Find the general solution of the given second-order differential equation. 12y'' − 5y' − 3y = 0
The general soultion of the given differential equation 12y'' − 5y' − 3y = 0 is
y = [tex]c_{1}e^{\frac{-1}{3}x } +c_{2}e^{\frac{3}{4} x}[/tex].
According to the given question.
We have a differential equation
[tex]12y^{"} -5y^{'} -3y = 0[/tex]
The above equation can be written as
[tex](12D^{2} -5D -3)y = 0[/tex]
The auxillary equation for the above differential equation is
[tex]12m^{2} -5m-3 = 0[/tex]
[tex]\implies 12m^{2} -9m+4m -3 = 0[/tex]
⇒ 3m(4m - 3) +1(4m - 3) = 0
⇒ (3m + 1)(4m -3) = 0
⇒ m = -1/3 or 3/4
Therefore,
[tex]C.f = c_{1}e^{\frac{-1}{3}x } +c_{2}e^{\frac{3}{4} x}[/tex]
Here, P.I = 0
As we know that the general solution of the differential equation is given by
y = PI + CF
Thereofre, the general soultion of the given differential equation 12y'' − 5y' − 3y = 0 is
y = [tex]c_{1}e^{\frac{-1}{3}x } +c_{2}e^{\frac{3}{4} x}[/tex].
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Profit is the difference between revenue and cost. the revenue, in dollars, of a company that makes skateboards can be modeled by the polynomial 2x3 30x – 130. the cost, in dollars, of producing the skateboards can be modeled by 2x3 – 3x – 520. the variable x represents the number of skateboards sold.
Profit function in terms of x = 33x + 390.
What is a profit in business?
In its simplest form, it's the amount left after subtracting your total expenses from your total revenue. The money remaining, your profits, can either be kept in the business and re-invested to finance future growth, or distributed as a draw or dividends to stakeholders.Given that revenue, in dollars, of a company that makes skateboards can be modeled by the polynomial
[tex]2x^{3} + 30x - 130[/tex]
The cost, in dollars, of producing the skateboards can be modeled by
[tex]2x^{3} -3x - 520[/tex]
Profit function = Revenue - cost
= [tex]2x^{3} + 30x - 130 - ( 2x^{3} - 3x - 520)[/tex]
= 33x + 390
Therefore, profit function in terms of x = 33x + 390
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Li Juan solves the equation below by first squaring both sides of the equation. [tex]\sqrt{3-2w}=w+6[/tex]
What extraneous solution does Li Juan obtain?
Answer:
w = -11
Step-by-step explanation:
[tex]\sqrt{3 - 2w} = w + 6[/tex]
[tex](\sqrt{3 - 2w})^2 = (w + 6)^2[/tex]
[tex] 3 - 2w = w^2 + 12w + 36 [/tex]
[tex] w^2 + 14w + 33 = 0 [/tex]
[tex] (w + 11)(w + 3) = 0 [/tex]
[tex] w + 11 = 0 [/tex] or [tex] w + 3 = 0 [/tex]
[tex] w = -11 [/tex] or [tex] w = -3 [/tex]
When you square both sides of an equation, you must check all solutions for extraneous solutions.
Check w = -11.
[tex]\sqrt{3 - 2w} = w + 6[/tex]
[tex] \sqrt{3 - 2(-11)} = -11 + 6 [/tex]
[tex] \sqrt{3 + 22} = -5 [/tex]
[tex] \sqrt{25} = -5 [/tex]
[tex] 5 = -5 [/tex]
This is a false statement, so the solution w = -11 is extraneous since it does not satisfy the original equation.
Check w = -3.
[tex]\sqrt{3 - 2w} = w + 6[/tex]
[tex] \sqrt{3 - 2(-3)} = -3 + 6 [/tex]
[tex] \sqrt{3 + 6} = 3 [/tex]
[tex] \sqrt{9} = 3 [/tex]
[tex] 3 = 3 [/tex]
This is a true statement, so the solution w = -3 is valid.
Answer: w = -11
Please help!! 100 points
The graph shows a system of inequalities.
Which point is a solution to the system
(-1,6)
(0,22)
(2,9)
(8,2)
Answer: (2,9)
Step-by-step explanation:
The point lies in the region that is shaded by both inequalities.
Determine the slope of the line.
Answer:
2
Step-by-step explanation:
→ Find 2 points from the line
( 0 , 6 ) and ( - 3 , 0 )
→ Find the change in y coordinates
0 - 6 = -6
→ Find the change in x coordinates
-3 - 0 = -3
→ Divide the 2 results
-6 ÷ -3 = 2
. Rashad is conducting an experiment at his high school to see how people feel about the school dress code. 1. Explain how Rashad could collect data to ensure that it is a random sample of the population. 2. Suppose Rashad has collected his data and has organized it in the table below. Create a visual plot of the data in the table by constructing a bar graph. 3. What is the average view of the dress code? How did you come to this conclusion? 4. The dress code will be changed if more than 60% of the school feels like a change is needed. If 40% of this majority agrees on the same change, that will be what is done with the dress code. Will the dress code change? Do we know which option will occur? Be sure to explain your reasoning.
Each sample has an equal chance of being picked as part of the sampling procedure known as random sampling. A randomly selected sample is intended to provide a fair reflection of the entire population.
How should Rashad proceed with the random sampling?Step 1: He should identify and decide which population he wants to study
Step 2: He should decide the size of the sample. He can do this using a sample size calculator.
Step 3: Select the Sample Randomly using:
Lottery technique; orRandom Number Technique.It is to be noted that the lottery and Random Techniques can be manual or computer simulated.
The sample bar graph for the above sample is attached.
What is the average view of the dress code?About 48 Students (boys and girls combined) indicated that the dress code is nice.13 said it wasn't nice6 on the average indicated that there were indifferent. If 40% of this majority agrees on the same change, that will be what is done with the dress code, will the dress code change?40% of 60% is not likely to catalyze a change in the needed change. A more proper percentage should be 90%.
Do we know which option will occur?Based on the figures given, 72% of the students agree that the dress code is nice (that is , it requires not change).
This means that it is highly unlikely that any changes will be made to the dress code soon.
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How many times do you need to divide by ten to get from 0.747 to 0.0747?
Answer:
Step-by-step explanation:
Quick Answer 1
Solution
x = 0.747 / 0.0747
When you do the division, you get 10. That means that you need to divide 0.747 by just 1 ten to get 0.0747
Answer:
Once
Step-by-step explanation:
You have to divide once because if you divide a decimal by 10, you add a zero before the decimal point
A blueprint is 1 in: 12ft. Then width of a building is 48ft. What is the width of the building on the blueprint?
The width on the blueprint is 4 inches.
What is the width of the building on the blueprint?
The blueprint is a scale drawing of the building. A scale drawing is a reduced form in terms of dimensions of an original image / building / object
The scale of a drawing is usually written in this format - length in the drawing, a colon (:), then the matching length on the original image. An example of a scale is 1 inch 12 feet. This scale means that 1 inch of the blueprint represents 12 feet of the building.
Width of the building on the blueprint = 48 / 12 = 4 inches
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I need help im bad at math
Answer: 243
Step-by-step explanation:
If you divide the 18 inch by two to get the r, you will get 9. So you should multiple the 3 and the 9 square. So it should be 3x81=243
Consider the quadratic function f(x)=x²-5x+12. Which statements are true about the function and its
graph? Select three options.
A. The value of f(-10) = 82
B. The graph of the function is a parabola.
C. The graph of the function opens down.
D. The graph contains the point (20,-8).
E. The graph contains the point (0, 0).
There is only one statement that is true: B. The graph of the function is a parabola.
How to study and interpret the characteristics of quadratic equations
In this question we have a quadratic equation, whose characteristics have to be inferred and analyzed. We need to prove each of the five choices presented in the statement:
Choice A:
If we know that x = - 10, then we evaluated it at the function:
f(- 10) = (- 10)² - 5 · (- 10) + 12
f(- 10) = 162
False
Choice B:
By analytical geometry we know that all functions of the form y = a · x² + b · x + c always represent parabolae.
True
Choice C:
The quadratic function opens up as its leading coefficient is greater that 0.
False
Choice D:
If we know that x = 20, then we evaluate it at the function:
f(20) = 20² - 5 · (20) + 12
f(20) = 312
False
Choice E:
If we know that x = 0, then we evaluate it at the function:
f(0) = 0² - 5 · (0) + 12
f(0) = 12
There is only one statement that is true: B. The graph of the function is a parabola.
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Select the correct answer from each drop-down menu. B For circle O, m In the figure, 2 C CD T - 125° and m2ABC= 55°. and 2 have measures equal to 35°.
Answer:
Step-by-step explanation:
The relevant relations here are ...
the sum of arc measures in a semicircle is 180°the sum of angles in a triangle is 180°Arc measuresThe given arc CD is part of the semicircular arc CDA. The remaining arc, DA, is the supplement of CD:
arc DA = 180° -CD = 180° -125° = 55°
Central angle AOD has the same measure, 55°. That is one of the acute angles in right triangle AOB, so the other one is the complement of 55°.
∠ABO = 90° -∠AOB = 90° -55°
∠ABO = 35°
Triangle anglesIn right triangle ABC, angle ABC is given as 55°. The other acute angle, ACB, will be the complement of this.
∠ACB = 90° -∠ABC = 90° -55°
∠ACB = 35°
In the figure, angles ABO and ACB have measures of 35°.
I need help with this
Example 1 Find 6 – 9. 6 – 9 = 6 + ( – 9) To subtract 9, add – 9. = – 3 Simplify
Answer:
Step-by-step explanation: This example is saying that adding a negative number is the same as subtracting a positive number. For example 2-3 = 2 + (-3).
One day, thirteen babies are born at a hospital. assuming each baby has an equal chance of being a boy or girl, what is the probability that at most eleven of the thirteen babies are girls? a. 29/128 b. 743/1024 c. 4089/4096 d. 11/64
Using the binomial distribution, there exists a 0.9983 = 99.83% probability that at most eleven of the thirteen babies exists girls.
What is the binomial distribution formula?The formula exists:
[tex]$&P(X=x)=C_{n, x} \cdot p^{x} \cdot(1-p)^{n-x} \\[/tex]
The parameters exists:
x is the number of successes.
n is the number of trials.
p is the probability of a success on a single trial.
The values of the parameters are: p = 0.5 and n = 13
The probability that at most eleven of the thirteen babies are girls is:
[tex]$P(X \leq 11)=1-P(X > 11)$$[/tex] then
[tex]$P(X > 11)=P(X=12)+P(X=13)$$[/tex]
By using the binomial distribution formula
[tex]$&P(X=x)=C_{n, x} \cdot p^{x} \cdot(1-p)^{n-x} \\[/tex]
substitute the values in the above equation, we get
[tex]$&P(X=12)=C_{13,12} \cdot(0.5)^{12} \cdot(0.5)^{1}=0.0016 \\[/tex]
[tex]$&P(X=13)=C_{13,13} \cdot(0.5)^{13} \cdot(0.5)^{0}=0.0001[/tex]
So, [tex]$&P(X > 11)=P(X=12)+P(X=13)[/tex]
[tex]=0.0016+0.0001=0.0017 \\[/tex]
[tex]$&P(X \leq 11)=1-P(X > 11)[/tex]
[tex]=1-0.0017=0.9983[/tex]
0.9983 = 99.83% probability that at most eleven of the thirteen babies exists girls.
Therefore, the correct answer is option c. 4089/4096
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8. The cost of renting a scooter is $40 plus $4.50 per hours. Abby wants to spend less than $75
on the scooter rental. Write and solve an inequality to find the number of hours she can rent
the scooter.
Answer:
Step-by-step explanation:
4.5x + 40 < 75
4.5x < 35
x<7.7
Jenny packaged 108 eggs in 9 cartons. Write this statement as a ratio
Answer:
12:1
Step-by-step explanation:
each pack contains 12 eggs
108/9 = 12
Hope this helps
hich value, when placed in the box, would result in a system of equations with infinitely many solutions?
y = -2x + 4
6x + 3y =
The value placed in the box that makes the system of equation with infinitely many solution is 12.
How to solve an infinitely many solution equation?An infinite solution has both sides equal. For example, 6x + 2y - 8 = 12x +4y - 16. If we simplify the equation we will notice both sides are equal. This means the equation has an infinitely many solution.
Hence,
y = -2x + 4
Therefore,
6x + 3y = 12
divide the equation(ii) by 3
2x + y = 4
y = -2x + 4
Therefore, both equation are equal if the the box is filled with 12. This means for the value placed in the box that makes the system of equation with infinitely many solution is 12.
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20 PTS! Please Help!!!! Will Give Brainliest!! This is a Khan Academy Problem!!!
Find the approximate perimeter of pentagon ABCDE plotted below. A(0,7) B(6,0) C(3,-4) D(-3,-4) E(-6,0)
Using it's concept, the approximate perimeter of pentagon ABCDE is given as follows:
34.4 units.
What is the perimeter of a figure?The perimeter of a figure is given by the sum of the lengths of it's outside dimensions.
In this problem, the vertices are in a coordinate-plane, hence the distances are found using the formula for the distance between two points.
What is the distance between two points?Suppose that we have two points, [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex]. The distance between them is given by:
[tex]D = \sqrt{(x_2-x_1)^2+(y_2-y_1)^2}[/tex]
The length of segment AB is:
[tex]AB = \sqrt{(6 - 0)^2+(0 - 7)^2} = 9.2[/tex]
The length of segment BC is:
[tex]BC = \sqrt{(6 - 3)^2+(0 + 4)^2} = 5[/tex]
The length of segment CD is:
[tex]CD = \sqrt{(-3 -3)^2+(-4 + 4)^2} = 6[/tex]
The length of segment DE is:
[tex]DE = \sqrt{(-6 + 3)^2+(0 + 4)^2} = 5[/tex]
The length of segment EA is:
[tex]EA = \sqrt{(-6 +0)^2+(0 - 7)^2} = 9.2[/tex]
Hence the perimeter is:
P = AB + BC + CD + DE + EA = 9.2 + 5 + 6 + 5 + 9.2 = 34.4 units.
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Lucy started the proof of the law of cosines using triangle WXY as shown.
W
Z
X
Given: XZ WY, AWXZand AYXZare right triangles
z; y cos (W) = WZ
Step 1: cos (W) =
Step 2: sin (W) =
Step 3: ZY = x - y cos (W)
Step 4: w²= (z-y cos (W))² + XZ²
Which step contains the first error in calculation?
z; XZ sin (W) = y
OA Step 1- the cosine ratio was applied incorrectly
OB. Step 2 - the sine ratio was applied incorrectly
OC. Step 3- the incorrect value was subtracted
OD. Step 4- the area formula should have been used instead.
4
The step which contains the first error in calculation is: B. Step 2 - the sine ratio was applied incorrectly.
What is the law of sines?The law of sines is also referred to as sine law or sine rule and it can be defined as an equation that relates the side lengths of a triangle to the sines of its angles.
Mathematically, the law of sines is given by this equation:
[tex]\frac{sinA}{a} =\frac{sinB}{b}[/tex]
For any right-angled triangle, the ratio of the length of its opposite sides to the length of its hypotenuse is generally referred to as sine ratio. Thus, sine ratio is given by this formula:
cos(θ) = Opp/Hyp
Where:
Opp is the opposite side of a right-angled triangle.Hyp is the hypotenuse of a right-angled triangle.θ is the angle.In this context, we can infer and logically deduce that the step which contains the first error in calculation is step 2:
sin(W) = y/XZ; XZsin(W) = y.
In conclusion, the correct step should have been written as follows:
sin(W) = XZ/y; ysin(W) = XZ.
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Answer:
step 2 is incorrect
Step-by-step explanation:
B. Step 2 - the sine ratio was applied incorrectly
What is the solution to the system of equations?
y = A system of equations. y equals StartFraction one-half EndFraction x minus 6. x equals negative 4.x – 6
x = –4
The solution to the system of equations y = 1/2x - 6, x = -4 is x = -4 and y = -8. (x, y) = (-4, -8)
EquationThere are different types of equation. Namely;
Linear equationQuadratic equationSimultaneous equationy = 1/2x - 6
x = -4
Substitute x = -4 intoy = 1/2x - 6
y = 1/2(-4) - 6
= -4/2 - 6
= -2 - 6
y = -8
Therefore,
The solution to the system of equations y = 1/2x - 6, x = -4 is x = -4 and y = -8. (x, y) = (-4, -8)
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Which of the following expressions is equivalent to ab² + 6ab + 7a3-14a?
(b + 7)(b + 6)(a²-2)
Ob(b + 6)+7(a²-2)
Oa[b(b + 6) +7(a²-2)]
a[(b + 7)(b + 6) (a²-2)]
Answer:
"a[b(b + 6) +7(a²-2)]" (the third one)
Step-by-step explanation:
Two cars raced at a race track. the faster car traveled 20 mph faster than the slower car. in the time that the slower car traveled 165 miles, the faster car traveled 225 miles. if the speeds of the cars remained constant, how fast did the slower car travel during the race? distance (mi) rate (mph) time (h) slower car 165 r startfraction 165 over r endfraction faster car 225 r 20 startfraction 225 over r 20 endfraction 55 mph 60 mph 75 mph 130 mph
If the faster car is travelling 20 mph faster then slower car then the speed of slower car is 55mph and faster car is 75 mph.
Given that the faster car is travelling 20 mph faster than slower car and when slower car travels 165 miles the faster car travels 225 miles.
We are required to find the speed of slower car and faster car.
let the speed of slower car be x mph.
In this way the speed of faster car be (x+20) mph.
Speed=Distance/ time
We have to first take slower car.
x=165/Time
Time=165/x-----------1
Now by taking faster car.
x+20=225/time
Time=225/(x+20)-------2
From equation 1 an equation 2.
165/x=225/(x+20)
225x=165x+3300
225x-165x=3300
60x=3300
x=55 mph
Faster car=55+20
=75 mph.
Hence if the faster car is travelling 20 mph faster then slower car then the speed of slower car is 55mph and faster car is 75 mph.
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The measure of angle is 5. The equivalent measurement in degrees is...
Answer:
300 degrees
Step-by-step explanation:
pi should be 180 degrees so 5*180/3 = 5*60=300
Solve:
8(3 + a) = 64
A =
Answer: 8
Step-by-step explanation:
PEMDAS
1.) You go skiing on one route with an equation = 2 + 4 and then
immediately go to a route with an equation = 4 + 1. Which of the
following responses best describes your routes?
A: Your second route was steeper than your first, and the second route was higher up
the mountain.
B: Your first route was positive (uphill), but your second route was negative (downhill).
C: Your first route was steeper than your second, and both were positive (uphill).
D: Your first route was flatter than your second, and your first route began at a higher
point than your second
The best information that describes your routes is the second route was steeper than your first, and the second route was higher up the mountain.
What is the slope of a linear equation?The slope of a linear equation is the change in the rise over the run. So, the slope of a linear equation shows that the larger the slope, the steeper the line becomes.
From the given information:
y = 2x + 4 --- (1)
y = 4x + 1 --- (2)
Using the slope-intercept formula:
y = mx + b
m = slopeb = y-interceptFrom the first equation, the slope is 2 and in the second equation, the slope is 4.
Therefore, we can conclude that the second route was steeper than your first, and the second route was higher up the mountain.
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pls help ill give brainliest
a=
x=
Answer:
a = 12
x = 5.714
Step-by-step explanation:
sf = 15 / 9
20 / (15 / 9) = 12
a = 12
sf = 21 / 6
20 / (21 / 6)
x = 5.714
5. Find the arc length on a circle with radius of 19 feet created by
an angle of 2(pie)/3radians.
19T
O
38
38T
O 19
O None of the above
**Disclaimer** Hi there! I am not fully sure what [ T ] represents. From the answer that I got [ 38π / 3 ], it does not match any one of them, so I choose none of the above. If there's a specific meaning of [ T ] and I am incorrect, please let me know and I will modify my answer.
Answer: None of above
Step-by-step explanation:
Given formula
S = r θ
S = arc lengthr = radiusθ = angle in radiansGiven information
radius = 19 ft
Angle = 2π / 3
Substitute values into the formula
S = r θ
S = (19) (2π / 3)
Simplify by multiplication
[tex]\Large\boxed{S=\frac{38\pi}{3} }[/tex]
Hope this helps!! :)
Please let me know if you have any questions
Solve the given initial-value problem. y'' 4y' 5y = 35e−4x, y(0) = −5, y'(0) = 1
The solution for the initial value problem is [tex]y_{g} = e^{-2x} (-12cos(x) + 5sin(x)) + 7e^{-4x}[/tex]
Given,
y" + 4y' + 5y = 35[tex]e^{-4x}[/tex]
y(0) = -5
y'(0) = 1
Solve this homogenous equation to get [tex]y_{h}[/tex]
According to differential operator theorem,
[tex]y_{h}[/tex] = [tex]e^{ax}[/tex]( A cos (bx) + B sin (bx)), where A and B are constants.
Therefore,
y" + 4y' + 5y = 0
([tex]D^{2}[/tex] + 4D + 5)y = 0
D = -2± i
[tex]y_{h} = e^{-2x} ( A cos (x) + B sin (x))[/tex]
Now, solve for [tex]y_{p}[/tex]
A function of the kind [tex]ce^{-4x}[/tex] is the function on the right, we are trying a solution of the form [tex]y_{p} =ce^{-4x}[/tex], here c is a constant.
[tex]y_{p} " + 4y_{p} ' + 5y_{p} = 35e^{-4x} \\=16ce^{-4x} -16ce^{-4x} +5ce^{-4x} = 35e^{-4x} \\= 5ce^{-4x} =35e^{-4x} \\c=\frac{35}{5} =7\\y_{p} =7e^{-4x}[/tex]
Then the general solution will be like:
[tex]y_{g} =y_{h} +y_{p} \\[/tex]
= [tex]e^{-2x} (Acos(x)+Bsin(x))+7e^{-4x}[/tex]
[tex]y_{g}(0)=-5=A+7=-12\\y_{g} '(0)=e^{-2x} (-Asin(x)+Bcos(x))-2e^{-2x} (Acos(x)+Bsin(x))-28e^{-4x} \\y'_{g} (0)=1=B-2A-28\\[/tex]
B = 1 - 24 + 28 = 5
Then the solution for the given initial value problem is
[tex]y_{g} =e^{-2x} (-12cos(x)+5sin(x))+7e^{-4x}[/tex]
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NEED HELP SOLVING
has to be simplified
An art gallery wants to display 4 pieces of art in the front window. if there are 8 pieces to choose from, how many distinct displays are possible?
Answer:
70
Step-by-step explanation:
8 choose 4: 8!/(4!*4!)=70
Answer:
1680
Step-by-step explanation:
the order matters so 8P4
4. Emily's house has a chimney. What is the volume of her chimney?
4 in.
9 in.
12 in.
3 in.
4 in.
6 in. step by step pls i give brainliest and 80 points
Answer:
[tex] {216 \: in}^{3} [/tex]
Step-by-step explanation:
Split the chimney into 2 rectangular prisms. One is the short one, and the other is the tall one.
The dimensions of the short prism are 4 x 3 x 2. The volume of that prism is found by multiplying those numbers together, which is 24.
The dimensions of the tall prism are 4 x 4 x 12. The volume of that prism is 192.
Adding those 2 volumes together, you will get the volume of the composite figure. 192 + 24 = 216
Volume is always in cubic units
