación, se presenta la siguiente expresión:
x = 389 · (1 - 12/100)
x = 389 - 46.68
x = 342.32
El precio resultante con descuento de 12 % es igual a 342.32 pesos.
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A sequence is defined by the recursive function f(n + 1) =1/3 f(n). if f(3) 9= , what is f(1)
Answer:
f(1) = 81
Step-by-step explanation:
f(n + 1) = 1/3 f(n)
⇔ f(n) = 3 × f(n + 1)
……………………………
if f(3) = 9 ⇒ f(2) = 3 × f(3) = 3 × 9 = 27
Then
f(1) = 3 × f(2) = 3 × 27 = 81
Step 2 - Fill in the missing number: A vertical line and horizontal line combine to make a L shape. There is one row of entries in the shape including 1, negative 3, negative 10, 24. On the outside to the left of the L shape is 2 and to the outside below 1 is a. a =
The synthetic division's representation of the dividend is 2x3 + 10x2 + x + 5.
Given that
An L shape is created when two lines intersect vertically and horizontally.
The shape has entries in two rows.
Entries in row 1 are 2, 10, 1, and 5.
Blank, -10, and 0 are the entries in row 2.
A simplified method of dividing a polynomial with another polynomial equation of degree one is known as synthetic division.
On the exterior, to the left of the form, is entry number 5.
The entry stands for the divisor's zero.
If the variable is x, then this entry to the variable is;
2x³+10x²+x+5
The dividend is thus represented by synthetic division as
2x³+10x²+x+5
The Question is incomplete And complete question is given below!!
What dividend is represented by the synthetic division below? A vertical line and horizontal line combine to make a L shape. There are two rows of entries within the shape. Row 1 has entries 2, 10, 1, 5. Row 2 has entries blank, negative 10, 0, negative 5. Entry negative 5 is on the outside to the left of the shape, and a third row of entries is outside and below the shape. Row 3 has entries 2, 0, 1, 0. Negative 10 x squared minus 5 2 x cubed 10 x squared x 5 2 x squared 1 2 x cubed x.
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Answer:
The answer on Edge is A= 1
Step-by-step explanation:
PLEASE HELP
Quentin is using paper folding to find the center of a circle. He draws triangle ABC on a sheet of patty paper and folds the paper so that vertex A is lined up on vertex B. He then repeats the process twice so that vertex A is lined up on vertex C and vertex B is lined up on vertex C.
Blank one: angle OR perpendicular
Blank two: lies on OR is the center of
blank three: inscribed OR circumscribed
The necessary solution that would be used to fill in the blanks would be
perpendicularis the center ofCircumscribedHow to find the center of the circleThe way that Quintin would be able to do this would be the center of the circumferential triangle is equidistant from the vertices.
The center is going to be the intersection of the bisectors of the three sides of the given triangle.
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perpendicular bisectors, is the center of, circumscribed
Answer:
In the image
Hope this helps!
Step-by-step explanation:
1. perpendicular
2. is the center of
3. circumscribed
Circle Q is shown. Secant L N and tangent P N intersect at point N outside of the circle. Secant L N intersects the circle at point M. Arc M P is y, arc L P is x, and arc M L is z.
Which equation is correct regarding the measure of ∠MNP?
m∠MNP = One-half(x – y)
m∠MNP = One-half(x + y)
m∠MNP = One-half(z + y)
m∠MNP = One-half(z – y)
By applying the Theorem of Intersecting Secant to circle Q, an equation which is correct about the measure of ∠MNP is: A. m∠MNP = One-half(x – y).
What is the Theorem of Intersecting Secant?The Theorem of Intersecting Secant states that when two (2) lines intersect outside a circle, the measure of the angle formed by these lines is equal to one-half (½) of the difference of the two (2) arcs it intercepts.
For this exercise, the following points should be noted:
x represents the major arc.y represents the minor arc.By applying the Theorem of Intersecting Secant to circle Q shown in the image attached below, we can infer and logically deduce that angle MNP will be given by this formula:
m∠MNP = One-half(x – y).
m∠MNP = ½(x – y)
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PLEASE HELP WILL GIVE YOU ALOT OF POINTS
How do you know that ABC is similar to BDC? Explain your answer.
Answer: No
Step-by-step explanation:
There is only one pair of congruent angles that can be determined, but to prove triangles similar, there needs to be two pairs of congruent angles.
Find the volume of the solid to the nearest whole cubic unit. use a calculator, if needed. yellow solid v = ______
The volume of the cuboid is 132 cubic inches.
Given length of cuboid 7 1/3 inches, breadth being 6 2/3 inches , height being 2 7/10 inches.
Volume is amount of liquid a container can hold in its capacity. Cuboid is a 3 dimensional figure having length , breadth and height.Volume of cuboid is the product of length, breadth and height of cuboid.
We have to calculate volume of cuboid.
We have to make a simple fraction showing length , breadth and height.
Length=(3*7+1)/3=22/3
Breadth=(3*6+2)/3=20/3
Height=(10*2+7)/10=27/10
Volume=22/3*20/3*27/10
=11880/90
=132 cubic inches.
Hence the volume of cuboid having length of 22/3 inches, breadth of 20/3 inches and height of 27/10 is 132 cubic inches.
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Question is incomplete as it should include figure.
-499" is to the __________ of "-500" on a number line
Answer: right
Step-by-step explanation:
[tex]-499 > -500[/tex], and larger numbers are to the right of numbers smaller than them on the number line.
give the volume and surface area of the sphere shown
Answer:
V≈3053.63
A≈1017.88
Step-by-step explanation:
V=[tex]\frac{4}{3}[/tex]π [tex]r^{3}[/tex]=[tex]\frac{4}{3}[/tex]·π·[tex]9^{3}[/tex] ≈3053.62806
A=4π[tex]r^{2}[/tex]=4·π·[tex]9^{2}[/tex] ≈1017.87602
Please help someone. {1,4,9,16...}
what are the next three members?
Answer:
The solution is 25, 36, 49
Step-by-step explanation:
a = 1
b = 3
c = 5-3
= 2
________________________________
Un = [tex]\sf a + (n-1) \times b + \frac{(n-1)(n-2)}{2} \times c[/tex]
Un = [tex]\sf 1 + (n-1) \times 3 + \frac{(n-1)(n-2)}{2} \times 2[/tex]
Un = [tex]\sf 1 + 3n-3 + (n-1)(n-2)[/tex]
Un = [tex]\sf 1 + 3n-3 + n^{2} - 2n - n + 2[/tex]
Un = [tex]\sf n^{2} + 3n - 2n - n + 2 + 1 - 3[/tex]
Un = [tex]\sf n^{2} + 0n + 0[/tex]
Un = [tex]\sf n^2[/tex]
________________________________
Un = n²
U5 = 5²
= 25
U6 = 6²
= 36
U7 = 7²
= 49
________________________________
So, the next three members are 25, 36, 49
Bob and Carol are teenagers. Bob is two years older than Carol. If the digits of Carol's age are reversed, the new number would be three times as large as Bob's age. Find Bob's age.
Bob is 2 years older than Carol and the reverse of Carol's age is 3 times Bob's age, which makes Bob's to be 17 years.
How can Bob's age be calculated?Let 1B represent Bob's age and let 1C represent Carol's age, we can write the following equations;
1B = 1C + 2Reversing Carol's age gives;
C1 = 3 × 1BThe multiples of 3 that have the form X1 have 7 as the rightmost number.
Given that 1B is a teenager, we have;
When;
1B = 171C = 17 - 2 = 15The reverse of Carol's age is therefore;
C1 = 51 = 3 × 17Therefore, from the given description, Bob's age 1B = 17 years
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Bd=16 and ac is the perpendicular bisector of bd. 2x-14 37-x d 2y-2 3 y=[?] enter
The value of y from the given expression is 5
Perpendicular Bisector
Given:
BD = 16
BC = 2y - 2
CD = y + 3
Since AC is a perpendicular bisector of BD:
BC = CD
2y - 2 = y + 3
2y - y = 3 + 2
y = 5
Hence, the value of y from the given expression is 5
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Sample response: there is a common ratio of 2/3 between the height of the ball at each bounce. so, the bounce heights form a geometric sequence: 27, 18, 12. two-thirds of 12 is 8, so on the fourth bounce, the ball will reach a height of 8 feet. what did you include in your response? check all that apply. there is a common ratio between bounce heights. multiply 12 by 2/3. the height on the fourth bounce is 8 feet.
Answer:
Step-by-step explanation:
A Geometric sequence can be used:
To Model this sequence you need to use this formula
A (subscript n) = Ar(n-1)
a = value of the first term
n = the # of the term you want to find (For example, if you want to find the term number 3, it is 12)
r = the common ratio, this is obtained by dividing the second term in the sequence by the first.
So the value of r is = 2/3 because 27 times 2/3 = 18 which is the second term
n = 4 since you want to find the 4th term in the sequence
Plug it in and the results are
4th term = 27(2/3)^(4-1) = 8
The answer is 8
Type the correct answer in each box. Use numerals instead of words. If necessary, use / for the fraction bar.
The focus of a parabola is (0, -1). The directrix is the line y=0. What is the equation of the parabola in vertex form?
Check the picture below, so the parabola looks more or less like so, with the vertex half-way from the focus point and the directrix.
Now, the distance from the directrix to the focus point is only 1 unit, so half that, or namely the "p" distance is 1/2 unit, and since the parabola is opening downwards, "p" is negative, so
[tex]\textit{vertical parabola vertex form with focus point distance} \\\\ 4p(y- k)=(x- h)^2 \qquad \begin{cases} \stackrel{vertex}{(h,k)}\qquad \stackrel{focus~point}{(h,k+p)}\qquad \stackrel{directrix}{y=k-p}\\\\ p=\textit{distance from vertex to }\\ \qquad \textit{ focus or directrix}\\\\ \stackrel{"p"~is~negative}{op ens~\cap}\qquad \stackrel{"p"~is~positive}{op ens~\cup} \end{cases} \\\\[-0.35em] ~\dotfill[/tex]
[tex]\begin{cases} h=0\\ k=-\frac{1}{2}\\[1em] p=-\frac{1}{2} \end{cases}\implies 4\left( -\frac{1}{2} \right)\left( ~~y-\left( -\frac{1}{2} \right) ~~ \right)~~ = ~~(x-0)^2 \\\\\\ -2\left( y+\frac{1}{2} \right)=x^2\implies y+\cfrac{1}{2} =\cfrac{x^2}{-2}\implies y = -\cfrac{1}{2}x^2-\cfrac{1}{2}[/tex]
now, we could also write that as either
[tex]y = -\cfrac{1}{2}(x^2+1)\qquad or\qquad y = \cfrac{1}{2}(-x^2-1)[/tex]
PLEASE HELP
Graph the triangle with the given vertices. Find the length and the slope of each side of the triangle. Then find the coordinates of the midpoint of each side. Is the triangle a right triangle? isosceles? Explain (Assume all variables are positive and m ≠ n)
D (0,n), E(m,n), F(m,0)
The given triangle is a right triangle
The length and slopeThe coordinates are given as:
D (0,n), E(m,n), F(m,0)
The length is calculated using:
[tex]l=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2[/tex]
So, we have:
[tex]DE=\sqrt{(0-m)^2 + (n-n)^2} = m[/tex]
[tex]DF=\sqrt{(0-m)^2 + (n-0)^2} = \sqrt{m^2 + n^2[/tex]
[tex]EF=\sqrt{(m-m)^2 + (n-0)^2} = n[/tex]
The slope is calculated using:
[tex]m = \frac{y_2 -y_1}{x_2 - x_1}[/tex]
So, we have:
[tex]DE = \frac{n -n}{0- m} = 0[/tex]
[tex]DF = \frac{n -0}{0- m} = -\frac nm[/tex]
[tex]EF = \frac{n -0}{m- m} = \mathbf{unde fined}[/tex]
The coordinates of the midpointsThis is calculated using
[tex]m = 0.5 * (x_1 + x_2, y_1 + y_2)[/tex]
So, we have:
[tex]DE = 0.5 * (0 + m, n+ n)=(0.5m, n)[/tex]
[tex]DF = 0.5 * (0 + m, n+ 0)=(0.5m, 0.5n)[/tex]
[tex]EF = 0.5 * (m + m, n+ 0)=(m, 0.5n)[/tex]
The type of triangleIn (a), we have:
Lengths
[tex]DE= m[/tex]
[tex]DF = \sqrt{m^2 + n^2[/tex]
[tex]EF= n[/tex]
Slope
[tex]DE = 0[/tex]
[tex]DF = -\frac nm[/tex]
[tex]EF = \mathbf{unde fined}[/tex]
The sides are not equal.
However, the 0 and the undefined slope implies that the triangle is a right triangle because the sides are perpendicular
It should be noted that the triangle cannot be graphed because the coordinates are not numeric
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Select the statement that describes the expression 5 + (3 x 6) − 4. 5 times the sum of 3 and 6, then subtract 4 Add 5 to the product of 3 and 6, then subtract 4 Add 5 to the product of 3 and 6, then add 4 Add 5 to the quotient of 3 and 6, then subtract 4
Answer: Add 5 to the product of 3 and 6, then subtract 4
Step-by-step explanation:
We will use the order of operations. Looking at 5 + (3 x 6) − 4, we would:
[1] Multiply 3 x 6
[2] Add 5 to that product
[3] Subtract 4 from that value
This means the answer to your question is:
Add 5 to the product of 3 and 6, then subtract 4
Answer: B
i DiD tHe TeSt
Graph and label each of the ordered pairs in the coordinate plane. Then state the quadrant
or axis in/on which the point is located.
55. A(2, 4)
56. B(0, -3)
57. C(1, -1)
59. E(-4,1)
61. G(-3,-2)
63. I(0, 2)
58. D(3, 3)
60. F(2,0)
62. H(-2, 3)
64. J(-1,-4)
and/or volume of the given figure.
d the perimeter & area:
The quadrants of the points that needs to be graphed are:
Quadrant 1- Points A and DQuadrant 2- Points E and HQuadrant 3- Points G and JQuadrant 4- Points Cx-axis- Point Fy-axis- Point B and IHow do you get the quadrants?Since the points that was given are:
A(2, 4) B(0, -3)
C(1, -1) E(-4,1)
G(-3,-2) I(0, 2)
D(3, 3) F(2,0)
H(-2, 3) J(-1,-4)
Then, we plot all the coordinate above on a graph (see image attached)
Hence, from the image attached graph, the categories that can be seen are:
Quadrant 1 is Points A and D
Quadrant 2 is Points E and H
Quadrant 3 is Points G and J
Quadrant 4 is Points C
x-axis- Point F
y-axis- Point B and I
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Describe a relationship that can be modeled by the function shown in the table, and explain how the function models the relationship.
x ------ f(x)
1 ------ 4
4 ------ 8
9 ------ 12
16 ------ 16
25 ------ 20
B: Identify and interpret the key features of the function in the context of the situation you described in part A
The relationship that can be modeled by the function shown in the table is y = 4·√x
How to illustrate the information?f(x) is defined as a product of 4 into √x
4 × √1 = 4
4 × √4 = 8
4 × √9 = 12
4 × √16 = 16
4 × √25= 20
Therefore, the relationship is defined as y is 4 times the square root of x that is, y = f(x) = 4√x
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A car dealer acquires a used car for $7,000, with terms fob shipping point. compute total inventory costs assigned to the used car if additional costs include
$290 for transportation-in.
$190 for shipping insurance.
$1,000 for car import duties.
$200 for advertising.
$1,400 for sales staff salaries.
$190 for trimming shrubs.
Shipping points are independent organizational entities within which processing and monitoring of the deliveries, as well as goods issue, is carried out. A delivery is processed by one shipping point only.
"FOB shipping point" or "FOB origin" means the buyer is at risk once the seller ships the product. The purchaser pays the shipping cost from the factory and is responsible if the goods are damaged while in transit. "FOB destination" means the seller retains the risk of loss until the goods reach the buyer.
Please check the attached file for a complete answer.
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The reference angle for is, which has a terminal point of (2).
What is the terminal point of ?
(-²)
(-4/2, 4/2)
○ B. (2,-²)
(-2/²2,-4/2)
OA
A.
O C.
○ D. (22)
Answer: C
Step-by-step explanation:
[tex]\frac{5\pi}{4}[/tex] is in the third quadrant, so both x and y are negative.
Therefore, the only possible answer is C.
use the intermediate value theorem to prove that there is a positive number c such that c2 = 2.
So lets try to prove it,
So let's consider the function f(x) = x^2.
Since f(x) is a polynomial, then it is continuous on the interval (- infinity, + infinity).
Using the Intermediate Value Theorem,
it would be enough to show that at some point a f(x) is less than 2 and at some point b f(x) is greater than 2. For example, let a = 0 and b = 3.
Therefore, f(0) = 0, which is less than 2, and f(3) = 9, which is greater than 2. Applying IVT to f(x) = x^2 on the interval [0,3}.
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What is the distance between the points (22,27) and (2,-10)
Answer:
42.1
Step-by-step explanation:
The formula to calculate distance is √[(x₂ - x₁)² + (y₂ - y₁)²]. Using (22,27) as our x1 and y1, and (2,-10) as our x2 and y2.
Therefore our formula is √[(2 - 22)² + (-10 - 27)²].
(2 - 22)² = 400
(-10 - 27)² = 1369
(Make sure for both of these you put the negative number in parenthesis and the exponent outside them, if you are using a calculator of some sort)
Then we add and square root
√[400 + 1369] = 42.0594816896
The distance between the points (22,27) and (2,-10) is 42.1 units
How to determine the distance between the points?The coordinates of the points are (22,27) and (2,-10)
The distance is calculated as:
[tex]d = \sqrt{(x_2 -x_1)^2 + (y_2 -y_1)^2[/tex]
So, we have:
[tex]d = \sqrt{(22 -2)^2 + (27 +10)^2[/tex]
Evaluate
[tex]d = \sqrt{1769[/tex]
Take the square root
d = 42.1
Hence, the distance between the points (22,27) and (2,-10) is 42.1 units
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On a balance scale, $3$ green balls balance $6$ blue balls, $2$ yellow balls balance $5$ blue balls, and $6$ blue balls balance $4$ white balls. How many blue balls are needed to balance $4$ green, $2$ yellow and $2$ white balls
The number of blue balls that are needed to balance four green, two yellow and two white balls is 16 blue balls.
Numbers of blue balls neededGreen(g)
Blue (b)
Yellow (y)
White (w)
First step is to formula an equation
3g=6b
g=2b
2y=5b
y=5/2b
4w=6b
w=3/2b
Second step is to substitute
4g+2y+2w
=4(2b)+2(5/2b)+2(3/2b)
=8b+5b+3b
=16b
Therefore 16 blue balls are needed.
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In a group of 75 student 20 like football only 30 like circket ony and 18 didn't like any of wo game
(i) 57students liked at least one game.
(ii) 7 students who liked both the games.
(iii) 27 students liked football.
(iv) 37 students liked cricket.
(v) Below we represented the result in a Venn diagram.
What is Venn diagram?
John Venn popularized the Venn diagram in the 1880s, which is a type of diagram that displays the logical relationship between sets. The illustrations are used in probability, logic, statistics, linguistics, and computer science to demonstrate basic set relationships and to teach rudimentary set theory.
Part (i)- We have to get number of people who like at least one game, there are total 75 students and only 18 did not like any of two games, so students who like at least one game are-
75-18 = 57
Part (ii)- There are 57 students who like at least one game, from the question statement we know that 20 students like football only and 30 liked cricket only, so student who like both games are-
57-20-30 = 7
Part (iii)- There are 7 students who liked both games and 20 who liked football only, so student who liked football are-
20+7 = 27
Part (iv)- There are 7 students who liked both games and 30 who liked cricket only, so student who liked cricket are-
30+7=37
Part (v)- Venn Diagram will look like-
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Complete Question- In a group of 75 students, 20 liked football only, 30 liked cricket only and 18 did not like any of two games? (i) How many of them liked at least one game? (ii) Find the number of students who liked both the games. (iii) How many of them liked football? (iv) How many of them liked cricket? (v) Represent the result in a Venn diagram.
Which of the following functions are solutions of the differential equation y'' + y = sin(x)? (Select all that apply.) y = − 1 2 x cos(x)
The function that is the solution of the differential y" + y = sin(x) is: y(x) = -1/2(x cos x)
What is a function?A function is an expression, rule, or law in mathematics that describes a connection between one factor (the independent variable) and another variable (the dependent variable).
What is the proof of the above function?Take a look at the the following differential equation:
y" + y = sin (x)
The auxiiliary equation is
m² + 1 = 0
m² + 1-1 = -1
m² + 0 = -1
m² = -1
m = ±√-1
m = ±i
So, the complimentary function is yₐ (x) = c₁ cos x + c₂ sin x
Let the particular integral be:
yₙ (x) = A cos x + B sin x
yₙ '(x) = - A sin x + B cos x
yₙ ''(x) = - A cos x + B cos x
yₙ ''(x) = - (A cos x + B cos x)
After we have substituted yₙ (x); and yₙ''(x) in the given differential equation
y'' + y = sin (x)
= - (A cos x + B cos x) + (A cos x + B cos x) = sin(x)
0 = sin (x)
If we take the particular integral to be:
yₙ (x) = x(A cos x + B cos x)
yₙ '(x) = x(-A sin x + B cos x) + A cos x + B cos x
yₙ ''(x) = x(-A cos x - B sin x) - A sin x + B cos x + -A sin x + B cos x
Substitute yₙ (x), yₙ''(x) into the stated differential equation
y'' + y = sin (x)
x (-Acosx - Bsin x) - Asinx + Bcosx + (-Asinx + Bcosx) - x (Acosx + Bsin x) = sin (x)
-Axcosx - Bxsin x - Asinx + Bcosx -Asinx + Bcosx - Axcosx + Bxsin x = sin (x)
-2Asinx + 2Bcosx = sin(x)
Compare the coefficients of like terms on both sides of the equation
-2A = 1, B = 0
A = -1/2, B = 0
Substitute A = -1/2, B =0 into the assumed solution.
yₙ(x) = x((-1/2)cosx + (0) sinx)
= -1/2xcosx +0
= -(1/2)xcosx
Now, the general solution for the given differential equation is:
y(x) = yₓ(x) +yₙ (x)
y (x) - c₁cosx + c₂sin x -1/2x cosx
Hence, the solution is:
y(x) = -1/2xcosx
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Full Question:
Which of the following functions are solutions of the differential equation y'' + y = sin x? (Select all that apply.)
A) y = 1 2 x sin x
B) y = cos x
C) y = x sin x − 5x cos x
D) y = − 1 /2 x cos x
E) y = sin x