for the variable A type the word lambda, fory type the word gamma, otherwise treat these as you would any other variable We will solve the heat equation -6, 0
Answers
The required heat equation is u(x, t) = (A×cos(λx) + B×sin(λx)) × exp(-λ²t)
To solve the heat equation in the given interval [-6, 0], we can use the separation of variables method. Let's denote the dependent variable as u(x, t), where x represents the spatial variable and t represents the temporal variable.
The heat equation in one dimension is given by:
∂u/∂t = α ∂²u/∂x²,
where α is the thermal diffusivity constant.
To solve this equation, we assume that the solution can be represented as a product of two functions, each depending on a single variable:
u(x, t) = X(x)T(t).
Substituting this into the heat equation, we have:
X(x)T'(t) = αX''(x)T(t),
where prime (') denotes differentiation with respect to the variable.
Dividing both sides by αX(x)T(t), we get:
T'(t)/T(t) = αX''(x)/X(x).
Since the left side depends only on t and the right side depends only on x, both sides must be equal to a constant, which we'll denote as -λ²:
T'(t)/T(t) = -λ² = αX''(x)/X(x).
Now, let's solve the temporal part of the equation:
T'(t)/T(t) = -λ²
This is a separable ordinary differential equation (ODE), and its general solution is given by:
T(t) = exp(-λ²t).
Next, let's solve the spatial part of the equation:
αX''(x)/X(x) = -λ².
This is also a separable ODE, and its general solution is given by:
X(x) = A×cos(λx) + B×sin(λx),
where A and B are arbitrary constants.
Therefore, the general solution to the heat equation is:
u(x, t) = (A×cos(λx) + B×sin(λx)) × exp(-λ²t).
Since we have the given interval [-6, 0], we can apply appropriate boundary conditions to determine the values of A, B, and λ that satisfy the problem.
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Related Questions
A triangle has side lengths of (10x+7y)(10x+7y) centimeters, (3x+3z)(3x+3z) centimeters, and (8z+10y)(8z+10y) centimeters. Which expression represents the perimeter, in centimeters, of the triangle?
Answers
Answer:
(109x²+140xy+18xz+49y²+160yz+73z²) centimeters
Step-by-step explanation:
perimeter = sum of all sides
side A = (10x+7y)(10x+7y)
= 100x² +70xy +70xy +49y²
= (100x² +140xy +49y²) centimeters
side B = (3x+3z)(3x+3z)
= 9x² +9xz +9xz +9z²
= (9x² +18xz +9z²) centimeters
side C = (8z+10y)(8z+10y)
= 64z² +80yz +80yz +100y²
= (64z² +160yz +100y²) centimeters
Perimeter = 100x²+140xy+49y² + 9x²+18xz+9z² + 64z²+160yz+100y²
collect like terms
= 100x² +9x²+140xy+18xz+49y²+160yz+9z²+64z²
= (109x²+140xy+18xz+49y²+160yz+73z²) centimeters
Answer: 13x+17y+11z
Step-by-step explanation:
A coach is buying snacks for 22 players at a soccer match. She pays a total of $77 to
buy each player a bottle of water and an energy bar. The price of one energy bar is $2.
Let w equal the price of a bottle of water. Write an equation that
represents the situation.
Answers
Answer: 22( w + 2 ) =77 and the amount of the water bottle would be 1.50
Step-by-step explanation: i dont know what to write??
Serenity filled up her car with gas before embarking on a road trip across the country. Let � G represent the number of gallons of gas remaining in her gas tank after driving for � t hours. A graph of � G is shown below. Write an equation for � G then state the � y-intercept of the graph and determine its interpretation in the context of the problem.
Answers
The equation is: G = -⁵/₄t + 15
The slope of the function represents that ⁵/₄ gallons of gas is consumed to drive the car for one hour.
How to find the linear equation of the graph?The formula for the equation of a line in slope intercept form is:
y = mx + c
where:
m is slope
c is y-intercept
From the graph, we see that:
y-intercept = 15 gallons
Now, the slope is gotten from the formula:
Slope = (y₂ - y₁)/(x₂ - x₁)
Slope = (10 - 5)/(4 - 8)
Slope = -⁵/₄
Thus, equation is:
G = -⁵/₄t + 15
The slope of the function represents that ⁵/₄ gallons of gas is consumed to drive the car for one hour.
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A rectangle has a height oft2+7t+6t2+7t+6t, squared, plus, 7, t, plus, 6 and a width of2t+12t
Answers
The area of the rectangle with a height of t² + 7t + 6 and a width of 2t + 1 is given by the expression 2t³ + 15t² + 19t + 6.
To find the area of the rectangle, we need to multiply the height by the width.
Given:
Height = t² + 7t + 6
Width = 2t + 1
Area = Height * Width
Substituting the given values:
Area = (t² + 7t + 6) * (2t + 1)
Expanding the expression:
Area = 2t * t² + 2t * 7t + 2t * 6 + 1 * t² + 1 * 7t + 1 * 6
Area = 2t³ + 14t² + 12t + t² + 7t + 6
Combining like terms:
Area = 2t³ + 15t² + 19t + 6
Therefore, the area of the rectangle with a height of t² + 7t + 6 and a width of 2t + 1 is given by the expression 2t³ + 15t² + 19t + 6.
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Given question is incomplete, the complete question is below
What is the area of a rectangle with a height of t² + 7t + 6 and a width of 2t + 1
Write 3g as a percentage of 20g
Answers
Answer:
15%
Step-by-step explanation:
3÷20=0.15
0.15×100=15
Lily and nada share silver coins in the ratio 7:3. if lily gives 3 silver coins to nada, the ratio becomes 5:3. how many silver coins did lily have initially?
Answers
The number of silver coins Lily initially had is 28.
What is ratios?A ratio is described as a comparing two quantities with the same unit to determine the amount of each quantity is existent in the other. Ratios are divided into two types.
The first is a part-to-part ratio, and the second is a part-to-whole ratio. The part-to-part ratio expresses the relationship between 2 separate entities or groups.Now according to the question;
Let 'x' be the share of silver coin of Lily.
Let 'y' be the share of silver coin of Nada.
Lily and Nada share silver coins in the ratio 7:3
Thus, x/y = 7/3
Cross-multiplying
3x = 7y (equation 1)
Now, if Lily gives 3 silver coins to Nada, the ratio becomes 5:3.
(x - 3)/(y + 3) = 5/3
cross multiplying;
3x - 9 = 5y + 15
Simplifying;
3x = 5y + 24 (equation 2)
Comparing equation 1 and 2.
7y = 5y + 24
2y = 24
y = 12 (Nada's share)
Substitute the value of y in equation 1;
3x = 7y
3x = 7×12
x = 28
Therefore, the initial share if Lily was 28.
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The algebra teacher said if you square my age then subtract 23 times my age, the result is 50. How old is he?
Answers
Answer:
25
Step-by-step explanation:
Answer:
She is 25
Step-by-step explanation:
25^2 = 625
23*25 = 575
625 - 575 = 50
See attached picture, is there a way to enter equations or algebraic symbols?
Answers
SOLUTIONS
Solve a given equations or algebraic symbols?
\(\begin{gathered} f(x)=x^2+2x \\ g(x)=1-x^2 \end{gathered}\)(A)
\(\begin{gathered} (f+g)(x)=(x^2+2x)+(1-x^2) \\ collect\text{ like terms} \\ x^2-x^2+2x+1 \\ (f+g)(x^)=2x+1 \end{gathered}\)(B)
\(\begin{gathered} (f-g)(x)=(x^2+2x)-(1-x^2) \\ =x^2+2x-1+x^2 \\ =x^2+x^2+2x-1 \\ (f-g)(x)=2x^2+2x-1 \end{gathered}\)(C)
\(\begin{gathered} fg(x)=(x^2+2x)(1-x^2) \\ =x^2-x^4+2x-2x^3 \\ fg(x)=-x^4-2x^3+x^2+2x \end{gathered}\)(D)
\(\begin{gathered} \frac{f}{g}(x)=(x^2+2x)\div(1-x^2) \\ =\frac{x^2+2x}{1-x^2} \end{gathered}\)kayla has a deck that measures 4 feet by 21 feet. she wants to increase each dimension by equal lengths so that the area is doubled. how much should she increase each dimension?
Answers
Answer:
Increase by 3.
Step-by-step explanation:
4 x 21 = 84
84 x 2 = 168
Begin to try numbers.
5 x 22 = 110
6 x 23 = 138
7 x 24 = 168
7 - 4 = 3
24 - 21 = 3
Increase each side by 3 ft. for a total dimension of 7 ft. x 24 ft.
A card is drawn from a well shuffled deck of 52 cards. Find P (drawing a face card or a 4). A face card is a king queen of jack
Answers
Answer:
The probability of drawing a face card or a 4 is approximately 0.2885, or 28.85%.
Step-by-step explanation:
To find the probability of drawing a face card or a 4 from a well shuffled deck of 52 cards, we need to count the number of cards that are either a face card or a 4, and divide that number by the total number of cards in the deck.
There are 12 face cards in a deck (4 kings, 4 queens, and 4 jacks) and 4 cards with the number 4, but the card with 4 is also a face card (the four of hearts), so we need to subtract one card from the total. Therefore, there are 15 cards in the deck that are either a face card or a 4.
The total number of cards in the deck is 52. Therefore, the probability of drawing a face card or a 4 from a well shuffled deck of cards is:
P = number of desired outcomes / total number of possible outcomes P = 15/52 P = 0.2885 (rounded to four decimal places)
So the probability of drawing a face card or a 4 is approximately 0.2885, or 28.85%.
brainliest Plssss
how do I solve this? 4-0,9+(-0,11)-(-1,3)
Answers
Answer:
Hope it will help a lot ask for more info
b) The average age of Kishan and Dolma is 15 years and that of Dolma and Shashwat is 14 years. If Kishan is 17 years old, find the age of Dolma and Shashwat.
Answers
Step-by-step explanation:
average means the sum of all values divided by a number of values
HELP MEEEEEEEE PLEASE
Answers
Answer:
Option A (3/4)
Step-by-step explanation:
Please mark Brainliest
Answer:
THATS CORREC T
Step-by-step explanation:
CAUSE A IS THE RIGHT ONE THE REST BIS WRONG BUT IT CAN E CONFUSING
I’d really appreciate if you’d also give a brief explanation:)
Answers
Answer:
Only area of PQR can be calculated.
Step-by-step explanation:
In PQR, we have both the base (24 cm) and the height (15cm) which is perpendicular to the base. Therefore using these two measurements, we can take
\( \frac{1}{2} \times 24 \times 15\)
This is equal to 180 cm².
Calculate the derivative of the following function. y=(secx+tanx) 12
dx
dy
=12(sec(x)+tan(x)) 11
(sec(x)tan(x)+sec 2
(x)) a. Use implicit differentiation to find dx
dy
. b. Find the slope of the curve at the given point. 9xy=3;(1, 3
1
) a. dx
dy
=− x
y
b. The slope at (1, 3
1
) is − 3
1
.
Answers
Given function:y = (sec x + tan x)12
On taking the derivative of y with respect to x, we get:dy/dx = d/dx(sec x + tan x)12
Use chain rule,dy/dx
= 12(sec x + tan x)11(d/dx(sec x + tan x)
Now, d/dx(sec x + tan x) can be written asd/dx(sec x + sin x/cos x)
= sec x tan x + sec² xWe get,dy/dx
= 12(sec x + tan x)11 (sec x tan x + sec² x)
Therefore, dy/dx = 12(sec x tan x + sec² x)(sec x + tan x)11
Now, let us solve for part a) of the question using implicit differentiation:9xy = 3
Taking the derivative of both sides with respect to x, we get:9(d/dx)(xy)
= (d/dx)3
Taking the derivative of xy with respect to x, we get:xy + x(dy/dx)y = 0
On substituting this value in the above equation, we get:9(xy + x(dy/dx)y)
= 0dy/dx
= -(9xy)/(xy)
= -9/1
= -9
Thus, the value of dx/dy is -9.
Now, let us solve for part b) of the question:9xy = 3
Let us first differentiate this expression with respect to x:9(d/dx)(xy)
= (d/dx)3xy + x(dy/dx)y
= 3y + x(dy/dx)Y
Substituting the value of dy/dx, we get:xy = 1
Solving these two equations for x = 1 and
y = 3,
we get:xy = 3
Now, the slope of the curve at point (1, 31) is given by:dy/dx = - x/y
= -1/3
Thus, the slope of the curve at point (1, 31) is -31.
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Answer this for 20 POINTS
Answers
Answer: C
Step-by-step explanation: im sorry if its wrong creati deleted all my points and im at -92 points
In March, Delphine's house had 40\%40%40, percent more snowfall than in February. Delphine's house had fff centimeters of snowfall in February.
Which of the following expressions could represent how much snowfall Delphine had at her house in March?
Choose 2 answers:
Choose 2 answers:
(Choice A)
A
40f40f40, f
(Choice B)
B
40+f40+f40, plus, f
(Choice C)
C
1.4f1.4f1, point, 4, f
(Choice D)
D
40f+f40f+f40, f, plus, f
(Choice E)
E
\dfrac4{10}f+f
10
4
f+f
Answers
2.4 f of snow would fall in March.
How to find centimeters of snowfall in February?Given that
Delphine's home received 40% 40% 40% 40% more snowfall in March compared to February.
now find the Delphine's house had fff centimeters of snowfall in February.
Let f represent the volume of snow that fell in February.
If there is 40% more snowfall in February than there was in March,
Consequently, the snowfall would be
\(f+( \frac{100+40}{100} )f\)
\(f+( \frac{140}{100} )f\)
f + 1.4f
2.4f
Therefore, 2.4f of snow would fall in march.
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HURRY I WILL GIVE U BRAINLYIST Renaissance artists learned about depth and ______, which they incorporated in their works of art.
perspective
shading
color
shapes
Answers
Answer:
persepctive
Step-by-step explanation:
have a great day<3
Solve.
10⁵
pls help meh
Answers
Answer:
100000
Step-by-step explanation:
10 x 10 x 10 x 10 x 10
if there is 1$ in taxes on 25$ worth of clothing, what percent is paid for the 25$ in clothing?
Answers
Answer:
Well it would be 4 percent
Step-by-step explanation:
Thinm of it this way, if you mutliply 25 times 4 it equals 100 then mulitply then 1 dollar by 4 getting 4 dollars worth of tax making it 4 percent of the total ckst
what is the equation of the major axis of y=1/x
Answers
The equation of the major axis of y = 1 / x would be y = x and y = -x.
How to find the equation ?The equation y = 1/x constitutes a rectangular hyperbola. Not like ellipses possessing broadly characterized principal axes, no finite line exists representing the major axis of the given hyperbola.
This is due to the symmetrical relationship around both x- and y-axes where its core remains positioned at (0, 0). In place of a definite line, the asymptotes operate in replaceable fashion, defined as the lines y = x and y = -x, which are values approached at near limit by this specific hyperbola yet never collided with.
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find the slope of this graph
Answers
Answer:
-3
Step-by-step explanation:
If I have $100 dollars and spend $99.99 dollars, how much money do I have left?
Answers
Answer:
You have $00.01
Step-by-step explanation:
100-99.99=0.01
Answer:
A cent.
Step-by-step explanation:
100 - 99.99 = 0.01
You would have $0.01, or a cent, left.
Hope this helps.
Karl spent $30 decorating for the annual holiday party. Each, x, table cover cost $2
and each, y, center piece cost $6. Karl bought 7 items total.
what are the first and second equation?
Answers
The two pair of linear equations are 2x+6y=30 and x+y=7.
What is a linear system of equations?A system of linear equations consists of two or more equations made up of two or more variables such that all equations in the system are considered simultaneously. The solution to a system of linear equations in two variables is any ordered pair that satisfies each equation independently.
Each, x, table cover cost $2 and each, y, center piece cost $6.
Karl spent $30 decorating for the annual holiday party.
So, the equation is 2x+6y=30 --------(I)
Karl bought 7 items total.
Now, x+y=7 --------(II)
From equation (II), x=7-y
Substitute, x=7-y in equation (I), we get
2(7-y)+6y=30
14-2y+6y=30
14+4y=30
4y=16
y=4
Substitute y=4 in equation (II), we get
x=3
Therefore, the two pair of linear equations are 2x+6y=30 and x+y=7.
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a store is having a 20% off sale on all merchandise. if mai buys one item and saves $13, what was the original price of her purchase
Answers
Will mark Brianliest please answer :)
Answers
Answer:
3.5×10^4=35000
Step-by-step explanation:
Consider the value
3.5
we are saying that we are counting in units (1's) in that we have
3
1
2
of 1's
'.....................................................................................
Consider the value
3.5
×
10
. Now we are saying we have
3
1
2
of tens. That is; 3 lots of 10 is 30 and
1
2
of 10 is 5 so we have
30
+
5
=
35
'.................................................................................
Note that
10
2
is 100
Consider the value
3.5
×
10
2
. Now we are saying we have
3
1
2
of
100
'
s
So if we are counting in 100's then the decimal point stays where
it is and we move the number of 3.5 to the left until the
3 becomes hundreds.
Note that we have to insert a place holder of 0 just to the left of the decimal point to make sure the three is viewed as in the hundreds.
3.5
×
10
2
−−−−−−−−−−−−−−−−−−→
slide to the left 2 places
350.0
1. Wesam buys an iPod for $120 and gets consumer surplus of $80.
a. What is her willingness to pay?
b. If she had bought the iPod on sale for $90, what would her consumer
surplus have been?
c. If the price of an iPod were $250, what would her consumer surplus
have been?
Answers
Consumer surplus is a valuable tool for measuring the satisfaction that consumers derive from a product. It helps to understand how much value consumers place on a product and how much they are willing to pay for it.
Consumer surplus is a measure of the difference between the maximum amount that a consumer is willing to pay for a product and the actual price paid. In this scenario, Wesam buys an iPod for $120 and gets a consumer surplus of $80. Therefore, her willingness to pay for the iPod would be $200 ($120 + $80).
If Wesam had bought the iPod on sale for $90, her consumer surplus would have been $110 ($200 - $90). This is because she would have paid less for the product, but her willingness to pay would remain the same at $200.
If the price of an iPod were $250, Wesam's consumer surplus would be zero. This is because her willingness to pay for the iPod is $200, and if the price of the iPod is higher than that, she would not purchase the product. Therefore, she would have no consumer surplus.
In summary, consumer surplus is a valuable tool for measuring the satisfaction that consumers derive from a product. It helps to understand how much value consumers place on a product and how much they are willing to pay for it.
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given eigenvalues, eigenvectors for matrix A are λ1 = 2, v1 = [-5 1]t and λ2 = 6, v2 = [5 4]t
(t means transpose ... means the row vector is actually a column vector)
Defn of Eigenvector, Eigenvalue:
Avi = λivi for all i, in this case, i = 1, 2
Answers
The eigenvectors v1 and v2 are such that when multiplied by matrix A, they result in scalar multiples of themselves (scaled by their respective eigenvalues).
An eigenvector is a non-zero vector that, when multiplied by a matrix, results in a scalar multiple of itself. In other words, if A is a square matrix and v is an eigenvector of A, then Av = λv, where λ is the eigenvalue associated with v.
Given the eigenvalues and eigenvectors for matrix A as λ1 = 2, v1 = [-5 1]ᵀ and λ2 = 6, v2 = [5 4]ᵀ, we can say that:
- Av1 = λ1v1 = 2[-5 1]ᵀ = [-10 2]ᵀ
- Av2 = λ2v2 = 6[5 4]ᵀ = [30 24]ᵀ
Therefore, the eigenvectors v1 and v2 are such that when multiplied by matrix A, they result in scalar multiples of themselves (scaled by their respective eigenvalues).
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I am politely asking for some assistance with, thy questions in quantities above 1. My very brain inside of my cranium is too little to answer thy questions. I am requesting some assistance.
Answers
Answer:
First one, true
Second one, 23/45
Third one, 11/29
Step-by-step explanation:
There's thy answer for thy self t-t
Answer:
1.)true
2.) 23/45
3.)11/29
Step-by-step explanation:
-3x (- -8x+52+:
+5+3)
A. 11x²8x-9
11x³8x² - 9x
B.
C. 24x³15x² - 9x
D.
24x²15x - 9
Answers
Answer:
To simplify the expression -3x(-8x+52+5+3), we can distribute the -3x to each term inside the parentheses:
-3x(-8x+52+5+3) = 24x² - 156x - 15x - 9x
Simplifying further by combining like terms, we get:
-3x(-8x+52+5+3) = 24x² - 180x - 9x
Therefore, the simplified expression is 24x² - 189x. None of the options given match this answer. Therefore, there seems to be an error in the original question.
Step-by-step explanation: