A tank contains 180 gallons of water and 15 oz of salt. water containing a salt concentration of 17(1+15sint) oz/gal flows into the tank at a rate of 8 gal/min, and the mixture in the tank flows out at the same rate.
the long-time behavior of the solution is an oscillation about a certain constant level. what is this level? what is the amplitude of the oscillation?
Let A(t) denote the amount of salt (in ounces, oz) in the tank at time t (in minutes, min).
Salt flows in at a rate of
[tex]\dfrac{dA}{dt}_{\rm in} = \left(17 (1 + 15 \sin(t)) \dfrac{\rm oz}{\rm gal}\right) \left(8\dfrac{\rm gal}{\rm min}\right) = 136 (1 + 15 \sin(t)) \dfrac{\rm oz}{\min}[/tex]
and flows out at a rate of
[tex]\dfrac{dA}{dt}_{\rm out} = \left(\dfrac{A(t) \, \mathrm{oz}}{180 \,\mathrm{gal} + \left(8\frac{\rm gal}{\rm min} - 8\frac{\rm gal}{\rm min}\right) (t \, \mathrm{min})}\right) \left(8 \dfrac{\rm gal}{\rm min}\right) = \dfrac{A(t)}{180} \dfrac{\rm oz}{\rm min}[/tex]
so that the net rate of change in the amount of salt in the tank is given by the linear differential equation
[tex]\dfrac{dA}{dt} = \dfrac{dA}{dt}_{\rm in} - \dfrac{dA}{dt}_{\rm out} \iff \dfrac{dA}{dt} + \dfrac{A(t)}{180} = 136 (1 + 15 \sin(t))[/tex]
Multiply both sides by the integrating factor, [tex]e^{t/180}[/tex], and rewrite the left side as the derivative of a product.
[tex]e^{t/180} \dfrac{dA}{dt} + e^{t/180} \dfrac{A(t)}{180} = 136 e^{t/180} (1 + 15 \sin(t))[/tex]
[tex]\dfrac d{dt}\left[e^{t/180} A(t)\right] = 136 e^{t/180} (1 + 15 \sin(t))[/tex]
Integrate both sides with respect to t (integrate the right side by parts):
[tex]\displaystyle \int \frac d{dt}\left[e^{t/180} A(t)\right] \, dt = 136 \int e^{t/180} (1 + 15 \sin(t)) \, dt[/tex]
[tex]\displaystyle e^{t/180} A(t) = \left(24,480 - \frac{66,096,000}{32,401} \cos(t) + \frac{367,200}{32,401} \sin(t)\right) e^{t/180} + C[/tex]
Solve for A(t) :
[tex]\displaystyle A(t) = 24,480 - \frac{66,096,000}{32,401} \cos(t) + \frac{367,200}{32,401} \sin(t) + C e^{-t/180}[/tex]
The tank starts with A(0) = 15 oz of salt; use this to solve for the constant C.
[tex]\displaystyle 15 = 24,480 - \frac{66,096,000}{32,401} + C \implies C = -\dfrac{726,594,465}{32,401}[/tex]
So,
[tex]\displaystyle A(t) = 24,480 - \frac{66,096,000}{32,401} \cos(t) + \frac{367,200}{32,401} \sin(t) - \frac{726,594,465}{32,401} e^{-t/180}[/tex]
Recall the angle-sum identity for cosine:
[tex]R \cos(x-\theta) = R \cos(\theta) \cos(x) + R \sin(\theta) \sin(x)[/tex]
so that we can condense the trigonometric terms in A(t). Solve for R and θ :
[tex]R \cos(\theta) = -\dfrac{66,096,000}{32,401}[/tex]
[tex]R \sin(\theta) = \dfrac{367,200}{32,401}[/tex]
Recall the Pythagorean identity and definition of tangent,
[tex]\cos^2(x) + \sin^2(x) = 1[/tex]
[tex]\tan(x) = \dfrac{\sin(x)}{\cos(x)}[/tex]
Then
[tex]R^2 \cos^2(\theta) + R^2 \sin^2(\theta) = R^2 = \dfrac{134,835,840,000}{32,401} \implies R = \dfrac{367,200}{\sqrt{32,401}}[/tex]
and
[tex]\dfrac{R \sin(\theta)}{R \cos(\theta)} = \tan(\theta) = -\dfrac{367,200}{66,096,000} = -\dfrac1{180} \\\\ \implies \theta = -\tan^{-1}\left(\dfrac1{180}\right) = -\cot^{-1}(180)[/tex]
so we can rewrite A(t) as
[tex]\displaystyle A(t) = 24,480 + \frac{367,200}{\sqrt{32,401}} \cos\left(t + \cot^{-1}(180)\right) - \frac{726,594,465}{32,401} e^{-t/180}[/tex]
As t goes to infinity, the exponential term will converge to zero. Meanwhile the cosine term will oscillate between -1 and 1, so that A(t) will oscillate about the constant level of 24,480 oz between the extreme values of
[tex]24,480 - \dfrac{267,200}{\sqrt{32,401}} \approx 22,995.6 \,\mathrm{oz}[/tex]
and
[tex]24,480 + \dfrac{267,200}{\sqrt{32,401}} \approx 25,964.4 \,\mathrm{oz}[/tex]
which is to say, with amplitude
[tex]2 \times \dfrac{267,200}{\sqrt{32,401}} \approx \mathbf{2,968.84 \,oz}[/tex]
help me please!!! i cant figure it out and its driving me insane
Answer:
y<_0
Step-by-step explanation:
The last answer.
Hope this helps! Have a great day :)
Holly is looking up at an angle at a helicopter. The direct distance from Holly to
the helicopter is 16 km. The helicopter is hovering 11 km above the ground.
Calculate the angle of elevation from Holly to the helicopter. Include a labelled
diagram in your solution
Answer:
43.43 degrees
Step-by-step explanation:
This is a right triangle with the opposite LEG = 11 km and the hypotenuse = 16 km
sin (angle) = opposite/hypotenuse
= 11/16
arc sin (11/16) = angle = 43.43 degrees
A fish tank is full of fish. There are 3 goldfish, 2 guppies, 6 clownfish, and 2 striped fish. If one of the goldfish dies and you remove it from the tank, what is the new probability of selecting a clownfish?
5/13
1/2
5/12
6/13
Answer:
1/2
Step-by-step explanation:
13 total fish
remove one fish
12 total fish
6 total clown fish
6/12 simplified is 1/2
There are 11 cookies in a jar, as listed below. The cookies are all the same size and shape.
•
6 chocolate chips
• 5 oatmeal
One cookie is randomly selected from the jar and not replaced. Then a second cookie is
randomly selected and not replaced. What is the probability they are both chocolate chips?
Answer:
6/11*5/10 = 3/11
Step-by-step explanation:
first we can get 6/11 which is the probability.
then we minus both sides by 1 to get 5/10
then we times them
Hello. Im stuck on this question. It says find 25% of 70. And I got 17.5 but it says it is wrong Could anyone help
Question: what is 25% of 70
= 17.50 or 25%
heyyy.... can someone pls help asap.
Rearrange the formula to write r in terms of P and pi.
[tex]p = 2r + \pi \: r[/tex]
Answer:
[tex]r = \dfrac{p}{2 + \pi}[/tex]
Step-by-step explanation:
[tex]~~~~~p = 2r +\pi r\\\\\implies p = r(2 + \pi)\\\\\implies r = \dfrac{p}{2 + \pi}[/tex]
What is the inverse of this function?
f(x) = − ½√x + 3, x ≥ −3
I DESPERATELY NEED HELP!!!!! IM GIVING 50 POINTS TO SOMEONE WHO CAN SOLVE THIS PLEASE!!!!!
Answer:
x=0 y=0
x=1 y=5
x=2 y=8
x=3 y=9
x=4 y=8
x=5 y=5
x=6 y=0
Step-by-step explanation:
2. Verify that cos(360° -0)=cos is an identity.
Answer:
it is an identity that was solved
Step-by-step explanation:
cosin - the a
The average starting salary for an editor at a textbook company is $39,800. The actual salaries can vary by less than $1,300. Which inequality can
be used to determine whether a salary, x, falls within this range? What is the range of starting salaries at the company?
O A 11,300-x) <39,800
The range of salaries is from $38,500 to $39,800.
OB.
|x-39,8001 <1,300
The range of salaries is from $38,500 to $41,100.
OC.
|x+39,800| <1,300
The range of salaries is from $38,500 to $41,100.
O D. x+1,3001 ≥ 39,800
The range of salaries is from $39,800 to $41,100.
The range of salary is (a) the range of salaries is from $38,500 to $39,800.
How to determine the salary range?The given parameters are:
Average salary = 39,800
Vary = Less than $1300
Let the salary be x.
So, we have:
x = 39,800
When the salary varies, the equation becomes
x = 39800 - 1300
Evaluate
x = 38500
This means that the salary is from 38500 to 39800
Hence, the range of salary is (a) the range of salaries is from $38,500 to $39,800.
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what is the name of these figures?
Answer:
1. Triangle 2. Pentagon 3. Hexagon 4. Parallelogram
Step-by-step explanation:
add or subtract the given percent from
71% subtracted from 100% is 29%
Complete questionAdd or subtract the given percent from 100%.
71%
How to solve the expression?The percentage (p) is given as:
p = 71%
When subtracted from 100%, we have:
100% - p
Substitute p = 71%
100% - 71%
71 subtracted from 100 is 29.
So, we have:
100% - 71% = 29%
Hence, 71% subtracted from 100% is 29%
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Find the length of the midsegment of each trapezoid.
12)
Answer: 25
Step-by-step explanation:
[tex]\frac{23+27}{2}=\boxed{25}[/tex]
please help me! will give the brainliest answer!
A furniture maker used 9/4 cans of paint to paint 2 chairs. He used the same amount of paint for each chair. How many cans of paint did he use for each chair?
Answer:
9/8 cans of paint
Step-by-step explanation:
if the furniture maker used the same amount of paint for each chair ,
we have to divide 9/4 by two to get the answer
(9/4)/(2/1) = (9*1)/(4*2) = 9/8
=> 9/8
hope this helps :)
Answer:
9/8 is the answer.
The point (-11,2) ____on the circle with radius 5 and center (-7,4)
Answer:
Does not lie
Step-by-step explanation:
The equation of the circle is
[tex] {(x + 7)}^{2} + {(y - 4)}^{2} = 25[/tex]
If we substitute x=-11, and y=2, we get that:
[tex] {( - 11 + 7)}^{2} + {(2 - 4)}^{2} = 25 \\ {4}^{2} + {2}^{2} = 25 \\ 20 = 25[/tex]
Which is false, meaning it does not lie on the circle
37.6 x ???=3760
this question is basically:
thirty seven point six times ?what? equals three thousand seven hundred and sixty
15 POINTS AVALIBLE + BRAINLIEST !!!!
thanks :)
Answer:
The "what" is 100
Step-by-step explanation:
Rewrite this as 37.6y = 3760 and set out to determine y:
y = 3760/37.6 = 100
The "what" is 100
Find the area of the triangle below.
Be sure to include the correct unit in your answer.
Answer:
area of a triangle =1/2 x b x h
which is half base x hieght
Which graph represents the function?
f(x)=√x + 1
Using translation concepts, it is found that graph 3 represents the function [tex]f(x) = \sqrt{x} + 1[/tex].
What is a translation?A translation is represented by a change in the function graph, according to operations such as multiplication or sum/subtraction in it's definition.
In this problem, the function is:
[tex]f(x) = \sqrt{x} + 1[/tex]
Which is a shift up of one unit of [tex]\sqrt{x}[/tex]. The square root function has y-intercept at (0,0), hence the translated function will have y-intercept at (0,1). Additionally, [tex]\sqrt{4} = 2[/tex], since f(4) = 3, which means that graph 3 is represents the function.
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What number needs to be added to 5 and 3 so that the ratio of the first number to the second becomes 3 : 2?
Answer:
1
Step-by-step explanation:
let x be the number to be added to 5 and 3
expressing the ratios in fractional form
[tex]\frac{5+x}{3+x}[/tex] = [tex]\frac{3}{2}[/tex] ( cross- multiply )
3(3 + x) = 2(5 + x) ← distribute parenthesis on both sides
9 + 3x = 10 + 2x ( subtract 2x from both sides )
9 + x = 10 ( subtract 9 from both sides )
x = 1
then the number to be added to both is 1
Twice during the assembly, a student is chosen at random to assist with the presentation. after the first student has finished assisting, the student returns to the group and can be chosen a second time. what is the probability that the first student chosen is a senior and the second student chosen is a sophomore?.
Using it's concept, it is found that the probability that the first student chosen is a senior and the second student chosen is a sophomore is given by:
[tex]p = \frac{11}{320}[/tex]
What is a probability?A probability is given by the number of desired outcomes divided by the number of total outcomes.
Researching this problem on the internet, we have that there are:
31 juniors, 10 sophomores, 17 juniors, 22 seniors.
Then:
Out of 31 + 10 + 17 + 22 = 80 students, 22 are seniors, hence the probability that the first student is a senior is given by [tex]p_1 = \frac{22}{80} = \frac{11}{40}[/tex].All students can be chosen again, hence the probability that the second student is a sophomore is given by [tex]p_2 = \frac{10}{80} = \frac{1}{8}[/tex].The events are independent, hence the probability of both is given by:
[tex]p = p_1p_2 = \frac{11}{40} \times \frac{1}{8} = \frac{11}{320}[/tex]
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Find the perimeter of ABCD
Find the area of Area
Step-by-step explanation:
A = (1, 3)
B = (3, 6)
C = (9, 2)
D = (7, -1)
the distance between 2 points is given by the Pythagoras equation based on the coordinate differences as legs of virtual right-angled triangles.
AD for example we get from
AD² = (7-1)² + (-1 - 3)² = 6² + (-4)² = 36 + 16 = 52
AD = sqrt(52) = sqrt(4×13) = 2×sqrt(13)
and AB we get from
AB² = (3-1)² + (6-3)² = 2² + 3² = 4 + 9 = 13
AB = sqrt(13)
the perimeter of the given rectangle is
2×sqrt(52) + 2×sqrt(13) = 2×2×sqrt(13) + 2×sqrt(13) =
= 6×sqrt(13) = 21.63330765...
and the area of the rectangle is
2×sqrt(13)×sqrt(13) = 2×13 = 26
Write an equation in slope-intercept form of the line that contains the point (-2, 2) and has the slope = -5.
Answer:
[tex]y=-5x-8[/tex]
Step-by-step explanation:
Given the following question:
To calculate the slope intercept of a line/point we need to use the formula for slope intercept, substitute, and solve for b.
Formula for slope intercept:
[tex]y=mx+b[/tex]
[tex]m=-5[/tex]
[tex]y=2[/tex]
[tex]x=-2[/tex]
[tex]2=-5(-2)+b[/tex]
Solve for b:
[tex]2=-5(-2)+b[/tex]
[tex]-5(-2)=10[/tex]
[tex]2=10+b[/tex]
[tex]10-10=0[/tex]
[tex]2-10=-8[/tex]
[tex]-8=b[/tex]
[tex]b=-8[/tex]
Substitute for the following:
[tex]y=mx+b[/tex]
[tex]b=-8[/tex]
[tex]m=-5[/tex]
[tex]y=-5x-8[/tex]
Your answer is "y = -5x - 8."
Hope this helps.
PLEASE HELP FOR THE LOVE OF GOD
The tree diagram represents an experiment consisting of two trials. p(a and c)
Answer:.2
Step-by-step explanation:.4/(.4+.3+.6+.7)
A museum groundskeeper is creating a semicircular statuary garden with a diameter of 27 feet. There will be a fence around the garden. The fencing costs $7.75 per linear foot. About how much will the fencing cost altogether? Round to the nearest hundredth. Use 3.14 for π.
The fencing will cost about $
.
To calculate the cost of the fence, we have to find the length of the semi-circle. The cost of fencing the garden would be $328.52 approximately
Circumference of a CircleTo find the length of the fence, we have to calculate the circumference of the semi-circle and divide it by 2
[tex]C = 2\pi r\\l = \frac{c}{2} \\l = \pi r[/tex]
Data;
diameter = 27ft r = diameter/2 = 13.5ft π = 3.14Let's substitute the values and calculate the length of the fence.
[tex]l = \pi r\\l = 3.14 * 13.5\\l = 42.39ft[/tex]
The length of the fence is equal to 42.39ft
The cost of fencing would be
[tex]42.39 * 7.75 = $328.5225[/tex]
The cost of fencing the garden would be $328.52 approximately
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Please help its very desperate and if i could get someone to help me on this whole assignment it would be awesome
Write -0.65 as a fraction or mixed number in simplest form. (you may use an improper or a mixed number as your final answer.)
also if someone could help me on this whole assignment my discord is goths#0001
Answer:
-13/20
Step-by-step explanation:
-0.65=-65/100
=13/20
If I give you an x-value of (x = 6), and a quardatic equation 2x²-3x+2=y, what is the y-value?
Please help me
Answer:
Step-by-step explanation:
2(6²)-3(6)+2=y
(2x36)-18+2=y
72-18+2=y
y=52 BODMAS
It may also be 56 tho im not sure if thats right
Put x in Equation
y=2x²-3x+2y=2(6)²-3(6)+2y=2(36)-18+2y=72-18+2y=54+2y=56A group of friends are playing a trivia game. players spin a spinner 2 times to find the 2 categories of trivia questions they will be asked. the spinner has 5 sections. the "music" and "sports" sections are each 1-half the size of the 3 larger sections.
to the nearest percentage, what is the probability that a player will be asked a math question ,begin emphasis,and then,end emphasis, a music question? enter the answer in the box.
The probability that a player will be asked a math question and then a music question is 0.03125
How to determine the probability?The size of the sections are given as:
Music = 0.5
Sport = 0.5
Others = 1
So, the total size is:
Total = 0.5 + 0.5 + 1 + 1 + 1
Evaluate
Total = 4
The probability of asking a math question is:
P(Math) = 1/4 = 0.25
The probability of asking a music question is:
P(Music) = 0.5/4 = 0.125
The required probability is:
P(Math and Music) = P(Math) * P(Music)
This gives
P(Math and Music) = 0.25 * 0.125
Evaluate
P(Math and Music) = 0.03125
Hence, the probability that a player will be asked a math question and then a music question is 0.03125
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What quadrant is point D located in?
A. quadrant I
B. quadrant II
C. quadrant III
D. quadrant IV
Answer:
Quadrant 4
Answer is D........