The homogeneous equation
y'' + y = 0
has characteristic equation
r² + 1 = 0
with roots at r = ± √(-1) = ± i, and so the characteristic solution is
y = C₁ cos(x) + C₂ sin(x)
Using the method of undetermined coefficients, suppose
y = a x cos(x) + b x sin(x)
(we would normally consider y = a cos(x) + b sin(x), but cos(x) and sin(x) are already part of the characteristic solution)
The derivatives are
y' = a cos(x) - a x sin(x) + b sin(x) + b x cos(x)
y' = (b - a x) sin(x) + (a + b x) cos(x)
and
y'' = -a sin(x) + (b - a x) cos(x) + b cos(x) - (a + b x) sin(x)
y'' = (-2a - b x) sin(x) + (2b - a x) cos(x)
Then
y'' + y = (-2a - b x) sin(x) + (2b - a x) cos(x) + a x cos(x) + b x sin(x)
7 sin(x) = -2a sin(x) + 2b cos(x)
and it follows that a = -7/2 and b = 0.
So, the general solution to the DE is
y = C₁ cos(x) + C₂ sin(x) - 7/2 x cos(x)
which means the only valid solutions from the given choices are
y = 7 sin(x)
y = 7 cos(x)
Find the dimensions of the rectangle with maximum area that can be inscribed in a circle of radius 10.
Answer:
Step-by-step explanation:
That will be a square with a diagonal of 20
side length of 20sin45
and area of (20sin45)² = 200 units²
prove it you say?
Area of a rectangle is base times height
A = bh
With a radius of 10, the diagonals of any rectangle inscribed will be 20 units
20² = b² + h²
h = [tex]\sqrt{400 - b^2}[/tex]
A = bh
A = b[tex]\sqrt{400 - b^2}[/tex]
Area will be maximized when the derivative is set to zero
dA/db = [tex]\sqrt{400 - b^2}[/tex] - b²/ [tex]\sqrt{400 - b^2}[/tex]
0 = [tex]\sqrt{400 - b^2}[/tex] - b²/ [tex]\sqrt{400 - b^2}[/tex]
b²/[tex]\sqrt{400 - b^2}[/tex] = [tex]\sqrt{400 - b^2}[/tex]
b² = 400 - b²
2b² = 400
b² = 200
b = [tex]\sqrt{200}[/tex]
h = [tex]\sqrt{400 - b^2}[/tex]
h = [tex]\sqrt{400 - \sqrt{200}^2 }[/tex]
h = [tex]\sqrt{200}[/tex]
A = bh
A = [tex]\sqrt{200}[/tex]•[tex]\sqrt{200}[/tex]
A = 200 units²
state whether each of the following equations represent growth or decay.
Answer:
ehhh Pfouto math ni
dali raka AA yooo
if fifteen families are coming to the picnic and they all split the cost evenly between them how much will each family contribute towards the cost of the shelter
What is 29−−√?
What 2 numbers does it fall between when placed on a number line?
Question 5 options:
Between 5 - 5.5
Between 4 - 4.5
Between 4.5 - 5
Between 5.5 - 6
someone help me with my assignments
Answer: Choice A
Step-by-step explanation: You are given the inequality g is less than or equal to. Greater/less than and equal to, you use solid circle. Greater/less than, you use open circle. From there, you just look at the inequality in which direction is it going at and from there, you should be able to select your choice which happens to be the first one.
I hope this explanation helps.
Best to you!
4.
(03.02)
Match each statement (term) with its value in relation to 0 (definition). (10 points)
1.
Oliver owes his sister $27.
2.
A building is 38 feet tall.
3.
Erica jumped 2.7 feet above ground.
4.
The scuba diver was 38 feet below sea level.
5.
The basement is 8.3 feet below ground.
a.
38
b.
-38
c.
-8.3
d.
-27
e.
2.7
Answer:
Below
Step-by-step explanation:
1.-d
2.-a
3.-e
4.-b
5.-c
Answer:
-8.3ithink so
Step-by-step explanation:
The original price of a Shirt was $45. In sales, the shirt is reduced to $32.50. Calculate the percentage reduction.
Answer:
27.8%
Step-by-step explanation:
[tex]\frac{32.5}{45} =0.722[/tex]
0.722 = 72.2%
The % reduction is:
(100-72.2)% = 27.8%
Hope this helps
The ages of Edna, Ellie, and Elsa are consecutive integers. The sum of their ages is 123. What are their ages? Show work.
Answer:
Ellie's age is 41, Edna's age is 40, Elsa's age is 42
Step-by-step explanation:
Let Ellie's age be x. Because they're consecutive integers, Edna's age is x-1 and Elsa's age is x+1. Therefore, 3x=123, x=41, so Ellie's age is 41, Edna's age is 40, Elsa's age is 42
Answer:
Edna is 40, Ellie is 41 and Elsa is 42.
Step-by-step explanation:
let Edna be m years old
m - Edna
m+1 - Ellie
m+2 - Elsa
m+m+1+m+2=123
Collect like terms
3m+3=123
3m=123-3
3m=120
Divide through by 3
m=40
Therefore, Edna is m=40 years old
Ellie=m+1=41 years old
Elsa=m+2=42 years old.
Find the common factors of the algebraic expression:- 7a²b, 14abc
please help me please it's important
factor of 7a²b, 14abc
7a²b, 14abc= 7ab
Can someone help me with his I will mark u brilliant
Answer: 1260 books
Step-by-step explanation:
2/5=0.4
19000x0.4=7600
12500x0.4=5000
7600+5000=12600
1/10=0.1
12600x0.1 = 1260
/Answer:
1,260 books (or C)
Step-by-step explanation:
1. 2/5 of fiction (12,500) are out, so 5,000 are out
2. 2/5 of non-fiction (19,000) are out, so 7,600 are out
3. Total books checked out are 12,600
4. 1/10 of checked out books are due this week, so 1,260 books (10%)
if Jenny buys 2 8-inch di|dos for 8$ then what is the total price
Answer:
$16
Step-by-step explanation 2 x 8 to 8 + 8 =16
Identify the slope and y-intercept of the line 4x + 2y < 3.
Answer:
Step-by-step explanation:
4x+2y<3
y<-2x + [tex]\frac{3}{2}[/tex]
In one serving of a fruit smoothie, the ratio of the number of ounces of water to the
number of ounces of fruit is 3 to 4.
Drag the numbers into the table to show how many ounces of water and fruit are
needed for different numbers of servings.
The ratios of ounces of water and fruits for different servings are:
1 serving = 3/4
2 servings = 6/8
3 servings = 9/12
4 servings = 12/16
5 servings = 15/20
What is a ratio?A ratio is an ordered pair of numbers a and b.
It is written as:
a / b
where b does not equal 0.
A proportion is an equation in which two ratios are set equal to each other.
We have,
In one serving:
= 3/4
The ratio of ounces of water to the ounces of fruit is 3/4.
In every number of servings, the ratios must be in proportion so,
We have the following cases:
2 servings:
= 3x2 / 4x2
= 6/8
3 servings:
= 3x3 / 4x3
= 9/12
4 servings:
= 3x4 / 4x4
= 12/16
5 servings:
= 3x5 / 4x5
= 15 / 20
Thus the ratios of ounces of water and fruits for different servings are:
1 serving = 3/4
2 servings = 6/8
3 servings = 9/12
4 servings = 12/16
5 servings = 15/20
Learn more about ratios and proportions here:
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1) -4/3
2) -3/4
3) 10/17
4)17/10
Answer:
boom bam ba boom bam ba
Answer: -3/4
Step-by-step explanation:
The y value is decreasing by -0.75
-3.75 - 0.75 = -4.5
-4.5 - 0.75 = -5.25
-5.25 - 0.75 = -6
ect.
-0.75 in fraction form is -3/4
PLEASE HELP I WILL GIVE BRAINLEST AND 20 POINTS PLEASEEEE
Answer:
Question 4: Which equation is parallel to the above equation and passes through the point (35, 30)
[tex]y=[/tex] [tex]\frac{5}{7} x + 5[/tex] is the correct answer, I found this by inputting the x and y value of the coordinate (35, 30) onto the equation and solving for y-intercept since the slope of all equations is the same (since it's traveling parallel)
[tex]30 = \frac{5}{7} (35) + y\\30 = 25 + y\\5 = y[/tex]
so the equation would be [tex]y=[/tex] [tex]\frac{5}{7} x + 5[/tex]
Question 5: Which equation is perpendicular to the above equation and passes through the point (35, 30)
[tex]y=\frac{-7}{5} x+79[/tex] is the correct answer, I found this using the same method as before, input coordinate values into the equation and solve for the y-intercept (The only thing changed from the last answer is the opposite reciprocal slope).
[tex]30 = \frac{-7}{5}(35) + y \\30 = -49 + y \\79 = y[/tex]
so the equation would be [tex]y=\frac{-7}{5} x+79[/tex]
Pls look at the picture for the question.
Answer:
10
Step-by-step explanation:
each tick mark is worth 5
there is the -10, then one to the right is -5, next is 0, next is 5, next one (the one where the x is) 10, next 15, next 20.
HELPP! PICTURE ATTACHED
find f(2)
Answer:
f(2) = Unidentified
Step-by-step explanation:
f(2) = ((2) +4)^2/(2)^2+2(2)-8
We can simplify the equation and get 36/4+4-8. After we simplify that we get 36/0. Since we have 0 as our denominator we know that our whole answer is unidentifiable.
Help help help help help m at h math math
Answer:X= 6X-14
Step-by-step explanation:
A college is selling tickets to the annual battle of the bands
competition. On the first day of ticket sales, the college sold 3
student tickets and 5 adult tickets for a total of $70. On the
second day, the college sold 12 student tickets and 12 adults
tickets for a total of $216.
Use elimination to solve the system of linear equations and
determine the price of each student ticket, x, and each adult
ticket, y. Write your answer as an ordered pair (x,y).
Answer:
Step-by-step explanation:
Answer:
(10,8)
Step-by-step explanation:
what is the value of x in m(x+n)=n
The equation m(x + n) = n si solved for x will be given as x = n / m (1 - m).
What is the linear system?A linear system is one in which the parameter in the equation has a degree of one. It might have one, two, or even more variables.
The equation is given below.
m(x + n) = n
By simplifying for x, then we have
x + n = n / m
x = n / m - n
x = (n - mn) / m
x = n / m (1 - m)
More about the linear system link is given below.
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32) Out of 525 people surveyed, 210 answered yes. Estimate how many people would answer yes if 900
people were surveyed.
Answer:
360 people
Step-by-step explanation:
[tex]\frac{ 210}{525}*900[/tex][tex]=360[/tex]
Hope this helped! :)
The term laissez-faire refers to an economic system that is characterized by
A. public ownership of the basic means of production
B. equal distribution of goods and services to all citizens
C. little governmental involvement in business and financial matters
D. government planning for production of consumer goods
Answer:
Answer: C.
Step-by-step explanation:
the term laissez faire refers to the economic policy of letting owners of industry and business set working conditions without interference
are in the ratio of 3:4. If the first number is decreased by 5, and
the second number is increased by 5, the new ratio will be 2 : 3. What are the two
numbers?
Answer:
the first number is 75 and the second number is 100
Solve the following system of equations.
y = 2x + 1
y = x2 – 5x + 7
The solution of the system of equation is (6, 13) and (1, 3).
How to solve system of equation?The system of equation can be solved using substitution method. Therefore, let's solve the system of equation by substitution method.
y = 2x + 1
y = x² - 5x + 7
substitute the value of y in equation(ii)
2x + 1 = x² - 5x + 7
x² - 5x + 7 - 2x - 1 = 0
x² - 7x + 6 = 0
x² - x - 6x + 6 = 0
x(x - 1) - 6(x - 1) = 0
(x - 6)(x - 1) = 0
x = 6 and x = 1
Hence,
when x = 6
y = 2(6) + 1
y = 12 + 1 = 13
when x = 1
y = 2(1) + 1
y = 3
learn more on system of equation here:https://brainly.com/question/20023128
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Tamika ran 90 meters in 16.1 seconds. Enrique ran 90 meters in 17.0 seconds. Who had the fastest time, Tamika or Enrique? A. Enrique B. Tamika
Which is equivalent to the expression below for a +5 minus a +2
Answer:
3
Step-by-step explanation:
Positive 5 minus 2 positive its just simple subtraction.
Bruce walks 3/4 mile in 1/5 what is his spewd miles per hour
Answer: Bruce's speed is 3.75 mph(miles per hour).
(-3,5) (3,7)
What is the slope of AB ?
A straight line has the point (4, 2) and has the slope 1. What is its y intercept?
================================================
Work Shown:
m = 1 is the given slope
The point (x,y) = (4,2) is on the line
Use slope intercept form to find the y intercept
y = mx+b
2 = 1*4 + b
2 = 4 + b
2-4 = b
-2 = b
b = -2 is the y intercept
Which of the following shows the prime factorization of 200 using exponential notation?
Answer:
As we use prime factorization for 200, we get 2 × 2 × 2 × 5 × 5 = 200.
