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
madam [21]
3 years ago
6

A 500-gallon tank initially contains 220 gallons of pure distilled water. Brine containing 5 pounds of salt per gallon flows int

o the tank at the rate of 5 gallons per minute, and the well stirred mixture flows out of the tank at the rate of 5 gallons per minute. Find the amount of salt in the tank after 8 minutes.
Mathematics
1 answer:
Wittaler [7]3 years ago
3 0

Answer: The amount of salt in the tank after 8 minutes is 36.52 pounds.

Step-by-step explanation:

Salt in the tank is modelled by the Principle of Mass Conservation, which states:

(Salt mass rate per unit time to the tank) - (Salt mass per unit time from the tank) = (Salt accumulation rate of the tank)

Flow is measured as the product of salt concentration and flow. A well stirred mixture means that salt concentrations within tank and in the output mass flow are the same. Inflow salt concentration remains constant. Hence:

c_{0} \cdot f_{in} - c(t) \cdot f_{out} = \frac{d(V_{tank}(t) \cdot c(t))}{dt}

By expanding the previous equation:

c_{0} \cdot f_{in} - c(t) \cdot f_{out} = V_{tank}(t) \cdot \frac{dc(t)}{dt} + \frac{dV_{tank}(t)}{dt} \cdot c(t)

The tank capacity and capacity rate of change given in gallons and gallons per minute are, respectivelly:

V_{tank} = 220\\\frac{dV_{tank}(t)}{dt} = 0

Since there is no accumulation within the tank, expression is simplified to this:

c_{0} \cdot f_{in} - c(t) \cdot f_{out} = V_{tank}(t) \cdot \frac{dc(t)}{dt}

By rearranging the expression, it is noticed the presence of a First-Order Non-Homogeneous Linear Ordinary Differential Equation:

V_{tank} \cdot \frac{dc(t)}{dt} + f_{out} \cdot c(t) = c_0 \cdot f_{in}, where c(0) = 0 \frac{pounds}{gallon}.

\frac{dc(t)}{dt} + \frac{f_{out}}{V_{tank}} \cdot c(t) = \frac{c_0}{V_{tank}} \cdot f_{in}

The solution of this equation is:

c(t) = \frac{c_{0}}{f_{out}} \cdot ({1-e^{-\frac{f_{out}}{V_{tank}}\cdot t }})

The salt concentration after 8 minutes is:

c(8) = 0.166 \frac{pounds}{gallon}

The instantaneous amount of salt in the tank is:

m_{salt} = (0.166 \frac{pounds}{gallon}) \cdot (220 gallons)\\m_{salt} = 36.52 pounds

You might be interested in
How do we get an<br> equation in slope-intercept form?
Viefleur [7K]

Answer:

y = mx + b

Step-by-step explanation:

1. m is the slope (lesson on slope ) mnemonic : 'm' means 'move'

2. b is the y -intercept ( lesson on the y-intercept ) mnemonic : 'b' means where the line begins.

7 0
3 years ago
Which of the following scenarios MUST represent two independent events?
egoroff_w [7]

Answer: C) P(blue) = 70%, P(green) = 30%, P(blue and green) = 21%

Step-by-step explanation:

Given probabilities P(A) AND P(B) are independent, the probability that both events to occur is given by:

P(A)∩P(B)

thus from the example, independent events will be:

P(blue)=70%, P(green)=30%, P(blue and green)=21%

Because:

P(blue)∩P(green)=0.7×0.3=0.21=21%

7 0
2 years ago
I Need Help Fast Please
rodikova [14]
No its 11 i just did it
3 0
3 years ago
The inverse of the function graphed below is a function.<br><br>A. True<br>B. False
Anuta_ua [19.1K]
False.  In order to have an inverse, your function must pass the horizontal line test.  The graph fails the test.
7 0
3 years ago
Read 2 more answers
Denise wants to cover a 21.75 square foot wall with weathered wood. If each box of weathered wood contains 1.5 square feet of wo
natka813 [3]

Answer: 15 boxes

Step-by-step explanation:

From the question, we are informed that Denise wants to cover a 21.75 square foot wall with weathered wood and that each box of weathered wood contains 1.5 square feet of wood.

The number of boxes of weathered wood that she need will be calculated by dividing 21.75 square feet by 1.5 square feet. This will be:

= 21.75/1.5

= 14.5

= 15 boxes approximately

6 0
3 years ago
Other questions:
  • Find the quotient of 1+7i/3i in standard form write your answer in standard form
    15·1 answer
  • Critical Thinking: Buses on your route run every 9 minutes from 6:00 AM to 10:00 AM. You get to the bus stop at 7:16 AM,…
    5·1 answer
  • A boat travels 45 miles upstream (against the current) in 5 hours. The boat travels the same distance downstream in 3 hours. Wha
    15·1 answer
  • In the figure, ray OB bisects ∠TOY.
    9·1 answer
  • Which of the following is the function represented by the graph?
    10·2 answers
  • 9=x+1<br><br><br>Find the answer of it ​
    9·2 answers
  • The length of the side of the cube is 5x units, what is the volume of the cube?
    14·1 answer
  • Solve for y: 2x+3y=15
    6·2 answers
  • Parents will kill me if I don't get this done in an hour (BRAINLIEST IF ANSWER IS CORRECT AND WORK IS SHOWN)
    10·1 answer
  • What is the answer to this?
    13·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!