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
STatiana [176]
3 years ago
11

A tank initially contains 1000 liters of salt solution with 70 grams of salt per liter. A solution containing 120 grams of salt

per liter enters the tank at the rate of 9 liters per minute, and the well mixed solution is pumped out at 10 liters per minute. Find the concentration of salt as a function of time.
Mathematics
2 answers:
Lera25 [3.4K]3 years ago
7 0

Answer:

10 - A(t)/(1000 + t)

Step-by-step explanation:

salt enters the tank at the rate of 9 liters per minute, and  is pumped out at 10 liters per minute.

\fracdv}{dt} = 10-9 = 1 liters/mins

V(0) = 1000 liters

V(t) = 1000 + t.

Let A(t) be the amount of salt in gram at time t

The concentration of salt entering the tank is 120 grams/liters.  

\frac{A(t)}{v(t)}   grams/liters is the concentration of the salt that is being pumped out of the tank at time t

\frac{da}{dt} = 1 * 10 - A(t)/(1000 + t)

= 10 - A(t)/(1000 + t)

gogolik [260]3 years ago
3 0

Answer:

In (1080 - 10C) = 10 In (1079.3 - 1.0793t)

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 = 9 L/min

Rate of flow out of the tank = F = 10 L/min

dV/dt = Fᵢ - F

dV/dt = (Fᵢ - F)

dV = (Fᵢ - F) dt

∫ dv = ∫ (Fᵢ - F) dt

Integrating the left hand side from 1000 litres (initial volume) to V and the right hand side from 0 to t

V - 1000 = (Fᵢ - F)t

V = 1000 + (Fᵢ - F)t

V = 1000 + (9 - 10) t

V = 1000 - t

Component balance for the concentration.

Let the initial concentration of salt in the tank be C₀ = 70/1000 = 0.07 g/L

Let the amount of salt coming into the tank be 120 g/L × 9 L/min = 1080 g/min

Concentration coming into the tank = Cᵢ = 1080/V (g/L.min)

Concentration of salt in the tank, at any time = C g/L

Rate at which that Concentration of salt is leaving the tank = (C × 10 L/min)/V = (10C/V) g/L.min

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

Rate of Change in the concentration of salt in the tank = dC/dt

Rate of flow of salt into the tank = (1080/V) g/L.min

Rate of flow of salt out of the tank = (10C/V) g/L.min

(dC/dt) = (1080/V) - (10C/V) = (1080 - 10C)/V

Recall, V = 1000 - t

(dC/dt) = (1080 - 10C)/(1000 - t)

dC/(1080 - 10C) = dt/(1000 - t)

∫ dC/(1080 - 10C) = ∫ dt/(1000 - t)

Integrating the left hand side from C₀ (initial concentration of salt) to C and the right hand side from 0 to t

- 0.1 In [(1080 - 10C)/(1080 - 10C₀)] = - In [(1000-t)] + K (where k = constant of integration)

At t= 0, C = C₀

- 0.1 In 1 = (- In 1000) + K

0 = (- In 1000) + K

K = In 1000

- 0.1 In [(1080 - 10C)/(1080 - 10C₀)] = - In [(1000-t)] + In 1000

0.1 In [(1080 - 10C)/(1080 - 10C₀)] = In [(1000-t)] - In 1000

In [(1080 - 10C)/(1080 - 10C₀)] = 10 In [(1000-t)/1000]

In (1080 - 10C) - In (1080 - 10C₀) = 10 In [(1000-t)/1000)

In (1080 - 10C) = 10 In [(1000-t)/1000)] + In (1080 - 10C₀)

C₀ = 0.07 g/L

In (1080 - 10C) = 10 In [(1000-t)/1000)] + In (1080 - 10(0.07))

In (1080 - 10C) = 10 In [(1000-t)/1000)] + In (1080 - 0.7)

In (1080 - 10C) = 10 In [(1000-t)/1000)] + In (1079.3)

In (1080 - 10C) = 10 In [1.0793(1000-t)]

In (1080 - 10C) = 10 In (1079.3 - 1.0793t)

You might be interested in
Simplify the following expression.
erica [24]

Answer:

B)  53x + 74

Step-by-step explanation:

(59x + 64) - (6x - 10) = 59x + 64 - 6x + 10

                                = 53x + 74

3 0
3 years ago
The following polynomial is defined on the interval [-2, 2]:
Bess [88]

Answer:

Using Matlab code for Fourier series to calculate for the function, see the attached

Step-by-step explanation:

Go through the picture step by step.

6 0
3 years ago
Solve the equation and enter the value of x below.<br> x+ (9) = -15<br> X=
Wittaler [7]

Answer:

x=-24 is the correct answer

3 0
3 years ago
Gloria earned $26 for babysitting for 4 hours. Write and Solve an equation to determine how much Gloria would earn after babysit
posledela

Answer:

Equation: ($26÷4 hours) × 25 hours

Answer: $156.25

Step-by-step explanation:

First, you would have to find out how much she earns per hour, so $26÷4. Then you multiply that by 25.

(26÷4) x 25=$156.25.

(hope this helps :P)

3 0
3 years ago
What is the area of 4 1/2 feet wide and 6 1/2 feet long
Vaselesa [24]

Answer: 29 1/4

Step-by-step explanation:

Multiply and simplify to 29 1/4

6 0
3 years ago
Other questions:
  • Benali's has a bag filled with marbles. There are 7 blue marbles and 1 green marble. she reaches into the bag and pulls out a ma
    8·1 answer
  • How to write 56/100 as a decimals
    13·1 answer
  • Which number line represents the solutions to lx + 4 = 2?
    8·1 answer
  • The ratio of french fries to packets of ketchup in Philip's bag is 8:1. What does this mean?
    11·1 answer
  • Solve for x.<br> s. 2+x=1
    6·1 answer
  • Explain how to change a negative exponent to a positive one
    12·2 answers
  • I’m new and I need help with this question.
    5·1 answer
  • Solve the system of equations by graphing. y=4x+2 y=7/2x+1 plz help
    10·1 answer
  • Please help me with my maths prep...........
    9·1 answer
  • Please help! I need this ASAP
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!