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
monitta
2 years ago
13

A 50-gal tank initially contains 10 gal of fresh water. At t = 0, a brine solution

Mathematics
1 answer:
scZoUnD [109]2 years ago
5 0

\huge \mathbb{SOLUTION:}

\begin{array}{l} \textsf{Let }A(t)\textsf{ be the function which gives the amount} \\ \textsf{of the salt dissolved in the liquid in the tank at} \\ \textsf{any time }t. \textsf{ We want to develop a differential} \\ \textsf{equation that, when solved, will give us an} \\ \textsf{expression for }A(t). \\ \\ \textsf{The basic principle determining the differential} \\ \textsf{equation is} \\ \\ \end{array}

\boxed{ \footnotesize \begin{array}{l} \qquad\quad \quad\Large{\dfrac{dA}{dt} = R_{in} - R_{out}} \\ \\ \textsf{where:} \\ \\ \begin{aligned} \bullet\: R_{in} &= \textsf{rate of the salt entering} \\ &= \left({\footnotesize \begin{array}{c}\textsf{Concentration of} \\\textsf{salt inflow}\end{array}}\right) \times \small(\textsf{Input of brine}) \\ \\ \bullet\: R_{out} &= \textsf{rate of the salt leaving} \\ &= \left({\footnotesize \begin{array}{c}\textsf{Concentration of} \\\textsf{salt outflow}\end{array}}\right) \times \small(\textsf{Output of brine}) \end{aligned} \end{array}} \\ \\

\begin{array}{l} \textsf{On the problem, the amount of salt in the tank,} \\ A(t), \textsf{changes overtime is given by the differential} \\ \textsf{equation}  \\ \\ \footnotesize A'(t) = \left(\dfrac{4\ \textsf{gal}}{1\ \textsf{min}}\right)\!\!\left(\dfrac{1\ \textsf{lb}}{1\ \textsf{gal}}\right) - \left(\dfrac{2\ \textsf{gal}}{1\ \textsf{min}}\right)\!\!\left(\dfrac{A(t)\ \textsf{lb}}{10 + (4 - 2)t\ \textsf{gal}}\right) \\ \\ \textsf{There's no salt in the tank (fresh water) at the} \\ \textsf{start, so }A(0) = 0. \textsf{ The amount of solution in the} \\ \textsf{tank is given by }10 + (4 -2)t, \textsf{so the tank will} \\ \textsf{overflow once this expression is equal to the total} \\ \textsf{volume or capacity of the tank.} \\ \\ 10 + (4 - 2)t = 50 \\ \\ \textsf{Solving for }t,\textsf{ we get} \\ \\ \implies \boxed{t = 20\textsf{ mins}} \\ \\ A'(t) = 4 - \dfrac{2A(t)}{10 + 2t} \\ \\ A'(t) = 4 - \dfrac{1}{5 + t} A(t) \\ \\ A'(t) + \dfrac{1}{5 + t} A(t) = 4 \\ \\ \textsf{This is a linear ODE with integrating factor} \\ \mu (t) = e^{\int \frac{1}{5 + t}\ dt} = e^{\ln |5 + t|} = 5 + t \\ \\ \textsf{Multiplying this to the ODE, we get} \\ \\ (5 + t)A'(t) + A(t) = 4(5 + t) \\ \\ [(5 + t)A(t)]' = 20 + 4t \\ \\ (5 + t)A(t) = 20t + 2t^2 + C \\ \\ \textsf{Since }A(0) = 0, \textsf{ we get } C = 0. \\ \\ A(t) = \dfrac{2t^2 + 20t}{t + 5} \\ \\ A(t) = 2t + 10 - \dfrac{50}{t + 5} \\ \\ \textsf{So the function that gives the amount of salt at} \\ \textsf{any given time }t,\textsf{ is given by} \\ \\ \implies A(t) = 2t + 10 - \dfrac{50}{t + 5} \\ \\ \textsf{The amount of salt in the tank at the moment} \\ \textsf{of overflow or at }t = 20\textsf{ mins is equal to} \\ \\ A(20) = 2(20) + 10 - \dfrac{50}{20 + 5} \\ \\ \implies \boxed{A = 48\ \textsf{gallons}} \end{array}

\Large \mathbb{ANSWER:}

\qquad\red{\boxed{\begin{array}{l} \textsf{a. }20\textsf{ mins} \\ \\ \textsf{b. }48\textsf{ gallons}\end{array}}}

#CarryOnLearning

#BrainlyMathKnower

#5-MinutesAnswer

You might be interested in
Peter went to his local zoo where 50% of its exhibits featured lions. If the zoo features 24 exhibits in total, then how many of
Sauron [17]
12 exhibits featured lions since half of 24 is 12.
3 0
3 years ago
Read 2 more answers
25% of what number is 9
Alexeev081 [22]
Keep in mind the the word "percent" translates to "for each 100." 25% can be read as "25 for each 100," which has the fractional representation \frac{25}{100} and the decimal representation 0.25. All of these representations are just different ways of writing the same value.

Note too that the fraction \frac{25}{100} can be reduced to \frac{1}{4}, so what the question is really asking is: 1/4 of what number is 9?

Well, if we know that 1/4 of the number is 9, we can easily find how large the whole number is by finding all 4/4 of it (the <em>whole</em> number)<em /><em />, or 4 x 9, which multiplies out to 36.
4 0
3 years ago
Kimiko works at the drive-thru window in a fast-food restaurant. At the end of a 2-hour shift, the display at her window shows t
Anastaziya [24]

Answer:

90 customers

Step-by-step explanation:

Total shift time  = 2 hours

1 hours = 60 minutes

Total shift time minutes = 2*60 minutes = 120 minutes

Average Time taken to serve 1 customer = 1 minute 20 seconds

lets convert  1 minute 20 seconds in fraction

60 seconds = 1 minute

20 seconds  = 20/60 minutes = 1/3 minutes

Thus,

Average Time taken to serve 1 customer = 1 minute + 1/3 minutes = 4/3 minutes

Lets assume she served x customer in her 2 hour shift

total time taken to serve x customer =x*Average Time taken to serve 1 customer = 4x/3 minutes

Given that she the customers for her shift time which is 120 minutes

4x/3 minutes = 120 minutes

x = 120*3/4 = 90

Thus,

Kimiko served 90 customers and this is the number of customer which came through the drive thru during those 2 hours.

3 0
3 years ago
a company rents out 18 food booths at the county fair. The fee for a food booth is $175 plus $7 per day. The fee for a game boot
aliya0001 [1]

Answer:

18 food booths initial costs are $(3,150 + 7d)

Step-by-step explanation:

8 0
2 years ago
Evaluate A² for A = -3.<br> O-6<br> O 9<br> O 6<br> O-9
Annette [7]

Answer:

A² = 9

Step-by-step explanation:

substitute A = - 3 into A² , that is

A² = (- 3)² = - 3 × - 3 = 9

3 0
1 year ago
Other questions:
  • Li needs to buy 2 kilograms of apples. If she buys 9 apples that each weigh approximately 150 grams, will she have enough? Expla
    14·1 answer
  • Describe the graph of of the equation x = -2. Is the equation a function?
    6·2 answers
  • Simplify the expression |ab| if a&lt;0 and b&gt;0
    9·1 answer
  • A sequence starts:
    9·1 answer
  • If anyone can help please I will do whatever you want Nancy found that 2 cups of strawberries weigh one-half pound, and 8 cups o
    11·2 answers
  • One banana and one peach weigh the same as 14 strawberries .Two peaches weigh the same as 10 strawberries how much does a banana
    8·1 answer
  • Help meeeeeeeeeeeeeeeeeeeeeeeeee
    7·2 answers
  • Please help me for urgent<br>​
    9·1 answer
  • What is the best way of choosing a sample to statistically represent a population? Why?
    11·1 answer
  • The following graph shows a proportional relationship.
    8·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!