The strength of an electrical current x flowing through the electric circuit shown is expressed as a function of time t and satisfies the following differential equation:
[tex]\displaystyle \large{L \frac{dx}{dt} + Rx = V}[/tex]
Find the strength of the electrical current x after switch S is closed at time t = 0. Assume that L, R and V are positive constants, and also that x = 0 when t = 0. Then, find [tex]\displaystyle \large{ \lim_{t \to \infty} x}[/tex]
Topic: Application of Differential Equation Reviews
Answer:
The current of the circuit at t = 0 is equal to 0.
If we take the limit as t approaches infinity, the current is equal to ε/R or V/R.
General Formulas and Concepts:
Calculus
Differentiation
DerivativesDerivative NotationDerivative Property [Multiplied Constant]:
[tex]\displaystyle (cu)' = cu'[/tex]
Derivative Property [Addition/Subtraction]:
[tex]\displaystyle (u + v)' = u' + v'[/tex]
Derivative Rule [Basic Power Rule]:
f(x) = cxⁿf’(x) = c·nxⁿ⁻¹Slope Fields
Separation of VariablesIntegration
IntegralsIntegration Rule [Reverse Power Rule]:
[tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Rule [Fundamental Theorem of Calculus 1]:
[tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]
Integration Property [Multiplied Constant]:
[tex]\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx[/tex]
Integration Method: U-Substitution
Electricity
Ohm's Law: V = IR
V is voltage (in Volts)I is current (in Amps)R is resistance (in Ohms)Circuits
Circuit SymbolsKirchhoff's Laws (Loop and Junction Rule)InductorsStep-by-step explanation:
*Note:
In the given equation, our variable of differentiation is x. I will rewrite this as current I for physics notation purposes.
Step 1: Define
Identify given.
[tex]\displaystyle L \frac{dI}{dt} + RI = V[/tex]
[Assuming switch S is closed] Recall that an inductor is used in a circuit to resist change. After a long period of time, when it hits steady-state equilibrium, we expect to see the inductor act like a wire.
Step 2: Find Current Expression Pt. 1
[Kirchhoff's Law] Rewrite expression:Step 3: Find Current Expression Pt. 2
Identify variables for u-substitution.
Set u:Step 4: Find Current Expression Pt. 3
[Kirchhoff's Law] Apply U-Substitution:Recall that our initial condition is when t = 0, denoted as u₀, and we go to whatever position u we are trying to find. Also recall that time t always ranges from t = 0 (time can't be negative) and to whatever t we are trying to find.
[Kirchhoff's Law] Integrate both sides:Recall that our initial condition u₀ (derived from Ohm's Law) contains only the voltage across resistor R, where voltage is supplied by the given battery. This is because the current is stopped once it reaches the inductor in the circuit since it resists change.
Back-Substitute in u and u₀:Step 5: Solve
If we are trying to find the strength of the electrical current I at t = 0, we simply substitute t = 0 into our current function:
[tex]\displaystyle\begin{aligned}I(t) & = \frac{\mathcal E}{R} - \frac{\mathcal E}{R} e^{- \frac{R}{L}t} \\I(0) & = \frac{\mathcal E}{R} - \frac{\mathcal E}{R} e^{- \frac{R}{L}(0)} \\& = \boxed{\bold{0}}\end{aligned}[/tex]
If we are taking the limit as t approaches infinity of the current function I(t), we are simply just trying to find the current after a long period of time, which then would just be steady-state equilibrium:
[tex]\displaystyle\begin{aligned}I(t) & = \frac{\mathcal E}{R} - \frac{\mathcal E}{R} e^{- \frac{R}{L}t} \\\lim_{t \to \infty} I(t) & = \frac{\mathcal E}{R} - \frac{\mathcal E}{R} e^{- \frac{R}{L}(\infty)} \\& = \boxed{\bold{\frac{\mathcal E}{R}}}\end{aligned}[/tex]
∴ we have found the current I at t = 0 and the current I after a long period of time and proved that an inductor resists current running through it in the beginning and acts like a wire when in electrical equilibrium.
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Topic: AP Physics C - EMAG
Unit: Induction
What was one of the main causes of the Great Depression?
a. The overuse of credit
b. The under-utilization of money
c. WWII
d. Excessive government borrowing
Answer:
B
Step-by-step explanation:
What are the solutions of 12 - x2 =0?
O x=2/3 and x = -2/3
O x= 3, and x = -373
O x= 4,3 and x = -4/3
O x=6 and x = -6
A radioactive element decays at a rate of 5.25% annually. There are 35 grams of the substance
present. How much of the substance remains after 12 years (Round to the nearest hundredth)?
TEKS A2.5(B)(D)
Answer:
18.32 gm
Step-by-step explanation:
= 35 ( 1-.0525)^12 = 18.32 gm remain
If x = 18, evaluate the following expression:
12 + 3
Help!! Khan academy geomtry practice
Answer:
33.51 units³
Step-by-step explanation:
Volume of a sphere is given by the formula :
V = [tex]\frac{4}{3} \pi r^{3}[/tex]
Because we have the radius we can plug it into the equation to give us :
V = (4/3)×π×(2)³
V = 4/3×8π
V ≈ 33.51
Units will be units³ since no measurement is given but it to the power of 3 since it is volume.
In a group, more than 1/2 are boys, but they are less than 2/3 of the group. Can there be:(In each case, if your answer is “yes”, find out how many boys there were. Explore all possible cases). Could there be 5 kids
Using proportions, it is found that Yes, there could be 5 kids, with 3 boys and 2 girls.
What is a proportion?A proportion is a fraction of a total amount.
In this problem, supposing a total of 5 kids:
More than 1/2 = 50% are boys, hence there are more than 2.5 boys, as 0.5 x 5 = 2.5.Less than 2/3 = 66.67% are boys, hence there are less than 3.33 boys, as 0.6667 x 5 = 3.33.Hence, Yes, there could be 5 kids, with 3 boys and 2 girls.
More can be learned about proportions at https://brainly.com/question/24372153
Factor each polynomial. 20 points
5a^2b^2 + 10ab +25a
8m^2n^2 - 24mn^3 +16mn
9xz^3 + 18yz^2 + 24z^2
16x^6y + 16x^2y^4 + 32x^3y^2
So, here we are to factor the polynomials :
[tex]\\ 1. \sf \qquad5a^2b^2 + 10ab +25a\\ \\[/tex]
[tex] \longrightarrow\sf \qquad5a(ab^2 + 2b +5)\\ \\[/tex]
[tex]\\ 2. \sf \qquad 8m^2n^2 - 24mn^3 +16mn\\ \\[/tex]
[tex] \longrightarrow\sf \qquad 8mn(mn - 3n^2 +2)\\ \\[/tex]
[tex]\\ 3. \sf \qquad 9xz^3 + 18yz^2 + 24z^2\\ \\[/tex]
[tex]\longrightarrow \sf \qquad 3z(3xz^2 + 6yz+ 8z)\\ \\[/tex]
[tex]\\ 4. \sf \qquad 16x^6y + 16x^2y^4 + 32x^3y^2\\ \\[/tex]
[tex] \longrightarrow \sf \qquad 16 {x}^{2} y(x^4 + y^3 + 2xy)\\ \\[/tex]
Some points to know :
Factorisation is the reverse process of multiplucation.The Factorisation is the process of finding two or more expressions such that their product is the given expression.*URGENT*
If you were at the Market Square and you wanted to get to the Doctor’s Office, according to the map, how would you get there? (Write your answer according to units.)
Answer: (2,-5) X:2. Y:-5
Move accross 2 units to the right and then 5 units down from 2
show how to rewrite 1/5 with the denominator of 10
Answer:
2/10
Step-by-step explanation:
Since the denominator is 10, we must find what the numerator is so that the fraction can be equal to 1/5. Since 10 is twice of 5, we multiply 2 by 2 as well. With that, we get a final answer of 2/10.
please help me, thank you if you do
Answer:
Step-by-step explanation:
1. The sequence is not arithmetic.
2. d=-30
dont mind my answers thanks for helping me out have a good day
Answer:
ACD and DCB are supplementary and others are correctly chosen ...
A baseball field in the shape of a square is shown. the corners are labeled home plate, first base, second base, and third base. the distance between home plate and first base is 90 feet. a baseball field is in the shape of a square. the distance between each pair of bases along the edge of the square is 90 feet. what is the distance between home plate and second base? startroot 2 endroot feet
Answer:
90√2 feet
Step-by-step explanation:
The geometry of the problem can be modeled by a right triangle with sides 90 feet. The length of the hypotenuse is the distance from home plate to 2nd base. The Pythagorean theorem tells you that distance (c) is ...
c² = a² +b²
c² = (90 ft)² +(90 ft)² = 8100 ft² +8100 ft²
c² = 16200 ft²
c = √16200 ft = 90√2 ft
The distance between home plate and second base is 90√2 feet.
Answer:
A baseball field is in the shape of a square. The distance between each pair of bases along the edge of the square is 90 feet. What is the distance between home plate and second base?
(90) StartRoot 2 EndRoot feet
Step-by-step explanation:
What is the equation of a line that passes through the hypotenuse of the right triangle in the graph?
A. 1/5x
B. y=5x
C. y=-5x
D. y=2/5x
Answer:
y = 5x
Step-by-step explanation:
Hey there!
To start with the hypotenuse of a triangle is the longest side of it
and to get an equation of a line you would use
y = mx + b
where m is the slope
and b is the y-intercept
to get the slope you would do rise over run
rise/run
take 2 points and see how much it rises and runs to the next point
you would then get 5
then we figure out the y-intercept where the line intercepts the y-axis
in this case 0
In conclusion the answer is y = 5x + 0
we can also remove the 0 because it is unnecessary
y = 5x
Help for the brainiest answer quickly please
Answer:
C
Step-by-step explanation:
A company makes concrete bricks that are shaped like rectangular prisms. Each brick is 5inches long, 4inches wide, and 2inches tall. How much concrete will they need to make 1600 bricks?
Answer:
The answer would be 64,000.
Step-by-step explanation:
To find the area of one concrete block you will multiple length x width x height. Using the measurements provided, one block will equal 40. You will then multiple this by 1,600 to equal 64,000.
f(3) if f(x) = 2x + 5 =
Answer:
11
Step-by-step explanation: Uh- if X=3 then it would be 2(3) +5. 2x3 is 6, and 6+5 is 11.
Answer:
11
Step-by-step explanation:
2(3) + 5
simplify
6 + 5 = 11
hope this helps
There are 100 students in a band. Forty percent are 12 years old, 1/2 are 13 years old, and the rest are 14 years old. Write the portion of the band for each age
Answer:
40x 12 year olds
50x 13 year olds
10x 14 year olds
Step-by-step explanation:
40% of 100 is 40
50% of 100 is 50
100%-90%=10%, 10% of 100 = 10
Students of 12 years are 40%, students of 13 years are 50% and students of 14 years are 10%.
What is percent?A percentage is a figure or ratio that can be stated as a fraction of 100. If we need to calculate the percentage of a number, we must divide it by its full value and then multiply it by 100. As a result, the percentage is one part in one hundred. "Per 100" is short for "per percent." It is represented by the symbol "%".
Given, total students in band = 100
40% are of 12 years
40% of 100 = 0.4 x 100 = 40
12 years student = 40 = 40%
1/2 of 13 years
1/2 of 100 = 1/2 x 100 = 50 =50%
13 years of student = 50
and rest are 14 years
100 - 50 - 40 = 10
14 years student = 10 = 10%
Hence 40% students are 12 years, 50% are of 13 years and 10% of students are of 14 years.
Learn more about percent;
https://brainly.com/question/28840349
#SPJ2
Find the perimeter of the parallelogram.
Will mark brainlyest!!
Answer:
58cm
Step-by-step explanation:
Answer:
58
Step-by-step explanation:
16 +16=32
13+13=26
32+26=58
Hope it Helps
4/9÷1/2 as a fraction (simplest form)
Answer:
8/9
Step-by-step explanation:
4/9 ÷ 1/2 = dividing by 1/2 is the same as multiplying by its reciprocal 2/1
4/9 × 2/1 = (4×2)/(9×1) multiply numerators (top numbers) and multiply denominators (bottom numbers)
The table shows the average bill totals for given numbers of diners at Simone’s restaurant. If the rate is constant and you graph this relationship with the number of people along the x-axis and bill amounts along the y-axis, which points will lie on the line you get?
Persons Bill Amount
1 ------------$40
2 $80
7 -------------$280
9 $360
11 $440
A.(12, 460
B.(20, 800)
C.(24, 1,020
D.(27, 1,080)
E.(28, 1,110)
HELPP
Answer:
(B) and (D)
Step-by-step explanation:
it's B and D because they are the only two points with constant rate
Find, from first principle the derivative of 1/x^2+1 with respect to x.
Answer:
d/dx = -2/x^3
Step-by-step explanation:
1/x^2 + 1
We can solve for the derivative using the power rule
d/dx = -2/x^3 + 0
d/dx = -2/x^3
I’ll give you brainliest for the first person with an answer
Answer:
1296mm^2
Step-by-step explanation:
Surface Area of the Rectangular Prism to the left on top of the box:
Area of square = lw, 9mm * 9mm = 81mm, then multiply by 2 because 2 of the squares are a part of the surface, giving 162mm^2
Area of a rectangle =lw, 9mm * 9mm = 81mm, then multiply by 2 again, because 2 rectangles are a part of the surface, giving 162mm^2
So, the total surface area of the rectangular prism to the left on top, is 324mm^2
Surface Area of the Triangular Prism:
Area of a triangle: 1/2bh, and our base length would be 12, because we have to subtract 9 from 21 since the base length of the triangle isn't stated. Anyways, A = 1/2bh, so A = 1/2(12)(9) = 54mm^2, but multiply by two, so we get 108mm^2.
Area of the rectangle: lw, so 15mm * 9mm = 135mm^2
So, the total surface area of the triangular prism is 243mm^2
Surface Area of the Rectangular Prism at the bottom:
Area of the long rectangles in front = lw, 21mm * 9mm = 189mm^2, multiply by 2, 378mm^2
Area of the rectangles to the side = lw, 9mm * 9mm = 81mm^2, multiply by 2, 162mm^2
Area of the rectangle at the very bottom = lw, 21mm * 9mm = 189mm^2
So, the total surface area of the rectangular prism at the bottom is 729mm^2
Add all the total surface areas of each shape to get the total surface area of the figure:
324mm^2 + 243mm^2 + 729mm^2 = 1296mm^2
The surface area of the figure above is 1296mm^2
Find the missing measurement for x.
Answer:
EASYYY 90°
Step-by-step explanation:
30 + 60 = 90
180-90=90
also
thats a right angle which is always 90°
ALSO
the angle opposite another angle is always the same measurement.
Two consecutive positive integers have a product of 156. Find the integers.
The integers are ___
So here this question is of concept linear equations. As in linear equations we assumes the unknown number as an variable after that an equation would be formed which we would solve and get the answer. Here also we would be doing same.!!
So let us assume the unknown positive integers as a and (a + 1).
★ Product of both integers :-
a (a + 1) = 156 [Equation formed]★ Solving the equation :-
[tex]: \longrightarrow \: \sf{a \: (a + 1) \: = \: 156} \\ \\ : \longrightarrow \: \sf{a \times (a + 1) \: = \: 156} \\ \\ : \longrightarrow \: \sf{a {}^{2} \: + \: a \: = \: 156} \\ \\ : \longrightarrow \: \sf{a {}^{2} \: + \: a \: - \: 156 = 0} \\ \\: \longrightarrow \: \sf{a {}^{2} \: - \: 12a + 13a\: - \: 156 = 0} \\ \\ : \longrightarrow \: \sf{a(a \: - \: 12) + 13(a\: - \: 12) = 0} \\ \\ : \longrightarrow \: \sf{a(a \: - \: 12) + 13(a\: - \: 12) = 0} \\ \\ : \longrightarrow \: \sf{(a \: - \: 12) (a\: + \: 13) = 0} \\ \\ : \longrightarrow \: \red{\bf{a \: = \: 12 }}[/tex]
Therefore,
First positive integer is 12★ Second positive integer :-
2nd positive integer = a + 1 2nd positive integer = 12 + 1 2nd positive integer = 13Use the regression equation to predict the sales of skin diving and scuba equipment in the year 2008
Answer:c
Step-by-step explanation:
What’s The Answer Of This? Please HELP FAST
Answer:
those are corrisponding angles which means they equal each other making <H and <G both 57 degrees
use an equation to find the value of k so that the line that passes through (1,3) and (5,k) has a slope of m=2.
[tex](\stackrel{x_1}{1}~,~\stackrel{y_1}{3})\qquad (\stackrel{x_2}{5}~,~\stackrel{y_2}{k}) ~\hfill \stackrel{slope}{m}\implies \cfrac{\stackrel{rise} {\stackrel{y_2}{k}-\stackrel{y1}{3}}}{\underset{run} {\underset{x_2}{5}-\underset{x_1}{1}}}~~ = ~~\stackrel{\stackrel{m}{\downarrow }}{2} \\\\\\ \cfrac{k-3}{4}=2\implies k-3=8\implies k=11[/tex]
Math Practices and Problem Sol
17. Number Sense Using three different
numerators, write an equation in which
three fractions, when added, have a
sum of 1.
Answer:
First choose a common denominator, here i picked 11 then we think of 3 diffrent numbers so that their sum is 11 such as 2+4+5=11 so a possible equasion would be
Step-by-step explanation:
MAn article is
bought and sold at a profit of one fourth of selling
price. Find the profit percent.
Answer:
25% profit
Step-by-step explanation:
Statement
An article is bought and sold at a profit of one fourth of selling price.
⇒ Buying price = x
⇒ Selling price = Buying price (1 + 1/4) = 5x/4
Profit %
Selling price - Buying price / Buying price(5x/4 - x) / x(x/4) / x⇒ 1/4⇒ 0.25⇒ 25% profit