A tank contains 210 liters of fluid in which 30 grams of salt is dissolved. Pure water is then pumped into the tank at a rate of 3 L/min; the well-mixed solution is pumped out at the same rate. Let the A(t) be the number of grams of salt in the tank at time t. Find the rate at which the number of grams of salt in the tank is changing at time t.

Answers

Answer 1

Answer:

[tex]dA(t)/dt = -\frac{3}{7}e^{-t/70}[/tex] g/min

Step-by-step explanation:

The mass flow rate dA(t)/dt = mass flowing in - mass flowing out

Since water is pumped in at a rate of 3 L/min, and it contains no salt, the concentration in is thus 0 g/L. the mass flow in is thus 0 g/L × 3 L/min = 0 g/min.

Let A(t) be the mass present at any time, t. The concentration at any time ,t is thus A(t)/volume = A(t)/210. Since water flows out at a rate of 3 L/min, the mass flow out is thus, A(t)/210 g/L × 3 L/min = A(t)/70 g/min.

So, dA(t)/dt = mass flowing in - mass flowing out

dA(t)/dt = 0 g/min - A(t)/70 g/min

dA(t)/dt =  - A(t)/70 g/min

 Since the tank initially contains 30 g of salt, the initial mass of salt in the tank is 30 g. So A(0) = 30

So, the initial value problem is thus  

dA(t)/dt =  - A(t)/70 , A(0) = 30

Separating the variables, we have

dA(t)/A(t) = -dt/70

Integrating both sides, we have

∫dA(t)/A(t) = ∫-dt/70

㏑A(t) = -t/70 + C

taking exponents of both sides, we have

A(t) = exp(-t/70 + C)

A(t) = exp(-t/70)expC

[tex]A(t) = e^{-t/70}e^{C}\\A(t) = Ce^{-t/70} whereC = e^{C}[/tex]

A(0) = 30, So

[tex]A(0) = Ce^{-0/70}\\30 = Ce^{0}\\C = 30[/tex]

[tex]A(t) = 30e^{-t/70}[/tex]

The rate at which the number of grams of salt in the tank is changing at time t is

[tex]dA(t)/dt = \frac{d30e^{-t/70} }{dt} \\dA(t)/dt = -\frac{30e^{-t/70} }{70}\\dA(t)/dt = -\frac{3}{7}e^{-t/70}[/tex]

So, the rate at which the number of grams of salt in the tank is changing at time t is

[tex]dA(t)/dt = -\frac{3}{7}e^{-t/70}[/tex] g/min


Related Questions

Can someone explain the washer method in calculus and help me solve this problem?

Answers

Answer:

Step-by-step explanation:


When Joe solved a system using substitution, he got 7 = 9. What can Joe
conclude? *
The solution to the system is (7.9).
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The work must be incorrect because it is not possible to get this equation
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Answers

Answer:

The system of equations does not have a solution

Step-by-step explanation:

Whenever you get a number = a different number, that means there's no solution.

Show how you found the slope and y-intercept

Answers

Answer:

Slope: [tex]\frac{5}{2}[/tex]

y-intercept: 0

Step-by-step explanation:

First, find the slope. Use two points given to you. In this case, I will use the following:

(x₁ , y₁) = (2 , 5)

(x₂ , y₂) = (4 , 10)

Use the following equation:

slope (m) = [tex]\frac{y_2 - y_1}{x_2 - x_1}[/tex]

Plug in the corresponding numbers to the corresponding variables:

slope (m) = [tex]\frac{10 - 5}{4 - 2}[/tex] = [tex]\frac{5}{2}[/tex]

Note the slope intercept form:

y = mx + b

y = (x , y)

m = slope

x = (x , y)

b = y-intercept

Plug in the slope into m for the equation:

[tex]y = \frac{5}{2} x + b[/tex]

Choose a point to use to find b. In this case, I will use (2 , 5). Plug in 2 for x, and 5 for y:

[tex]5 = \frac{5}{2}(2) + b[/tex]

Simplify. Isolate the variable, b. Note the equal sign, what you do to one side, you do to the other:

[tex]5 = \frac{5}{2}(2) + b\\5 = 5 + b\\5 (-5) = 5 (-5) + b\\b = 5 - 5\\b = 0[/tex]

Y-intercept: 0

Factorise
7lm - 21lmn​

Answers

Answer:

7lm - 21lmn

taking common 7lm

7lm(1-3n)

I cut a pizza in half. Then I cut each half into 3 equal pieces and each piece into 2 equal slices. I sold 6 slices. What fraction of the pizza did I sell?​

Answers

Answer:

6/12

Step-by-step explanation:

im pretty sure, i did it on a paper correct me if im wrong sorry if i am

in a classroom 1/6 of the students are wearing blue shirts and 2/3 of wearing white shirts there are 36 students in the class how many students are wearing blue shirts​

Answers

Answer:

6 students are wearing blue shirts

Step-by-step explanation:

you divide 36 by 6 and that equals 1/6 of the students.

What is the answer to -5+(-5r)+10

Answers

Answer:

=−5r+5

Step-by-step explanation:

Let's simplify step-by-step.

−5−5r+10

=−5+−5r+10

Combine Like Terms:

=−5+−5r+10

=(−5r)+(−5+10)

=−5r+5

Answer: -5r+5

Step-by-step explanation: 1rst: Eliminate redundant parentheses

                                               -5+(-5r)+10

                                               -5-5r+10

                                             2nd: Add the numbers

                                                -5-5r+10

                                                5-5r

                                             3rd: Rearrange terms

                                                5-5r

                                                -5r+5

Please Help me, what the answer

Answers

It’s either 1. Or 2. I’m really not sure which one tho so sorry! There’s not enough information given but if I had to guesstimate I would say 1.

A single six-sided die is rolled. Find the probability of rolling a seven.

Answers

Answer: there is no probability to roling a 7

Step-by-step explanation: there are  only 6 sides to the die that is rolled

Laura had a toy robot collection. 4/7 of her 28 robots make noise. How many of Laura’s robots make noise? If you guys answer can you please do show you work.

Answers

Answer:

4/7 of 28 robots is 16.

Step-by-step explanation:

This is because 4/7 is equal to 16/28 when you simplify it. multiply 7 by four and you get 28 as your denominator and 16 as your numerator.

A bus can hold a total of 60 students. The bus is 73% full how many students are on the bus?​

Answers

Answer:

44

Step-by-step explanation:

so youre looking for 73% of students rite? of means multiplication , you just times 73% by 60

73%*60=43.8

but there cant be point 8 of a person unless theyre missin a couple limbs, so you round up to 44!

hope it helps :)

44
---------------------

Use the bar graph to find the probability of the event

Not spinning a 1

Answers

10+8 you add all the data to get your answer

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Answers

Answer:

translation.........

u={1,2,3,4,5,6,7,8,9}
a={2,4,6,8}
b={3,6,9}

What is the set AUB?

Answers

Answer:

AUB = {2,3,4,6,8,9}

Step-by-step explanation:

The set u should not be included in set AUB

When 6 is added to four times a number, the result is 50​

Answers

x=the number
4x+6=-50
4x=-50-6
4x=-56
x=-56/4
x=-14

Answer:

50-6= 54

54÷4= 13.5

editing for clarity.

first you subtract the 6 from the 50 to find out what the "4 times a number" came to.

in this case, 50-6, which equals 54.

then, you divide the 54 by 4 to get 13.5.

so, the whole math problem would be

4 times 13.5 plus 6 equals 50.

A polynomial function has zeros at 5/2 (multiplicity 2), 3 (multiplicity 1), and 0 (multiplicity 4). Write a function in standard form that could represent this function

Answers

Answer:

-75 x^4 + 85 x^5 - 32 x^6 + 4 x^7

Step-by-step explanation:

[tex](x-5/2)^{2}(x-3)(x-0)^4 =0\\(2x-5)^{2}(x-3)(x)^4 =0[/tex]

Expand and get

-75 x^4 + 85 x^5 - 32 x^6 + 4 x^7

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The given expression is

[tex](x-\dfrac{5}{2})^2(x-1)^2(x-0)^4=0[/tex]

[tex](x^2-5x+\frac{25}{4} )(x^2-2x+1)(x^4)=0[/tex]

[tex](x^4-2x^3+x^2-5x^3+10x^2-5x+\dfrac{25x^2}{4} -\dfrac{50}{4} x+\dfrac{25}{4} ) x^4=0[/tex]

[tex]X^4(x^4-7x^3+12x^2-\dfrac{35x}{2} +\dfrac{25x^2}{4} +\dfrac{25}{4} =0[/tex]

Thus the standard form will be  

[tex]X^4(x^4-7x^3+12x^2-\dfrac{35x}{2} +\dfrac{25x^2}{4} +\dfrac{25}{4} =0[/tex]

To know more about Polynomials follow

https://brainly.com/question/15702527

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Answers

Answer:

8 hours

Step-by-step explanation:

The equation is 2.25X + 3.50 = 21.50

X represents the unknown amount of hours she worked. 2.25 is the cost per hour. 3.50 is the fee she charges for every gig. That fee only happens once a gig so naturally it wasn't the number x was placed with.

2.25X + 3.50 = 21.50

2.25X + 3.50 - 3.50 = 21.50 - 3.50

2.25X = 18

2.25X = 18

2.25 2.25

X = 8

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Answers

Answer:B

Step-by-step explanation:

There are 4 oranges, 7 bananas, and 5 apples in a fruit basket.
Ignacio selects a piece of fruit at random and then Terrance
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Answers

                                                                                                                                                   

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Is my Answers Correct? Very easy Question!

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Answer:

Step-by-step explanation:

Correct

Step-by-step explanation:

the 5th one on the right is standard form,

the second one is perfect square form.

HELPPPPP PLSSSSSS !!!!!!!


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Answers

Answer:

Top is 18, Bottom goes 6 for 12, then 7 for 14, then 10 for 20

Step-by-step explanation:

2 guests = 1 table

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Answers

Answer:

I think the question is incorrect

Step-by-step explanation:

they boost the economy why is this in mathematics

‍♀️‍♀️

PLS HELP I DONT KNOW THIS ONE ITS URGENT

Answers

Answer:

A. [tex] \frac{UV}{VW} = \frac{ZY}{YX} [/tex]

Step-by-step explanation:

WX, VY, and UZ are parrelel to each other, therefore, base on triangle proportional theorem, the two transversal lines are divided proportionally by the 3 parallel lines.

Thus, we would have the following equation which is true:

[tex] \frac{UV}{VW} = \frac{ZY}{YX} [/tex]

What is the measure of angle A?
Enter your answer as a decimal in the box. Round your final answer to the nearest hundredth.

Answers

Answer: 45 degrees

Step-by-step explanation:

Answer is 45 degrees

NEEd help quickly please!!! The basketball game had 500 people in attendance. If the ratio of Pioneer fans to Lion fans is 15:5, how many MORE Pioneer fans were there? Pls use tape diagram.

Answers

Answer:

Step-by-step explanation:

15(500/20)-5(500/20)

375-125

250

There were 250 more Pioneer fans than Lion fans.

5th grade math. correct answer will be marked brainliest. what are the last 2 boxes?

Answers

Answer:

you forgot (2,28) :)

Step-by-step explanation:

the answer should be (5,37)

Please!!!!! help!!!!!

Answers

Answer:

x = 6, x = 8, and x = 10

Step-by-step explanation:

Since 44 > 35, that means that

[tex]2x + 3 > x + 8\\x + 3 > 8\\x > 5[/tex]

so that means the ones that work are x = 6, x = 8, and x = 10

Look at the image. Could someone please help me out? Only question B Thank you! WILL GIVE BRAINLYEST​

Answers

Answer:

y = 3

Step-by-step explanation:

2x+6y=5x+4y   if x=2

2(2)+6y=5(2)+4y

4+6y=10+4y

-4     =-4

6y=6+4y

-4y=   -4y

2y=6

/2=/2

y=3

Answer and explain ILL MARK BRAINLIEST I DONT UNDERSTAND

Answers

First let's reduce the feet to miles

there are 5280 feet in a mile therefore

26400 feet=5 miles

31680 feet=6 miles

Jet A(the first jet) descends 5 miles in 96 miles

Jet B(the second jet) descends 6 miles in 120 miles

We can compare these as fractions to see which is steeper. This can be viewed as slope and the origin (0,0) is the airport.

slope: 6/120=?=5/96

1/20=?=5/96

Now we know that 5/100 =1/20 so 5/96 must be bigger than 5/100 because you are dividing by a smaller number.

so 1/20<5/96

So Jet B is descending steeper than Jet A.

As for linear model, I don't exactly know what your teacher means but I think I actually used the linear model when I'm thinking of steepness as slope in the coordinate plane, I will include a picture.

In this extremely zoomed out graph, you can see the blue line is just slighly higher than the red line(slope as in explanation is way easier to tell) this could be seen as the linear model) :) Hope it helped!

Please help I'm confused with what to put in those boxes

Answers

It is asking how many layers which is 5 because in order to build it it went up 5
and the numbers in each layer is 3 going across
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