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
Jet001 [13]
3 years ago
9

A 400 gallon tank initially contains 100 gal of brine containing 50 pounds of salt. Brine containing 1 pound of salt per gallon

enters the tank at a rate o 5 gal/s, and the well-mixed brine in the tank flows out at the rate of 3 gal/s. How much salt will the tank contain when it is full of brine?

Mathematics
1 answer:
posledela3 years ago
6 0

Answer:

The amount of salt in the tank when it is full of brine is 393.75 pounds.

Step-by-step explanation:

This is a mixing problem. In these problems we will start with a substance that is dissolved in a liquid. Liquid will be entering and leaving a holding tank. The liquid entering the tank may or may not contain more of the substance dissolved in it. Liquid leaving the tank will of course contain the substance dissolved in it. If Q(t) gives the amount of the substance dissolved in the liquid in the tank at any time t we want to develop a differential equation that, when solved, will give us an expression for Q(t).

The main equation that we’ll be using to model this situation is:

Rate of change of <em>Q(t)</em> = Rate at which <em>Q(t)</em> enters the tank – Rate at which <em>Q(t)</em> exits the tank

where,

Rate at which Q(t) enters the tank = (flow rate of liquid entering) x

(concentration of substance in liquid entering)

Rate at which Q(t) exits the tank = (flow rate of liquid exiting) x

(concentration of substance in liquid exiting)

Let y<em>(t)</em> be the amount of salt (in pounds) in the tank at time <em>t</em> (in seconds). Then we can represent the situation with the below picture.

Then the differential equation we’re after is

\frac{dy}{dt} = (Rate \:in)- (Rate \:out)\\\\\frac{dy}{dt} = 5 \:\frac{gal}{s} \cdot 1 \:\frac{pound}{gal}-3 \:\frac{gal}{s}\cdot \frac{y(t)}{V(t)}  \:\frac{pound}{gal}\\\\\frac{dy}{dt} =5\:\frac{pound}{s}-3 \frac{y(t)}{V(t)}  \:\frac{pound}{s}

V(t) is the volume of brine in the tank at time <em>t. </em>To find it we know that at time 0 there were 100 gallons, 5 gallons are added and 3 are drained, and the net increase is 2 gallons per second. So,

V(t)=100 + 2t

We can then write the initial value problem:

\frac{dy}{dt} =5-\frac{3y}{100+2t} , \quad y(0)=50

We have a linear differential equation. A first-order linear differential equation is one that can be put into the form

\frac{dy}{dx}+P(x)y =Q(x)

where <em>P</em> and <em>Q</em> are continuous functions on a given interval.

In our case, we have that

\frac{dy}{dt}+\frac{3y}{100+2t} =5 , \quad y(0)=50

The solution process for a first order linear differential equation is as follows.

Step 1: Find the integrating factor, \mu \left( x \right), using \mu \left( x \right) = \,{{\bf{e}}^{\int{{P\left( x \right)\,dx}}}

\mu \left( t \right) = \,{{e}}^{\int{{\frac{3}{100+2t}\,dt}}}\\\int \frac{3}{100+2t}dt=\frac{3}{2}\ln \left|100+2t\right|\\\\\mu \left( t \right) =e^{\frac{3}{2}\ln \left|100+2t\right|}\\\\\mu \left( t \right) =(100+2t)^{\frac{3}{2}

Step 2: Multiply everything in the differential equation by \mu \left( x \right) and verify that the left side becomes the product rule \left( {\mu \left( t \right)y\left( t \right)} \right)' and write it as such.

\frac{dy}{dt}\cdot \left(100+2t\right)^{\frac{3}{2}}+\frac{3y}{100+2t}\cdot \left(100+2t\right)^{\frac{3}{2}}=5 \left(100+2t\right)^{\frac{3}{2}}\\\\\frac{dy}{dt}\cdot \left(100+2t\right)^{\frac{3}{2}}+3y\cdot \left(100+2t\right)^{\frac{1}{2}}=5 \left(100+2t\right)^{\frac{3}{2}}\\\\\frac{dy}{dt}(y \left(100+2t\right)^{\frac{3}{2}})=5\left(100+2t\right)^{\frac{3}{2}}

Step 3: Integrate both sides.

\int \frac{dy}{dt}(y \left(100+2t\right)^{\frac{3}{2}})dt=\int 5\left(100+2t\right)^{\frac{3}{2}}dt\\\\y \left(100+2t\right)^{\frac{3}{2}}=(100+2t)^{\frac{5}{2} }+ C

Step 4: Find the value of the constant and solve for the solution y(t).

50 \left(100+2(0)\right)^{\frac{3}{2}}=(100+2(0))^{\frac{5}{2} }+ C\\\\100000+C=50000\\\\C=-50000

y \left(100+2t\right)^{\frac{3}{2}}=(100+2t)^{\frac{5}{2} }-50000\\\\y(t)=100+2t-\frac{50000}{\left(100+2t\right)^{\frac{3}{2}}}

Now, the tank is full of brine when:

V(t) = 400\\100+2t=400\\t=150

The amount of salt in the tank when it is full of brine is

y(150)=100+2(150)-\frac{50000}{\left(100+2(150)\right)^{\frac{3}{2}}}\\\\y(150)=393.75

You might be interested in
Select all equations whose graphs have a vertex with x-coordinate 2
iren2701 [21]

Answer:

C

Step-by-step explanation:

because you have to foil then do -b/2a

8 0
3 years ago
A car dealership has 240 cars in the parking lot and 17.5% of them are red. Of the other 6 colors in the lot, each color has the
Nonamiya [84]

Answer:

Step-by-step explanation:

Ok guys I’m really confused

4 0
3 years ago
Help how do and is help?
Dimas [21]

Answer:

x = 12

Step-by-step explanation:

The total of all the angle is 180 degrees so just divide 180 by the total number of x's which is 5+4+3+2+1 = 15

180/15 = 12

If you wanted to know how much each angle is just plug in 12 for x.

So first angle is just X so it is 12 degrees.

Second angle is 2X so 2x12 = 24 degrees.

and so on for all the angles.

6 0
3 years ago
Devra soled the
tester [92]

Answer:

What’s the problem? If you could attach the problem we can help u out!

Step-by-step explanation:

7 0
3 years ago
Point What is the value of x2 – 3x + 2 when x=4?​
andrey2020 [161]

Answer:

Answer is -2

Step-by-step explanation:

6 0
3 years ago
Other questions:
  • How much would you need to deposit in an account each month in order to have $50,000 in the account in 8 years? Assume the accou
    14·1 answer
  • Find the slope of the line that passes through the pair points (4,5) (10,0)
    8·1 answer
  • The ticket office at Orchestra Center estimates that if it charges x dollars for box seats for a concert, it will sell 50 - x bo
    9·2 answers
  • Can some one help knock a math sheet out? Model and solver addition and subtraction inequalities?
    8·1 answer
  • Which of these sets of ordered pairs below does not represent a function?
    14·1 answer
  • Giving Brainliest if correct!!
    12·2 answers
  • The New England Revolutions are having a charity soccer event for which tickets cost $18 per person. However, the ticket agency
    7·1 answer
  • I BEGGG U TO HELP Me PLZZ ITS tESt<br> I WILl GiVE U BRAINLIESTTTT),:
    15·1 answer
  • What is the answer <br> -48&gt;6x =
    8·2 answers
  • Miguel and three of his friends went to the movies. They originally had a total of $40. Each boy had the same amount of money an
    6·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!