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3 years ago

The owner of a small store buys coats for $55.00 each. Answer parts a and b.

Mathematics

1 answer:

kenny6666 [7]3 years ago

5 0

The percent of the purchased price to the sale price is 40%.

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Solve the following differential equation using using characteristic equation using Laplace Transform i. ii y" +y sin 2t, y(0) 2

kifflom [539]

Answer:

The solution of the differential equation is $y(t)= - \frac{1}{3} Sin(2t)+2 Cos(t)+\frac{5}{3} Sin(t)$

Step-by-step explanation:

The differential equation is given by: y" + y = Sin(2t)

<u>i) Using characteristic equation:</u>

The characteristic equation method assumes that y(t)= $e^{rt}$ , where "r" is a constant.

We find the solution of the homogeneus differential equation:

y" + y = 0

$y'=re^{rt}$

$y"=r^{2}e^{rt}$

$r^{2}e^{rt}+e^{rt}=0$

$(r^{2}+1)e^{rt}=0$

As $e^{rt}$ could never be zero, the term (r²+1) must be zero:

(r²+1)=0

r=±i

The solution of the homogeneus differential equation is:

$y(t)_{h}=c_{1}e^{it}+c_{2}e^{-it}$

Using Euler's formula:

$y(t)_{h}=c_{1}[Sin(t)+iCos(t)]+c_{2}[Sin(t)-iCos(t)]$

$y(t)_{h}=(c_{1}+c_{2})Sin(t)+(c_{1}-c_{2})iCos(t)$

$y(t)_{h}=C_{1}Sin(t)+C_{2}Cos(t)$

The particular solution of the differential equation is given by:

$y(t)_{p}=ASin(2t)+BCos(2t)$

$y'(t)_{p}=2ACos(2t)-2BSin(2t)$

$y''(t)_{p}=-4ASin(2t)-4BCos(2t)$

So we use these derivatives in the differential equation:

$-4ASin(2t)-4BCos(2t)+ASin(2t)+BCos(2t)=Sin(2t)$

$-3ASin(2t)-3BCos(2t)=Sin(2t)$

As there is not a term for Cos(2t), B is equal to 0.

So the value A=-1/3

The solution is the sum of the particular function and the homogeneous function:

$y(t)= - \frac{1}{3} Sin(2t) + C_{1} Sin(t) + C_{2} Cos(t)$

Using the initial conditions we can check that C1=5/3 and C2=2

<u>ii) Using Laplace Transform:</u>

To solve the differential equation we use the Laplace transformation in both members:

ℒ[y" + y]=ℒ[Sin(2t)]

ℒ[y"]+ℒ[y]=ℒ[Sin(2t)]

By using the Table of Laplace Transform we get:

ℒ[y"]=s²·ℒ[y]-s·y(0)-y'(0)=s²·Y(s) -2s-1

ℒ[y]=Y(s)

ℒ[Sin(2t)]= $\frac{2}{(s^{2}+4)}$

We replace the previous data in the equation:

s²·Y(s) -2s-1+Y(s) = $\frac{2}{(s^{2}+4)}$

(s²+1)·Y(s)-2s-1= $\frac{2}{(s^{2}+4)}$

(s²+1)·Y(s)= $\frac{2}{(s^{2}+4)}+2s+1=\frac{2+2s(s^{2}+4)+s^{2}+4}{(s^{2}+4)}$

Y(s)= $\frac{2+2s(s^{2}+4)+s^{2}+4}{(s^{2}+4)(s^{2}+1)}$

Y(s)= $\frac{2s^{3}+s^{2}+8s+6}{(s^{2}+4)(s^{2}+1)}$

Using partial franction method:

$\frac{2s^{3}+s^{2}+8s+6}{(s^{2}+4)(s^{2}+1)}=\frac{As+B}{s^{2}+4} +\frac{Cs+D}{s^{2}+1}$

2s^{3}+s^{2}+8s+6=(As+B)(s²+1)+(Cs+D)(s²+4)

2s^{3}+s^{2}+8s+6=s³(A+C)+s²(B+D)+s(A+4C)+(B+4D)

We solve the equation system:

A+C=2

B+D=1

A+4C=8

B+4D=6

The solutions are:

A=0 ; B= -2/3 ; C=2 ; D=5/3

So,

Y(s)= $\frac{-\frac{2}{3} }{s^{2}+4} +\frac{2s+\frac{5}{3} }{s^{2}+1}$

Y(s)= $-\frac{1}{3} \frac{2}{s^{2}+4} +2\frac{s }{s^{2}+1}+\frac{5}{3}\frac{1}{s^{2}+1}$

By using the inverse of the Laplace transform:

ℒ⁻¹[Y(s)]=ℒ⁻¹[ $-\frac{1}{3} \frac{2}{s^{2}+4}$ ]-ℒ⁻¹[ $2\frac{s }{s^{2}+1}$ ]+ℒ⁻¹[ $\frac{5}{3}\frac{1}{s^{2}+1}$ ]

$y(t)= - \frac{1}{3} Sin(2t)+2 Cos(t)+\frac{5}{3} Sin(t)$

3 0

3 years ago

A gallon of gas costs $4.34.Greg puts 12 gallons of gas in his car,He has a 50-dollar bill.Tell how much Greg will have left,or

Lerok [7]

Greg will need $2.08 more money to buy 12 gallons of gas.

<h3>Further explanation</h3>

A gallon of gas costs $4.34

Greg puts 12 gallon of gas

Greg has $50

First, we are going to calculate how much money does Greg needs for 12 gallon of gas:

= 12 gallon of gas x cost/gallon

= 12 x 4.34

= 52.08

To multiply the decimal it's just the same way as the whole number multiplication, except for the placement of the decimal point in the answer. For the example

4.34 ⇒ 2 decimal place

<u> 12 x</u> ⇒ 0 decimal place

868

52.08 ⇒ 2 decimal place

By doing the same process as the whole number multiplication we got the answer is 52.08, but don't forget to add the decimal point/decimal place. The first factor has 2 decimal place, the second factor has 0 decimal place, so the answer will have two decimal place (2+0 = 2), by adding the decimal place.

The next step , 12 gallon of gas cost $52.08, if he has $50 so it cost more than the money he has, We are going to do a subtraction:

$52.08

<u>$50.00 </u>- ⇒ <em>we put zero so we can the borrowing (regrouping)</em>

$ 2.08

To subtract a decimal number, we need to line up the decimal point, add trailing zero as a placeholder and then do the subtraction

<h3>Learn more</h3>

Total spend brainly.com/question/12203917

Decimal multiplication brainly.com/question/2124592

Decimal multiplication brainly.com/question/1254534

Keywords: multiplication, decimal multiplication, two digits number additional, additional, subtraction decimal, regrouping

3 0

3 years ago

Read 2 more answers

Solve the following. 5/8 divided by 2 <br><br> O 15/6<br> O 5/16<br> O 1 1/5<br> O 5/6

lesantik [10]

Answer:

The correct answer is, B.

$\frac{5}{16}$

3 0

3 years ago

Select the correct answer from each drop-down menu.

Arada [10]

Answer:

AC is 13.054

AB is 8.390996

3 0

3 years ago

Tomika uses 619 inches of wire to make a necklace and 313 inches of wire to make a bracelet.

Lisa [10]

Answer:

2,796.

Step-by-step explanation:

First, I added 619 + 313, which is 932. Since she is able to use more, I multiplied it by two to get me 1,864. If I tried multiplying it by three, I would get 2,796, which is a perfect amount.

4 0

3 years ago

The owner of a small store buys coats for ​$55.00 each. Answer parts a and b.

The owner of a small store buys coats for $55.00 each. Answer parts a and b.