1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
Salsk061 [2.6K]
3 years ago
9

Suppose that a large mixing tank initially holds 500 gallons of water in which 50 pounds of salt have been dissolved. Another br

ine solution is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at a slower rate of 2 gal/min. If the concentration of the solution entering is 2 lb/gal, determine a differential equation (in lb/min) for the amount of salt A(t) (in lb) in the tank at time t > 0. (Use A for A(t).)
Mathematics
1 answer:
aliya0001 [1]3 years ago
4 0

Answer:

A=1500-1450e^{-\dfrac{t}{250}}

Step-by-step explanation:

The large mixing tank initially holds 500 gallons of water in which 50 pounds of salt have been dissolved.

Volume = 500 gallons

Initial Amount of Salt, A(0)=50 pounds

Brine solution with concentration of 2 lb/gal is pumped into the tank at a rate of 3 gal/min

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

=(2\frac{lbs}{gal})( 3\frac{gal}{min})\\R_{in}=6\frac{lbs}{min}

When the solution is well stirred, it is then pumped out at a slower rate of 2 gal/min.

Concentration c(t) of the salt in the tank at time t

Concentration, C(t)=\dfrac{Amount}{Volume}=\dfrac{A(t)}{500}

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

=(\frac{A(t)}{500})( 2\frac{gal}{min})\\R_{out}=\dfrac{A}{250}

Now, the rate of change of the amount of salt in the tank

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

\dfrac{dA}{dt}=6-\dfrac{A}{250}

We solve the resulting differential equation by separation of variables.  

\dfrac{dA}{dt}+\dfrac{A}{250}=6\\$The integrating factor: e^{\int \frac{1}{250}dt} =e^{\frac{t}{250}}\\$Multiplying by the integrating factor all through\\\dfrac{dA}{dt}e^{\frac{t}{250}}+\dfrac{A}{250}e^{\frac{t}{250}}=6e^{\frac{t}{250}}\\(Ae^{\frac{t}{250}})'=6e^{\frac{t}{250}}

Taking the integral of both sides

\int(Ae^{\frac{t}{250}})'=\int 6e^{\frac{t}{250}} dt\\Ae^{\frac{t}{250}}=6*250e^{\frac{t}{250}}+C, $(C a constant of integration)\\Ae^{\frac{t}{250}}=1500e^{\frac{t}{250}}+C\\$Divide all through by e^{\frac{t}{250}}\\A(t)=1500+Ce^{-\frac{t}{250}}

Recall that when t=0, A(t)=50 (our initial condition)

50=1500+Ce^{-\frac{0}{250}}50=1500+Ce^{0}\\C=-1450\\$Therefore the amount of salt in the tank at any time t is:\\A=1500-1450e^{-\dfrac{t}{250}}

You might be interested in
3.6x107<br> (2.5x106) (1.6x103)
denis23 [38]

Answer:

16822454.4 is the answer

6 0
3 years ago
I need<br> Help on this one
elixir [45]
The answer is
X=7
Have a nice day <3
5 0
2 years ago
Read 2 more answers
Jose can read 7 pages of his book in 5 minutes.At this rate, how long will it take him to read the entire 175 page book
Naily [24]

\dfrac{175 \textrm{ pages}}{ \frac{7 \textrm{ pages}}{5 \textrm{ minutes}}} = 125 \textrm{ minutes}

Answer: 125 minutes


8 0
3 years ago
Original price:$30; markdown:33%
Travka [436]
30 x 0.33 = 9.90 And that's the answer
3 0
3 years ago
I don't understand this type of stuff so plz if u know what the answer is otherwise don't comment
Shalnov [3]

I think it's A  becaouse the answer I got was very close to the answer of A so it's A I think. Im just tying to help c:

3 0
2 years ago
Read 2 more answers
Other questions:
  • <img src="https://tex.z-dn.net/?f=72%20%5Cdiv%20what%20%5C%3A%20%20%3D%203" id="TexFormula1" title="72 \div what \: = 3" alt="7
    7·2 answers
  • 4 donuts cost 6.20 how much would 12 donuts cost?
    9·2 answers
  • Can I get some help with this just tell me where they go lol thanks
    11·1 answer
  • Product -6 and sum 1 what is the factors
    6·1 answer
  • The Great Speedster roller coaster runs 3 trains every 10 minutes. At that rate, how many trains does the roller coaster run eve
    9·1 answer
  • Shorts are on sale for 10% off. If Kate buys 10 pairs of shorts at s dollars apiece, which expression is NOT equivalent to the s
    9·2 answers
  • Graph the function. Please include the graph thank you.
    15·1 answer
  • The curved part of this figures is a semicircle.
    6·2 answers
  • What is the slope of the line represented by the equation y = -1/2x + 1/4?
    10·2 answers
  • The sales manager gathered information on the numbers of sales calls made and the number of copiers sold for a random sample of
    11·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!