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
saul85 [17]
3 years ago
7

A 200-gal tank contains 100 gal of pure water. At time t = 0, a salt-water solution containing 0.5 lb/gal of salt enters the tan

k at the rate of 5 gal/min, and the mixture, which is kept uniform by stirring, is withdrawn at the rate of 3 gal/min. (1) Write down a differential equation for the amount of salt in the tank at a time t. (2) Find the amount of salt and its concentration in the tank at a time t. (3) At the time the tank is full, how many pounds of salt will it contain? (4) What would be the limiting concentration of salt at infinity time if the tank had infinity capacity?
Mathematics
1 answer:
Artyom0805 [142]3 years ago
6 0

Answer:

1) \frac{dy}{dt}=2.5-\frac{3y}{2t+100}

2) y(t)=(50+t)- \frac{12500\sqrt{2} }{(50+t)^{\frac{3}{2} }}

3) 98.23lbs

4) The salt concentration will increase without bound.

Step-by-step explanation:

1) Let y represent the amount of salt in the tank at time t, where t is given in minutes.

Recall that: \frac{dy}{dt}=rate\:in-rate\:out

The amount coming in is 0.5\frac{lb}{gal}\times 5\frac{gal}{min}=2.5\frac{lb}{min}

The rate going out depends on the concentration of salt in the tank at time t.

If there is y(t) pounds of  salt and there are 100+2t gallons at time t, then the concentration is: \frac{y(t)}{2t+100}

The rate of liquid leaving is is 3gal\min, so rate out is =\frac{3y(t)}{2t+100}

The required differential equation becomes:

\frac{dy}{dt}=2.5-\frac{3y}{2t+100}

2) We rewrite to obtain:

\frac{dy}{dt}+\frac{3}{2t+100}y=2.5

We multiply through by the integrating factor: e^{\int \frac{3}{2t+100}dt }=e^{\frac{3}{2} \int \frac{1}{t+50}dt }=(50+t)^{\frac{3}{2} }

to get:

(50+t)^{\frac{3}{2} }\frac{dy}{dt}+(50+t)^{\frac{3}{2} }\cdot \frac{3}{2t+100}y=2.5(50+t)^{\frac{3}{2} }

This gives us:

((50+t)^{\frac{3}{2} }y)'=2.5(50+t)^{\frac{3}{2} }

We integrate both sides with respect to t to get:

(50+t)^{\frac{3}{2} }y=(50+t)^{\frac{5}{2} }+ C

Multiply through by: (50+t)^{-\frac{3}{2}} to get:

y=(50+t)^{\frac{5}{2} }(50+t)^{-\frac{3}{2} }+ C(50+t)^{-\frac{3}{2} }

y(t)=(50+t)+ \frac{C}{(50+t)^{\frac{3}{2} }}

We apply the initial condition: y(0)=0

0=(50+0)+ \frac{C}{(50+0)^{\frac{3}{2} }}

C=-12500\sqrt{2}

The amount of salt in the tank at time t is:

y(t)=(50+t)- \frac{12500\sqrt{2} }{(50+t)^{\frac{3}{2} }}

3) The tank will be full after 50 mins.

We put t=50 to find how pounds of salt it will contain:

y(50)=(50+50)- \frac{12500\sqrt{2} }{(50+50)^{\frac{3}{2} }}

y(50)=98.23

There will be 98.23 pounds of salt.

4) The limiting concentration of salt is given by:

\lim_{t \to \infty}y(t)={ \lim_{t \to \infty} ( (50+t)- \frac{12500\sqrt{2} }{(50+t)^{\frac{3}{2} }})

As t\to \infty, 50+t\to \infty and \frac{12500\sqrt{2} }{(50+t)^{\frac{3}{2} }}\to 0

This implies that:

\lim_{t \to \infty}y(t)=\infty- 0=\infty

If the tank had infinity capacity, there will be absolutely high(infinite) concentration of salt.

The salt concentration will increase without bound.

You might be interested in
Wich is the greatest six digit number 860,553 or 865,530
Serjik [45]
865,530 is the greatest because the other number, 860,530 has '0' in the place of '5' in the 865,530.
8 0
3 years ago
Read 2 more answers
What is the answer of 10 to the power of -4<br><br> [tex] 10^-4
malfutka [58]
Remember
x^{-m}= \frac{1}{x^m}

so

10^{-4}= \frac{1}{10^4}= \frac{1}{10000}=0.0001


3 0
3 years ago
Someone please help
nika2105 [10]

Answer:

b

Step-by-step explanation:

i dont kno lol

6 0
3 years ago
Find the value of L. Perimeter =22. Width is 5. So what is L. On a square
SSSSS [86.1K]
P= L+W 
P= Perimeter, L= Length  and W=Width
Length + Width = Perimeter
So, then P(22) - W(5) = L(17)
Hope that helps.
 

4 0
3 years ago
Someone plz help Which of the following equations represent f(x)=x^4 stretched vertically by a factor of 2 and horizontally by a
anzhelika [568]

Step-by-step explanation:

the answer is in the image above

5 0
3 years ago
Other questions:
  • Write an equation for the translation of the function. y = cos x; translated 6 units up
    12·2 answers
  • The tail of a 1-mile long train exits a tunnel exactly 3 minutes after the front of the train entered the tunnel. If the train i
    5·2 answers
  • Write the standard form equation that passes through (0,-1) and (-6,-9)
    8·1 answer
  • The faces of a triangular pyramid have a base of 5 cm and a height of 11 cm. what is the lateral area of the pyramid?
    5·1 answer
  • Classify the numbers as prime or composite 31 42 89 93
    6·2 answers
  • Show if lines are parallel, perpendicular, or neither<br><br>Y=10x-2<br>-10x+y=4
    13·2 answers
  • If a string that is 15.5 inches long is cut into equal peices that are 1.25 inches long how many prices of ribbon can be created
    12·1 answer
  • Find the value of f(5) for the function f(x) = 6 +3x
    13·2 answers
  • Can someone help me
    11·1 answer
  • The point (12.-5) is<br> how many units away from the origin
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!