A 500500-gallon tank initially contains 200200 gallons of brine containing 100100 pounds of dissolved salt. brine containing 11 pounds of salt per gallon flows into the tank at the rate of 44 gallons per minute, and the well-stirred mixture flows out of the tank at the rate of 11 gallon per minute. set up a differential equation for the amount of salt a(t)a(t) in the tank at time tt. how much salt is in the tank when it is full? (round your answer to the 2 decimal places).

Answers

Answer 1

Let [tex]A(t)[/tex] denote the amount of salt in the tank at time [tex]t[/tex]. We're given that the tank initially holds [tex]A(0)=100[/tex] lbs of salt.

The rate at which salt flows in and out of the tank is given by the relation

[tex]\dfrac{\mathrm dA}{\mathrm dt}=\underbrace{\dfrac{11\text{ lb}}{1\text{ gal}}\times\dfrac{44\text{ gal}}{1\text{ min}}}_{\text{rate in}}-\underbrace{\dfrac{A(t)}{200+(44-11)t}\times\dfrac{11\text{ gal}}{1\text{ min}}}_{\text{rate out}}[/tex]
[tex]\implies A'(t)+\dfrac{11}{200+33t}A(t)=484[/tex]

Find the integrating factor:

[tex]\mu(t)=\exp\left(\displaystyle\int\frac{11}{200+33t}\,\mathrm dt\right)=(200+33t)^{1/3}[/tex]

Distribute [tex]\mu(t)[/tex] along both sides of the ODE:

[tex](200+33t)^{1/3}A'(t)+11(200+33t)^{-2/3}A(t)=484(200+33t)^{-1/3}[/tex]
[tex]\bigg((200+33t)^{1/3}A(t)\bigg)'=484(200+33t)^{-1/3}[/tex]
[tex]A(t)=484\displaystyle\int(200+33t)^{-1/3}\,\mathrm dt[/tex]
[tex]A(t)=22(200+33t)^{2/3}+C[/tex]

Since [tex]A(0)=100[/tex], we get

[tex]100=22(200)^{2/3}+C\implies C\approx-652.39[/tex]

so that the particular solution for [tex]A(t)[/tex] is

[tex]A(t)=22(200+33t)^{2/3}-652.39[/tex]

The tank becomes full when the volume of solution in the tank at time [tex]t[/tex] is the same as the total volume of the tank:

[tex]200+(44-11)t=500\implies 33t=300\implies t\approx9.09[/tex]

at which point the amount of salt in the solution would be

[tex]A(9.09)\approx733.47\text{ lb}[/tex]

Related Questions

Alice says she can find the area of the room by dividing the floor plan into two trapezoids. show how she can divide the floor plan. then find the area using her method

Answers

To find the room's area, Alice can divide the floor plan into two trapezoids, calculate each trapezoid's area using the formula for a trapezoid's area (A = (a + b)/2 × h), and then sum the two areas.

Alice can find the area of the room by first dividing the floor plan into two trapezoids, which can be done by drawing a line across the room that connects two non-adjacent vertices. Once divided, you can use the formula for the area of a trapezoid to find the area of each part and then sum the two areas to get the total area of the room.

Calculating the Area of a Trapezoid

The formula for the area of a trapezoid is:

A = (a + b)/2 × h

Where a and b are the lengths of the two parallel sides, and h is the height. Once you calculate the area of both trapezoids using this formula, you can add them together to find the total area of the room.

Evaluate the expression 28C14

Answers

For choose notation, you...
xCy = x! / (x-y)! y!
! means factorial

(28 x 27 x 26...x 3 x 2 x 1)/(14 x 13 x 12...x 3 x 2 x 1)(14 x 13 x 12...x 3 x 2 x 1) = 40116600

About 70% of Earth's surface is covered by water. If the diameter of earth is about 12,750km, find the area not covered by water to the nearest 100,000km^2.

Answers

A ≈ 0.30πd² ≈ 153,200,000 km²

write an expression that is,equivalent tp 7/8

Answers

Hello!

[tex] \frac{7}{8} [/tex] = [tex] \frac{3.5}{4} [/tex]

Enjoy.
~Isabella

Express the number .0957 as a percent rounded to the hundredth place. (please include the % symbol.)

Answers

The answer would be 9.57%. You would move the decimal 2 places to the right and add the percent.

Which is a better buy? 20-ounce box of cereal for $3.60 or 28 ounce for $4.20?

Answers

28 oz for $4.20. Hope this helps :)

Divide the price for each by the number of ounces to find the price per ounce.

PRODUCT 1:
=$3.60 ÷ 20
= $0.18 per ounce

PRODUCT 2:
= $4.20 ÷ 28
= $0.15 per ounce

ANSWER: The better buy is product two at $0.15 an ounce.

Hope this helps! :)

if 100 envelopes cost 70 cents, how much would 250 cost?

Answers

1.75! If you add together 200, and then half of 100, it’s easy!

an investment is expected to pay 5 per year compounded continuously. If you want the value of the investment to be $700,000 after 25 years, how much should you invest initially? Round to the nearest dollar

Answers

The multiplier for continuous compounding is e^(rt). You want
700,000 = P·e^(0.05·25) ≈ 3.490343P
Then
700,000/3.490343 = P

You should invest $200,553 initially.

A rectangular pool is 12m long and 6m wide. A walkway of uniform width surrounds the pool. If the area of the walkway is 9m squared less than that of the pool, how wide is the walkway?

Answers

The walkway is 1.5 m wide.

The area of the pool is 12(6) = 72 m².

Adding a walkway of unknown width, x, around all 4 sides of the pool increases the width by 2x and the length by 2x; thus the area of the entire pool and walkway together would be given by

(12+2x)(6+2x)

We know that the area of just the walkway is 9 m² less than the area of the pool. This means that:

(12+2x)(6+2x)-72 = 72-9

Multiplying through we have:
12*6+12*2x+2x*6+2x*2x - 72 = 63
72 + 24x + 12x + 4x² - 72 = 63
24x + 12x + 4x² = 63
36x + 4x² = 63

Writing in standard form we have:
4x² + 36x = 63

We want to set it equal to 0 to solve, so subtract 63 from both sides:
4x² + 36x - 63 = 63 - 63
4x² + 36x - 63 = 0

Using the quadratic formula,

[tex]x=\frac{-b\pm \sqrt{b^2-4ac}}{2a} \\ \\=\frac{-36\pm \sqrt{36^2-4(4)(-63)}}{2(4)} \\ \\=\frac{-36\pm \sqrt{1296--1008}}{8}=\frac{-36\pm \sqrt{1296+1008}}{8} \\ \\=\frac{-36\pm \sqrt{2304}}{8}=\frac{-36\pm 48}{8}=\frac{-36+48}{8}\text{ or }\frac{-36-48}{8} \\ \\=\frac{12}{8} \text{ or }\frac{-84}{8}=1.5 \text{ or }-10.5[/tex]

Since a negative width makes no sense, the walkway is 1.5 m wide.

math teacher doesn't explain this very well, please help solve.

Answers

a) Since we know they have equal perimeters, we can set them equal to each other. First, let's recap the perimeters of squares and triangles (s = side):
[tex]P_s = 4s/tex]
[tex]P_t = 3s=s+s+s[/tex]

Now, we can set up our equation:
[tex]4(x-3) = 8 + x + x[/tex]
[tex]4x-12 = 8+2x[/tex]

We can solve for x:
[tex]2x-12=8[/tex]
[tex]2x = 20[/tex]
[tex]x = \frac{20}{2} = 10[/tex]

So, x = 10.

b) We known the perimeter formulas and we have what x is. So, let's just plug and chug.
Square:
[tex]4(x-3) = 4(10-3) = 40-12 = 28[/tex]
So, the perimeter of the square is 28 units².
Since their perimeters are equal the perimeter of the triangle is also 28 units²

Number of dinners = 125 Individual portion = 8 oz. Size of can = 32 oz. Number of cans required = _____. 2

Answers

To get total of oz for the dinner:

125 * 8 = 1000 oz

To get number of cans:

1000 ÷ 32 = 3.125 cans or at least 4 cans to served dinner

Answer:

Number of cans required = 32

Step-by-step explanation:

Number of dinners = 125

Individual portion size of each dinner = 8 oz.

Total portion size of all the dinners = 125 × 8

⇒ Total portion size of all the dinners = 1000 oz.

Size of one can = 32 oz.

[tex]\text{Number of cans required = }\frac{\text{Total portion size of dinner}}{\text{Size of each can}}\\\\\implies\text{Number of cans required = }\frac{1000}{32}\approx 31.25[/tex]

Hence, Number of cans required = 32

which of the following must be true?
check ALL that apply,

Answers

B and D are true.

_____
The Law of Cosines tells you
.. c^2 = a^2 +b^2 -2ab*cos(θ)
Your problem statement provides a relation involving c^2, so we can substitute to get
.. a^2 +b^2 > a^2 +b^2 -2ab*cos(θ)
.. 0 > -2ab*cos(θ)
.. cos(θ) > 0
In the range 0 ≤ θ ≤ 180°, cos(θ) will be positive for θ < 90°, an acute angle.
Thus A and C are false, B and D are true.

Can someone help me please!!

Answers

To factor a trinomial, first we should factor out the GCF, or greatest common factor, shared by all of the coefficients and constants. The greatest common factor of -3, -21, and -30 is -3, therefore, this is the number we must factor out first.

When we factor out the GCF of -3, we get:
-3(q^2 + 7q + 10)

Next, we must look for two numbers that will serve as coefficients for the variables in factored form. These two numbers must add to 7 (the middle term) and multiply to 10 (the constant or last term). The two numbers that satisfy these parameters are 5 and 2.

Therefore, we must use these numbers to create factors of the innermost factor that we have left, to get:

-3(q+5)(q+2)

The above expression is your final answer, because you cannot factor any more. And remember: you can always check to make sure that your factoring is correct by multiplying out the factors and making sure you get your original trinomial.

I hope this helps!

PLZ HELP.ONLY ANSWER IF YOUR 100% SURE. which equation represents a non proportional situation? A. y = 4x B. y = 5x - 1 C. y = 16x D. y = -8x

Answers

y = 5x - 1

-1 is a y-intercept

if there is a y-intercept, it is non proportional

Hope I helped

Let me know if you need anything else!

~ Zoe

the answer is b because the y-intercept is -1

what is the surface area of the following figure and I need work to please and thank you

Answers

[tex]\bf \textit{total surface area of a cylinder}\\\\ SA=2\pi r(h+r)~~ \begin{cases} h=height\\ r=radius\\ -----\\ r=4\\ h=11 \end{cases}\implies SA=2\pi (4)(11+4) \\\\\\ SA=8\pi (15)\implies SA=120\pi [/tex]

the answer is C. 376.8 cm²

What is the approximate volume of a cylinder with a radius of 2 centimeters and height of 10 centimeters? Drag and drop the answer into the box.

A. 125.6

B. 251.3

C. 502.7

D. 1005.3

Answers

Formula
V = pi * r^2 h

where
V = ??
pi = 3.14
r = 2 cm
h = 10 cm
V = 3.14 * 2*2 * 10
V = 3.14 * 4 * 10
V = 125.6 cm^3
Choice A <<<<====

Answer:

125.6

Step-by-step explanation:

PLZ HELP
The price of a technology stock was 9.83
yesterday. Today, the price fell to 9.72
. Find the percentage decrease. Round your answer to the nearest tenth of a percent.

@nilbert

Answers

The percentage change can be found from
percent change = ((new value)/(original value) - 1) × 100%
For your numbers, that is
percent change = ((9.72/9.83) - 1) × 100% = -0.01119... × 100%

The decrease in value is about 1.12%

Tom and Sonya were comparing their answers on a math problem. It was an order of operations problem, and they each had a different answer. Tom had an answer of 11, and Sonya had an answer of 104. Which student correctly solved the order of operations problem given by the teacher?

(52 +6)–(7•4)+8

*Challenge: Can you nd the mistake that Sonya made in completing her problem?

Answers

Order of operations is PEMDAS (parentheses, exponents, multiplication, division, addition, subtraction)

= (52 + 6) - (7 * 4) + 8
work inside parentheses first
= 58 - 28 + 8
work left to right now
= 30 + 8
= 38

SONYA'S MISTAKE
= (52 + 6) - (7 * 4) + 8
I think she added the 52, 6+4, multiplied 7*4 to equal 28, then added the 6+8 to equal 14
= 52 + 28 + 14 + 10
= 104

ANSWER: 38

Hope this helps! :)

the angles formed by two perpendicular lines have a measure of

Answers

Hi there!

Angles of perpendicular lines are always 90 degrees.

Hope this helps!

What is the value of X?
Write as decimal.

Answers

we have that
dMN/dMP=dAM/x
dAM=46.2-14=32.2 cm

then
x=dAM*[dMP/dMN]=32.2*[72.6/46.2]=50.60 cm

the answer is x=50.60 cm

At Central High School, there are 60 football players and 25 cheerleaders. What is the ratio of cheerleaders to football players? please slimlfy

Answers

the simplified ratio is 5/12

Answer:

The answer is 5:12

Step-by-step explanation:

Mark’s salary is $900 per month. What is his yearly salary? $

Answers

His yearly salary is 10,800

900$*12months=10,800$

Jean adds 35 and 9. How can she solve using only equations to model her thinking?

Answers

since she only has a two digit number and a one digit number she will be able to create one or two equations.

35+9=44
Or she may want to break up the two digit number into two or more parts, for example: (5×6) + (1×5) + 9 = 44 or (7×5) + 9 = 44

FIND.......16 2/3 % of 96 lamps

Answers

Convert percentage to fraction by dividing by 100
16 2/3 = 16 2/3 ÷ 100 = 1/6

16 2/3 % of 96 = 1/6 x 96 = 16

Ans: 16 2/3 % of 96 lamps = 16

32% of 50 please answer fast need help ............

Answers

The answer for 32% of 50 is 16.

hello,

the answer is 16.

32% needs to change to 0.32. then, you multiply 0.32 by 50, and the sum is 16.

hope you have a great day / night :)

Savanna had 40 iris blooms last year. This year she had 15% more iris blooms. How many more iris blooms did Savanna have this year?

Answers

40*15%=6

40+15%=46

hope one of these helped

To solve this problem what you can do is the following rule of three:
40 iris blooms -----> 100%
x iris blooms -------> 115%
Clearing the value of x we have:
x = (115/100) * (40)
x = 46
Then, comparing with last year we have:
40-46 = 6
Answer:
Savanna had this year 6 more iris blooms than last year.

Which of the following uses absolute value correctly to show the distance between -40 and 18

Answers

The third selection is appropriate.
|-40 -18| = |-58| = 58 . . . . units

_____
In the first selection, it is trying to tell you |-58| = -58. That is not so. The result of taking the absolute value is that the sign is never negative.

the average number of minutes that it took jermaine to complete his homework assignments for a week in september was 27 minutes, with a mean absolute deviation of 4 minutes. with this informatiom, which statement is true. Pls someone help me

Answers

Answer:

On average, Jermaine spent between 23 and 31 minutes on his homework.

On average, Jermaine spent between 23 and 31 minutes on his homework.

What is Mean absolute deviation?

This is defined as the average of the absolute deviations from a central point.

In this scenario, the mean absolute deviation

= 27 ± 4 minutes

= 27+4 or 27 - 4

= 31 or 23 minutes.

Therefore we can infer that Jermaine spent between 23 and 31 minutes on his homework.

Read more about Mean absolute deviation here https://brainly.com/question/447169

what is the probability of randomly picking a girl without brown hair

Answers

6 girls have brown hair, and you want to know the probability of finding one without brown hair, so you take the population and subtract brown hair, or 22. You have a 22/28 chance of picking a girl without brown hair, simplified to 11/14.

A cell phone uses a 3.0 V battery. The circuit board it uses needs a 0.05 A current. What size resistor is needed to generate this current?

___ Ω

Answers

Ohm's law equation used to solve this problem is;
V = IR
where V - voltage
I - current passing
R - resistance
voltage is the potential difference between 2 points and current passing through these 2 points is proportional to the potential difference, in other words voltage.
V∞I
Resistance is the constant in the equation
the current required to pass through is 0.05 A (amperes)
the battery voltage is 3.0 V (volts)
substituting these values
3 V = 0.05 A * R
R = 3/0.05
R = 60 Ω (ohms)

Answer:

the answer is 60.

Step-by-step explanation:

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