Suppose brine containing 0.2 kg of salt per liter runs into a tank initially filled with 500 L of water containing 5 kg of salt. The brine enters the tank at a rate of 5L/min. The mixture, kept uniform by stirring, is flowing out at the rate of 5 L/min. (The same conditions as in the previous problem). After 10 min, a leak develops in the tank and an additional liter per minute of mixture flows out of the tank.

(a) Find the concentration, in kilograms per liter, of salt in the tank after 10 min. [Hint: Let A denote thenumber of kilograms of salt in the tank at t minutes after the process begins and use the fact that rate of increase in A = rate of input - rate of exit.

(b) After 10 min, a leak develops in the tank and an additional liter per minute of mixture flows out of the tank. What will be the concentration, in kilograms per liter, of salt in the tank 20 min after the leak develops?

Answer:

(a) 0.288 kg/liter

(b) 0.061408 kg/liter

Step-by-step explanation:

(a) The mass of salt entering the tank per minute, x = 0.2 kg/L × 5 L/minute = 1 kg/minute

The mass of salt exiting the tank per minute = 5 × (5 + x)/500

The increase per minute, Δ/dt, in the mass of salt in the tank is given as follows;

Δ/dt = x - 5 × (5 + x)/500

The increase, in mass, Δ, after an increase in time, dt, is therefore;

Δ = (x - 5 × (5 + x)/500)·dt

Integrating with a graphing calculator, with limits 0, 10, gives;

Δ = (99·x - 5)/10

Substituting x = 1 gives

(99 × 1 - 5)/10 = 9.4 kg

The concentration of the salt and water in the tank after 10 minutes = (Initial mass of salt in the tank + Increase in the mass of the salt in the tank)/(Volume of the tank)

∴ The concentration of the salt and water in the tank after 10 minutes =  (5 + 9.4)/500 = (14.4)/500 = 0.288

The concentration of the salt and water in the tank after 10 minutes = 0.288 kg/liter

(b) With the added leak, we now have;

Δ/dt = x - 6 × (14.4 + x)/500

Δ = x - 6 × (14.4 + x)/500·dt

Integrating with a graphing calculator, with limits 0, 20, gives;

Δ = 19.76·x -3.456 = 16.304

Where x = 1

The increase in mass after an increase in = 16.304 kg

The total mass = 16.304 + 14.4 = 30.704 kg

The concentration of the salt in the tank then becomes;

Concentration = 30.704/500 = 0.061408 kg/liter.

Calculating the salt concentration per liter of water we will find:

(a) [tex]0.288 kg/liter[/tex]

(b) [tex]0.061408 kg/liter[/tex]

The mass of salt entering the tank per minute:

[tex]X= (0.2)( 5) = 1 kg/minute[/tex]

The mass of salt exiting the tank per minute:

 [tex]\frac{5 * (5 + x)}{500}[/tex]

The increase per minute, Δ/dt, in the mass of salt in the tank is given as follows;  

[tex]\frac{\Delta}{dt} = \frac{x - 5 * (5 + x)}{500}[/tex]

The increase, in mass, Δ, after an increase in time, dt, is therefore;

[tex]\Delta = (x - 5 * (5 + x)/500)dt[/tex]

Integrating with a graphing calculator, with limits 0 at 10, gives;  

[tex]\Delta = (99X - 5)/10[/tex]

Substituting x = 1 gives:

[tex](99)( 1 - 5)/10 = 9.4 kg[/tex]

The concentration of the salt and water in the tank after 10 minutes = (Initial mass of salt in the tank + Increase in the mass of the salt in the tank)/(Volume of the tank). The concentration of the salt and water in the tank after 10 minutes:

[tex](5 + 9.4)/500 = (14.4)/500 = 0.288[/tex]

The concentration of the salt and water in the tank after 10 minutes = 0.288 kg/liter.

(b) With the added leak, we now have;

[tex]\frac{\Delta}{dt} = \frac{ (x - 6 ) (14.4 + x}{500} \\\Delta = \frac{(x - 6)(14.4 + x)}{500} dt[/tex]

Integrating with a graphing calculator, with limits 0 at 20, gives;

[tex]\int\limits^{20} _0 {\frac{(x - 6)(14.4 + x)}{500} dt}[/tex]

Where x = 1:

[tex]\Delta = 19.76(1) -3.456 = 16.304[/tex]

The increase in mass after an increase in = 16.304 kg. The total mass:

[tex]16.304 + 14.4 = 30.704 kg[/tex]

The concentration of the salt in the tank then becomes;

[tex]30.704/500 = 0.061408 kg/liter[/tex]

Learn more: brainly.com/question/11762787

The scale of a map is 1.5cm : 75km. What is the actual distance between two
towns that are 3.5cm apart on the map?

Planes X and Y are perpendicular. Points A, E, F, and G are points only in plane X. Points R and S are points in both planes X and Y. Lines EA and FG are parallel. check all that aply

If m<1=105 and m<4=115, then m<6=

A.75 B.40 C.50 D.60​

kai is making 96 cupcakes for a schoolg event. how many more teaspoons of vanilla are needed to make all 96 cupcakes?

Hannah has 2 hamsters named Bob and Fran. Bob weighs 3.62
ounces, and Fran weighs 2.79 ounces. How much heavier is Bob
than Fran?

1634 divided by 12 round it to the nearst tenth

Bill has candy from trick-or-treating. He has 15 Reese's and 20 Kit-Kat's.
What is the ratio of Reese's to all candy?

. What are the slope and y-intercept of the line y = -6

Answerr this. Will also give branliest. helpp

Solve the equation
below. Show all of your
work.
5x = 45

Will give brainliest A school chess club needs to raise at least $750 to attend a state competition. The club has already raised $400 and there are 4 months remaining until the competition. Which inequality and statement can be used to determine the dollar amount the club will need to raise during the remaining months?

An electronics manufacturer has 175 capacitors and the same number of capacitors is needed for each circuit board made. The manufacturer uses the capacitors to make 35 circuit boards. Identify and interpret the intercepts, then use the intercepts to graph the line

Help T-T I’m a bit stuck

Salon he runs 1/5 of a mile in 1/20 of an hour he swims 2/3 of a mile and 1/9 of an hour he bikes for miles in a quarter of an hour at these rights how many minutes does it take being to run 2 miles swim 2 miles in bike eight

Which of the following is true about the lines defined by the equations 4x−2y=6 and x+2y=14? .
A.The lines share the same y-intercept.
B.Both lines intersect at point (1,−1).
C.The lines are perpendicular.
D.The lines are parallel.

subtract. write your answer in scientific notation.
(7.15×105)–(5.964×105)

Leeza is 6 years older than two times her brother Sammy’s age. Let s represent Sammy’s age.

Which expression represents Leeza’s age?

a.6s + 2


b.2s


c.s + 6


d.2s + 6

Given AABC ADEF, AB = 4x-1, FD = 7, and DE = 8x-33.
Find the value of x and the length of DE.
x=8 and DE = 31
x=8 and DE = 9
Ox=2 and DE = 10
x = 2 and DE = 7

Compare 2 4/5 ? 2.325

BRAINLIEST ANSWER IF CORRECT!!!!!
The function F has a removable discontinuity at?

I need to turn that into an equation but I cant figure it out.​

Home Algebra 1 Support Final Exam Term 1
y = 3x - 1
y = -3x - 1
Which describes the number of solution(s) to the system of linear equations shown above?
C
Exactly two
None
A
An infinite number
D
B
Exactly one

It was 7 degrees below zero in the morning. By afternoon the temperature rose 15 degrees. How warm was it?

Is the side length √ 23+3 rational or irrational? Explain your reasoning.
100 points but If you answer something random I will report.

10x- 2y=16 and x+5y=-20
are perpendicular,parallel or neither

Which list shows all the factors of 18?

Una tableta contiene 500 mg de paracetamol, ¿cuántas tabletas se obtendrán con 5kg del mismo medicamento? xfavor ayudenme con explicación y todo lo necesito para el sabado

When a coordinate grid is superimposed on a map of lanesville the middle school is located at (15,13) and the librart is located at (27,9). If each unit represents 1 mile how many miles apart are the middle school in the liebrary round your answer to the nearest 10th

Student ID number consists of 6 characters. The first four
characters can be any number 1-9 without restrictions. The
last two characters are letters and cannot repeat. How
many student IDs are possible?
a) 6,760,000
b) 4,264,650
O C) 6,500,000
d) 78
Please hurry

What are the two words that allow you to check your division problem backward?