A tank contains 70 lb of salt dissolved in 400 gallons of water. A brine solution is pumped into the tank at a rate of 4 gal/min; it mixes with the solution there, and then the mixture is pumped out at a rate of 4 gal/min. Determine A(t), the amount of salt in the tank at time t, if the concentration of salt in the inflow is variable and given by cin(t)
Answers
Answer:
\(A(t)=400C_{in}(t)+[70-400C_{in}(t)]\cdot e^{-\frac{t}{100}}\)
Step-by-step explanation:
Volume of water in the Tank =400 gallons
Let A(t) be the amount of salt in the tank at time t.
Initially, the tank contains 70 lbs of salt, therefore:
A(0)=70 lbs
Amount of Salt in the Tank
\(\dfrac{dA}{dt}=R_{in}-R_{out}\)
=(concentration of salt in inflow)(input rate of brine)
\(=(C_{in}(t))( 4\frac{gal}{min})\\=4C_{in}(t)\frac{lbs}{min}\)
=(concentration of salt in outflow)(output rate of brine)
\(=(\frac{A(t)}{400})( 4\frac{gal}{min})=\frac{A}{100}\)
Therefore:
\(\dfrac{dA}{dt}=4C_{in}(t)-\dfrac{A}{100}\)
We then solve the resulting differential equation by separation of variables.
\(\dfrac{dA}{dt}+\dfrac{A}{100}=4C_{in}(t)\\$The integrating factor: e^{\int \frac{1}{100}}dt =e^{\frac{t}{100}}\\$Multiplying by the integrating factor all through\\\dfrac{dA}{dt}e^{\frac{t}{100}}+\dfrac{A}{100}e^{\frac{t}{100}}=4C_{in}(t)e^{\frac{t}{100}}\\(Ae^{\frac{t}{100}})'=4C_{in}(t)e^{\frac{t}{100}}\)
Taking the integral of both sides
\(\int(Ae^{\frac{t}{100}})'=\int [4C_{in}(t)e^{\frac{t}{100}}]dt\\Ae^{\frac{t}{100}}=4*100C_{in}(t)e^{\frac{t}{100}}+C, $(C a constant of integration)\\Ae^{\frac{t}{100}}=400C_{in}(t)e^{\frac{t}{100}}+C\\$Divide all through by e^{\frac{t}{100}}\\A(t)=400C_{in}(t)+Ce^{-\frac{t}{100}}\)
Recall that when t=0, A(t)=70 lbs (our initial condition)
\(A(t)=400C_{in}(t)+Ce^{-\frac{t}{100}}\\
70=400C_{in}(t)+Ce^{-\frac{0}{100}}\\
C=70-400C_{in}(t)\\$Therefore, the amount of salt in the tank at any time t is:
\\\\A(t)=400C_{in}(t)+[70-400C_{in}(t)]\cdot e^{-\frac{t}{100}}\)
Related Questions
Maria wants to go to the movies with her friends. Each ticket is $9.25. How much would 5 tickets cost?
Answers
Answer:
$46.25
Step-by-step explanation:
9.25 times 5 = 46.25
Need help please please
Answers
Answer:
B
Step-by-step explanation:
Reflects over x axis then translates down and left.
Solve for y. 2(y-2) -6y=-28
Answers
Answer:
y = -8
Step-by-step explanation:
2(y-2)- 6y-28 = 0
2y - 4 -6y - 28 = 0
-4y = 32
y = -8
Please help me with this proof.
Answers
Answer:
See below
Step-by-step explanation:
For the second step, \(\angle T\cong\angle R\) by Alternate Interior Angles. The rest of the steps appear to be correct.
A line passes through the point (-4,6) and has a slope of 5/4
Answers
Answer:
4y-5x-44=0
Step-by-step explanation:
It passes through one point with the formula being
y-y1=m(x-x1)
y-6=5/4(x-(-4))
y-6=5/4(x+4)
cross multiply
4(y-6)=5(x+4)
4y-24=5x+20
collect like terms
4y-5x=20+24
4y-5x=44
4y-5x-44=0
Answer:
Step-by-step explanation:
I will ASSUME you want an equation
In slope/intercept form it is
6 = (5/4)(-4) + b
b = 11
y = (5/4)x + 11
or in slope/point form is
(y - 6) = (5/4)(x + 4)
help me plsssssssssssss
Answers
Answer:
The radius is 5 cm
Step-by-step explanation:
The formula for the area of a circle is : \(\p*radius^2\)
To work out the length of the radius you would first need to divide the area of 78.5 cm by pi (pi=3.14), this gives you \(25cm^2\\\). This is because by dividing the area by pi we are isolating the value of the radius squared.
The final step is to work out the radius. You can do this by finding the square root of the radius squared which is 25, this gives you 5 cm. This means that the radius is 5 cm. This is because the square root is a number that when squared equals that number.
1) Divide 78.5 by pi.
\(78.5/\pi=25cm^2\)
2) Find the square root of 25 cm squared.
\(\sqrt{25} =5 cm\)
In this question pi is 3.14
Isandro and Yvette each have a calculator. Yvette starts at 100 and subtracts 7 each time. Isandro starts at 0 and adds 3 each time. If they press the keys at the same time, will their displays ever show the same number? If so, what is the number?
Answers
Answer: 30
Step-by-step explanation:
Yvette- 100-7
Isandro- 0+3
1.) I'm going to speed up the process and multiply by 5.
Yvette: 100-7
7 × 5 = 35
100-35= 65
65-35= 30
Isandro: 0+3
3 × 5 = 15
0+15= 15
15+15= 30
2.) If they keep going on this pattern they will have the same number on the 10 round.
Bound
Figure 2.
Jenny would like to buy a new air conditioner for his home. As seen in Figure 2 above he has
3 types of brands that he can consider. The cash price for Daewoo is RM1,899, Daikin at a
price of RM1,699 and last choice is Mitsubishi at a price of RM1.999. He plans to make 18
monthly instalments, if he pays RM300 as a down payment for the air conditioner that he would
like to buy, find the interest charge for each brand if the interest rate is 4% based on originall
balance if he chooses to buy from Daikin, find the monthly instalment price and instalment
price that he needs to pay.
Answers
if Jenny chooses to buy from Daikin, the monthly installment price would be approximately RM98.89, and the total installment price to be paid would be approximately RM1,779.98 (18 monthly installments * RM98.89).
How to determine the monthly instalment price and instalment price that Daikin needs to pay.Calculating the loan amount for each brand by subtracting the down payment from the cash price:
Loan amount for Daewoo = RM1,899 - RM300 = RM1,599
Loan amount for Daikin = RM1,699 - RM300 = RM1,399
Loan amount for Mitsubishi = RM1,999 - RM300 = RM1,699
Calculating the interest charge for each brand:
Interest charge for Daewoo = Loan amount for Daewoo * Interest rate = RM1,599 * 0.04 = RM63.96
Interest charge for Daikin = Loan amount for Daikin * Interest rate = RM1,399 * 0.04 = RM55.96
Interest charge for Mitsubishi = Loan amount for Mitsubishi * Interest rate = RM1,699 * 0.04 = RM67.96
To find the monthly installment price, divide the total amount (cash price + interest charge) by the number of monthly installments:
Monthly installment price for Daewoo = (Cash price of Daewoo + Interest charge for Daewoo) / Number of monthly installments = (RM1,899 + RM63.96) / 18 ≈ RM107.22
Monthly installment price for Daikin = (Cash price of Daikin + Interest charge for Daikin) / Number of monthly installments = (RM1,699 + RM55.96) / 18 ≈ RM98.89
Monthly installment price for Mitsubishi = (Cash price of Mitsubishi + Interest charge for Mitsubishi) / Number of monthly installments = (RM1,999 + RM67.96) / 18 ≈ RM116.72
Therefore, if Jenny chooses to buy from Daikin, the monthly installment price would be approximately RM98.89, and the total installment price to be paid would be approximately RM1,779.98 (18 monthly installments * RM98.89).
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The American Association of Individual Investors (AAII) On-Line Discount Broker Survey polls members on their experiences with electronic trades handled by discount brokers. As part of die survey, members were asked to rate members were asked to rate their satisfaction with the trade price and the speed of execution, as well as provide an overall satisfaction rating. Possible responses (scores) were no opinion (0), unsatisfied (1), somewhat satisfied (2), satisfied (3), and very-satisfied (4). For each broker, summary scores were computed by computing a weighted average of die scores provided by each respondent. A portion the survey results follow (AAII Web site. February 7, 2012). Develop an estimated regression equation using trade price and speed of execution to predict overall satisfaction with the broker. Interpret die coefficient of determination. Use the F test to determine die overall significance of the relationship. What is your conclusion at the 0.05 level of significance? Use the t test to determine the significance of each independent variable? What are your conclusions at the 0.05 level of significance? Interpret the estimated regression parameters. Are die relationships indicated by these estimates what you would expect? Finger Lakes Investments has developed a new electronic trading system and would like to predict overall customer satisfaction assuming they can provide satisfactory service levels (3) for both trade price and speed of execution. Use the estimated repression equation developed in part a to predict overall satisfaction level for Finger Lakes Investments if they can achieve these performance levels. What concerns (if any) do you have with regard to die possible responses the respondents could select on the survey.
Previous question
Answers
The estimated regression equation developed in part (a) was found to be an effective predictor of overall customer satisfaction, as indicated by the significant F and t test results.
The estimated regression equation developed in part (a) to predict overall customer satisfaction is given by:
Overall Satisfaction = 0.3151*Trade Price + 0.3132*Speed of Execution + 1.5026
The coefficient of determination (R2) is a measure of how well the equation fits the data and is calculated as the proportion of the variation in overall satisfaction that is explained by the model. In this case, the R2 = 0.39, which means that 39% of the variation in overall satisfaction can be explained by the trade price and speed of execution variables.
To determine the overall significance of the relationship, the F test was used. The F statistic was calculated to be 17.874, which was significant at the 0.05 level (p-value = 0.000), indicating that the model was an effective predictor of overall customer satisfaction.
The t tests were then used to determine the significance of each independent variable. The t statistic for Trade Price was 2.639 (p-value = 0.009), indicating that the variable was significant at the 0.05 level, while the t statistic for Speed of Execution was 2.565 (p-value = 0.012), indicating that the
The estimated regression equation developed in part (a) was found to be an effective predictor of overall customer satisfaction, as indicated by the significant F and t test results.
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5b.) Use this equation to write the area A in terms of the perimeter p.
Answers
And the perimeter is 4x
We know that x = p/4
As area is x^2 , we can substitute the value of p/4 in x place to get:
(p/4)^2
= p^2/4^2
=p^2/16
So this is the area A In terms of the perimeter p
Help pls!!!!!!!!!! :<<<<
Answers
The answer to this question is D 2/4
Jessica needs to know how much water her new fish tank can hold:
A rectangular prism with a length of 8 inches, a width of 4 inches, and a height of 9 inches.
Determine the total volume of the fish tank.
Answers
The fish tank has a total volume of 288 inch³. As a result, Jessica's new fish tank has a capacity of 288 inch³ for water.
The volume of a rectangular prism can be calculated using the formula:
V = l x b x h..........(i)
where,
V ⇒ Volume
l ⇒ length
b ⇒ width
h ⇒ height
From the question, we are given the values,
l = 8 inches
b = 4 inches
h = 9 inches
Putting these values in equation (i), we get,
V = 8 x 4 x 9
⇒ V = 288 in³
Therefore, the fish tank has a total volume of 288 inch³. As a result, Jessica's new fish tank has a capacity of 288 inch³ for water.
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Given that cos x = 3/5 and 0
Answers
Answer:
What do we have to do in this? Find x? Let me know I'll explain it in the comment section
I'll brainlist!!
Plant A is 18 inches tall after one week, 36 inches tall after two weeks, 56 inches tall after three weeks. Plant B is 18 inches tall after one week, 36 inches tall after two weeks, 54 inches tall after three weeks. Which situation represents a proportional relationship between the plants?
Height and number of weeks?
Answers
Answer:
There are 2 given situations:
Plant A and Plant B growing in 3 weeks. Find which plant grows proportional
Plant A
=> 1st week = 18 inches
=> 2nd week = 36 inches
=> 3rd week = 56 inches
=> 18 + 18 = 36 + 20 = 56
Thus, this is not proportional
Plant B
=> 1st week = 18 inches
=> 2nd week = 36 inches
=> 3rd week = 54 inches
=> 18 + 18 = 36 + 18 = 54 inches
thus, the growth of this plant is proportional
a person deposits rs 55000 in bank p for 2 years at the rate of 10% per annum compounded annually .but after one year , bank has changed the policy and decided to pay semi annual compounded interest at the same rate .what is the percentage difference between the compound interests of the first year and second year
Answers
Answer:
The interest generated the first year will be 10% of the principal value, while the second year said interest will be 10.25%.
Step-by-step explanation:
Given that a person deposits Rs 55000 in Bank P for 2 years at the rate of 10% per annum compounded annually, but after one year, bank has changed the policy and decided to pay semi annual compounded interest at the same rate, to determine what is the percentage difference between the compound interests of the first year and second year, the following calculation must be performed:
Year 1 =
55,000 x (1 + 0.1 / 1) ^ 1x1 = X
55,000 x 1.1 = X
60,500
Year 2 =
60,500 x (1 + 0.1 / 2) ^ 1x2 = X
60,500 x 1.05 ^ 2 = X
66,701.25 = X
55,000 = 100
60,500 = X
60,500 x 100 / 55,000 = X
110 = X
60,500 = 100
66,701.25 = X
66,701.25 x 100 / 60,500 = X
110.25 = X
Therefore, the interest generated the first year will be 10% of the principal value, while the second year said interest will be 10.25%.
Write the point-slope form of an equation of the line through the points (-2, -3) and (-7, 4).
71
Answers
7x+5y +29= 0
Step-by-step explanation: first we have to find slop of the line that is.( Y2-Y1)/x2-x1
M = slope = 4+3)/-7+2
M= 7/-5
Equation of line y - y 1 = M (X - X1)
Y+3 = 7/-5 (x+2)
7x + 5y +29 =0 Ans .
Using the decimal 0.00 what is 2% as a decimal.
Answers
the expression is
\(2\text{ \%=}\frac{2}{100}=0.02\)so, the answer is 0.02
For the equation f(x) = 1.1 * (0.44) ^ x state the initial value C, the growth or decay factor a, and percent change R for each unit increase in x
c = (Type an integer or a decimal)
a = (Type an integer or a decimal)
R = % (Simplify your answer. Type an integer or a decimal)
Answers
The parameters of the exponential function in this problem are given as follows:
c = 1.1.a = 0.56.R = -56%.What is an exponential function?The standard format of an exponential function is given as follows:
y = c(1 - a)^t.
This is the case for a decaying exponential function, and the meaning of each parameter is given as follows:
c is the initial value, value assumed by y when t = 0.a is the decay rate.In this problem, the function is given as follows:
f(x) = 1.1(0.44)^x.
Hence the values for the parameters are given as follows:
c = 1.1, which is the initial value.a = 0.56, as 1 - a = 0.44 -> a = 0.56.Then the percent of change is of -56%, as it is a decaying exponential function with a = 0.56.
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Molly wants to buy a dress that costs $20. She has a coupon for 40% off, and there is a 10% sales tax. How much will she pay altogether?
Answer the questions to find out.
1. What is 40% of $20? Show your work. (2 points)
2. After using the coupon, what is the sale price of the dress? (2 points)
3. How much is the 10% tax on the sale price? Explain how you found your answer. (3 points)
4. After tax is added to the sale price, how much does Molly pay for the dress? (3 points)
Answers
Answer:
She will pay $13.20 altogether.
Step-by-step explanation:
1. 40% of 20 is 8.
Write 40% as 40/100.
40/100 of 20 = 40/100 × 20.
2. Sale Price = $12. This means the cost of the dress to Molly is $12. She will pay $12 for a dress with an original price of $20 when discounted 40%.
3. The 10% tax will be $1.20. When a store-issued coupon is redeemed, the sales tax is based on the discounted price (the cost of the item after the coupon is applied). So multiply 12 x 0.10 and you will get 1.20.
4. After tax is added to the sale price, she will pay $13.20.
12.00 + 1.20 = 13.20.
Find the volume of each cylinder. Use 3.14 for Pi
Round your answer to the nearest tenth.
7 mm
Your answer
8 mm
Answers
Answer: 5 mm 6 mm
Step-by-step explanation:
A pool has some initial amount of water in it. Then it starts being filled so the water level rises at a rate of 666 centimeters per minute. After 202020 minutes, the water level is 220220220 centimeters.
Answers
Answer:
y = mx + b
slope(m) = 6
(20,220)....x = 20 and y = 220
now sub and find b, the y int
220 = 6(20) + b
220 = 120 + b
220 - 120 = b
100 = b
so ur equation is : y = 6x + 100
to graph :
y = 6x + 100.....u have a slope of 6....u have a y int of 100 or (0,100)...
I would say plot (0,100) and since ur slope is 6, go up 6 and to the right 1...but that is not to helpful because ur graph is not in intervals of 1.
So we will find some points on ur line.
y = 6x + 100....when x = 5
y = 6(5) + 100
y = 30 + 100
y = 130....so another point on ur line is (5,130)
y = 6x + 100...subbing in 10 for x
y = 6(10) + 100
y = 60 + 100
y = 160....another point is (10,160)
keep finding points this way...here are some more...
(15,190) , (20,220), (25,250)
Step-by-step explanation:
Answer:
y = 6x + 100
Step-by-step explanation:
Find the vertex and focus of the
parabola.
x² + 8x - 12y + 88 = 0
Answers
For the given quadratic the vertex is (-4, 6) and the focus (-4, 6.02)
How to find the vertex of the quadratic equation?
Here we have the quadratic equation:
x^2 + 8x - 12y + 88 = 0
Isolating y, we get:
x^2 + 8x + 88 = 12y
(x^2 + 8x + 88)/12 = y
(1/12)*x^2 + (8/12)*x + (88/12) = y
Now, using the formula for the x-value of the vertex we get:
x = -(8/12)/(2*1/12) = -4
To get the y-value of the vertex we evaluate in x = -4, we will get:
y = ((-4)^2 + 8*-4 + 88)/12 = 6
So the vertex is (-4, 6)
Now we want to find the focus, the x-value is the same one as the vertex, and the y-value is:
y = k + 1/4a
Here we have k = y-value o the vertex = 6
a = leading coefficient = 1/12
Then:
y = 6 + 1/4*12 = 6.02
Then the focus is (-4, 6.02)
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The weight, in pounds, of dogs at the Humane Society are listed:[15, 45, 50,62,66,68,70,74,81] Determine what the shape of the data would be, and identify the median.
Answers
To find the median, add all the numbers and divide the result by the number of numbers.
\(\begin{gathered} 15+45+50+62+66+68+70+74+81 \\ =531 \end{gathered}\)\(\begin{gathered} \frac{531}{9} \\ =59 \end{gathered}\)so, the median is 59
A bank's loan officer rates applicants for credit. The ratings are normally distributed with a mean of 200 and a standard deviation of 50. If an applicant is randomly selected, find the probability of a rating that is between 200 and 275. Round to four decimal places. A. 0.4332 B. 0.0668 C. 0.5000 D. 0.9332
Answers
Find the midpoint of segment PQ
Answers
Answer:
Step-by-step explanation:
hello :
P(2 ; 6) and Q(-6 ; -8)
the midpoint of segment PQ is : H ( (2-6)/2 ; (6-8)/2)
H(-4/2 ; -2/2)
H(-2 ; -1)
What is the Analysis of Cramer’s rule?
Answers
11 is the quotient of a number y and 6.
Answers
Answer:
y = 66
Step-by-step explanation:
11 = y/6
11*6 = y
66 = y
Which statement is false
Answers
A recipe requires 1 cup of milk for every 4 cups of flour. Choose a linear equation that describes the relationship.
A. y=4x
B. y=4x+1/4
C. y=1/4x+4
D. y=1/4x
Answers
Answer:
i think it is a but i am not shore. Hope this helps
Step-by-step explanation:
Find the measure of each angle in the problem.
Answers
Answer:
C
Step-by-step explanation:
70+20=90 and thats a right angle which is 90°