Solve the exponential equation using the method of relating the bases by first rewriting the equation in the form of e^u=e^v

A tank contains 240 liters of fluid in which 20 grams of salt is dissolved. Brine containing 1 gram of salt per liter is then pu

Novosadov [1.4K]

Answer:

$A(t)=240-220e^{-\frac{t}{40}}$

Step-by-step explanation:

A tank contains 240 liters of fluid in which 20 grams of salt is dissolved.

Volume of the tank = 240 liters
Initial Amount of Salt in the tank, A(0)=20 grams

Brine containing 1 gram of salt per liter is then pumped into the tank at a rate of 6 L/min

$R_{in}=$ (concentration of salt in inflow)(input rate of fluid)

$R_{in}=(1\frac{gram}{liter})( 6\frac{Liter}{min})=6\frac{gram}{min}$

$R_{out}=$ (concentration of salt in outflow)(output rate of fluid)

$R_{out}=(\frac{A(t)}{240})( 6\frac{Liter}{min})\\R_{out}=\frac{A}{40}$

Rate of change of the amount of salt in the tank:

$\dfrac{dA}{dt}=R_{in}-R_{out}$

$\dfrac{dA}{dt}=6-\dfrac{A}{40}$

We then solve the resulting differential equation by separation of variables.

$\dfrac{dA}{dt}+\dfrac{A}{40}=6\\$The integrating factor: e^{\int \frac{1}{40}dt} =e^{\frac{t}{40}}\\$Multiplying by the integrating factor all through\\\dfrac{dA}{dt}e^{\frac{t}{40}}+\dfrac{A}{40}e^{\frac{t}{40}}=6e^{\frac{t}{40}}\\(Ae^{\frac{t}{40}})'=6e^{\frac{t}{40}}$

Taking the integral of both sides

$\int(Ae^{\frac{t}{40}})'=\int 6e^{\frac{t}{40}} dt\\Ae^{\frac{t}{40}}=6*40e^{\frac{t}{40}}+C, $(C a constant of integration)\\Ae^{\frac{t}{40}}=240e^{\frac{t}{40}}+C\\$Divide all through by e^{\frac{t}{40}}\\A(t)=240+Ce^{-\frac{t}{40}}$

Recall that when t=0, A(t)=20 (our initial condition)

$20=240+Ce^{-\frac{0}{40}}\\20-240=C\\C=-220\\$Therefore, the number A(t) of grams of salt in the tank at time t\\A(t)=240-220e^{-\frac{t}{40}}$

3 0

3 years ago

I really need help please!!!!!

RideAnS [48]

Answer:

Y= 2 · 4x

Step-by-step explanation:

6 0

2 years ago

What should you do to both sides of the inequality in order to solve x + (-2) > 3?

lilavasa [31]

x+(-2) >3

open the bracket

x-2>3

collect the like terms by taking 2 to the other side

x>3+2

x>6

3 0

2 years ago

So this has to do with permutations and honestly I need an explanation on how I can figure these out.

AnnZ [28]

Well, the answer quite simple ....there are 4! ways to arrange these numbers...and as 4! = 4*3*2*1 = 24

hence 24 is the correct answer....

u can also remember it by theorem of multiplication as.....in first place (I.e. first code can be any no. out of 4,5,2&7 ....so 4*.....

as first place is acquired by a certain no. that leaves three no. to fill third place and when third place I'd occupied it leaves 2 numbers to fill second place and lastly only one no. to fill the last place .....so it's result will be like 4*3*2*1.

I know this is pretty much confusing ....but still I tried my best....

if anything troubles u here feel free to ask me

6 0

3 years ago

2142-MTH_06_14_FND.pdf-A4-Identify whether the given value of the variable makes the equation true.p/2 + 9 = 12; p = 6

Arturiano [62]

First answer is A(true), second one is B(false), and the last one is A (xx).

8 0

3 years ago

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