In the differential equation
[tex]\dfrac{dy}{dx} + 12y = \cos\left(\dfrac{\pi x}{12}\right)[/tex]
multiply on both sides by the integrating factor
[tex]\mu = \exp\left(\displaystyle\int12\,dx\right) = e^{12x}[/tex]
Then the left side condenses to the derivative of a product.
[tex]e^{12x} \dfrac{dy}{dx} + 12 e^{12x} y = e^{12x} \cos\left(\dfrac{\pi x}{12}\right)[/tex]
[tex]\dfrac{d}{dx}\left[e^{12x}y\right] = e^{12x}\cos\left(\dfrac{\pi x}{12}\right)[/tex]
Integrate both sides with respect to [tex]x[/tex], and use the initial condition [tex]y(0)=0[/tex] to solve for the constant [tex]C[/tex].
[tex]\displaystyle \int \frac{d}{dx} \left[e^{12x}y\right] \, dx = \int e^{12x} \cos\left(\dfrac{\pi x}{12}\right) \, dx[/tex]
As an alternative to integration by parts, recall
[tex]e^{ix} = \cos(x) + i \sin(x)[/tex]
Now
[tex]e^{12x} \cos\left(\dfrac{\pi x}{12}\right) = e^{12x} \mathrm{Re}\left(e^{i\pi x/12}\right) = \mathrm{Re}\left(e^{(12+i\pi/12)x}\right)[/tex]
[tex]\displaystyle \int \mathrm{Re}\left(e^{(12+i\pi/12)x}\right) \, dx = \mathrm{Re}\left(\int e^{(12+i\pi/12)x} \, dx\right)[/tex]
[tex]\displaystyle. ~~~~~~~~ = \mathrm{Re}\left(\frac1{12+i\frac\pi{12}} e^{(12+i\pi/12)x}\right) + C[/tex]
[tex]\displaystyle. ~~~~~~~~ = \mathrm{Re}\left(\frac{12 - i\frac\pi{12}}{12^2 + \frac{\pi^2}{12^2}} e^{12x} \left(\cos\left(\frac{\pi x}{12}\right) + i \sin\left(\frac{\pi x}{12}\right)\right)\right) + C[/tex]
[tex]\displaystyle. ~~~~~~~~ = \frac{12}{12^2 + \frac{\pi^2}{12^2}} e^{12x} \cos\left(\frac{\pi x}{12}\right) + \frac\pi{12} e^{12x} \sin\left(\frac{\pi x}{12}\right) + C[/tex]
[tex]\displaystyle. ~~~~~~~~ = \frac1{12(12^4+\pi^2)} e^{12x} \left(12^4 \cos\left(\frac{\pi x}{12}\right) + \pi (12^4+\pi^2) \sin\left(\frac{\pi x}{12}\right)\right) + C[/tex]
Solve for [tex]y[/tex].
[tex]\displaystyle e^{12x} y = \frac1{12(12^4+\pi^2)} e^{12x} \left(12^4 \cos\left(\frac{\pi x}{12}\right) + \pi (12^4+\pi^2) \sin\left(\frac{\pi x}{12}\right)\right) + C[/tex]
[tex]\displaystyle y = \frac1{12(12^4+\pi^2)} \left(12^4 \cos\left(\frac{\pi x}{12}\right) + \pi (12^4+\pi^2) \sin\left(\frac{\pi x}{12}\right)\right) + C[/tex]
Solve for [tex]C[/tex].
[tex]y(0)=0 \implies 0 = \dfrac1{12(12^4+\pi^2)} \left(12^4 + 0\right) + C \implies C = -\dfrac{12^3}{12^4+\pi^2}[/tex]
So, the particular solution to the initial value problem is
[tex]\displaystyle y = \frac1{12(12^4+\pi^2)} \left(12^4 \cos\left(\frac{\pi x}{12}\right) + \pi (12^4+\pi^2) \sin\left(\frac{\pi x}{12}\right)\right) - \frac{12^3}{12^4+\pi^2}[/tex]
Recall that
[tex]R\cos(\alpha-\beta) = R\cos(\alpha)\cos(\beta) + R\sin(\alpha)\sin(\beta)[/tex]
Let [tex]\alpha=\frac{\pi x}{12}[/tex]. Then
[tex]\begin{cases} R\cos(\beta) = 12^4 \\ R\sin(\beta) = 12^4\pi+\pi^3 \end{cases} \\\\ \implies \begin{cases} (R\cos(\beta))^2 + (R\sin(\beta))^2 = R^2 = 12^8 + (12^4\pi + \pi^3)^2 \\ \frac{R\sin(\beta)}{R\cos(\beta)}=\tan(\beta)=\pi+\frac{\pi^3}{12^4}\end{cases}[/tex]
Whatever [tex]R[/tex] and [tex]\beta[/tex] may actually be, the point here is that we can condense [tex]y[/tex] into a single cosine expression, so choice (D) is correct, since [tex]\cos(x)[/tex] is periodic. This also means choice (C) is also correct, since [tex]\beta=\cos(x)\implies\beta=\cos(x+2n\pi)[/tex] for infinitely many integers [tex]n[/tex]. This simultaneously eliminates (A) and (B).
Y₁ = 10x-4, y2 = 6x +17, and y₁ exceeds y₂ by 3.
Answer:
y1+3=y2
10x-4=10x-4+3
The number π is WHAT IS THE ANSWER I LOVE IT
Answer:
what's your question?
Step-by-step explanation:
For the pie chart, how many people aged 31-50 years old are cell phone users in the tri-County area if the number of people who use
cell phones is 72,345,000?
The number of people aged 31-50 years old are cell phone users in the tri-County area are 26,767,650.
What is defined as the pie chart?A pie chart, also known as a circle chart, is a method of summarizing a set of numeric variables or displaying the variables of a given factor (e.g. percentage distribution).
A circle is divided into segments in this type of chart. Each section represents a different category.A pie chart (also known as a circle chart) is a radial statistical graphic that is divided in to other slices to show numerical proportion. The arc length of each slice in a pie chart (and thus its central angle but also area) is proportional to the amount it represents.For the given question;
The total population = 72,345,000.
The total percentage between 31-50 years old = 23% + 14% (see from graph).
Let x be the total number of people between aged between 31 -50 years who are cell phone users.
(23% + 14%) of 72,345,000 = x
x = 37× 72,345,000 / 100
x = 26,767,650.
Therefore, the number of cell phone users aged between 31 to 50 are 26,767,650.
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A billboard painter is using a pulley system to hoist a can of paint up to a scaffold at a rate of half a meter per second. The height of the can of paint as a function of time is given by h(t) = 0.5t + 8. Five seconds after he starts raising the can of paint, his partner accidentally kicks a paint brush off of the scaffolding, which falls to the ground. The height of the falling paint brush can be represented by h(t) = −4.9(t − 5)2 + 38. When does the brush pass the paint can? If necessary, round your answer to the nearest hundredth.
The value of the t in the linear equation is 10 approximately.
According to the statement
We have to find that the value of the t.
So, For this purpose, we know that the
The definition of a linear equation is an algebraic equation in which each term has an exponent of one and the graphing of the equation results in a straight line.
From the given information:
The equations are:
h(t) = 0.5t + 8.
h(t) = −4.9(t − 5) + 38.
Then
= 0.5t + 8
= −4.9(t − 5) + 38.
And
= 0.5t + 8
= −4.9t + 24.5 + 38.
And
= 0.5t + 8
= −4.9t + 62.5
Now,
Subtact the equations then
= 0.5t + 8 - (−4.9t + 62.5)
= 0.5t + 8 + 4.9t - 62.5
= 0.5t + 4.9t - 62.5+8
= 5.4t -54.5
Then
5.4t = 54.5
t = 54.5/5.4
t = 10.092.
So, The value of the t in the linear equation is 10 approximately.
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Given the function g(x) = |x-5|, describe the graph of the function, including the vertex, domain, and range.
Answer:
Step-by-step explanation:
The graph will open upwards
Shifted to the right by 5
Vertex: (5,0)
Domain: x= All Real Numbers
Range: y[tex]\geq[/tex]0
There are 135 people in a sport centre.
73 people use the gym.
73 people use the swimming pool.
67 people use the track.
36 people use the gym and the pool.
35 people use the pool and the track.
32 people use the gym and the track.
14 people use all three facilities.
A person is selected at random.
What is the probability that this person uses at least 2 facilities?
Probability that this person uses at least 2 facilities is; 0.556
How to Interpret Venn Diagrams?
We are told that 14 people use all three facilities. This means:
36 − 14 = 22 represents the number of people that are using the gym and the pool but no the track.
35 − 14 = 21 represents the number of people that are using the pool and the track but no the gym.
32 − 14 = 18 represents the number of people that are using the gym and the track but no the pool.
Thus, total number of people that use at least 2 facilities = 14 + 22 + 21 + 18 = 75
Since total number of people at the sports center is 135, then it means that;
Probability that this person uses at least 2 facilities = 75/135 = 0.556
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Determine the range of f(x) = |x + 5|.
{y | −∞ < y < ∞}
{y | −5 < y < ∞}
{y | 0 ≤ y < ∞}
{y | 5 ≤ y < ∞}
The range for the given function f(x) = | x + 5 | is {y | 0 ≤ y < ∞}
What is a function?A function defines a relationship between one variable (the independent variable) and another variable (the dependent variable).
The independent variable = domain
The dependent variable = Range
We have a function,
f(x) = | x + 5 |
Let,
f(x) = y
x = -7, y = | -7 + 5 | = | -2 | = 2
x = -5, y = | -5 + 5 | = 0
x = -6, y = | -6 + 5 | = | -1 | = 1
x = -4, y = | -4 + 5 | = 1
x = -3, y = | -3 + 5 | = 2
x = -2, y = | -2 + 5 | = 3
x = -1, y = | -1 + 5 | = 4
x = 0, y = | 0 + 5 | = | 5 | = 5
x = 1, y = | 1 + 5 | = | 6 | = 6
x = 2, y = | 2 + 5 | = | 7 | = 7
x = 3, y = | 3 + 5 | = | 8 | = 8
..
..
..
We see that as x takes any value y increases.
x = domain, and y = range
y = 0, 1, 2, 3, 4, 5, ,,,,,,,∞
Thus the range is {y | 0 ≤ y < ∞}
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Answer:
c
Step-by-step explanation:
Finx, inc, purchased a truck for $40,000. The truck is expected to be driven 15,000 miles per year over a five-year period and then sold for approximately$5000.
Determine depreciation for the first year of the trucks useful life by the street wine in units of output methods if the truck is actually driven 16,000 miles.
Round to two decimal places.
The depreciation for the first year of the truck's useful life is $32,000.
The cost of the truck is $40000.
The useful time is 5 years.
The residual value is $5000.
Now, the depreciation is given as:
SLM depreciation = (Cost - Residual Value) / 5 years
Depreciation expense per year = ($40,000 - $5,000) / 5
Depreciation expense for the first year = $7,000
Now,
The depreciation expense per mile is the difference between cost and residual value divided by expected miles over the truck's life.
Depreciation expense per mile = (Cost - Residual Value) / Expected miles over truck's life.
Depreciation expense = Depreciation expense per mile × Miles driven
Therefore,
Depreciation expense per mile will be:
= ($40,000 - $5,000) / 15,000 miles = $2.33 or $2
The first year depreciation will be:
= 16,000 miles × $2 per mile = $32,000
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Find the polynomial of degree 3, with constant coefficient -12 and zeros -3, -1, and 2.
The equation of the polynomial equation is P(x) = -12(x + 3))(x + 1)(x - 2)
What are polynomial expressions?Polynomial expressions are mathematical statements that are represented by variables, coefficients and operators
How to determine the polynomial equation?The given parameters are
Degree of polynomial = 3Zeros = -3, -1, 2Constant coefficient = -12The sum of multiplicities of the polynomial equation must be equal to the degree.
This means that the multiplicity of each zero is 1
The equation of the polynomial is then calculated as
P(x) = Coefficient * (x - zero)^ multiplicity
So, we have
P(x) = -12(x - (-3))(x - (-1))(x - 2)
This gives
P(x) = -12(x + 3))(x + 1)(x - 2)
Hence, the equation of the polynomial equation is P(x) = -12(x + 3))(x + 1)(x - 2)
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An oceanside surf shop sells a bodyboard for $34.95. The wholesale cost of the bodyboard is $21.18. What is the percent of markup? Round to the nearest percent.
Answer: 14%
Step-by-step explanation:
There are 35 green, 22 white, 30 purple and 14 blue gumballs in the gumball machine. Sharee wants to get a white or green gumball. What is the probability of getting a Green gum ball
The probability of getting a green gum ball is 0.35 (35/101)
What is probability?Probability is the measure of chance that some event would happen.
Probability that an event will occur = (No of favorable possibilities of the event) / (total possible possibilities)
Given that there are 35 green,22 white, 30 purple and 14 blue gumballs in the gumball machine.
Therefore, Total gum balls = 35+22+30+14 = 101
The probability of getting a green gumball = no of green gumballs/ total gumballs
Therefore, probability of getting a green gumball = 35/101 = 0.35
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Justify Reasoning The quotient of two negative integers results in an integer. How does the value of the quotient compare to the value of the original two integers? Explain.
The quotient compares to the value of the original two integers as it's a positive number.
How to calculate the value?The result of dividing two numbers is a quotient; the divided number is referred to as the dividend, and the number by which it is divided is referred to as the divisor.
Zero, a positive natural number, or a negative integer denoted by a minus sign are all examples of integers.
From the information, the quotient of two negative integers results in an integer.
Fur example, -4 divided by -2 will give a value of 2. Therefore, the quotient of the negative numbers is a positive number.
Therefore, the quotient compares to the value of the original two integers as it's a positive number.
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Can someone help me with geometry
Answer:
Angle A = Right angleAngle B = Obtuse angleAngle C = Acute angleStep-by-step explanation:
The little square symbol made in intersection point of two rays of angle A represents a right angle.
Angle B looks greater than Angle A but less than 180° (straight line) And Angle C looks smaller than Angle A but greater than 0 in measure.
Angle that lies between 0° and 90° = acuteAngle that measure 90° = rightAngle that lies between 90° and 180° = obtuseAnswer:
A. Right angle - it's is a right angle bcuz it's of 90° and perpendicular.
B. Obtuse angle - it's an obtuse angle bcuz it's more than 90°
C. Acute angle - it's an acute angle bcuz it's less than 90°
is q=2p+1/2 proportional?
Answer:
Step-by-step explanation:
a(i) q = 2p + (1/2)
(ii) v = (1/10)u
(iii) t = 15d
(iv) m = 0.75d - 2
2p is added by the same constant so it's not proportional
27) A magician has a bag of colored scarves for a magic trick that he performs. The bag contains 3
blue scarves, 1 red scarf, 5 green scarves, and 2 orange scarves. If the magician removes scarves at
random and the first scarf he removes is red, what is the probability that the next scarf will be orange?
28) A card is drawn at random from a deck of cards. Find the probability of getting the ace of spades.
29) A card is drawn at random from a deck of cards. Find the probability of getting a king, queen, or
jack.
a. The probability that the next scarf will be orange will be 1/5.
b. The probability of getting the ace or spades will be 17/52.
c. The probability of getting a king, queen, or jack will be 3/14.
How to calculate the probability?a. The bag contains 3 blue scarves, 1 red scarf, 5 green scarves, and 2 orange scarves. The probability that the second scarf will be orange will be:
= 2/(3 + 5 + 2)
= 2/10
= 1/5
B. When a card is drawn at random from a deck of cards, the probability of getting the ace or spades will be:
Total number of cards withe the ace = 52
Total number of ace = 4
Number of spade = 13
The probability will be:
= 4/52 + 13/52
= 17/52
C. The probability of getting a king, queen, or
jack will be:.
= 4/52 + 4/52 + 4/52
= 12/52
= 3/14
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Fill in the blank with the missing number that makes this a true equation.
9/10 − 11/__ = 1/6
8. Fiona buys different amounts of gas at $2.25. She has a graph which shows the
different amounts she should pay. What constraints are there on the domain of
the function?
Fiona has a function that shows the amount of money she has to pay ( f(.) as a function of the gas the buys (g).
So, the function can be written as: f(g).
As each litter costs $2.25 we can say that;
f(g) = 2.25g
Lets try to find put which constraints the function f(g) has in its domain, it is, what are the restrictions for the values of g.
Is evident that g cannot be negative, as one cannot buy negative liters of gasoline. We can buy any positive amount of gasoline, doesn't matter if it is an integer number or not. it is possible to buy 5 liters and also to buy 5.45454545545454 liters. A zero amount of liters is also possible, having zero cost.
Thus, as any positive or zero value is positive, but negative values are not, we can restrict the domain to every non-negative real number:
Therefore ,Domain(f) = [0, infinite ) or
Domain(f) = Real non negative.
Domains have to be purchased, and they are registered where they are purchased and paid for. You have a Domain Registrant, much like a house has a Home Owner. This person pays the mortgage to keep the house, just like a Domain Registrant has to pay for the domain yearly to keep it.
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ill give brainly and 10 points
Answer: D
Step-by-step explanation:
What is the scale in first graph? In the second? Image above
Please help I’ll give you brainlist answer
Answer:
The first one is a scale by 1
The second one is a scale of 2
The third one is scale of 5 starting at 100
Step-by-step explanation:
Find and y.
x
A
8
||
y=
B
»
yx
18
D
The solution of the given equations that is x = 7 and x + y = 18 is :
x = 7
y = 11
What is a parallelogram ?
A parallelogram is a two-dimensional geometrical shape, whose sides are parallel to each other. It is a type of polygon having four sides (also called quadrilateral), where the pair of parallel sides are equal in length. The Sum of adjacent angles of a parallelogram is equal to 180 degrees.
The figure shown is an parallelogram, where length of opposite sides are equal to each other
From the figure,
x = 7, as opposite side of x is given as 7
x + y = 18,
Putting the value of x in the above equation,
7 + y = 18
y = 18 - 7
y = 11
Therefore, we get after solving the given equations :
x = 7
y = 11
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LOTS OF POINTS PLS ANSWER
Answer:
continuous it is continuous
7) Juan makes a base monthly salary of $3600. As a vendor, he must sell $35000 worth of items per month. He also makes a 9% commission on all sales beyond the monthly quota. There is also an additional 9% bonus on top of the normal commission rate for any sales beyond $52500. If Juan sold $56000 worth of items this month, what is his total salary for the month including base salary and commission to the nearest dollar?
The salary for the month for Juan including base salary and commission is $5805.
How to calculate the value?It should be noted that that Juan makes a base monthly salary of $3600 and he must sell $35000 worth of items per month. He also makes a 9% commission on all sales beyond the monthly quota.
Also, there is also an additional 9% bonus on top of the normal commission rate for any sales beyond $52500.
The total salary will be:
= $3600.+ (9% × $56000 - $35000) + (9% × $56000 - $52500)
= $3600 + $1890 + $315
= $5805
Therefore, the salary for the month for Juan including base salary and commission is $5805.
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There is a proportional relationship between the number of quarts and number of gallons for liquid measurements. What constant of proportionality relates gallons to quarts?
The value of constant of proportionality that relates gallons to quarts is 0.25 .
A constant of proportionality may be a constant worth for the magnitude relation of 2 proportional quantities. If the magnitude relation or product is constant, the 2 variables area unit proportional. the worth of the constant of quotient depends on the kind of relationship between the 2 given quantities (positive versus inverse variation).As we all know that one gallon have four quartz.
The proportional relation is written as y = kx.
We know that y varies proportionately with x.
Substitute within the such that x = four and y = one values and solve for k.
1 = k(4)
k =1 /4 = 0.25
Therefore the constant of proportionality is 0.25
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The graph represents the number of hours that Walter sleeps in terms of the number of days.
Graph shows a ray plotted in quadrant 1 of a coordinate plane with Days on X-axis, and Hours of Sleep on Y-axis. The ray begins at the origin and it extends upward, slanting right, through (2, 16) and (4, 24).
Which function represents Walter’s situation?
A.
y = 6x
B.
y = 6
C.
y = 8
D.
y = 8x
Answer:
I believe D is the answer to the question above.
Step-by-step explanation:
Answer: pretty sure its D
Step-by-step explanation:
a desk is on sale for $380, which is 40% off the original price. which equation can be used to determine the amount of money saved, s, in dollars, when purchasing this desk on sale?
you save $253.33 if you purchase this desk on sale.
How to find the money saved?
We know that the price in sale of the desk is $380, and this is given by a discount of the 40% of the original price.
So, if the original price is defined by the variable P, the amoun you save by buying the discounted deck is:
P - $380
Now, remember that the discounted price should be written in terms of P as:
$380 = P*(1 - 0.4) = P*0.6
Where 0.4 is the decimal form of 40%.
With that we can find the value of P as:
P = $380/0.6
Then the amount that you save is given by:
P - $380 = $380/0.6 - $380 = $253.33
We conclude that you save $253.33 if you purchase this desk on sale.
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Divide 2 5/16 ÷ 7/8
helpppppppppp
Answer:
When dividing fractions we "invert and multiply".
First, we should change 2 (5/16) to 37/16.
So, 37 / 16 divided by 7 / 8 equals
37 / 16 times 8 / 7
which equals 296 / 112 = 37 / 14 which equals
2 (9/14)
Step-by-step explanation:
Julia studied math for 3 and 1/3 hours during the 4 days before her last math test. What was the average amount of time she studied each day? Hours
Answer:50 min a day
Step-by-step explanation: julia studied 50 min a day
Which of the following is a factor of x3 − 1331?
x − 11
x2 − 11x + 121
x2 + 22x + 121
None of the above
Based on the information, the option that is a factor of x³ - 1331 is A. x - 11.
What are factors?It should be noted that factor simply means the number that divides another number. In contrast to multiples, which are multiplied by another number to produce particular numbers, factors are the numbers that split the supplied number exactly.
It should be noted that when we factor out 1331, we will get the values of 11 × 11 × 11. Therefore, it should be noted that the expression will become x³ - 11³.
Therefore, it should be noted that the formula that will be applied will be:
a³ - b³ = (a - b)(a² + an + b²)
Therefore, applying the formula illustrated will give:
x³ - 11³
= (x - 11)(x² + 11x + 11²)
= (x - 11)(x² + 11x + 121
Therefore, the option that is a factor of x³ - 1331 is x - 11.
Based on the information, the option that is a factor of x³ - 1331 is A. x - 11. In conclusion, the correct option is A.
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Answer:
A
Step-by-step explanation:
John wrote the numbers 678901 and 67890. How many times greater is the value of 7 in 678901 than the value of 7 in 67890?
Answer:In the number, 687,760, there are two 7s; one is at a place value of hundred and the other is at thousand.
Hence, the higher place value 7 is 10 times greater than lower place value 7. Every consecutive place value is 10 times greater than the right one.
Step-by-step explanation:
Diego is making a stack of Pennie’s. He starts with 5 and then adds them 1 at a time. A penny is 1.52 mm thick.
The stack of pennies is an illustration of linear equation.
The equation for n pennies is: h(n) = 7.6 + 1.52nh(n)=7.6+1.52n .
h(1.52) does not make sense
(a) Complete the table
From the question, we have:
Start = 5Start=5 ---- the number of penny he started with
Thick = 1.52Thick=1.52 --- the thickness of a penny
The height of n pennies is calculated using:
h(n) = (Start + n) \times Thickh(n)=(Start+n)×Thick
So, we have:
h(n) = (5 + n) \times 1.52h(n)=(5+n)×1.52
h(n) = 7.6 + 1.52nh(n)=7.6+1.52n
When n = 0;
h(0) = 7.6 + 1.52 \times = 7.6h(0)=7.6+1.52×=7.6
When n = 1;
h(1) = 7.6 + 1.52 \times 1 = 9.12h(1)=7.6+1.52×1=9.12
When n = 2
h(2) = 7.6 + 1.52 \times 2= 10.64h(2)=7.6+1.52×2=10.64
When n = 3
h(2) = 7.6 + 1.52 \times 3= 12.16h(2)=7.6+1.52×3=12.16
b) h(1.52) does not make sense.
This is so because:
n will always be a whole number because it represents the number of pennies
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