The general solution of the differential equation is 4A + 2B = 1 (coefficient of t²e³ᵗ).
To find the general solution of the given differential equation y'' + 4y = t²e³ᵗ + 3, we can use the method of undetermined coefficients.
The homogeneous equation associated with the given equation is y'' + 4y = 0, which has the characteristic equation r² + 4 = 0. The roots of this equation are r = ±2i, indicating that the homogeneous solution is of the form y_h(t) = c₁cos(2t) + c₂sin(2t), where c₁ and c₂ are constants.
To find the particular solution, we assume that the particular solution has the form y_p(t) = A(t) + B, where A(t) represents the particular solution related to the term t²e³ᵗ and B represents the particular solution related to the constant term 3.
Differentiating y_p(t), we have:
y'_p(t) = A'(t)
y''_p(t) = A''(t)
Substituting these derivatives into the original differential equation, we get:
A''(t) + 4(A(t) + B) = t²e³ᵗ + 3
To match the right-hand side, we set A''(t) + 4A(t) = t²e³ᵗ and 4B = 3.
The solution to the equation A''(t) + 4A(t) = t²e³ᵗ can be found using the method of undetermined coefficients. Since the right-hand side includes t²e³ᵗ, we assume a particular solution of the form A_p(t) = (At² + Bt + C)e³ᵗ, where A, B, and C are constants.
Differentiating A_p(t), we have:
A'_p(t) = (2At + B)e³ᵗ + (At² + Bt + C)3e³ᵗ
A''_p(t) = (2A + 2A + 2B)e³ᵗ + (2At + B)3e³ᵗ + (2At + Bt + C)9e³ᵗ
= (4A + 2B)e³ᵗ + (6At + 3B + 9A + 3Bt + 9C)e³ᵗ
= (4A + 2B + 6At + 3Bt + 9A + 9C)e³ᵗ
Substituting these derivatives into the equation A''(t) + 4A(t) = t²e³ᵗ, we get:
(4A + 2B + 6At + 3Bt + 9A + 9C)e³ᵗ + 4(At² + Bt + C)e³ᵗ = t²e³ᵗ
Matching the coefficients of like terms on both sides, we have:
(4A + 2B) + 6A = 0 (coefficient of e³ᵗ)
(3B + 9C) = 0 (coefficient of e³ᵗ)
4C = 0 (coefficient of e³ᵗ)
4A + 2B = 1 (coefficient of t²e³ᵗ)
From the first equation, we get A = -B/2, and substituting this into the fourth equation, we get B = 1/14. Substituting these values of A and B into the second equation
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The J.R. Simplot Company produces frozen French fries that are then sold to customers such as McDonald's. The "prime" line of fries has an average length of 6.00 inches with a standard deviation of 0.50 inch. To make sure that Simplot continues to meet the quality standard for "prime" fries, they plan to select a random sample of n = 100 fries each day. The quality analysts will compute the mean length for the sample. They want to establish limits on either side of the 6.00 inch mean so that the chance of the sample mean falling within the limits is 0.99. What should these limits be?
Answer:
(5.871 `; 6.127)
Step-by-step explanation:
Given :
Mean = 6
Standard deviation. σ = 0.5
Samplr size, n = 100
Zcritical at 99% confidence = 2.58
The confidence interval :
Mean ± margin of error
Margin of Error = Zcritical * σ / sqrt(n)
Margin of Error = 2.58 * 0.5/sqrt(100) = 0.129
Confidence interval :
Lower boundary = 6.00 - 0.129 = 5.871
Upper boundary = 6.00 + 0.129 = 6.129
(5.871 `; 6.127)
what are the answers to a, b, c, d?
Answer:
Step-by-step explanation:
Find the area of the figure
Answer:
A=153.94
Step-by-step explanation:
Please help I’m struggling
Show step-by-step solution. Compute manually.
1. Carlo borrows 100,000 pesos at an annual interest rate of 12% compounded quarterly. The loan is to be repaid by equal quarterly payments for 2 years. Determine each payment. Then make an amortization schedule for this loan.
Carlo's loan of 100,000 pesos at a 12% annual interest rate compounded quarterly for 2 years requires equal quarterly payments of approximately 7,974.51 pesos.
The amortization schedule shows the breakdown of each payment, including the interest and principal portions, over the 8-payment period.
To compute the equal quarterly payments for Carlo's loan, we can use the formula for the equal payment amount in an amortizing loan:
Payment = (Principal * Interest Rate) / (1 - (1 + Interest Rate)^(-n))
Where:
Principal = 100,000 pesos (loan amount)
Interest Rate = 12% per year (convert to quarterly rate by dividing by 4: 0.12/4 = 0.03)
n = number of payments (2 years * 4 quarters per year = 8 payments)
Let's calculate the payment amount:
Payment = (100,000 * 0.03) / (1 - (1 + 0.03)^(-8))
Payment = 7,974.51 pesos
Therefore, each quarterly payment for Carlo's loan is 7,974.51 pesos.
To create an amortization schedule, we can calculate the interest and principal portion of each payment for each quarter:
Quarter | Beginning Balance | Payment | Interest | Principal | Ending Balance
1 | 100,000 | 7,974.51| 3,000 | 4,974.51 | 95,025.49
2 | 95,025.49 | 7,974.51| 2,851.27 | 5,123.24 | 89,902.25
3 | 89,902.25 | 7,974.51| 2,697.07 | 5,277.44 | 84,624.81
4 | 84,624.81 | 7,974.51| 2,537.87 | 5,436.64 | 79,188.17
5 | 79,188.17 | 7,974.51| 2,373.66 | 5,600.85 | 73,587.32
6 | 73,587.32 | 7,974.51| 2,204.37 | 5,769.14 | 67,818.18
7 | 67,818.18 | 7,974.51| 2,029.89 | 5,944.62 | 61,873.56
8 | 61,873.56 | 7,974.51| 1,850.13 | 6,124.38 | 55,749.18
This amortization schedule shows the payment number, beginning balance, payment amount, interest portion, principal portion, and ending balance for each quarter.
Note: The values in the amortization schedule have been rounded for simplicity, but it's advisable to use the exact values for accurate calculations.
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(q11) Find the center of mass of the system of objects that have masses 2 , 3 and 5 at the point (-1,2),(1,1) and (3,3) respectively.
The center of mass of the system of objects that have masses 2 , 3 and 5 at the point (-1,2),(1,1) and (3,3) respectively is located at the point (1.6, 2.2).
To find the center of mass of the system of objects, we need to consider the masses and positions of each object. Let's go through the steps to calculate the center of mass:
Assign variables to the masses and coordinates of the objects:
Object 1: Mass = 2, Coordinates = (-1, 2)
Object 2: Mass = 3, Coordinates = (1, 1)
Object 3: Mass = 5, Coordinates = (3, 3)
Calculate the total mass of the system by summing the masses of the objects:
Total mass = 2 + 3 + 5 = 10
Calculate the weighted average of the x-coordinates and y-coordinates to find the center of mass:
Center of mass (x-coordinate) = (m1 * x1 + m2 * x2 + m3 * x3) / Total mass
Center of mass (y-coordinate) = (m1 * y1 + m2 * y2 + m3 * y3) / Total mass
Substituting the values:
Center of mass (x-coordinate) = (2 * -1 + 3 * 1 + 5 * 3) / 10 = ( -2 + 3 + 15) / 10 = 16 / 10 = 1.6
Center of mass (y-coordinate) = (2 * 2 + 3 * 1 + 5 * 3) / 10 = (4 + 3 + 15) / 10 = 22 / 10 = 2.2
Therefore, the center of mass of the system of objects is located at the point (1.6, 2.2).
The center of mass represents the average position of the system, taking into account the masses and positions of the objects. It is a point where the system can be considered to be balanced. Calculating the center of mass is important in various areas of physics, such as studying the motion, stability, and collisions of objects.
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The graph solves this system of equations. Enter the solution in the boxes. –2y = –4x + 8 x + y = 5
Answer:
x=2-y/2
Step-by-step explanation:
The solution of system of equations –2y = –4x + 8 and x + y = 5 is (3, 2).
What is Graph?Graph is a mathematical representation of a network and it describes the relationship between lines and points.
The given system of equations are –2y = –4x + 8
x + y = 5
x=5-y
Nu=ow plug in this in the first equation
-2y=-4(5-y)+8
-2y=-20+4y+8
Take the variable terms on one side and constant on other side.
-6y=-12
Divide both sides by 6
y=2
Now put in the equation of x
x=5-2=3
Hence, the solution of system of equations –2y = –4x + 8 and x + y = 5 is (3, 2).
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On a certain hot summer's day, 344 people used the public swimming pool. The daily prices are $1.50 for children and $2.00 for adults. The receipts for admission totaled $537.00. How many children and how many adults swam at the public pool that day?
Answer: 302 children and 42 adults
Step-by-step explanation:
x= Children y= Adults
x+ y = 344
1.50x + 2y= 537
Then I would use elimination and multiply the first equation by -2.
-2x-2y= -688
+1.5x+2y= 537
Add the equations and get
-0.5x= -151
Divide by -0.5
x= 302
Then plug x back into the equation.
302+y=344
Subtract 302 from both sides.
y=42
What is the volume of this sphere? Use a ~ 3.14 and round your answer to the nearest hundredth. 20 mm
Answer:
4186.67 cubic mm=V
Step-by-step explanation:
The volume of a sphere can be found using the equation, [tex]\frac{4}{3} \pi r^{3}[/tex]. It gives us the diameter but we need the radius. To find the radius just divde the diameter by 2, so the radius of the sphere is 10mm.
[tex]\frac{4}{3} (3.14) (10^{3})[/tex]=V
[tex]\frac{4}{3} (3.14) (1000)[/tex]=V
[tex]\frac{4}{3}[/tex](3140)=V
4186.67 cubic mm=V
Simplify
(2½)² x (0,5)²
Answer:
1.5625
Step-by-step explanation:
2½ = 5/2
(5/2)² = 25/4
25/4 = 6.25
0.5² = 0.25
6.25 x 0.25 = 1.5625
Mario is buying a number of hamburgers from the local store that cost $2.90 each. He is also buying one packet of hamburger rolls at a cost of $4.75. He has $39.55 to spend at the store. Write and solve an inequality that shows how many hamburgers, h, Mario can afford to buy.
Write the inequality.
solve the inequality
Answer:
I think hamburgers=12
Step-by-step explanation:
Please help.
Is algebra.
Answer:
1 is C
2 is B
3 is C
Step-by-step explanation:
Answer:
give em brainliest because i have no idea
Step-by-step explanation:
8^x=1 Please solve for x.
Answer:
[tex] {8}^{x} = 1[/tex]
The exponent 0 makes the number as 1.
[tex]x = 0[/tex]
Answer:
the answer for x is = 0
Step-by-step explanation:
hope this helps
could someone help pls
Answer:
40%
Step-by-step explanation:
6/15=favorable amount/total amount
6/15=2/5
2/5*20
Answer: 40%
6/15 can be simplified to 2/5. multiply by 20 on both sides to get 40/100 (40%)
What Statistic information we can get from a histogram?
A-Standard Deviation
B-Interquartile Range
C-Mean
The volume of a square pyramid is 48 in^3 with a height of 4 inches. Find the length of one of the side of the base.
Answer:
6 inches
Step-by-step explanation:
area of square = 48 x 3 / 4 = 36 in^2
side = 6 in
Answer:
6 in
Step-by-step explanation:
The volume formula for a square-base pyramid of side length x and height h is V = (1/3)x²h. We want to solve this for the base side length x:
Multiplying both sides by (3/h) results in:
3V
----- = x² and so the side length, x, is √(3V/h).
h
Substituting 4 in for h and 48 in³ for V, we get: x = √(144 in³/4 in) = 6 in
Check: Does the volume formula given above result in 48 in³ when x = 6 in and h = 4 in?
V = (1/3)x²h = (1/3)(6 in)²(4 in) = 48 in³ YES: x = 6 in is correct.
SOMEONE PLS HELP SHEHSJHSHS
Answer:
1. Quadrant II
2. Quadrant IV
3. Y-axis
Step-by-step explanation:
Prove the following:
(2cos x) / (cos 2x + 1) = sec x
The equation (2cos x) / (cos 2x + 1) = sec x is proven to be true.
We start with the expression on the left-hand side and simplify it step by step to show that it is equivalent to sec x.
(2cos x) / (cos 2x + 1) = (2cos x) / ((2cos^2 x - 1) + 1) [Using the double-angle formula cos 2x = 2cos^2 x - 1]
= (2cos x) / 2cos^2 x [Simplifying the numerator and denominator]
= cos x / cos^2 x [Cancelling out the factor of 2]
Now, we can simplify the expression further using the identity sec x = 1 / cos x:
cos x / cos^2 x = 1 / cos x = sec x.
Therefore, we have proved that (2cos x) / (cos 2x + 1) is equal to sec x.
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7 ⅓ ÷ 225 =
answer this question plz
Answer:
0.03259..
Step-by-step explanation:
Which statement explains the type of function that is represented by the equation
y=x2 +9?
5. What might be the dimensions of a
cylindrical container that holds 750 mL
of juice?
Answer:
radius of 10.9254843 and height of 10.9254843
Step-by-step explanation:
The equation for the volume of a cylinder is V = 2 pi r h
If V (volume) = 750, find for r and h
(I'm just going to make the radius and height the same thing)
750 = 2 pi r r
375 = pi r^2
119.366207 = r^2
10.9254843 = r
Describe the attributes of An acute triangle: An obtuse triangle: A right triangle: How many equal sides does an equilateral triangle have: isosceles: scalene pls help I will give you 15 pointss
Answer:
An acute triangle - interior angles are always less than 90 degrees with different side measures.
An obtuse triangle - one of the interior angles is more than 90 degrees. if one angle measures more than 90 degrees, then the sum of the remaining two angles is less than 90 degrees.
An right triangle - a triangle with a right angle (90 degrees)
Equilateral triangle has three equal sides. Isosceles triangles has two equal side lengths. Scalene triangles has no equal side lengths
To finish a board game Yanis needed to land on the last square rolling a sum of 6 with two dice. She was dismayed that it took her eight tries. Should she have been surprised?
Yanis should not have been surprised that it took her eight tries to roll a sum of six with two dice to finish the board game.
When rolling two dice, the total number of possible outcomes is 36 (6 sides on the first die multiplied by 6 sides on the second die). Out of these 36 possible outcomes, there are five ways to obtain a sum of six: (1, 5), (2, 4), (3, 3), (4, 2), and (5, 1). This means that the probability of rolling a sum of six is 5/36.
Since each roll is independent of the previous rolls, the probability of not rolling a sum of six in a single roll is 31/36 (36 possible outcomes minus the 5 favorable outcomes). To calculate the probability of not rolling a sum of six in eight consecutive rolls, we raise this probability to the power of eight: (31/36)^8 ≈ 0.282.
Therefore, there was approximately a 28.2% chance that Yanis would not roll a sum of six in eight tries. This is a significant probability, indicating that it was not unlikely for her to take eight attempts to land on the last square. Thus, she should not have been surprised by the outcome.
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Which expression represents the perimeter of the rectangle below?
A. 13x - 31
B. 13x - 55
C. 26x - 62
D. 26x-124
The perimeter of the rectangle given is P = 26x - 62.
What is the perimeter of a rectangle?
The perimeter of a rectangle is the sum of its sides. Mathematically -
Perimeter = [P] = 2(L + B)
Given is a rectangle with its length and breadth.
We can write the perimeter of the rectangle as follows -
[P] = 2(L + B)
L = 5x + 12
B = 8x - 43
Therefore -
P = 2(L + B)
P = 2(5x + 12 + 8x - 43)
P = 2(13x - 31)
P = 26x - 62
Therefore, the perimeter of the rectangle given is P = 26x - 62.
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1. A package of sticky notes is in the shape of
a parallelogram. The dimensions of one
sticky note are shown. What is the area of
one sticky note?
A. 248.4 cm?
B. 124.2 cm2
C. 62.1 cm2
D. 22.7 cm2
Answer:
B. 124.2
Step-by-step explanation:
Area= b*h
Area= 13.5*9.2
Area = 124.2
Please Note* = Multiply
My Samsung charger broke so I went to Five Below to get a new one. They were on sale for $15,35 (not so five below). There
was a 6% tax too. How much did the new charger cost?
( I don’t need the explanation, only the right answer)
Answer:
The answer to your question is 16.27
Explanation:
You would start with the equation,
15.35 + 6% = C (Cost)
15.35 · 6% = 0.921
15.35 + 0.921 = 16.271
Round the one down, since it is less than four, and you get 16.27
Hope this helps.
waffletowne
A right triangle has a 30° angle and the adjacent side to
the angle of length 6, draw the triangle and label all
sides and angles. Round to the nearest tenth.
Answer
let the triangle be supposed ABC
where angle BAC is 30 degree and angle ACB is 60 degree and where we know that angle ABC is right angled triangle which means angle ABC = 90 degree
the base side of triangle is 6cm which means ;
By taking angle A as reference angle
Hypotenuse = AC = x(let)
perpendicular = BC
base = AB = 6 cm
then by taking base and hypotenuse
b/h = AB/AC
or, cos (30 degree)= 6/ x
x =[tex]4\sqrt{x} 3[/tex] cm
that concluded that AC = [tex]4\sqrt{3}[/tex] cm
Now,
by phythagorous theorem,
[tex]h^{2} = p^{2} + b^{2}[/tex]
[tex]AC =\sqrt{AB^{2} + BC^{2} } AC = \sqrt{6^{2} + 4\sqrt{3} ^{2}} \\AC = 2\sqrt{21 } cm[/tex]
so, the length of side AB is 6 cm , BC is 4[tex]\sqrt{3}[/tex] cm and AC is 2[tex]\sqrt{21}[/tex] cm
Step-by-step explanation:
Simplify the following expressions to have fewer terms (MIDDLE SCHOOL)
5x-3+(4x-6)+2
4y+3-(8x-2)
Answer:
Install math calculator
Thank you
Answer:
x=1\9
Step-by-step explanation:
first part:
5x-3+4x+2
5x+4x-3+2
9x-1
9x=1
x=1\9 answer.
A student government organization is interested in estimating the proportion of students who favor a mandatory "pass-fail" grading policy for elective courses. A list of names and addresses of the 645 students enrolled during the current quarter is available from the registrar's office. Using three-digit random numbers in row 10 of table 7. 1 and moving across the row from left to right, identify the first 10 students who would be selected using simple random sampling. The three-digit random numbers begin with 816, 283, and 610
Simple random sampling is a statistical method in which every member of the population has an equal chance of being chosen as a subject for the survey. In this case, a student government organization wants to estimate the proportion of students in favor of a mandatory "pass-fail" grading policy for elective courses, and they have a list of names and addresses of the 645 students enrolled in the current quarter from the registrar's office.
They can use simple random sampling to select a sample of students to participate in the survey. The first 10 students who would be selected using simple random sampling using three-digit random numbers in row 10 of table 7.1 and moving across the row from left to right are as follows:816283610752991768275354233410 The procedure for selecting a simple random sample of size n from a population of N subjects is as follows: Assign a unique identification number to every member of the population Obtain a list of identification numbers of the population. Use a random number generator to select n random numbers from 1 to N, without replacement, to identify the members of the sample. Identify the members of the sample using the randomly selected identification numbers. Simple random sampling is the most straightforward sampling method, and it produces samples that are unbiased and representative of the population. It is important to note that the size of the sample chosen should be large enough to make accurate inferences about the population.For such more question on organization
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Find the volume of each composite figure.
Answer:
630 in^3
Step-by-step explanation:
6 x 6 x 14 = 504
(0.5 x 3 x 6) x 14 = 126
504 + 126 = 630