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
Artist 52 [7]
4 years ago
9

Initially 15 grams of salt are dissolved into 25 liters of water. Brine with concentration of salt 4 grams per liter is added at

a rate of 6 liters per minute. The tank is well mixed and drained at 6 liters per minute. a) Let x be the amount of salt, in grams, in the solution after t minutes have elapsed. Find a formula for the rate of change in the amount of salt, dx/dt, in terms of the amount of salt in the solution x. dxdt= grams/minute b) Find a formula for the amount of salt, in grams, after t minutes have elapsed. x(t)= grams c) How long must the process continue until there are exactly 20 grams of salt in the tank? minutes
Mathematics
1 answer:
Alla [95]4 years ago
4 0

Answer:

a) dx/dt = 600 - 6x

b) x = 100 - 4.12((e⁻ ⁽⁶ᵗ ⁻ ³•⁰²⁾)

c) The mass of salt in the tank attains the value of 20 g at time, t = 0.227 min = 13.62 s

Explanation:

Taking the overall balance, since the total Volume of the setup is constant, then flowrate in = flowrate out

Let the concentration of salt in the tank at anytime be C

Let the concentration of salt entering the tank be Cᵢ

Let the concentration of salt leaving the tank be C₀ = C (Since it's a well stirred tank)

Let the flowrate in be represented by Fᵢ

Let the flowrate out = F₀ = F

Fᵢ = F₀ = F = 6 L/min

a) Then the component balance for the salt

Rate of accumulation = rate of flow into the tank - rate of flow out of the tank

dx/dt = Fᵢxᵢ - Fx

Fᵢ = 6 L/min, C = 4 g/L, F = 6 L/min

dC/dt = 24 - 6C

dx/dt = 25 (dC/dt), (dC/dt) = (1/25) (dx/dt) and C = x/25

(1/25)(dx/dt) = 24 - (6/25)x

dx/dt = 600 - 6x

b) dC/dt = 24 - 6C

dC/(24 - 6C) = dt

∫ dC/(24 - 6C) = ∫ dt

(-1/6) In (24 - C) = t + k (k = constant of integration)

In (24 - 6C) = -6t - 6k

-6k = K

In (24 - 6C) = K - 6t

At t = 0, C = 15 g/25 L = 0.6 g/L

In (24 - 6(0.6)) = K

In 20.4 = K

K = 3.02

So, the equation describing concentration of salt at anytime in the tank is

In (24 - 6C) = K - 6t

In (24 - 6C) = 3.02 - 6t

24 - 6C = e⁻ ⁽⁶ᵗ ⁻ ³•⁰²⁾

6C = 24 - (⁻⁽⁶ᵗ ⁻ ³•⁰²⁾)

C = 4 - ((e⁻⁽⁶ᵗ ⁻ ³•⁰²⁾)/6

C = 4 - (e⁻ ⁽⁶ᵗ ⁻ ³•⁰²⁾)/6)

But C = x/25

x/25 = 4 - (e⁻ ⁽⁶ᵗ ⁻ ³•⁰²⁾)/6)

x = 100 - 4.12((e⁻ ⁽⁶ᵗ ⁻ ³•⁰²⁾)

c) when x = 20 g

20 = 100 - 4.12(e⁻ ⁽⁶ᵗ ⁻ ³•⁰²⁾)

80 = (e⁻ ⁽⁶ᵗ ⁻ ³•⁰²⁾)

- (6t - 3.02) = In 80

- (6t - 3.02) = 4.382

(6t - 3.02) = -4.382

6t = -4.382 + 3.02

t = 1.362/6 = 0.227 min = 13.62 s

You might be interested in
What is the range of the relation below?
Greeley [361]
[6, -infinity)

have a good night
5 0
3 years ago
What is this plz help me
iren [92.7K]
20 is the answer for x
8 0
3 years ago
Read 2 more answers
How do i express x^2-3x into the form of (x-m)^2+n. I will give brainliest to correct answer
mel-nik [20]

Step-by-step explanation:

let's simply do the multiplications and then compare with the original.

(x-m)² + n

right ?

or is it

{(x - m)}^{2 + n}

let's go for the first.

x² - 2mx + m² + n = x² - 3x

-2mx + m² + n = -3x

fun there we see two things :

-2m = -3

m = 3/2

and

m² + n = 0

(3/2)² = -n

9/4 = -n

n = -9/4

so our transformed expression looks like

(x - 3/2)² - 9/4

7 0
3 years ago
Can someone help me?
BabaBlast [244]
Both are equal because the distance from AB is equal to RS.
6 0
3 years ago
Mark is treating his family to ice cream. He buys 4 sundaes and 3 cones for a total of $26. Brian also buys ice cream for his fa
Elina [12.6K]
<span>Jesse loves to make cookies. She makes chocolate chip cookies 4 out of every 5 times she makes cookies. If she makes cookies 55 times, how many times will she not make chocolate chip cookies?</span>
4 0
3 years ago
Other questions:
  • Which of the following function types exhibit the end behavior (PICTURE INCLUDED)
    14·1 answer
  • Maria will spin the arrow on the spinner 2 times. What is the probability that the arrow will stop on the same letter twice?
    9·2 answers
  • A compact disc is designed to last an average of 4 years with a standard deviation of 0.8 years.
    13·1 answer
  • I don’t understand inverses and I need step by step for each one. Please help me
    14·1 answer
  • Write and solve an equation to find the value of x and the missing angle measures.​
    12·1 answer
  • Please I need help ASAP
    10·1 answer
  • A science teacher taught classes of 29, 27, and 24 students. In these classes, a total of 32 students said they eat 7 servings o
    12·1 answer
  • There are 150 children attending a summer camp. Students were provided with the option to sign up for swimming and canoeing. The
    10·2 answers
  • The number of quarters required to play an arcade game varies directly with the number of levels played. The game requires 9 qua
    6·1 answer
  • Please helppp quicklyyy!!
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!