Answer:
a.)Q' = -kQ + r
b.)Q = [tex]\frac{r}{k} [ 1 - e^{-kt} ][/tex]
c.) Limiting long run value of Q = [tex]\frac{r}{k}[/tex]
Step-by-step explanation:
a.)
The rate of change is directly proportional to the quantity, Q
⇒[tex]\frac{dQ}{dt}[/tex] ∝ Q
⇒[tex]\frac{dQ}{dt}[/tex] = -kQ ( because it is decreasing )
Also given, quantity is increasing with a constant rate r
⇒[tex]\frac{dQ}{dt}[/tex] = -kQ + r
⇒Q' = -kQ + r
b.)
As we have
[tex]\frac{dQ}{dt}[/tex] = -kQ + r
⇒[tex]\frac{dQ}{-kQ + r}[/tex] = dt
⇒∫[tex]\frac{dQ}{-kQ + r}[/tex] = ∫dt
⇒-[tex]\frac{1}{k}log(-kQ + r) = t + C[/tex]
⇒log(-kQ + r) = -kt -kC
Taking exponential both side, we get
⇒-kQ + r = [tex]e^{-kt +A}[/tex]
⇒-kQ = -r + [tex]e^{-kt +A}[/tex]
⇒Q = [tex]\frac{r}{k}[/tex] - [tex]\frac{1}{k}[/tex][tex]e^{-kt +A}[/tex]
⇒Q = [tex]\frac{r}{k}[/tex] - [tex]\frac{1}{k}[/tex][tex]e^{-kt}.e^{A}[/tex]
⇒Q = [tex]\frac{r}{k}[/tex] - [tex]\frac{1}{k}[/tex][tex]Be^{-kt}[/tex] .......(1)
Now,
At t = 0, Q = 0
0 = [tex]\frac{r}{k}[/tex] - [tex]\frac{1}{k}[/tex]B
⇒[tex]\frac{1}{k}B = \frac{r}{k}[/tex]
⇒B = r
∴ equation (1) becomes
Q = [tex]\frac{r}{k}[/tex] - [tex]\frac{r}{k}[/tex][tex]e^{-kt}[/tex]
⇒Q = [tex]\frac{r}{k} [ 1 - e^{-kt} ][/tex]
c.)
for limiting long run value of Q
[tex]\lim_{n \to \infty} Q = \lim_{n \to \infty}[/tex] [tex]\frac{r}{k} [ 1 - e^{-kt} ][/tex]
= [tex]\frac{r}{k}[/tex][tex]\lim_{n \to \infty} [ 1 - e^{-kt} ]= \frac{r}{k} [ 1 - e^{\infty} ] = \frac{r}{k}[ 1-0][/tex]
= [tex]\frac{r}{k}[/tex]
⇒Limiting long run value of Q = [tex]\frac{r}{k}[/tex]
Q' = -kQ + r
Q = r/k (1- [tex]\rm e^{-kt}[/tex])
The limiting long-run value of Q = [tex]\rm \frac{r}{k}[/tex]
A drug is administered intravenously at a constant rate of r mg/hour and is excreted at a rate proportional to the quantity present, with a constant of proportionality k > 0.
What is a differential equation?The differential equation is an equation that contains the derivative of the unknown function.
a) It is given that the rate of change is directly proportional to the quantity Q
[tex]\rm \frac{dQ}{dt}[/tex] ∝ Q
[tex]\rm \frac{dQ}{dt}[/tex] = -kQ
So, Q' = -kQ + r
where quantity Q is increasing with constant rate r and k is unknown constant.
b) Q' = -kQ + r
[tex]\rm \frac{dQ}{dt}[/tex] = -kQ + r
[tex]\rm \frac{dQ}{-kQ+r}=dt\\[/tex]
[tex]\int\limits {\rm \frac{dQ}{-kQ+r} = \int\limits dt[/tex]
log(-kQ + r) = -kt -kC
by taking exponential both side, we get
-kQ + r = [tex]e^{-kt+A}[/tex]
-kQ = -r + [tex]e^{-kt+A}[/tex]
-Q /k= -r/k + [tex]e^{-kt+A}[/tex]/k
Q = -r/k -[tex]\rm \frac{1}{k}Be^{-kt}[/tex]
At t = 0, Q = 0
[tex]\rm \frac{1}{k} B=\frac{r}{k}[/tex]
B = r
by substituting the value in the above equation
Q = r/k (1- [tex]\rm e^{-kt}[/tex])
c) The limiting long-run value of Q
[tex]\rm \lim_{n \to \infty} Q= \lim_{n \to \infty} \frac{r}{k} [1-e^{-kt} ]\\\rm= \frac{r}{k}\lim_{n \to \infty} [1-e^{-kt} ]\\\rm =\frac{r}{k}[1-0]\\\rm=\frac{r}{k}[/tex]
The value Q =[tex]\rm \frac{r}{k}[/tex]
Learn more about differential equations here:
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65% of what number is 39? 30% of what number is 15?
Please answer both questions.
Answer:
65% of what number is 39=25.35
30% of what number 15=4.5
Step-by-step explanation:
Pleaaase help this is due today
Answer:
umm look it up
Step-by-step explanation:
The right triangle on the right is a scaled copy of the right triangle on the left. Identify the scale factor. Express your answer as a whole number or fraction in simplest form.
Answer:
1/3
Step-by-step explanation:
the one on the left is 3 times larger than the one on the right. in other words, the right triangle's dimensions times 1/3 is equal to the left triangle's dimensions.
Find the value of X if H is the midpoint of JG, GJ equals 4X -6 and GH equals 27.
Answer:
27
Step-by-step explanation:
Judith made 4 stacks of wooden blocks. The first stack was 9 blocks tall. The second stack was 5 blocks taller than the first. The third stack was 5 blocks shorter than the second stack, and the last stack was 10 blocks taller than the third stack. How many blocks did Judith use in all?
Answer: the answer is 42
the size of a rectangular photo frame is 12 11/16cm by 8 3/4cm. Which of the following gives the product of these dimensions rounded to the nearest whole number of centimeters
answers
A) 12 cm x 8 cm
B) 12 cm x 9 cm
C) 13 cm x 8 cm
D) 13 cm x 9 cm
Answer:
D
Step-by-step explanation:
the answer can't be A or C because we know that 8 3/4 rounds to 9
so we have to figure out if 12 11/16 is closer to 12 or 13
anything above 8/16 should round up; therefore, 11/16 should round the 12 to be 13
how long would the toy car be?...
Answer:
6.37 cm.
Step-by-step explanation:
Since Hot Wheels uses a scale ratio of 1:64 (1 cm to scale = 64 cm actual) for their toys, knowing that the actual length of a VW Beetle is 160.6 inches, to determine the length of the toy car is you must perform the following calculation:
1 inch = 2.54 cm
(160.6 x 2.54) / 64 = X
407.924 / 64 = X
6.373 = X
Therefore, the size of the toy car will be 6.37 cm long.
Please help, I have no idea how to do this one
PLS I NEED HELP: Which one of the following is an exponential growth function?
A. y = (1.5)^x
B.y = (2/3)^x
C. y = (1/2)^x
D. y = (0.5)^x
According to the National Weather Service, the average monthly high temperature in the Dallas/Fort Worth, Texas area from the years of 2006-2008 is given by the following table:
Month
Average Maximum Monthly Temperature
Jan
48.1
Feb
50.9
Mar
62.4
Apr
67.0
May
76.5
Jun
83.9
Jul
86.8
Aug
88.1
Sep
79.2
Oct
69.9
Nov
59.5
Dec
49.6
Plot the data on a scatter plot. Produce a sine regression model for the data. Round the values for a, b, c, and d to the nearest 0.001.
a.
y = 67.577 sine (0.500 x minus 1.959) + 20.077
b.
y = 20.077 sine (0.500 x minus 1.959) + 67.577
c.
y = 20.077 sine (1.959 x minus 0.500) + 67.577
d.
y = 20.077 sine (67.577 x minus 1.959) + 0.500
Answer:
b. y = 20.077 sin(0.500x - 1.959) + 67.577Step-by-step explanation:
The easiest way to solve
1. Plot the points for one year2. Add the functions to same graph and compareThis gave us the option B as the best fit line
See the graph below
Step-by-step explanation:
the dude above me is right
find the slope of the line passing through the points (7,-6) and (-7,-6)
Answer:
1.1667
Step-by-step explanation:
Slope = change in y / change in x
Slope = (-6 - -6) / (-7 - 7)
Slope = 0/-14
Because the change in y is 0 the slope would be 0
Slope = 0
[10 • 32 - 3] + 10 I need help nowww!
Answer:
Its 327!
Step-by-step explanation:
[10x32-3]+10 10x32=320 320-3=317 317+10=327
Answer:
327
Step-by-step explanation:
PEMDAS - parenthesis exponent multiply divide add subtract, the order to solve.
solve the parenthesis first, inside the parenthesis do the multiplication 1st so 10 times 32 which = 320. then subtract 3 = 317. then, lastly, add the 10 to get 327.
3 x 10 ^4 as an ordinary number
Answer:
30,000
Step-by-step explanation:
First exponents first so 10^4 is 10,000
10,000 times 3 is 30,000
The ordinary number for the given scientific notation is 30000.
What is scientific notation?Scientific notation is a method for expressing a given quantity as a number having significant digits necessary for a specified degree of accuracy, multiplied by 10 to the appropriate power such as 1.56×10⁷.
The given scientific notation is 3×10⁴.
To write scientific notation in ordinary number, multiply the number of 10 powers.
Now, 10⁴ can be expanded as 10×10×10×10=10000
So, 3×10⁴=3×10000
= 30000
Therefore, the ordinary number is 30000.
To learn more about the Scientific notation visit:
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What is the value of x in the equation 3x-4y=65, when y=4?
O x=134
O x=21 1 / 3
O x=23
O x=27
Answer:
27
Step-by-step explanation:
3x-4y=65
3x-4(4)=65
3x-16=65
3x=65+16
3x=81
x=81/3
x=27
In the jone school library,10/20 of the computers have scanners.in simplest form, which fraction of the computers have scanners? 1/2 1/4 5/10 6/12
Answer:
1/2
Step-by-step explanation:
simplify 12/18 into an equivalent fraction
Answer:
2/3
Step-by-step explanation:
Andres bought 3 shirts that each cost the same amount and a pair of jeans that cost $40. The items he
bought cost a total of $106 before tax was added.
PART A: Create an equation to find s, the cost of each shirt.
Reminder! Your equation should not have any spaces included in it. Ex. 4x+2=10
PART B: What is the cost of each shirt?
shirt-x
a.3s+40=106
b. 3s= 66
s=22
if my answer helps please mark as brainliest.
Ms. Rios washed 2 5 of her laundry. Her son washed 1 3 of it. Who washed most of the laundry? How much of the laundry still needs to be washed?
Answer:
The son has washed most of the laundry.
Step-by-step explanation:
What’s the right answer ?
Answer:
B
Step-by-step explanation:
the answer is in the attachment
x
8
17
Find the exact value of x.
X=
Answer:
621
629
Step-by-step explanation:
We know that
sin
(
x
+
y
)
=
sin
x
cos
y
+
sin
y
cos
x
If
cos
x
=
8
17
and
sin
y
=
12
37
We can use,
cos
2
x
+
sin
2
x
=
1
and
cos
2
y
+
sin
2
y
=
1
To calculate
sin
x
and
cos
y
sin
2
x
=
1
−
cos
2
x
=
1
−
(
8
17
)
2
=
225
17
2
sin
x
=
15
17
cos
2
y
=
1
−
sin
2
y
=
1
−
(
12
37
)
2
=
1225
37
2
cos
y
=
35
37
so,
sin
(
x
+
y
)
=
15
17
⋅
35
37
+
12
37
⋅
8
17
=
621
629
Answer link
Shwetank Mauria
Nov 22, 2016
sin
(
x
+
y
)
=
621
629
or
−
429
629
depending on the quadrant in which sine and cosine lie.
Explanation:
Before we commence further, it may be mentioned that as
cos
x
=
8
17
,
x
is in
Q
1
or
Q
4
i.e.
sin
x
could be positive or negative and as
sin
y
=
12
37
,
y
is in
Q
1
or
Q
2
i.e.
cos
y
could be positive or negative.
Hence four combinations for
(
x
+
y
)
are there and for
sin
(
x
+
y
)
=
sin
x
cos
y
+
cos
x
sin
y
, there are four possibilities.
Now as
cos
x
=
8
17
,
sin
x
=
√
1
−
(
8
17
)
2
=
√
1
−
64
289
=
√
225
289
=
±
15
17
and
as
sin
y
=
12
37
,
cos
y
=
√
1
−
(
12
37
)
2
=
√
1
−
144
1369
=
√
1225
1369
=
±
35
37
Hence,
(1) when
x
and
y
are in
Q
1
sin
(
x
+
y
)
=
15
17
×
35
37
+
8
17
×
12
37
=
525
+
96
629
=
621
629
(2) when
x
is in
Q
1
and
y
is in
Q
2
sin
(
x
+
y
)
=
15
17
×
−
35
37
+
8
17
×
12
37
=
−
525
+
96
629
=
−
429
629
(3) when
x
is in
Q
4
and
y
is in
Q
2
sin
(
x
+
y
)
=
−
15
17
×
−
35
37
+
8
17
×
12
37
=
525
+
96
629
=
621
629
(4) when
x
is in
Q
4
and
y
is in
Q
1
sin
(
x
+
y
)
=
−
15
17
×
35
37
+
8
17
×
12
37
=
−
525
+
96
629
=
−
429
629
Hence,
sin
(
x
+
y
)
=
621
629
or
−
429
629
Classify the following triangle. Check all that apply.
98
419
O A. isosceles
I B. scalene
C. obtuse
U
D. equilateral
I E. acute
F. right
Answer:
obtuse and isosceles
angle greater than 90° and 2 equal sides
Suppose you have a student loan of $45,000 with an APR of 9% for 30 years. Complete parts (a) through (c) below.
The medical practice you are working at has seen an average of 22.4 patients a day for the past 3 months. 3/4ths of those patients have insurance in one form or another. how to analyze and what the process you will use Please express to the nearest whole number how many of the patients on average per day do not have insurance, and thus, pay cash price for their visits.
Answer:
5.6 patients
Step-by-step explanation:
Given
[tex]Average = 22.4[/tex]
[tex]Insurance = \frac{3}{4}[/tex]
Required
How many do not have insurance (average)
First, calculatet he fraction of those without insurance (x).
[tex]x = 1 - Insurance[/tex]
Substitute value for Insurance
[tex]x = 1 - \frac{3}{4}[/tex]
[tex]x = \frac{1}{4}[/tex]
The number of visits is:
[tex]Visits =x * Average[/tex]
[tex]Visits = \frac{1}{4} * 22.4[/tex]
[tex]Visits = 5.6[/tex]
X 1. Solve the equation
Atun
(x + 2) = 20
Answer:
hi
Step-by-step explanation:
Answer:
18 ?
Step-by-step explanation: What is Atun ?
PLEASE I WILL MARK BRAINLIEST 22. Find the measure of angle HJK.
Answer:
i think its 38 but i could be wrong sorry
Step-by-step explanation:
calcular la altura que podemos alcanzar una escalera de 3 metros apoyada sobre ma pared si la parte inferior la situamos a .07 metros de esta.
I need some homework help please.
Answer:
x^2-5x=6
x^2-5x-6=0
x^2+x-6x-6=0
x(x+1)-6(x+1)=0
(x-6)(x+1)=0
x=6, -1
-5x= -30, 5
Step-by-step explanation:
long division: Problem type 2 Divide, (12x3 +14x2+24x+9) = (4x2+2x) Your answer should give the quotient and the remainder Quotient: 3 3 Remainder: 1 X ?
9514 1404 393
Answer:
quotient: 3x +2remainder: 20x +9Step-by-step explanation:
Just as with numerical long division, the product of the partial quotient and the divisor is subtracted from the dividend to get the new dividend. The partial quotient is the ratio of the highest-degree terms of dividend and divisor. The process ends when the degree of the dividend is less than the degree of the divisor.
Use synthetic substitution to find f(3) and f(-4) for f(x)= x^4 - 8x- 11.
Answer:
f(3)=56, f(-4)=277
Step-by-step explanation:
f(x)=x^4-8x-11
f(3)=3^4-8(3)-11=3*3*3*3-24-11=9*3*3-35=27*3-25=81-25=56
f(-4)=(-4)^4-8(-4)-11=(-4)(-4)(-4)(-4)+32-11=(16)(16)+32-11=256+32-11=288-11=277
Answer:
227
Step-by-step explanation:
3
8
8
1
.
2. I need help!!!!!
Answer:
3 7/8
Step-by-step explanation:
8 3/8 - 4 1/2 =
8 3/8 - 4 4/8=
7 11/8 - 4 4/8 =
3 7/8
Answer:
The person above is correct
Step-by-step explanation:
When subtracting fraction you should make sure the deniminators (bottom number of fraction) of both fractions are the same number so you can subtract evenly. Also note whatever you do to the deniminator, you do to the numerator (top number). In this example you turn 1/2 to 4/8 in order to subtract.