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k at the rate of 5 gal/min, and the mixture, which is kept uniform by stirring, is withdrawn at the rate of 3 gal/min.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
WHY IS MATH SO COMPLICATED?!
Tomtit [17]
Closed dot on 53, arrow going to the right
6 0
3 years ago
Plz help me!!!!!!!!!!!!!!!!!!!!
miss Akunina [59]
There are 5 dimes and 13 nickels
5 0
3 years ago
What is 5d=470 and what is 8a=47
Igoryamba
5d=470:

in this case we have to divide both sides by 5:

d=94

8a=47:

in this case we have to divide both sides by 8:

a=5.875

as a general rule, if you want to find out a value of a in
ab=c

you have to divide both sides of the equation by b (if it's not zero); then you will have left:

a=c/b

and if you calculate c/b, this will give you the answer


8 0
3 years ago
Read 2 more answers
What's bigger 17.5 or 1/6
bearhunter [10]
17.5 is greater than 1/6
4 0
3 years ago
Read 2 more answers
What is the perimeter of the triangle shown on the coordinate plane, to the nearest tenth of a unit?
Akimi4 [234]

Answer:

25.6 units

Step-by-step explanation: From the figure we can infer that our triangle has vertices A = (-5, 4), B = (1, 4), and C = (3, -4).

First thing we are doing is find the lengths of AB, BC, and AC using the distance formula:

d=\sqrt{(x_2-x_1)^{2} +(y_2-y_1)^{2}}

where

(x_1,y_1) are the coordinates of the first point

(x_2,y_2) are the coordinates of the second point

- For AB:

d=\sqrt{[1-(-5)]^{2}+(4-4)^2}

d=\sqrt{(1+5)^{2}+(0)^2}

d=\sqrt{(6)^{2}}

d=6

- For BC:

d=\sqrt{(3-1)^{2} +(-4-4)^{2}}

d=\sqrt{(2)^{2} +(-8)^{2}}

d=\sqrt{4+64}

d=\sqrt{68}

d=8.24

- For AC:

d=\sqrt{[3-(-5)]^{2} +(-4-4)^{2}}

d=\sqrt{(3+5)^{2} +(-8)^{2}}

d=\sqrt{(8)^{2} +64}

d=\sqrt{64+64}

d=\sqrt{128}

d=11.31

Next, now that we have our lengths, we can add them to find the perimeter of our triangle:

p=AB+BC+AC

p=6+8.24+11.31

p=25.55

p=25.6

We can conclude that the perimeter of the triangle shown in the figure is 25.6 units.

Read more on Brainly.com - brainly.com/question/12560433#readmore

8 0
3 years ago
Other questions:
  • How do you do -9(x+5-9)=9--9
    14·1 answer
  • If 3x - 1 = 11, what is the value of x2 + x?
    11·2 answers
  • You are interested in determining which of two brands of light bulbs, brand A and brand B, lasts longer under differing conditio
    13·2 answers
  • ) Picture frames are produced such that the four sides of a picture frame consist of two pieces from a population whose mean len
    11·1 answer
  • 12 obreros se demoran 3 días para realizar un trabajo, cuántos días se demorarán si se tienen la mitad de 8 obreros
    10·1 answer
  • I don’t get this help
    7·2 answers
  • A researcher posts a radio advertisement offering $35 in exchange for participation in a short study. The researcher accepts the
    14·1 answer
  • What is the answer please
    10·1 answer
  • WILL BE MARKED AS BRAINLIEST IF ANSWER IS CORRECT!! <br> PRE CAL
    11·1 answer
  • The method of completing the square was used to solve the equation 2x^2 – 12x + 6 = 0. Which equation is a correct step when usi
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!