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
Zigmanuir [339]
3 years ago
14

A 500500-gallon tank initially contains 200200 gallons of brine containing 100100 pounds of dissolved salt. brine containing 11

pounds of salt per gallon flows into the tank at the rate of 44 gallons per minute, and the well-stirred mixture flows out of the tank at the rate of 11 gallon per minute. set up a differential equation for the amount of salt a(t)a(t) in the tank at time tt. how much salt is in the tank when it is full? (round your answer to the 2 decimal places).
Mathematics
1 answer:
mixer [17]3 years ago
4 0
Let A(t) denote the amount of salt in the tank at time t. We're given that the tank initially holds A(0)=100 lbs of salt.

The rate at which salt flows in and out of the tank is given by the relation

\dfrac{\mathrm dA}{\mathrm dt}=\underbrace{\dfrac{11\text{ lb}}{1\text{ gal}}\times\dfrac{44\text{ gal}}{1\text{ min}}}_{\text{rate in}}-\underbrace{\dfrac{A(t)}{200+(44-11)t}\times\dfrac{11\text{ gal}}{1\text{ min}}}_{\text{rate out}}
\implies A'(t)+\dfrac{11}{200+33t}A(t)=484

Find the integrating factor:

\mu(t)=\exp\left(\displaystyle\int\frac{11}{200+33t}\,\mathrm dt\right)=(200+33t)^{1/3}

Distribute \mu(t) along both sides of the ODE:

(200+33t)^{1/3}A'(t)+11(200+33t)^{-2/3}A(t)=484(200+33t)^{-1/3}
\bigg((200+33t)^{1/3}A(t)\bigg)'=484(200+33t)^{-1/3}
A(t)=484\displaystyle\int(200+33t)^{-1/3}\,\mathrm dt
A(t)=22(200+33t)^{2/3}+C

Since A(0)=100, we get

100=22(200)^{2/3}+C\implies C\approx-652.39

so that the particular solution for A(t) is

A(t)=22(200+33t)^{2/3}-652.39

The tank becomes full when the volume of solution in the tank at time t is the same as the total volume of the tank:

200+(44-11)t=500\implies 33t=300\implies t\approx9.09

at which point the amount of salt in the solution would be

A(9.09)\approx733.47\text{ lb}
You might be interested in
Hello everyone (this is 6th grade math)​
PtichkaEL [24]

Answer:

7 batches

Step-by-step explanation:

4 0
3 years ago
Evaluate the expression<br><br>​
Aleks04 [339]

Answer:

1

Step-by-step explanation:

(-1) × (-1) × (-1) × (-1)

And - × - = +

+ × + = +

- × + = -

: = 1

7 0
3 years ago
Plss help will give brainliest and 5star
irina1246 [14]

Answer:

<h3><em>isosceles</em></h3>

An isosceles triangle is a triangle that has two sides of equal length.

An acute triangle is a triangle with three acute angles (less than 90°).

A scalene triangle has all its sides of different lengths. Equivalently, it has all angles of different measure.

An equilateral triangle is a triangle in which all three sides are equal.

8 0
3 years ago
Each set of ordered pairs represents a function. Write a rule that represents the function.
givi [52]

The function rule is : y=x^2


I hope this helped

4 0
3 years ago
Can somebody help me !
Ugo [173]

Answer:

\angle 1

Step-by-step explanation:

\angle 1 is not congruent to \angle 6

3 0
3 years ago
Other questions:
  • Simplify and show your work please. 3x+6y+x+3(2x+3y
    14·2 answers
  • Help help help help
    13·1 answer
  • If the length and breadth of a rectangle are 10cm and 20cm respectively,find the length of its diagonal
    6·2 answers
  • X – 7 = -2. how do i do this​
    14·2 answers
  • You usually buy a 5.4 ounce bottle of lotion there is a new bottle that says it gives you 20% more free what is the equation
    5·1 answer
  • Is it Line s,q ,or r?
    15·1 answer
  • Using your own words, explain direct and inverse proportionality. (Be sure to use complete sentences.)
    8·2 answers
  • GUYS PLEASE HELP ME PLEASE HELP PLEASE PLEASE!!!
    13·2 answers
  • The heights of the men age 18 and over in HANES5 averaged 69 inches; the SD was 3 inches. Suppose the histogram of their heights
    13·1 answer
  • Is borrowing money a <br><br> Delegated power <br> Reserved power <br> Concurrent power
    5·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!