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
posledela
3 years ago
8

6. A tank contains 100-gallon of pure water. At time t = 0, a solution containing 2 lb of salt per gallon flows into the tank at

a rate of 3 gallons per minute, and the well-stirred mixture flows out at a rate of 4 gallons per minute. Find the amount (in lb) of salt Q in the solution as a function of t in minutes.
Mathematics
1 answer:
SOVA2 [1]3 years ago
7 0

Answer:

The answer is below

Step-by-step explanation:

Let Q represent the amount of salt in the tank at time t.

\frac{dQ}{dt}= flow\ in - flow\ out\\ \\flow\ in=3\ gal/min*2\ lb/gal=6\ lb/min\\\\net\ gain\ in\ tank\ volume=3-4=-1, henceflow\ out= \frac{4Q}{100-t} \\\\\frac{dQ}{dt}= 6-\frac{4Q}{100-t} \\\\\frac{dQ}{dt}+ \frac{4Q}{100-t}=6\\\\The \ integrating\ factor\ is:\\\\IF=e^{\int\limits {\frac{4}{100-t} } \, dt }=e^{-4\int\limits {\frac{-1}{100-t}}=e^{-4ln(100-t)}=(100-t)^{-4}}\\\\Multiplying\ through  \ by\ IF: \\\\(100-t)^{-4}\frac{dQ}{dt}+ (100-t)^{-4}\frac{4Q}{100-t}=6(100-t)^{-4}\\\\

Integrating:\\\\A(100-t)^{-4}=-2(100-t)^{-3}+c\\\\A=-2(100-t)+\frac{c}{(100-t)^{-4}} \\\\at, t=0,A=0\\\\0=-2(100-0)+\frac{c}{(100-0)^{-4}}\\\\c=0.02\\\\A=-2(100-t)+\frac{0.02}{(100-t)^{-4}}

You might be interested in
A lawyer has an annual income of $173,482 and will pay a 5.5% income tax rate. What is the amount of income tax in dollars and c
gayaneshka [121]

Answer:

173,482-5.5=173,476.5

4 0
3 years ago
Which of the following functions are solutions of the differential equation y'' + y = 3 sin x? (select all that apply)
DiKsa [7]

Answer:

C

Step-by-step explanation:

We must compute the derivatives and check if the equation is satisfied.

A. y'=(3\sin x)'=3\cos x. Differentiate again to get y''=(3\cos x)'=-3\sin x, then y''+y=-3\sin x+3\sin x=0\neq 3\sin x so this choice of y doesn't solve the equation.

B. y'=3\sin x+3x\cos x-5\cosx +5x\sin x=(5x+3)\sin x+(3x-5)\cos x and y''=5\sin x+(5x+3)\cos x+3\cos x-(3x-5)\sin x=(10-3x)\sin x+(5x+6)\cos x, then y''+y=10\sin x+6\cos x\neq 3\sin x so y is not a solution

C. y'=\frac{-3}{2}\cos x+\frac{3}{2}x\sin x hence y''=\frac{3}{2}\sin x+\frac{3}{2}\sin x+\frac{3}{2}x\cos x=3\sin x+\frac{3}{2}x\cos x. Then y''+y=3\sin x+\frac{3}{2}x\cos x-\frac{3}{2}x\cos x=3\sin x so y is a solution.

D.y'=-3\sin x and y''=-3\cos x, then y''+y=0 thus y isn't a solution

E. y'=\frac{3}{2}\sin x+\frac{3}{2}x\cos x hence y''=\frac{3}{2}\cos x+\frac{3}{2}\cos x-\frac{3}{2}x\sin x=3\cos x-\frac{3}{2}x\cos x. Then y''+y=3\cos x-\frac{3}{2}x\cos x-\frac{3}{2}x\cos x=3\cos x\neq 3\sin x then y is not a solution.

3 0
3 years ago
Jada has 12 pencils and 24 erasers. She wants to create identical groups. What is the
nekit [7.7K]

Answer:

12

Step-by-step explanation:

6 0
3 years ago
Pls help me! I need help asap!<br><br>​
valentinak56 [21]

1.

18 units

10 units right and 8 units up.

2.

7 units

3 units right and 4 units down.

3.

8 units

1 units right and 7 units down

4.

8 units

5 units and 3 units down

7 0
3 years ago
Got It? Do these problemst
dedylja [7]
I did where you subtract 4 from both sides then divide 2 from 4 and you get 0.5 as x for your answer but im not sure
6 0
2 years ago
Other questions:
  • Every night, John spends 2 hours doing homework, 30 minutes eating, and 45 minutes playing outdoors. How many minutes does he sp
    15·2 answers
  • If a circle is inscribed in a square, which of the following must be true? Check<br> all that apply
    14·2 answers
  • I was wondering how do you solve this?
    7·1 answer
  • The dollar value v(t) of a certain car model that is t years old is given by the following exponential function.
    6·1 answer
  • I really need help with b and c
    10·1 answer
  • I need Help with Functions ​
    8·1 answer
  • Pls answer :c ! I need help ASAP ! (Math)
    8·1 answer
  • Meal A has 170 calories in 3 servings. Meal B has 160 calories
    8·1 answer
  • Write an equation in slope-intercept form of the line that passes through the given points
    14·1 answer
  • Which of the following is true about the relation shown below?
    12·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!