A large tank is filled to capacity with 600 gallons of pure water. brine containing 4 pounds of salt per gallon is pumped into the tank at a rate of 6 gal/min. the well-mixed solution is pumped out at the same rate. find the number a(t) of pounds of salt in the tank at time t.

Answers

Answer 1

If [tex]A(t)[/tex] is the amount of salt in the tank at time [tex]t[/tex], then the rate at which the amount of salt in the tank changes is given by

[tex]\dfrac{\mathrm dA(t)}{\mathrm dt}=\dfrac{4\text{ lbs}}{1\text{ gal}}\dfrac{6\text{ gal}}{1\text{ min}}-\dfrac{A(t)\text{ lbs}}{600\text{ gal}}\dfrac{6\text{ gal}}{1\text{ min}}[/tex]
[tex]\dfrac{\mathrm dA}{\mathrm dt}=24\dfrac{\text{lb}}{\text{min}}-\dfrac{A(t)}{100}\dfrac{\text{lb}}{\text{min}}[/tex]

Let's drop the units for now. We have

[tex]\dfrac{\mathrm dA(t)}{\mathrm dt}+\dfrac{A(t)}{100}=24[/tex]
[tex]e^{t/100}\dfrac{\mathrm dA(t)}{\mathrm dt}+e^{t/100}\dfrac{A(t)}{100}=24e^{t/100}[/tex]
[tex]\dfrac{\mathrm d}{\mathrm dt}\left[e^{t/100}A(t)\right]=24e^{t/100}[/tex]
[tex]e^{t/100}A(t)=\displaystyle24\int e^{t/100}\,\mathrm dt[/tex]
[tex]e^{t/100}A(t)=2400e^{t/100}+C[/tex]
[tex]A(t)=2400+Ce^{-t/100}[/tex]

We're given that the water is pure at the start, so [tex]A(0)=0[/tex], giving

[tex]A(0)=0=2400+Ce^{-0/100}\implies C=-2400[/tex]

So the amount of salt in the tank (in lbs) at time [tex]t[/tex] is

[tex]A(t)=2400\left(1-e^{-t/100}\right)[/tex]

Answer 2

Final answer:

To find the amount of salt in the tank at a given time, one can use the equation a(t) = Q - Qe^(-rt). In this case, Q (the quantity of salt at a steady state) equals the pump rate multiplied by the salt concentration (24lb/min), and r (the rate of inflow and outflow of the solution) is the rate at which water is pumped out divided by the volume of the tank (1/100 per min). Substituting these values into the equation gives the salt content at any given time.

Explanation:

The quantity of salt in the tank at any given time can be determined by the equation of the form a(t) = Q - Qe^(-rt), in which Q is the quantity of salt that would be in the tank at a steady state (i.e., if enough time had passed that the quantity of salt in the tank stopped changing), r is the rate of inflow and outflow of the solution, and t is the time at which you're trying to determine the number of pounds of salt in the tank.

In this case, Q = rate of inflow x concentration of the inflow, which is 6 gal/min x 4 lb/gal = 24 lb/min. This amount is reached after a sufficient amount of time has passed and the tank has reached a steady state.

The rate, r, is the rate at which the water is pumped out of the tank. In this situation, that's 6 gallons per minute. Since there are 600 gallons of water in the tank at the start, r = 6 gal/min ÷ 600 gallons = 1/100 min^-1.

Therefore, the number of pounds of salt in the tank at any time t is a(t) = Q - Qe^(-rt) = 24 lb/min - 24 lb/min * e^[-(1/100 min^-1)*t].

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Related Questions

Consider a company that selects employees for random drug tests. The company uses a computer to randomly select employee numbers that range from 1 to
6857
6857. Find the probability of selecting a number divisible by 1000. Find the probability of selecting a number that is not divisible by 1000.

Answers

A. 999 numbers under 1000, so prob of picking one is 999/6296 = 0.1587
B. 6 numbers divisible by 1000, so prob 6/6296 = 0.000953
C. (6296 - 6)/6296 = 1 - 0.000953 = 0.999047

Final answer:

The probability of selecting a number divisible by 1000 from the range of 1 to 6857 is approximately 0.0008763. The probability of selecting a number that is not divisible by 1000 is approximately 0.9991237.

Explanation:

To find the probability of selecting a number divisible by 1000, we need to determine the total number of possible outcomes and the number of favorable outcomes. There are a total of 6857 possible outcomes since the company selects employee numbers ranging from 1 to 6857. For a number to be divisible by 1000, it must end with three zeros, so we need to find how many numbers satisfy this condition between 1 and 6857.

Since 1000 is the smallest number divisible by 1000, we can calculate the number of favorable outcomes by finding the largest integer that satisfies the condition. In this case, it is 6000. Therefore, there are 6 favorable outcomes.

The probability of selecting a number divisible by 1000 is calculated by dividing the number of favorable outcomes by the total number of possible outcomes:

Probability = Favorable outcomes / Total outcomes = 6 / 6857 ≈ 0.0008763.

The probability of selecting a number that is not divisible by 1000 can be calculated as the complement of the probability of selecting a number divisible by 1000:

Probability(not divisible by 1000) = 1 - Probability(divisible by 1000) ≈ 1 - 0.0008763 ≈ 0.9991237.

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Information about movie ticket sales was printed in a movie magazine. in a sample of of fifty pg-rated movies, 36% had ticket sales in excess of $30,000,000. in a sample of thirty-five r-rated movies, 23% grossed over $30,000,000. suppose that this data is used to test the claim that the proportion of movies with ticket sales in excess of $30,000,000 is the same for pg movies as it is for r-rated movies. what would be the test statistic for this test? z = 1.29 z = 2.07 z = 4.005 z = 2.58

Answers

Let us say that the samples for fifty movies is 1 and samples for thirty five is 2. To solve the test statistic z, we can use the formula:

z = (p1 – p2) / sqrt [p (1 – p) (1 / n1 + 1 / n2)]

Where,

p1 = is the proportion for sample 1 = 0.36

p2 = is the proportion for sample 2 = 0.23

n1 = number of samples = 50

n2 = number of samples = 35

while p can be calculated using the formula:

p = [p1 * n1 / (n1 + n2)] + [p2 * n2 / (n1 + n2)]

p = [0.36 * 50 / (50 + 35)] + [0.23 * 35 / (50 + 35)]

p = 0.211764705 + 0.094705882

p = 0.306470587

Going back for the calculation of z:

z = (0.36 – 0.23) / sqrt [0.306 (1 – 0.306) (1 / 50 + 1 / 35)]

z = 1.279

Therefore the nearest answer is:

z = 1.29

value for X^2 left for 95% confidence interval when n=18

Answers

Given that n = 18, then the degree of freedom is n - 1 = 17.

From the table of Chi Square [tex](X^2)[/tex]

The value of [tex](X^2)[/tex] to the left for 95% confidence interval given a degree of freedom of 17 is 8.672

g(x) = x2 + 2, find g(3).

Answers

Plug in 3 for x on the right side
3^2+2
3 squared equals 9 plus 2 equals 11
Final answer: 11

Answer:

Plug in 3 for x on the right side

3^2+2,

3 squared equals 9 plus 2 equals 11

Final answer: 11

Step-by-step explanation:

EASY 5 POINTS!! Which expression gives the correct volume of the figure?

Answers

The correct answer is:
(3×2×1)+(3×1×1)

Answer:

(3×2×1)+(3×1×1)

Step-by-step explanation:

AS you can se ein the figure, the figure is not a solid full rectangula prism, it is made out of two prisms the first one is the 3x2x1 rectangular prism, and the second is the 3x1x1 square prism, so by adding the volume of those two you can figure out the volume of the whole figure that is why the answer to the problem is (3×2×1)+(3×1×1)

What is the sum of the arithmetic series? ∑[i=1,15,Fn=5i-1]

Answers

[tex]\bf \sum\limits_{i=1}^{15}\ 5i-1\implies 5\sum\limits_{i=1}^{15}\ i-\sum\limits_{i=1}^{15}\ 1\implies 5\left[ \cfrac{15(15+1)}{2} \right]-(15\cdot 1) \\\\\\ 5\cdot (15\cdot 8)-15\implies 585[/tex]

Given that x has a Poisson distribution with
mu
μ
equals
=
13
13, what is the probability that x
equals
=
5
5?
P(
5
5)
almost equals
≈

0.9930
0.9930 (Round to four decimal places as needed.)

Answers

The Poisson probability distribution function is
[tex]P(x;\mu)= \frac{e^{-\mu}\mu ^{x}}{x!} [/tex]
where
μ = mean number of successes
x = actual or expected number of successes

Given:
μ = 13
x = 5

Therefore the probability that x=5 is
P(x = 5) = (e⁻¹³*13⁵)/5!
= 0.8392/120
= 0.006994
= 0.0070 (to 4 dec. places)

Answer: 0.0070 (to 4 decimal places)

It is interesting to observe P(x) as x varies, as in the graph shown below.

simplify the expression below. show your work. -(-7y+12)

Answers

[tex]-(-7y+12) = 7y -12[/tex]

A taxi cab charges $0.55 per mile in addition to a $1.75 flat rate fee. Susie has $10 to spend on a taxi cab ride. The taxi driver will not give anyone a ride unless they are going somewhere that is more than 2 miles away. Model Susie’s situation with a system of inequalities.

Answers

The taxi driver doesn't give anyone a ride unless it's 2 miles away, as such you need to add at least $0.55x2 in addition to the $1.75 at the start
1.75+(0.55x2)=2.85
As Susie only has 10 dollars to spend, she can't exceed that. So Susie must spend more than $2.85(minimum from the taxi) and less than $10(her money)
so,
$2.85<Susie<$10

Which figure has the correct lines of symmetry drawn in?

Answers

The figure which has the correct lines of symmetry drawn is figure D.

What is line of symmetry?

A line of symmetry refers to the line that divides a shape or an object into two equal and symmetrical parts.

The figure given is a five sided shape referred to as a pentagon.

A pentagon has five lines of symmetry.

Therefore, the correct line of symmetry is 5 lines drawn from each angle respectively.

Answers

110,000 is your answer

hope this helps

110,000. it is that number because 5 is over 0,1,2,3,and 4 why end in 4 because to make it fair that have to make both 5numbera and 0 counts as a number so 5,6,7,8,and 9 are the other 5 why not 10 because 10 is the 0 in the other line so 5 is over the count and that means change the 5 to 0 and all the numbers following it to 0 and in the ten thousands place change 0 to 1. that was a lot to type.

You have a 20-foot ladder that you are using as you paint your house. You place the ladder so that it forms an angle of 30 at the point of contact between the ladder and the ground. How high will the top of the ladder be above the ground

Answers

First draw out the picture it will help. Then you can see that you want the opposite segment of the angle, which leads you to use a sine. So you have 20sin30 or 20(1/2) or 10ft.

Answer:

The Ladder is 10 ft high.

Step-by-step explanation

According to the question we have a 20 ft ladder inclined against a wall at a 30° angle, and are asked to find how high the top of the ladder is from the ground. I have added a visual representation to help you better understand the situation. As you can see we need to solve for x. Since we are given one angle and the hypotenuse we can solve for x using the SIN operator.

[tex]sin(a) = \frac{opposite}{hypotenuse}[/tex]

Since we have the angle (a) and the hypotenuse we can just plug the information into the sin equation and solve for the opposite (x).

[tex]sin(30) = \frac{x}{20ft}[/tex]

[tex]sin(30)*20ft = x[/tex]

[tex]0.5*20ft = x[/tex]

[tex]10ft = x[/tex]

So now we can see that the height of the top of the ladder to the ground is 10 ft

I hope this answered your question. If you have any more questions feel free to ask away at Brainly.

The graph of a limacon curve is given. Without using your graphing calculator, determine which equation is correct for the graph.[-5, 5] by [-5, 5] (1 point)

Answers

The general form for this curve is:
[tex]r = Acos \theta + B[/tex]
Now find A,B given 2 points:
[tex]r(0) = 5, r(\pi) = 1[/tex]
[tex]A +B = 5 \\ \\ -A+B = 1[/tex]
Solving this system yields:
[tex]A = 2, B = 3 \\ \\ r = 2cos \theta + 3[/tex]

Answer:

r = 2 - 3 cos θ

Step-by-step explanation:

This is the best I could come up with.

2w+2l=24 what is the value of l

Answers

If the question is 2w+2L=24 than the answer is 6-w=L
but if the question is 2w+21=24 the answer is w=3/2

Write an equation for the function. Tells what each variable you use represents. A plant's height is 1.6 times its age in months.

Answers

Y=1.6x+b (B can be anything or you won't need to add it into the equation)

This is function can be defined as a linear equation, in the standard form of y = mx + b. X is the independent variable, y is the dependent variable, m is the slope (rate of change/growth), and b is the constant (starting point).

Since the growth rate of height for the plant is 1.6 times the independent variable and there is no starting point (so assume 0), our equation should look like this:

y = 1.6x

The plant's height is y and its age in months is x.

Hope my answer helped clear things up a bit.

A teacher has 3 hours to grade all the papers submitted by the 35 students in her class. She gets through the first 5 papers in 30 minutes. How much faster does she have to work to grade the remaining papers in the allotted time?

Answers

At first, she was grading papers at 10 papers per hour. To meet her time limit, she would have to begin grading them at 12 papers per hour, or 20% faster than her previous rate

so, she has 3hrs to grade all papers, for 35 students.. alrite.

the first 5, she does them in 30minutes.. what's the speed rate? well, 5/30 or 1/6

now, she has still 2 hours and a half, or 150 minutes, to do the remaining 30 papers... she has to work at a rate of 30/150 then... which is 1/5 simplified.

now if we take 1/6 as the 100%, what is 1/5 in percentage then?

[tex]\bf \begin{array}{ccllll} rate&\%\\ \text{\textemdash\textemdash\textemdash}&\text{\textemdash\textemdash\textemdash}\\ \frac{1}{6}&100\\\\ \frac{1}{5}&x \end{array}\implies \cfrac{\frac{1}{6}}{\frac{1}{5}}=\cfrac{100}{x}\implies \cfrac{1}{6}\cdot \cfrac{5}{1}=\cfrac{100}{x}\implies \cfrac{5}{6}=\cfrac{100}{x} \\\\\\ x=\cfrac{6\cdot 100}{5}\implies x=120[/tex]

so 1/5 is 120% in relation to 1/6... meaning the rate of 1/5 she needs to move through, namely 1 paper every 5 minutes, is 20% faster than 1/6.

A geometry class has a total of 34 students. The number of males is 14 more than the number of females. How many males and how many females are in the class?

Answers

m=14+f, m+f=34. Solve for the equations simultaneously.

one way is to substitute teh first into the second to get, 14+2f=34, or 7+f=17, or f=10. Since f is 10 and m+f=34, m=34-10=24.

Good luck!

James deposits $600 in an account that pays 4% simple interest, and $1200 in a second account which has a higher interest rate but is more risky. What interest rate must he get on the second account in order to earn at least $132 in interest for the year?

Answers

1) $600 deposited at 4% simple interest for one year generates:

$600 * 4% = $600 * 0040 = $24

2) If he wants to earn $132 in interest for the year, he yet needs to generate $132 - $24 = $108 (additional)

3) The interest rate to generate $108 on a deposit of $1200 in a year, at simple interest is:

$1200 * (r) = $108 => r = 108 / 1200 = 0.09 = 9/100 = 9%

Answer: 9%

What is the value of -2 (7-15) over 4

Answers

The answer to the question is: 4

Steve can complete the 100m dash in 10 seconds while Paul can run it in 12 seconds. How does Steve's time compare to Paul's?

Answers

Steve is faster than Paul by 2 seconds after 100m

Suppose a basketball player has made 184 out of 329 free throws. If the player make the next two free throws, I will pay you $24. Otherwise you pay me $12. Find the expected value of the proposition

Answers

Final answer:

Expected value of proposition will be 20.134.

Explanation:

Given that the player made 184 out of 329 throws, the probability of making the next throw will be:

P(x)=[Number of shots made]/[Total number of throws]

=184/329

=0.559

Thus the expected value of proposition will be:

0.599 x 24+0.559 x 12

=20.134

7/2x-2=28-4x solve for x

Answers

According to a calculator, the value of x is 4

The coordinates of △ABC△ABC are A(12,8), B(10,18), C(4,16)A(12,8), B(10,18), C(4,16). After a dilation, the coordinates are A'(6,4), B'(5,9), C'(2,8)A′(6,4), B′(5,9), C′(2,8). Find the scale factor.

Answers

The scale factor is 1/2 because when you multiply the x and y-coordinates of each point by 1/2, you will get the coordinates of the new dilated points.

Answer: [tex]\dfrac{1}{2}[/tex]

Step-by-step explanation:

The dilation is a transformation which makes similar shapes.

We know that In two similar geometric figures, the ratio of their corresponding corresponding x-coordinate is known as the scale factor.

For the given situation, the x-coordinate of A in pre-image = 12

The x-coordinate of A' in image = 6

Then , the scale factor for the dilation is given by :-

[tex]k=\dfrac{6}{12}=\dfrac{1}{2}[/tex]

Hence, the scale factor = [tex]\dfrac{1}{2}[/tex]

The total interest paid on a 3-year loan at 9% interest compounded monthly is $1505.82 determine the monthly payment for the loan.

Answers

number of compounding periods is
n=12months×3years=36
I assume that

The total interest=
monthly payment×number of compounding periods - the amount of the present value of an annuity ordinary
I=x×n-pv

Let monthly payment be X

I =Total interest is 1505.82

The present value of an annuity ordinary is
Pv=X [(1-(1+0.09/12)^(-36))÷(0.09/12)]

now plug those in the formula of the total interest above
I=x×n-pv
1505.72=36X-X [(1-(1+0.09/12)^(-36))÷(0.09/12)]
Solve for X using Google calculator to get the monthly payment which is
X=330.72

Check your answer using the interest formula
36×330.72−330.72×((1−(1+0.09
÷12)^(−12×3))÷(0.09÷12))
=1,505.83

Which pairs of triangles are similar? Check all that apply.

Answers

Answer:

option (2) and (5) are correct.

ΔABC ≅ ΔJLK

ΔDEF ≅ ΔGHI

Step-by-step explanation:

Given four right angled triangle with measure of sides.

We have to check for the pairs of triangle to be similar.

Two triangles are said to be similar if their corresponding angles are equal or their corresponding sides are same ratio.

Consider, ΔABC and ΔJLK.

∠C = ∠L = 90° (given)

Also ratio of corresponding sides are same ratio, that is

[tex]\frac{AC}{LJ}=\frac{14}{7}=\frac{2}{1}[/tex]

Also, [tex]\frac{CB}{KL}=\frac{20}{10}=\frac{2}{1}[/tex]

Thus, ΔABC ≅ ΔJLK.

Option (5) is correct.

Consider, ΔDEF and ΔGHI.

∠I = ∠F = 90° (given)

Also ratio of corresponding sides are same, that is

[tex]\frac{DF}{GI}=\frac{8}{12}=\frac{2}{3}[/tex]

Also, [tex]\frac{EF}{HI}=\frac{10}{15}=\frac{2}{3}[/tex]

Thus, ΔDEF ≅ ΔGHI.

Option (2) is correct.

Thus, option (2) and (5) are correct.

Each figure is filled with unit cubes. Drag 22 figures whose combined volume is equal to 45 unit cubes.

Answers

To solve this problem, what we have to do first is to calculate for the volumes of each figure. Let us start from the figure on the left going to the right. Let us call the figures 1 to 4. Formula for volume is:

V = l * w * h

Figure 1:

V1 = 4 * 3 * 1 = 12 unit cubes

Figure 2:

V2 = 3 * 2 * 3 = 18 unit cubes

Figure 3:

V3 = 2 * 2 * 4 = 16 unit cubes

Figure 4:

V4 = 3 * 3 * 3 = 27 unit cubes

The two combinations that would result in 45 total unit cubes would be:

V = V2 + V4 = 18 unit cubes + 27 unit cubes = 45 unit cubes

So the answers are:

Figure 2 and Figure 4

100 meter long Christmas train needs 30 seconds to cross a 400nmetrr long bridge assuming the train goes at a steady speed how fast is it

Answers

Since it's 100m long, to cross the front needs to go 500m. 500/30= (16+2/3)meters per second

6x2=(3×2)×___=___

6×4=2×(6×___)=___

2×(6×4)=____×8=____

Answers

Let me try.

6×2 = (3×2)×2 = 12

6×4 = 2×(6×2) = 24

2×(6×4) = 6×8 = 48

Can the polynomial below be factored into a perfect square? If not, select the answer that best describes why not.
64x^2+49x+8
A.

The x^2 coefficient does not permit the factoring.

B.

The x^2 coefficient does permit the factoring, but the x coefficient does not permit the factoring.

C.

The constant value does not permit the factoring.

D.

The polynomial may be factored into a perfect square.

Answers

Option B;
A suitable x-coefficient for a quadratic function that factors into a perfect square has to be even.

Answer:

Option B is correct.

Step-by-step explanation:

We will work with the formula : [tex](a+b)^{2}[/tex]

= [tex]a^{2}+2ab+b^{2}[/tex]

Given polynomial is :

[tex]64x^{2} +49x+8[/tex]

here a = [tex]\sqrt{64x^{2} } =8x[/tex]

b = [tex]\sqrt{8}= 2\sqrt{2}[/tex]

2ab = [tex]2*8x*2\sqrt{2} =32\sqrt{2}x[/tex]

Now, we can see that the middle term should be [tex]32\sqrt{2} x[/tex] but in the question, it is given 49x

So, option B is true that - The x^2 coefficient does permit the factoring, but the x coefficient does not permit the factoring.

a local charity held a crafts fair selling donated,handmade items. Total proceeds from the sale were $1,875. A total of 95 items were sold,some at $15 each and the rest at $25 each. Let x be the number of $15 items and y the number of $25 items. How many items sold at $25?

Answers

15x+25y=1875
Let
y=95-x
15x+25 (95-x)=1875
Solve for x
15x+2375-25x=1875
15x-25x=1875-2375
-10x=-500
X=500/10
X=50 the number of $15 items

Y=95-50=45 the number of $25 items

Answer:

It’s 30

Step-by-step explanation:

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