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A tank initially has 300 gallons of a solution that contains 50 lb. of dissolved salt. A brine solution with a concentration of

belka [17]

Let s(t) be the amount of salt in the tank at time t. Then s(0) = 50 lb.

Salt flows into the tank at a rate of

(2 gal/min) (6 lb/gal) = 12 lb/min

and flows out at a rate of

(2 gal/min) (s(t)/300 lb/gal) = s(t)/150 lb/min

so that the net rate at which salt is exchanged through the tank is

ds(t)/dt = 12 - s(t)/150 … (lb/min)

Solve for s(t). The DE is separable, so we have

ds/dt = 12 - s/150

150 ds/dt = 1800 - s

150/(1800 - s) ds = dt

Integrate both sides to get

-150 ln|1800 - s| = t + C

Solve for s :

ln|1800 - s| = -t/150 + C

1800 - s = exp(-t/150 + C )

1800 - s = C exp(-t/150)

s = 1800 - C exp(-t/150)

Now given that s(0) = 50, we solve for C :

50 = 1800 - C exp(-0/150)

50 = 1800 - C

C = 1750

Then the amount of salt in the tank at any time t ≥ 0 is

s(t) = 1800 - 1750 exp(-t/150)

To find the time it takes for the tank to hold 100 lb of salt, solve for t in

100 = 1800 - 1750 exp(-t/150)

1700 = 1750 exp(-t/150)

34/35 = exp(-t/150)

ln(34/35) = -t/150

t = -150 ln(34/35) ≈ 4.348

So it would take approximately 4.348 minutes.

By the way, we didn't have to solve for s(t), we could have instead stopped with

-150 ln|1800 - s| = t + C

Solve for C - this C is not the same as the one we found using the other method. s(0) = 50, so

-150 ln|1800 - 50| = 0 + C

C = -150 ln|1750|

==> t = 150 ln(1750) - 150 ln|1800 - s|

Then s(t) = 100 lb when

t = 150 ln(1750) - 150 ln(1700)

t = 150 ln(1750/1700)

t = 150 ln(35/34)

t ≈ 4.348

5 0

3 years ago

There are 8 volunteers during the weekday and only 5 volunteers on the

MaRussiya [10]

Answer:

Monday is a weekday, with which there would be 8 volunteers.

In turn, Saturday is a weekend day, with which there would be 5 volunteers.

2 people will not be helped if the ratio of volunteers to the person needing help is 1:1.

Step-by-step explanation:

Given that there are 8 volunteers during the weekday and only 5 volunteers on the weekend, to determine how many volunteers do you have on Monday and Saturday the following logical reasoning must be carried out:

Monday is a weekday, with which there would be 8 volunteers.

In turn, Saturday is a weekend day, with which there would be 5 volunteers.

In turn, if you have 10 people who need help from the volunteers on Monday, to determine how many will not get helped if the ratio of volunteers to the person needing help is 1: 1 the following logical reasoning must be carried out:

10: 8

1: 1

10 - 8 = 2

Thus, 2 people will not be helped if the ratio of volunteers to the person needing help is 1:1.

6 0

3 years ago

Read 2 more answers

What is the average rate of change from x = −3 to x = −4?

Vika [28.1K]

Answer:

-2

Step-by-step explanation:

Since it would be immensely helpful to know the equation of this parabola, we need to figure it out before we can continue. We have the work form of a positive upwards-opening parabola as

$y=a(x-h)^2+k$

where a is the leading coefficient that determines the steepness of lack thereof of the parabola, x and y are coordinates of a point on the graph, and h and k are the coordinates of the vertex. We know the vertex: V(-3, -3), and it looks like the graph goes through the point P(-2, -1). Now we will fill in the work form equation and solve for a:

$-1=a(-2-(-3))^2-3$

which simplifies a bit to

$-1=a(1)^2-3$

and

-1 = a(1) - 3. Therefore, a = 2 and our parabola is

$y=2(x+3)^2-3$

Now that know the equation, we can find the value of y when x = -3 (which is already given in the vertex) and the value of y when x = -4. Do this by subbing in the values of x one at a time to find y. When x = -3, y = -3 so the coordinate of that point (aka the vertex) is (-3, -3). When x = -4, y = -1 so the coordinate of that point is (-4, -1). The average rate of change between those 2 points is also the slope of the line between those 2 points, so we will use the slope formula to find it:

$m=\frac{-1-(-3)}{-4-(-3)} =\frac{2}{-1}=-2$

And there you have it! I'm very surprised that this question sat unanswered for so very long! I'm sorry I didn't see it earlier!

5 0

2 years ago

Beth wants to paint the four walls of her rectangular bedroom. Her room is 10 feet wide, 13 feet long, and 8 feet high. She will

steposvetlana [31]

Answer:

D; 1019 square feet

Step-by-step explanation:

10 x 13 x 8 = 1040 square feet

3 x 7 = 21 square feet

1040 - 21 = 1019 square feet (:

6 0

3 years ago

Read 2 more answers

Determine the angles. There’s only two answers.

LuckyWell [14K]

Answer:

its would be a and d

Step-by-step explanation:

6 times 5

3 0

2 years ago