This looks like a system of differential equations.
[tex]\begin{cases}x' + 2y' + x = 0 \\ x' - y' + y = 0 \\x(0)=0, y(0)=1\end{cases}[/tex]
Eliminating x' gives
[tex](x'+2y'+x)-(x'-y'+y) = 0 - 0 \implies 3y' - y + x = 0[/tex]
and eliminating y' gives
[tex](x'+2y'+x) + 2(x'-y'+y)=0+2\times0 \implies 3x' + x + 2y = 0[/tex]
so that we can rewrite the system as
[tex]\begin{cases}3x' = -x - 2y \\ 3y' = -x + y\end{cases}[/tex]
or equivalently in matrix form as
[tex]\begin{bmatrix}x\\y\end{bmatrix}' = \dfrac13 \begin{bmatrix}-1&-2\\-1&1\end{bmatrix} \begin{bmatrix}x\\y\end{bmatrix}[/tex]
Compute the eigenvalues for the coefficient matrix:
[tex]\det\begin{bmatrix}-1-\lambda&-2\\-1&1-\lambda\end{bmatrix} = (-1-\lambda)(1-\lambda) - 2 = \lambda^2 -3 = 0 \implies \lambda=\pm\sqrt3[/tex]
Compute the corresponding eigenvectors:
[tex]\lambda=\sqrt3 \implies \begin{bmatrix}-1-\sqrt3&-2\\-1&1-\sqrt3\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix} \\\\ \implies v_1=(1-\sqrt3)v_2 \implies \vec v = \begin{bmatrix}1-\sqrt3\\1\end{bmatrix}[/tex]
[tex]\lambda=-\sqrt3 \implies \begin{bmatrix}-1+\sqrt3&-2\\-1&1+\sqrt3\end{bmatrix}\begin{bmatrix}v_1\\v_2\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix} \\\\ \implies v_1 = (1+\sqrt3)v_2 \implies \vec v = \begin{bmatrix}1+\sqrt3\\1\end{bmatrix}[/tex]
We end up multiplying the matrix by 1/3, so the eigenvalues also get scaled by 1/3 and λ = ±1/√3. The eigenvectors stay the same.
Then the characteristic solution to the system is
[tex]\begin{bmatrix}x\\y\end{bmatrix} = C_1 e^{t/\sqrt3} \begin{bmatrix}\frac{1-\sqrt3}3\\\frac13\end{bmatrix} + C_2 e^{-t/\sqrt3} \begin{bmatrix}\frac{1+\sqrt3}3\\\frac13\end{bmatrix}[/tex]
Use the initial conditions to solve for the constants.
[tex]\begin{bmatrix}0\\1\end{bmatrix} = C_1 \begin{bmatrix}\frac{1-\sqrt3}3\\\frac13\end{bmatrix} + C_2 \begin{bmatrix}\frac{1+\sqrt3}3\\\frac13\end{bmatrix}[/tex]
[tex]\implies \begin{cases}(1-\sqrt3) C_1 + (1+\sqrt3) C_2 = 0 \\ C_1 + C_2 = 3\end{cases}[/tex]
[tex]\implies C_1=\dfrac{3+\sqrt3}2, C_2=\dfrac{3-\sqrt3}2[/tex]
Then the particular solution is
[tex]\begin{bmatrix}x\\y\end{bmatrix} = \dfrac{3+\sqrt3}6 e^{t/\sqrt3} \begin{bmatrix}1-\sqrt3\\1\end{bmatrix} + \dfrac{3-\sqrt3}6 e^{-t/\sqrt3} \begin{bmatrix}1+\sqrt3\\1\end{bmatrix}[/tex]
or
[tex]\begin{cases}x(t) = -\dfrac1{\sqrt3} e^{\sqrt3 \, t} + \dfrac1{\sqrt3} e^{-\sqrt3\,t} \\\\ y(t) = \dfrac{3+\sqrt3}6 e^{\sqrt3\,t} + \dfrac{3-\sqrt3}6 e^{-\sqrt3\,t}\end{cases}[/tex]
a chair costs rupees 250 and a table costs rupees 400 . If housewife purchased a certain number of chair and two tables fo rupees 2800,find the number of chairs she purchased
Answer:
The housewife purchased 8 chairs
The slope is 3 and the line passes through point (5, 17).
Answer:
[tex]\sf y=3x+2[/tex]
Step-by-step explanation:
Point-slope form of linear equation: [tex]\sf y-y_1=m(x-x_1)[/tex]
(where m is the slope and (x₁, y₁) is a point on the line)
Given:
slope (m) = 3point (x₁, y₁) = (5, 17)Substituting given values into the equation to find the equation of the line:
[tex]\implies \sf y-17=3(x-5)[/tex]
[tex]\implies \sf y-17=3x-15[/tex]
[tex]\implies \sf y=3x+2[/tex]
What are the solutions of this quadratic equation? 9x^2+10=5
Answer:
No solution
Step-by-step explanation:
9x² + 10 = 5
⇔ 9x² = 5 - 10
⇔ 9x² = -5
This is impossible because 9x² > 0 and -5 < 0
A rectangle has sides that are 7 and 2/5 inches wide and 11 and 2/3 inches long. What is the perimeter of the rectangle?
Answer:
38.1333333334
Step-by-step explanation:
Perimeter of rectangle:
2 × (length + width) inches
2 × (7 2/5 + 11 2/3) inches
2 × 19.0666666667 inches
38.1333333334
PLEASE HELP PLEASE PLEASE
Answer:
Sublimation - is the transformation from a solid to gas.
Hope this helps!
Which of the following is an arithmetic sequence?
O A. -2,3, 8, 13, 18,...
O B. 1,1, 2, 3, 5, 8, ...
O C. 4,-4, 4, -4, 4, ...
O D. 1, 1/2,1/4,1/16
ANSWERS THESEEESS!!!!
read below thanks :D
Answer:
(d) 112 m²
Step-by-step explanation:
The area of a parallelogram is given by the formla ...
A = bh . . . . . . where b is the base length and h is the height
__
Here, the base is divided into parts, but their total length is 5m+9m = 14m. The height is given as 8m, so the area is ...
A = (14 m)(8 m) = 112 m²
The area of the parallelogram is 112 square meters.
__
Alternate solution
You can also figure this as the sum of the areas of the two triangles and that of the center rectangle.
A = 2(1/2bh) +bh
A = 2(1/2(5 m)(8 m)) +(9 m)(8 m) = 40 m² +72 m² = 112 m²
A repair company charges a one-time service fee and an hourly rate for the time needed to complete each job. The
table shows the amounts the company used for these charges over four consecutive years.
Charges for Repair Jobs
Year
12 3 4
Service Fee $20 $25 $30 $35
Hourly Rate $52$55 $58 $61
Part A
Using the data in the table, what would the company's service fee and hourly rate, in dollars, be in year 8? Enter
your answers in the first and second response boxes.
Part B
What total amount, in dollars, would the company charge for a job that requires 3 hours of work in year 9? Enter
your answer in the third response box.
Part A: Service Fee
Part 1: Hourly Rate
Answer:
A: Service Fee = $55
Hourly Rate = $ 73
B: $288
Step-by-step explanation:
A: the service rate increases by $5 per year, so year 8 would be $20 more than year 4 (5*4)
The hourly rate increases by $3 per year, so year 8 would be $12 more than year 4 (3*4)
B. Service fee for year 9 would be $60 ($5 more than year 8) and the hourly rate would be $76 per hour ($3 more than year 8). This gives you Service Fee + (Hourly rate * 3) = $60 + ($73*3) = $60 + $228 = $288
Circle the exponent
need answer asap thank you
Answer:
56
Step-by-step explanation:
First of all, if the width of the frame is 2, then add 2 to both sides of 7 to get the frame length7 + 2 + 2 = 11Do the same for 33 + 2 + 2 = 7Area of frame and white rectangle combined = 11 × 7 = 77Area of white rectangle is 7 × 3 = 21Now to get the area of the blue bit, subtract 21 from 77This gives you 56I WILL GIVE YOU 20 POINTS TO THOSE WHO ANSWER THIS QUESTION RIGHT NOOOO SCAMS AND EXPLAIN WHY THAT IS THE ANSWER
please help!
Matrices
[tex]\mathfrak{\huge{\orange{\underline{\underline{AnSwEr:-}}}}}[/tex]
Actually Welcome to the concept of Scalar factor Multiplication in Matrix.
So from the above matrix, -3A , is just simply multiplying the -3 to every element of the matrix, so we get as,
===> -3A = | -9 -18 |
| -24 -21 |
| -6 6 |
hence the correct option is B.
33
The base of a right rectangular prism has
an area of 172.5 square centimeters and
a height of 8 centimeters. What is the
volume, in cubic centimeters, of the
right rectangular prism?
A 180
B 280
C 1,280
D 1,380
3
Answer:
D 1,380
Step-by-step explanation:
V = area of base × height
V = 172.5 cm² × 8 cm
V = 1380 cm³
Write the answer in simplest form
Answer:
57/40
Step-by-step explanation:
So we need to find the LCD, or the least common denominator. The LCD is 40. Now, we do a transition to find the new numerator.
[tex]\frac{4}{5} =\frac{x}{40} \\\\\\frac{5}{8} = \frac{x}{40}[/tex]
The new equation is this:
[tex]\frac{32}{40} +\frac{25}{40}[/tex]
Now we use our common sense. 32+25=57
57/40.
You can't simplify this...
Find the amount of tax and the tax rate. Round to nearest hundredth of a percent.
Cost of item: $76
Selling price: $89.83
since the selling price is 89.83 and the cost of is 76, the tax amount is 89.83 - 76 = 13.83.
if we take 76 to be the 100%, what is 13.83 off of it in percentage?
[tex]\begin{array}{ccll} amount&\%\\ \cline{1-2} 76 & 100\\ 13.83& x \end{array} \implies \cfrac{76}{13.83}=\cfrac{100}{x} \\\\\\ 76x=1383\implies x=\cfrac{1383}{76}\implies x\approx 18.20[/tex]
help with geometry pleaseeee will mark the brainiest!
Answer:
384.29 in²
Step-by-step explanation:
The radius of these circles is 5 in.
- As the radii are equal, and 4 of them are used to make up the side of the square. [20 ÷ 4 = 5]
To find the area of circle, you use the formula [tex]\pi r^{2}[/tex]
[tex]\pi[/tex] · (5)² = 25[tex]\pi[/tex]
The area of the square equals the length x width
20 · 20 = 400
To find the area of the shaded region, subtract these values.
400 - 25[tex]\pi[/tex] ≈ 389.29 in²
Which polygon has two sets of parallel sides?
Answer:
C
Step-by-step explanation:
C is rectangle
Rectangles have 2 sets of parallel lines
Identify the terms and like terms in the question 3n+7-n-3
Answer:
like terms: (3n,-n),(7,-3)
Step-by-step explanation:
Answer:=2n+2
Step-by-step explanation:
3n+7-n-3=2n+2
1. a movie Theater sold 74 tickets of these tickets. 32 Where tickets for children, and the rest where tickets for adults.
which equation should be used to determine the number of tickets that were for adults.
A. 32+a=14
B. 32a=74
C. 74a=32
D. 32-a=74
Answer:
Its d
Step-by-step explanation:
why did you delete my answer??
Huuuurrry plllzzzzzz. Determine the equation of the line that passes through the given points (2,6) and (4,16)
Linear equations are typically organized in slope-intercept form:
[tex]y=mx+b[/tex]
m is the slope of the lineb is the y-intercept (the value of y when the line passes through the y-axis)To find linear equations in slope-intercept form:
Determine the slopePlug the slope into the general formDetermine the y-intercept by isolating bPlug the b back into the equationSolving the QuestionWe're given:
The line passes through the points (2,6) and (4,16)First, determine the slope of the line.
[tex]m=\dfrac{y_2-y_1}{x_2-x_1}[/tex] where [tex](x_1,y_1)[/tex] and [tex](x_2,y_2)[/tex] are points that fall on the line
⇒ Plug in the given points (2,6) and (4,16):
[tex]m=\dfrac{16-6}{4-2}\\\\m=\dfrac{10}{2}\\\\m=5[/tex]
⇒ Therefore, the slope of the line is 5. Plug this back into the general form:
[tex]y=5x+b[/tex]
Now, determine the y-intercept.
[tex]y=5x+b[/tex]
⇒ Plug in one of the given points:
[tex]6=5(2)+b\\6=10+b\\b=-4[/tex]
⇒ Therefore, the y-intercept is -4. Plug this back into our original equation:
[tex]y=5x-4[/tex]
Answer[tex]y=5x-4[/tex]
Mary swam 3/4 mile each day for 8 days
Step-by-step explanation:
I'm assuming the question is how many miles in total did mary swim.
if this is the case then we just multiply the number of days by the number of miles per day.
Hence,
3/4 × 8 = 6
so 6 miles in total.
make sure to ask if you need any further help
You ask four people to the prom. If X is the number of "yes" responses, what is E(X)? Assume equal probability for
"yes" and "no" response.
Answer:
e is the amount of people
Step-by-step explanation:
4 x so let’s say you have two x you have 50% of the people that said yes.
if you were to flip the coin 15, how many times will the coin land on tails based on experimental probability? jordan flipped a count of 45 times.His result are below: heads: 24 tails: 21
Answer:
It will land 7.5 times on tails, which is rounded up to 8 times
Step-by-step explanation:
there are 2 possibilities, heads or tails
so you divide 15 in 2 and get 7.5
The coin will land 7 times tail based on the experimental probability.
What is Probability?Probability refers to potential. A random event's occurrence is the subject of this area of mathematics.
The range of the value is 0 to 1. Mathematics has incorporated probability to forecast the likelihood of various events.
The degree to which something is likely to happen is basically what probability means.
Given:
When the coin tossed 45 times.
Head= 24, tail = 21
So, fraction of appearing tail will be
= 21 / 45
= 0.466667
Now, if the coin tossed 15 times then the coin land on tails
= 15 x 0.466667
= 7
Learn more about Probability here:
brainly.com/question/11234923
#SPJ2
what is the sum of 9+10?
Answer:
19.
Step-by-step explanation:
i need help on this 6th grade math question:
its for 20 points
Answer:
Quadrant I.Step-by-step explanation:
Quadrant I examples:
(2, 5)(3, 4)(7, 1)Quadrant II examples:
(-1, 5)(-6, 2)(-4, 7)Quadrant III examples:
(-3, -4)(-7, -1)(-8, -5)Quadrant IV examples:
(5, -6)(1, -4)(8, -2)
8. When solving the equation (2 - x)(x + 1) = 11, Priya graphs y=(2 - x)(x + 1) - 11 and then looks to find where the graph crosses the x-axis.
Tyler looks at her work and says that graphing is unnecessary and Priya can set up the equations 2 - x = 11 and x + 1 = 11, the solutions
are x = -9 or x = 10.
A) Do you agree with Tyler?
yes
no
Answer:
No
Step-by-step explanation:
Tyler is incorrect. The only way you can set individual terms equal to the result is when the result is zero. This is due to zero times anything equalling zero. If one of the terms equals zero, then the other term does not matter because the whole thing will be zero. Let's say:
[tex](2 - x)(x + 1) = 0[/tex]
If the first term equals zero, the entire left side is zero. For Priya's case, the right side is not zero, it is 11. If one term is 11, the other term will contribute. You can check this by plugging in Tyler's solutions. The correct way to solve this equation is to use FOIL on the two terms, then subtract 11 on both sides to get zero. Then use the quadratic formula.
Identify the lateral area and surface area of a regular square pyramid with base edge length 11 cm and slant height 15 cm, rounded to the nearest tenth.
The surface area of the square pyramid is: 451 cm²
The lateral surface area of the square pyramid = 330 cm²
What is the Lateral Surface Area and Surface Area of a Square Pyramid?Surface area = a² + 2al, where a is the base edge and l is the slant height.
Lateral Surface Area = 2al.
Given the following:
a = 11 cml = 15 cmSurface area = a² + 2al = 11² + 2(11)(15)
Surface area = 451 cm²
Lateral Surface Area = 2al = 2(11)(15)
Lateral Surface Area = 330 cm²
Learn more about square pyramid on:
https://brainly.com/question/3688277
The lateral and surface area of a square base pyramid with side length of 11 cm and slant height of 15 cm are 330 cm² and 451 cm².
Surface area of a pyramidsurface area = A + 1 / 2 ps
where
A = area of the basep = perimeter of the base s = slant heightTherefore,
surface area = 11² + 1 / 2 × (11 × 4) × 15
surface area = 121 + 1 / 2 × 44 × 15
surface area = 121 + 330
surface area = 451 cm²
Lateral area of a square pyramidlateral area = 1 / 2 ps
lateral area = 1 / 2 × 44 × 15
lateral area = 330 cm²
learn more on lateral and surface area here; https://brainly.com/question/23857644
I WILL GIVE 20 POINTS TO THOSE WHO ANSWER THIS QUESTION RIGHT NOOOO SCAMS PLEASE AND PLEASE EXPLAIN WHY THAT IS THE ANSWER
Answer:
x = 108 , y = 72
Step-by-step explanation:
x and 108 are alternate exterior angles and are congruent , then
x = 108
x and y are same- side exterior angles and sum to 180° , that is
x + y = 180
108 + y = 180 ( subtract 108 from both sides )
y = 72
9+ – 12(34) + 27 = 0, A= 3*
Use substitution
Answer:
9+ – 12(34) + 27 = 0, A= 3*
Use substitutionStep-by-step explanation: