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
Oliga [24]
2 years ago
8

A tank contains 100 L of water. A solution with a salt con- centration of 0.4 kg/L is added at a rate of 5 L/min. The solution i

s kept mixed and is drained from the tank at a rate of 3 L/min. If y(t) is the amount of salt (in kilograms) after t minutes, show that y satisfies the differential equationdy/dt=2-3y/100+2tSolve this equation and find the concentration after 20 minutes.

Mathematics
1 answer:
Fantom [35]2 years ago
8 0

Answer:

a) (dy/dt) = 2 - [3y/(100 + 2t)]

b) The solved differential equation gives

y(t) = 0.4 (100 + 2t) - 40000 (100 + 2t)⁻¹•⁵

c) Concentration of salt in the tank after 20 minutes = 0.2275 kg/L

Step-by-step explanation:

First of, we take the overall balance for the system,

Let V = volume of solution in the tank at any time

The rate of change of the volume of solution in the tank = (Rate of flow into the tank) - (Rate of flow out of the tank)

The rate of change of the volume of solution = dV/dt

Rate of flow into the tank = Fᵢ = 5 L/min

Rate of flow out of the tank = F = 3 L/min

(dV/dt) = Fᵢ - F

(dV/dt) = (Fᵢ - F)

dV = (Fᵢ - F) dt

∫ dV = ∫ (Fᵢ - F) dt

Integrating the left hand side from 100 litres (initial volume) to V and the right hand side from 0 to t

V - 100 = (Fᵢ - F)t

V = 100 + (5 - 3)t

V = 100 + (2) t

V = (100 + 2t) L

Component balance for the amount of salt in the tank.

Let the initial amount of salt in the tank be y₀ = 0 kg

Let the rate of flow of the amount of salt coming into the tank = yᵢ = 0.4 kg/L × 5 L/min = 2 kg/min

Amount of salt in the tank, at any time = y kg

Concentration of salt in the tank at any time = (y/V) kg/L

Recall that V is the volume of water in the tank. V = 100 + 2t

Rate at which that amount of salt is leaving the tank = 3 L/min × (y/V) kg/L = (3y/V) kg/min

Rate of Change in the amount of salt in the tank = (Rate of flow of salt into the tank) - (Rate of flow of salt out of the tank)

(dy/dt) = 2 - (3y/V)

(dy/dt) = 2 - [3y/(100 + 2t)]

To solve this differential equation, it is done in the attached image to this question.

The solution of the differential equation is

y(t) = 0.4 (100 + 2t) - 40000 (100 + 2t)⁻¹•⁵

c) Concentration after 20 minutes.

After 20 minutes, volume of water in tank will be

V(t) = 100 + 2t

V(20) = 100 + 2(20) = 140 L

Amount of salt in the tank after 20 minutes gives

y(t) = 0.4 (100 + 2t) - 40000 (100 + 2t)⁻¹•⁵

y(20) = 0.4 [100 + 2(20)] - 40000 [100 + 2(20)]⁻¹•⁵

y(20) = 0.4 [100 + 40] - 40000 [100 + 40]⁻¹•⁵

y(20) = 0.4 [140] - 40000 [140]⁻¹•⁵

y(20) = 56 - 24.15 = 31.85 kg

Amount of salt in the tank after 20 minutes = 31.85 kg

Volume of water in the tank after 20 minutes = 140 L

Concentration of salt in the tank after 20 minutes = (31.85/140) = 0.2275 kg/L

Hope this Helps!!!

You might be interested in
Super easy!!<br><br> 3 1/2 x 5 =
bearhunter [10]

Answer:

35/2 or 17.5 or 17 1/2

Step-by-step explanation:

4 0
2 years ago
Read 2 more answers
Identify a set of parallel lines and a set of perpendicular lines in this image. Please help
ddd [48]

Answer:

line AB and line CD are parallel

line AB and line EF are perpendicular

Step-by-step explanation:

it is the third option

4 0
2 years ago
14. a house-painting company charges 376$ plus 12$ an hour. Another painting company charges 280$ plus 15$ an hour.
Art [367]
One company charges:
$376 + $12*h

Other company charges:
$280 + $15*h

They will both charge the same when:

$376 + $12*h = $280 + $15*h

96 = 3*h

hours = 32
Cost will be $760

Double Check
$376 + $12 * 32 hours = $760

$280 + $15 * 32 hours = $760


5 0
3 years ago
What is the value of x in the equation 1.8-3.7x=4.2x+0.3<br><br>​
wlad13 [49]

Answer:

15/79

Step-by-step explanation:

or in Alternate form is X= 0.1898734177215

5 0
2 years ago
Z1 and Z2 are vertical angles. Z2 has measure of 93° what is the measure of Z1? enter your answer in the box.
Soloha48 [4]

Answer:

m∠Z1 = 93°

Step-by-step explanation:

Vertical Angles are the angles opposite each other when two lines cross.

They are congruent (the same as each other).

Therefore, m∠Z1 = m∠Z2, so if  m∠Z2 = 93° then m∠Z1 = 93°

6 0
2 years ago
Other questions:
  • Identify which table shows a direct variation? A. Table 1 only B. Table 2 only C. Both tables D. Neither table
    15·1 answer
  • What is 1578 divided 2
    14·2 answers
  • Help plz.......................!
    11·1 answer
  • Which factor does the width of the peak of a normal curve depend on?
    8·2 answers
  • The ratio of two numbers is 3 to 5. Their sum is 136. Find the two numbers. how do I write it into an equation?
    14·1 answer
  • If f(x) = x3 – x2 – 3, which of the following is equal to g(x) = f(2 – x)?
    9·1 answer
  • Share 660kg in ratio of 5:7<br>​
    13·1 answer
  • PLEASE HELP!
    6·1 answer
  • Look at the graph shown below:
    12·2 answers
  • PLEASE HELP, What is the "weight" or value of one circle in this hanger? In other words, what does one "W" have to be for this t
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!