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
13. Find the equations of the straight lines which passes
ipn [44]

9514 1404 393

Answer:

  • y = 1/2x +2
  • y = -2x +7

Step-by-step explanation:

The slope of a line is the tangent of the angle it makes with the x-axis. The given line has a slope of -1/3, so the lines we want will have slopes of ...

  m1 = tan(arctan(-1/3) +45°) = 0.5 . . . . . using a calculator

  m2 = tan(arctan(-1/3) -45°) = -2

Of course, these two lines are perpendicular to each other, so their slopes will have a product of -1: (0.5)(-2) = -1.

__

We can use the point-slope form of the equation for a line to write the desired equations:

  y = m(x -h) +k . . . . . line with slope m through point (h, k)

<u>Line 1</u>:

  y = 1/2(x -2) +3

  y = 1/2x +2

<u>Line 2</u>:

  y = -2(x -2) +3

  y = -2x +7

3 0
3 years ago
Help me pleasee I really<br> Need help
DanielleElmas [232]
We can’t measure these. How long is each of them
6 0
3 years ago
Which pairs of angles are supplementary? A) ∠BCD and ∠BAD B) ∠BCD and ∠ABC C) ∠BCD and ∠CDA D) ∠ABC and ∠BAD
Bogdan [553]
<span>∠ABC ∠BAD is the correct answer

</span>
4 0
3 years ago
Alec and Kiara went to Freddy's Yogurt Shop for frozen yogurt. Alec got 8.6 ounces of strawberry frozen yogurt, and Kiara got 8.
LiRa [457]

Answer:

$8.40

Step-by-step explanation:

8.6 plus 8.2 is 16.8 so they had a total of 16.8 ounces.  $0.50 per ounce so you need to multiply 16.8 by 0.50 which is 8.40.  you can also do 8.6(0.5) plus 8.2(0.5)=  4.3 plus 4.1 = 8.4

6 0
3 years ago
Read 2 more answers
When working overtime, an auto mechanic gets paid 1.5 times the normal hourly wage. Overtime is the amount of time worked beyond
DedPeter [7]
So

x=hours
y=pay

use a piecewise function maybe

for x≤40, use y=21x


for x>40, use 840+1.5(21(x-40))


k, sso 46>40 so

840+1.5(21(46-40))
840+1.5(21(6))
840+1.5(126)
840+189
1029



A. 840+1.5(21(x-40))
B. $1029
7 0
3 years ago
Other questions:
  • (2 - 3)4(4) - 9 ÷ 3.
    5·2 answers
  • john is knitting a blanket. he records the length he knits each day. how many more inches does john need to knit so the blanket
    5·1 answer
  • Last week you worked 24 hours, and earned $240.
    14·2 answers
  • A nursery has $60,000 of inventory in dogwood trees and red maple trees. The profit on a dogwood tree is 29% and the profit on a
    11·1 answer
  • What is the solution to the equation?
    14·1 answer
  • Rewrite using exponents:<br> A.<br> 4.4.4.4<br> B.<br> a.a.a
    13·1 answer
  • Y varies inversely as x. If y=5 and x=12, find y if x=4
    7·1 answer
  • How to find surface area of a triangular prism
    8·1 answer
  • Hi how are you guys doing today
    7·2 answers
  • Question 3(Multiple Choice Worth 2 points) (Adding and Subtracting Linear Expressions MC) Write the following expression in simp
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!