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

Wittaler [7]

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:

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

3 0

3 years ago

Find the slope of the line through the given points (1,6),(-8,-9)

cestrela7 [59]

The answer to the question

6 0

2 years ago

Suppose a population is growing by 8% each year. How many years will it take the population to double?

Doss [256]

ANSWER

$\text{9 years}$

EXPLANATION

We want to find the number of years that it will take the population to double.

To do this, we have to apply the exponential growth function:

$y=a(1+r)^t$

where y = final value

a = initial value

r = rate of growth

t = time (in years)

For the population to double, it means that the final value must be 2 times the initial value:

$y=2a$

Substitute the given values into the function above:

$\begin{gathered} 2a=a(1+\frac{8}{100})^t \\ \frac{2a}{a}=(1+0.08)^t=1.08^t \\ 2=1.08^t \end{gathered}$

To solve further, convert the function from an exponential function to a logarithmic function as follows:

$\log _{1.08}2=t$

Solve for t:

$\begin{gathered} \frac{\log _{10}2}{\log _{10}1.08}=t \\ \Rightarrow t=9\text{ years} \end{gathered}$

It will take 9 years for the population to double.

4 0

9 months ago

Two more than the Quotient of twelve and a number

chubhunter [2.5K]

Answer:

x/12+2

Step-by-step explanation:

the answer to the question

4 0

3 years ago

On a map, 1 inch represents 100 feet. How many inches would

ohaa [14]

Answer:

3.5

Step-by-step explanation:

6 0

2 years ago

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