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
klio [65]
3 years ago
13

A swimming pool with a volume of 30,000 liters originally contains water that is 0.01% chlorine (i.e. it contains 0.1 mL of chlo

rine per liter). Starting at t = 0, city water containing 0.001% chlorine (0.01 mL of chlorine per liter) is pumped into the pool at a rate of 20 liters/min. The pool water flows out at the same rate. Let A(t) represent the amount of chlorine (in mL) in the tank after t minutes. Write a differential equation for the rate at which the amount of chlorine in the pool is changing with respect to time. Then solve the DE to state a model representing the amount of chlorine in the pool at time t.
Be sure to remember to state the initial conditions for this DE clearly.
Rin =____________
Concentration of chlorine in the tank: c(t) =_________
Rout = _________
Differential equation:
Mathematics
1 answer:
SpyIntel [72]3 years ago
3 0

Answer:

R_{in}=0.2\dfrac{mL}{min}

C(t)=\dfrac{A(t)}{30000}

R_{out}= \dfrac{A(t)}{1500} \dfrac{mL}{min}

A(t)=300+2700e^{-\dfrac{t}{1500}},$  A(0)=3000

Step-by-step explanation:

The volume of the swimming pool = 30,000 liters

(a) Amount of chlorine initially in the tank.

It originally contains water that is 0.01% chlorine.

0.01% of 30000=3000 mL of chlorine per liter

A(0)= 3000 mL of chlorine per liter

(b) Rate at which the chlorine is entering the pool.

City water containing 0.001%(0.01 mL of chlorine per liter) chlorine is pumped into the pool at a rate of 20 liters/min.

R_{in}=(concentration of chlorine in inflow)(input rate of the water)

=(0.01\dfrac{mL}{liter}) (20\dfrac{liter}{min})\\R_{in}=0.2\dfrac{mL}{min}

(c) Concentration of chlorine in the pool at time t

Volume of the pool =30,000 Liter

Concentration, C(t)= \dfrac{Amount}{Volume}\\C(t)=\dfrac{A(t)}{30000}

(d) Rate at which the chlorine is leaving the pool

R_{out}=(concentration of chlorine in outflow)(output rate of the water)

= (\dfrac{A(t)}{30000})(20\dfrac{liter}{min})\\R_{out}= \dfrac{A(t)}{1500} \dfrac{mL}{min}

(e) Differential equation representing the rate at which the amount of sugar in the tank is changing at time t.

\dfrac{dA}{dt}=R_{in}-R_{out}\\\dfrac{dA}{dt}=0.2- \dfrac{A(t)}{1500}

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

\dfrac{dA}{dt}+\dfrac{A}{1500}=0.2\\$The integrating factor: e^{\int \frac{1}{1500}dt} =e^{\frac{t}{1500}}\\$Multiplying by the integrating factor all through\\\dfrac{dA}{dt}e^{\frac{t}{1500}}+\dfrac{A}{1500}e^{\frac{t}{1500}}=0.2e^{\frac{t}{1500}}\\(Ae^{\frac{t}{1500}})'=0.2e^{\frac{t}{1500}}

Taking the integral of both sides

\int(Ae^{\frac{t}{1500}})'=\int 0.2e^{\frac{t}{1500}} dt\\Ae^{\frac{t}{1500}}=0.2*1500e^{\frac{t}{1500}}+C, $(C a constant of integration)\\Ae^{\frac{t}{1500}}=300e^{\frac{t}{1500}}+C\\$Divide all through by e^{\frac{t}{1500}}\\A(t)=300+Ce^{-\frac{t}{1500}}

Recall that when t=0, A(t)=3000 (our initial condition)

3000=300+Ce^{0}\\C=2700\\$Therefore:\\A(t)=300+2700e^{-\dfrac{t}{1500}}

You might be interested in
If you make a circle and one part is 100 the other part is 120 what is the other part​
drek231 [11]

Answer:

140

Step-by-step explanation:

circle = 360°

360° - 120° - 100° = 140°

If my answer is incorrect, pls correct me!

If you like my answer and explanation, mark me as brainliest!

-Chetan K

3 0
3 years ago
Gary is building a new L-shaped desk for his room. He wants to make sure it
Bingel [31]

Answer:

30ft2.  so its c

Step-by-step explanation:

8 0
3 years ago
Read 2 more answers
Teresas restraunt bill comes to $29.99 before tax. If the sales tax is 6.25% and she tips the waiter 20%. What is the total cost
alisha [4.7K]

Answer:

$37.86

Step-by-step explanation:

Multiply 29.99 by 0.0625 to get the sales tax

29.99x0.0625=1.87

Multiply 29.99 by 0.20 to get the tip

29.99x0.20=5.998

Round 5.998 to 6

Add 1.87 to 29.99

29.99+1.87=31.86

Add 6 to 31.86

31.86+6=37.86


3 0
3 years ago
Write a rate that represents the situation
Alika [10]
Pretty sure it's 10 liters to every 3 min. (10:3 or 10 over 3)
i got u fam
3 0
3 years ago
Read 2 more answers
What equation(s) represent joint variations? check all that apply c=2pi z=3x/y y=3x+2 w=abc/4 v=lwh
Alinara [238K]

Answer:

Step-by-step explanation:

Joint variations occurs when one variable depends on the value of two or more variables. The variable varies directly or indirectly with the other variables combined together. The other variables are held constant. From the given examples, the equation(s) that represent joint variations are

1) z = 3x/y

z varies directly with x and inversely with y.

2) w = abc/4

w varies inversely with a,b and c. 4 is the value of the constant of variation.

7 0
3 years ago
Other questions:
  • Order the following measures of weight from least to greatest, going from top to bottom.
    10·1 answer
  • The first few steps in deriving the quadratic formula are shown.
    7·2 answers
  • The Woodward family went to dinner. The cost of the dinner was $60.00. They wanted to leave a 15% tip. How much did they leave f
    13·1 answer
  • In a local park,a few children were on the monkey bars.the number of children occupying the sides is 1/8 of the number of childr
    13·1 answer
  • Given the point (-3,5) and a slope of 1/4, find 3 other points
    7·1 answer
  • What is 15 divided by 9/2?
    10·2 answers
  • Which of the following expressions has three factors?
    9·1 answer
  • MATH PROBLEM! PLEASE HELP ME IT'S TTM!!!
    6·1 answer
  • What is the equatiin of the linr that passes through tue pionts (-3,2) and (-5,8)?
    13·2 answers
  • IF YOU ANSWER THIS QUESTION CORRECTLY I GIVE BRAIN AND I GIVE HEARTS AND 5 STARS
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!