Please help thanks so much

Suppose that a large mixing tank initially holds 500 gallons of water in which 50 pounds of salt have been dissolved. Another br

aliya0001 [1]

Answer:

$A=1500-1450e^{-\dfrac{t}{250}}$

Step-by-step explanation:

The large mixing tank initially holds 500 gallons of water in which 50 pounds of salt have been dissolved.

Volume = 500 gallons

Initial Amount of Salt, A(0)=50 pounds

Brine solution with concentration of 2 lb/gal is pumped into the tank at a rate of 3 gal/min

$R_{in}$ =(concentration of salt in inflow)(input rate of brine)

$=(2\frac{lbs}{gal})( 3\frac{gal}{min})\\R_{in}=6\frac{lbs}{min}$

When the solution is well stirred, it is then pumped out at a slower rate of 2 gal/min.

Concentration c(t) of the salt in the tank at time t

Concentration, $C(t)=\dfrac{Amount}{Volume}=\dfrac{A(t)}{500}$

$R_{out}$ =(concentration of salt in outflow)(output rate of brine)

$=(\frac{A(t)}{500})( 2\frac{gal}{min})\\R_{out}=\dfrac{A}{250}$

Now, the rate of change of the amount of salt in the tank

$\dfrac{dA}{dt}=R_{in}-R_{out}$

$\dfrac{dA}{dt}=6-\dfrac{A}{250}$

We solve the resulting differential equation by separation of variables.

$\dfrac{dA}{dt}+\dfrac{A}{250}=6\\$The integrating factor: e^{\int \frac{1}{250}dt} =e^{\frac{t}{250}}\\$Multiplying by the integrating factor all through\\\dfrac{dA}{dt}e^{\frac{t}{250}}+\dfrac{A}{250}e^{\frac{t}{250}}=6e^{\frac{t}{250}}\\(Ae^{\frac{t}{250}})'=6e^{\frac{t}{250}}$

Taking the integral of both sides

$\int(Ae^{\frac{t}{250}})'=\int 6e^{\frac{t}{250}} dt\\Ae^{\frac{t}{250}}=6*250e^{\frac{t}{250}}+C, $(C a constant of integration)\\Ae^{\frac{t}{250}}=1500e^{\frac{t}{250}}+C\\$Divide all through by e^{\frac{t}{250}}\\A(t)=1500+Ce^{-\frac{t}{250}}$

Recall that when t=0, A(t)=50 (our initial condition)

$50=1500+Ce^{-\frac{0}{250}}50=1500+Ce^{0}\\C=-1450\\$Therefore the amount of salt in the tank at any time t is:\\A=1500-1450e^{-\dfrac{t}{250}}$

4 0

3 years ago

I’ll give brainliest <br> Completed a full explanation of statistics with explanation and examples ?

slamgirl [31]

Answer: Hope This Helps!

Step-by-step explanation:

Statistic Definition: A mathematical science concerned with data collection, presentation, analysis, and interpretation

Examples: "Statistics is the only mathematical field required for many social sciences." and "The statistics from the Census for apportionment are available."

3 0

3 years ago

0.524 as a percentage

brilliants [131]

Answer:

52.4%

Step-by-step explanation:

Simply multiply the value with 100 to find the percentage

thus 0.524× 100 = 52.4%

3 0

4 years ago

The area of one field is 16 acres, and the area of the second field is 1 1/4 times bigger. A barn of wheat on the first field ma

Elodia [21]

Answer:

858 thousand pounds

Step-by-step explanation:

The first field has an area of 16 acres; the second has an area "1 1/4 times bigger".

Most people will interpret that incorrectly, saying the area of the second field is 16 time 1 1/4, which is 20. But the grammatically correct interpretation of "1 1/4 times bigger" is the size of the first field, plus 1 1/4 more times the size of the first field.

So the size of the second field in acres is 16+%2B+1.25%2A16+=+36

So the first field has an area of 16 acres and the second has an area of 36 acres.

The first field yields 37.5 thousand pounds of wheat per acre. The total amount of wheat from the first field is 16%2A37500+=+600000

The yield from the second filed is 1 2/15 times as much per acre as the first field. The yield in pounds per acre for the second field is 37500%2A%2817%2F15%29+=+40500.

So the total amount of wheat from the second field is 36%2A40500+=+1458000

The difference in the total yield from the two fields is 1458000-600000+=+858000

The second field yielded 858 thousand pounds more of wheat.

4 0

3 years ago

Read 2 more answers

The vertices of a figure are FC -9,- 9),G-3, -6),H(-3,-9). Find the coordinates of the image after a dilation with a scale facto

Zanzabum

Answer:

Math is really really

7 0

4 years ago