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]
i donnt know ..............
Answer:
7
Step-by-step explanation:
[tex]\dfrac 13(15+6)\\\\\\=\dfrac{21}3\\\\\\=7[/tex]
Explain how the expression 32 relates to the diver's depth
Answer:
See below
Step-by-step explanation:
Each 32 feet of depth is 1 more atmosphere of pressure on the diver
Please help me, greatly appreciate it!
Answer: See the screenshots below
I'm answering in this format because the questions aren't numbered, so its hard to reference them consistently (so we're both on the same page).
All decimal values are approximate. I decided to round to 4 decimal places, but feel free to use another level of precision.
1 soda + 1 cupcake+1popsicle=$22.00
1 soda +3cupcakes+3popsicle=$40.00
if a cupcake cost,$2:00 more than a popsicle,find the cost of the following
a)1 cupcake
b)1 popsicle
Answer:
a) 1 cupcake = $5.50
b) 1 Popsicle = $3.50
Step-by-step explanation:
Let s = cost of one soda
Let c = cost of one cupcake
Let p = cost of one Popsicle
From the given information we can create 3 equations:
Equation 1: s + c + p = 22Equation 2: s + 3c + 3p = 40Equation 3: c = p + 2Subtract Equation 1 from Equation 2 to eliminate the variable s:
s + 3c + 3p = 40
- s + c + p = 22
2c + 2p = 18
Substitute Equation 3 into this new equation and solve for p:
⇒ 2c + 2p = 18
⇒ 2(p + 2) + 2p = 18
⇒ 2p + 4 + 2p = 18
⇒ 4p = 14
⇒ p = 3.5
Substitute the found value of p into Equation 3 and solve for c:
⇒ c = p + 2
⇒ c = 3.5 + 2
⇒ c = 5.5
To find the cost of a soda (s), substitute the found values of p and c into Equation 1:
⇒ s + c + p = 22
⇒ s + 5.5 + 3.5 = 22
⇒ s = 13
Therefore,
1 soda costs $131 cupcake costs $5.501 Popsicle costs $3.50find the angle of elevation of the sun if a building 325 feet tall casts a shadow 296 feet long
Answer:
Step-by-step explanation:
SOH CAH TOA
we have the (opposite) --> 325
we have the (adjacent) --> 296
so do the inverse tan of opposite over adjacent...
[tex]tan^{-1} (\frac{325}{296} ) = 47.67[/tex] degrees
Solve- x/x+1 - 1/2 = -1/x+1
Answer:
Step-by-step explanation:
Remark
x/(x+1) - 1/2 = -1 /(x + 1) is the equation to be solved. Add 1/(x + 1 to both sides
x/(x+1) + 1/(x+1) - 1/2 = -1/(x + 1) + 1/(x + 1) Combine
x/(x+1) + 1/(x + 1) - 1/2 = 0 Combine x/(x+1) + 1/(x+1)
(x + 1)/(x + 1) - 1/2 = 0 Cancel (x+1)/(x+1) = 1
1 - 1/2 = 0 No solution
Answer: No solution
what is the solution to the inequality below 14-2(x+1)< -4
Answer:
x > 8
Step-by-step explanation:
14 - 2(x + 1) < - 4 ← distribute parenthesis and simplify left side
14 - 2x - 2 < - 4
12 - 2x < - 4 ( subtract 12 from both sides )
- 2x < - 16
divide both sides by - 2 , reversing the symbol as a result of dividing by a negative quantity.
x > 8
Segment GH is shown on the graph.
What are the coordinates of the point that divides the
segment into a 3:2 ratio?
O (-7.5, 0.5)
O (0.5, -7.5)
O (-1.2,-2.2)
O (-2.2,-1.2)
The coordinates of the point that divides the segment into a 3:2 ratio is (-2.2, -1.2)
How to determine the coordinate?The points are given as:
G = (-4,3)
H = (-1,-4)
Ratio, m : n = 3 : 2
The coordinate of the point is calculated using:
[tex](x,y) = \frac{1}{m + n} * (mx_2 + nx_1, my_2 + ny_1)[/tex]
This gives
[tex](x,y) = \frac{1}{3 + 2} * (3 * -1 + 2 * -4, 3 * -4 + 2 * 3)[/tex]
Evaluate
[tex](x,y) = \frac{1}{5} * (-11, -6)[/tex]
Evaluate the product
(x,y) = (-2.2, -1.2)
Hence, the coordinate of the point is (-2.2, -1.2)
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Which is the standard form of the equation of a parabola with a focus of (8, 0) and directrix x = –8?
Answer:
Hi,
Step-by-step explanation:
directrix: x=-8
focus=(8,0)
[tex]x=\dfrac{(y-0)^2}{2*(8+8)}+\dfrac{8-8}{2}\\\\\boxed{x=\dfrac{y^2}{32}}[/tex]
how to find the volume of a room
multiply the height x the length x the width to find the volume of a room (or another 3d object)
A new amusement park presold discounted tickets for the opening day as well as upon arrival at the park. the opening day ticket sales for the park, including presales, is represented by this function, where is the number of tickets sold h hours after opening at 7:00 a.m. what is the approximate rate of change in the number of tickets sold between 10:00 a.m. and 1:00 p.m.? a. tickets per hour b. tickets per hour c. tickets per hour d. tickets per hour
The approximate rate of change in the number of tickets sold between 10:00 a.m. and 1:00 p.m. -21 tickets per hour
How to determine the rate of change?The missing part of the question is represented by the following function:
[tex]T(h) = \frac{386 + 110h}{h}[/tex]
10:00 a.m is 3 hours after 7:00 am and 1:00 p.m is 6 hours after 7:00 am.
This means that:
h = 3 at 10:00 am
h = 6 at 1:00 pm
Next, we calculate T(3) and T(6)
[tex]T(3) = \frac{386 + 110*3}{3} =238.7[/tex]
[tex]T(6) = \frac{386 + 110*6}{6} =174.3[/tex]
The average rate of change is then calculated as:
[tex]T'(h) = \frac{T(3) - T(6)}{3 - 6}[/tex]
This gives
[tex]T'(h) = \frac{238.7 - 174.3}{-3}\\[/tex]
Evaluate
T'(h) = -21
This means that the approximate rate of change in the number of tickets sold between 10:00 a.m. and 1:00 p.m. -21 tickets per hour
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what’s the area of the base
Answer:
16 square units
Step-by-step explanation:
16 square units
the prom committee is made up of 8 boys and girls. If there are 12 boys and 15 girls in the group to pick the committee from, how many ways can the committee be organized?
1). 1860 ways
2). 3185325 ways
3). 2220075 ways
4). 675675 ways
Thanks guys!!
The number of ways can the committee be organized will be 2220075. Then the correct option is 3.
What are permutation and combination?A permutation is an act of putting things or elements in the right sequence. Combinations are a method of picking things or pieces from a collection of objects or sets when the sequence of the objects is irrelevant.
The prom committee is made up of 8 boys and girls.
If there are 12 boys and 15 girls in the group to pick the committee from.
Then the number of ways can the committee be organized will be
Then the number of ways is given by the combination formula. Then we have
Number of ways = ¹²C₀ x ¹⁵C₈ + ¹²C₁ x ¹⁵C₇ + ¹²C₂ x ¹⁵C₆ + ¹²C₃ x ¹⁵C₅ + ¹²C₄ x ¹⁵C₄ + ¹²C₅ x ¹⁵C₃ + ¹²C₆ x ¹⁵C₂ + ¹²C₇ x ¹⁵C₁ +¹²C₈ x ¹⁵C₀
Number of ways = 6435 + 77220 + 330330 + 660660 + 675675 + 360360 + 97020 + 11880 + 495
Number of ways = 2220075
Thus, the correct option is 3.
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The circumference of the hub cap of a tire is 80.07 centimeters. find the area of this hub cap. use
3.14 for it. if the circumference of the hub cap were smaller, explain how this
would change the area of the hub cap.
the area of this hub cap is about how many
square centimeters?
The area of this hub cap is about 510.44625 cm².
What is Area of Circle?The area of circle is pi times the square of the radius.
Area of circle =π r²
Circumference (C) = 2π r
80.07 = 2 (3.14) r
76.87/ (2x 3.14) =r
r = 12.75 cm
now, Area be
=π r²
= 3.14*12.5*12.75
=510.44625 cm²
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If you pull a card from a standard deck of 52 and then pull a second card without replacing the first card, you have about a ____ chance(round to nearest percent) of pulling out a face card, followed by on ace.
The probability that you pull a card from a standard deck of 52 and then pull a second card without replacing the first card is 2%
How to determine the probability?The given parameters are:
Cards = 52
Face card = 12
Ace = 4
The probability of selecting a face card is:
P(Face card) = 12/52
Now, there are 51 cards.
The probability of selecting an ace is:
P(Ace) = 4/51
The required probability is:
P = 12/52 * 4/51
Evaluate
P = 0.0181
Express as percentage
P = 1.81%
Approximate
P = 2%
Hence, the probability is 2%
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For his phone service, Chris pays a monthly fee of $25, and he pays an additional $0.06 per minute of use. The least he has been charged in a month is $86.74.
What are the possible numbers of minutes he has used his phone in a month?
Use M for the number of minutes, and solve your inequality for M.
Answer:
1029 minutes or 17.15 hours
Step-by-step explanation:
25+0.06M=86.74
0.06M=61.74
M=61.74/0.06=1029
A lake near the Arctic Circle is covered by a thick sheet of ice during the cold winter months. When spring arrives, the warm air gradually melts the ice, causing its thickness to decrease at a rate of 0.2 meters per week. After 7 weeks, the sheet is only 2.4 meters thick.
Let y represent the ice sheet's thickness (in meters) after x weeks.
Complete the equation for the relationship between the thickness and number of weeks.
y=_____
The equation for the relationship between the thickness and number of weeks is y = 3.8 - 0.2x
How to write equation?Ice lost per week = 0.2 metersIce remaining after 7 weeks = 2.4 metersNumber of weeks = xRelationship between the thickness and number of weeks = yIce lost at 7 weeks = 7(0.2) = 1.4 meters
Total thickness of ice before loss = Ice remaining after 7 weeks + Ice lost at 7 weeks
= 2.4 + 1.4
= 3.8 meters
The equation:
y = Total thickness of ice before loss - (Ice lost per week × Number of weeks)
= 3.8 - (0.2 × x)
y = 3.8 - 0.2x
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100 POINTS PLEAS please help
(-2u3 + 5u - 1) + (7u3 - 2u2 + 9)
A) 5 u3 - 2 u2 + 5 u + 8
B) -5 u3 - 2 u2 + 5 u + 10
C)5 u3 + 3 u + 8
Answer:
[tex]\textsf{A)} \quad 5u^3-2u^2+5u+8[/tex]
Step-by-step explanation:
Given expression:
[tex](-2u^3+5u-1)+(7u^3-2u^2+9)[/tex]
Remove parentheses:
[tex]\implies -2u^3+5u-1+7u^3-2u^2+9[/tex]
Collect like terms:
[tex]\implies -2u^3+7u^3-2u^2+5u-1+9[/tex]
Combine like terms:
[tex]\implies 5u^3-2u^2+5u+8[/tex]
Option A
A light bulb consumes 1440 watt-hours per day. How long does it take to consume 6480 watt-hours?
The number of days it will take to consume the given power = 4.5 days
Calculation of energy consumptionThe amount of power consumed per day = 1440watt-hours
Therefore X days = 6480 watt-hours?
Make X the subject formula;
X = 1 × 6480/1440
X = 4.5 days
Therefore, The number of days it will take to consume the given power = 4.5 days.
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The area A of a circle is the product of π and the radius r squared
i need written in a algebraic equation
Answer:
A=πr^2
Step-by-step explanation:
A=π*r^2
Where, A is area of circle
r is radius of circle
π is constant
.
Hope it's help and I think I have give brainleist answer
Answer:
πr^2
Step-by-step explanation:
it's just πr^2.
easy
Omar paid $8.64 for 5.4 pounds of grapes. what was the cost of 1 pound of grapes?
Sherri solved the equatjon 1/5(x-20)=45. For each step, write the name of the property sherri used
Answer: The awnser is linear.
I hope this helped.
Step-by-step explanation:
Part E
Use the table showing Manuel's and Gretchen's data to determine the mean, median, standard deviation, and
interquartile range for each data set. Use this information to complete the second table. Use the graphing tool to
determine the value of standard deviation.
The mean, median, standard deviation, and interquartile range for each data set is given below.
What is the definition of Interquartile Range (IQR)?When arranged from lowest to highest, the IQR represents the central 50% of values.
Find the median (middle value) of the lower and higher half of the data to get the interquartile range (IQR).
These are the quartile 1 (Q1) and quartile 3 (Q3) values (Q3). The difference between Q3 and Q1 is the IQR.
What are the measures of central tendency for Manuels Data?Using the statistical tool, the following were arrived at:
Standard Deviation = 3.13961Mean = 8Median = 8IQR = Q3-Q1 = 10 - 6 = 4What are the measures of central tendency for Grethcen's Data?Using the statistical tool, the following were arrived at:
Standard Deviation = 4.68737.Mean = 9.6Median = 9IQR = IQR = Q3-Q1 = 12 - 7 = 5.See the respective Histograms attached.
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Given: ΔABC (to the right)
AB=15
BD=9
AD⊥BC
m∠C=30°
Find Perimeter of ABC
Applying the Pythagorean theorem and the trigonometric ratios, the perimeter of triangle ABC is: 68.8 units.
What is the Perimeter of a Triangle?The perimeter of a triangle = sum of its 3 sides.
Given the following:
AB = 15 BD = 9m∠C=30°Use the Pythagorean theorem to find AD considering triangle ADB as a right triangle:
AD = √(15² - 9²)
AD = 12 units
Use the trigonometric ratios to find DC and AC:
sin 30 = AD/AC
sin 30 = 12/AC
AC = 12/sin 30
AC = 24 units
tan 30 = AD/DC
tan 30 = 12/DC
DC = 12/tan 30
DC ≈ 20.8 units
BC = BD + DC = 9 + 20.8
BC = 29.8 units.
Perimeter of triangle ABC = BC + AB + AC
Perimeter of triangle ABC = 29.8 + 15 + 24
Perimeter of triangle ABC = 68.8 units.
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Select the correct answer.
To solve this system of equations using substitution, what could be substituted in place of y in the first equation?
4x=5-2y
y-2x=7
Answer: y=2x+7
(this is a rearranged version of y-2x=7 from question)
Step-by-step explanation:
4x=5-2y
y-2x=7
i would rearrange the 2nd value into y intercept form y=mx+b
so move the -2x to the other side of the = sign by doing the opposite, +2x on both sides to isolate the y
y=2x+7
you didn't give any options for "correct answer" so i can't give
you A B C or D answers
Evaluate the surface integral ∫sf⋅ ds where f=⟨2x,−3z,3y⟩ and s is the part of the sphere x2 y2 z2=16 in the first octant, with outward normal orientation away from the origin
Parameterize S by the vector function
[tex]\vec s(u,v) = \left\langle 4 \cos(u) \sin(v), 4 \sin(u) \sin(v), 4 \cos(v) \right\rangle[/tex]
with 0 ≤ u ≤ π/2 and 0 ≤ v ≤ π/2.
Compute the outward-pointing normal vector to S :
[tex]\vec n = \dfrac{\partial\vec s}{\partial v} \times \dfrac{\partial \vec s}{\partial u} = \left\langle 16 \cos(u) \sin^2(v), 16 \sin(u) \sin^2(v), 16 \cos(v) \sin(v) \right\rangle[/tex]
The integral of the field over S is then
[tex]\displaystyle \iint_S \vec f \cdot d\vec s = \int_0^{\frac\pi2} \int_0^{\frac\pi2} \vec f(\vec s) \cdot \vec n \, du \, dv[/tex]
[tex]\displaystyle = \int_0^{\frac\pi2} \int_0^{\frac\pi2} \left\langle 8 \cos(u) \sin(v), -12 \cos(v), 12 \sin(u) \sin(v) \right\rangle \cdot \vec n \, du \, dv[/tex]
[tex]\displaystyle = 128 \int_0^{\frac\pi2} \int_0^{\frac\pi2} \cos^2(u) \sin^3(v) \, du \, dv = \boxed{\frac{64\pi}3}[/tex]
help me please I will give you brainliest
Answer:
-17, -10, -6, -5, 3, 8 only change is between 3 and 8
The endpoints of wx are w (5,-3) and x (-1,-9) what is the length of wx ?
Using the distance formula, the length of WX is approximately: 8.5.
How to Find the Length of a Segment Using the Distance Formula?Distance formula is, [tex]d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}[/tex].
Given the endpoints:
W(5,-3) = (x1, y1)
X(-1,-9) = (x2, y2)
Plug in the values:
WX = √[(−1−5)² + (−9−(−3))²
WX = √[(−6)² + (−6)²]
WX = √72
WX ≈ 8.5
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Answer:
6 2 with the little check mark line in the midddle
Step-by-step explanation:
What is the slope of the line graphed below?
(-2,2)
(3, 1)
Answer:
-1/5
Step-by-step explanation:
To find the slope, the formula is y2-y1/x2-x1
1-2/3-(-2) = -1/5
Answer:
‐1/5Step-by-step explanation:
hope this helps!!!!
What is 95x9 divided by 14?
Answer:
213.75
Step-by-step explanation:
95×9=855
855÷4=213.75